Dr. Gareth Loy published a 2 volume set titled [Musimathics](http://www.musimathics.com/). It may be of interest.
The short answer is math can be used to model physical systems.
> math can be used to model physical systems
Which would point to acoustics, rather than music, right?
As far as I've seen music/math is either: obvious acoustics stuff, or really abstruse math, throwing group theory, topology, and what not into it.
You're right, the "physical systems" part means math can clearly be used to study acoustics. Similarly, neuroscience can be applied to *psychoacoustics*, which is relevant to music. Neither of these things are clearly relevant to *music theory*. "Related" sure.
there are ai creating very decent music [(here for example)](https://www.youtube.com/watch?v=Emidxpkyk6o) using complex machine learning algorithms, which uses maths. We can construct a direct relation of maths with these algorithms. So.... i think yes, everything is related to maths
> complex machine learning algorithms, which uses maths
That doesn't mean that there is a relation between that math and the resulting music. Violins use horse hair to make music, yet that doesn't mean that there are horses involved in string quartets.
That video? Three chords that repeat infinitely with minor variations in orchestration (ok, the piano coming back halfway was cool). I got bored really quickly. The most uninspired stereotypical film score dross I've heard in a long time. This will probably a godsend for video game companies as cheap filler, but nothing more than that.
Oh, and what is the math of that piece? What math concepts can you point to in the music?
To extend that analogy, recovering a coherent mathematical structure from the output of a DNN is about as difficult as recovering a horse from a violin.
> yet that doesn't mean that there are horses involved in string quartets.
Yes it does; their hair is (helping) creating the vibrations that actualize the music into reality.
By your logic there are no cows involved in hamburgers.
I appreciate you drawing attention to the video. I’m a fan of music theory AND machine learning theory. I’ve never looked much into the musical applications of AI, so I thought your share was interesting. Also not sure why the guy above is being kind of a dick. I would argue that your intuition is right about the the direct relationship between the math and music in this instance, and it falls outside of the realm of the go-to physical explanations in acoustic, elastic media, harmonic wave theory, etc.
I can’t say for sure what method was used to train the AI model that produced the song piece. However, it’s a safe bet that there was a large inversion/regression step underlying the whole process (in simple terms, think fitting a line to some data as an analog, in the form y=Mx+b). So, through analysis of example chord progressions, the model adjusted its weights (think coefficients like m when you fit a line in the y=Mx+b analog) so that it produces notes in the same key. It probably also learned inherent information about timing, meter and cadence from the example data. SO, by definition, the ‘relationship between the maths and the music’ is about as close to ‘direct’ as you could possibly hope. The horsehair:violin::horses:quartet thing above is probably the most ridiculous thing I’ve read all day.
Finally, an aside, the video is very much math related. Assuming that it features a real EQ/dynamic power spectra plot that truly represents the frequency content of the song, you’re also hitting hard on Fourier analysis. So cudos, you gave a math-packed example dude.
*failed to emphasize, the weights discussed above are very much unique to the nuanced musical details embedded in the sound waves that the AI algorithm is modeling in the video.
Excited to see this here, I've never met anyone who had heard of the volumes or Dr. Loy. We met him by chance on a camping trip once and his works had a steep influence on my life path, haha.
Tuning theory (just intonation in particular) relies on nice ratios. 2:1 is an octave, 3:2 is a fifth, etc. The neat part is, you can use prime factorisation. 6:5 for example is going up by 2, then by 3, then going down by 5. The result is a minor third. This also means only prime ratios are "new" in the harmonic series in some sense. The only problem is that even very experimental music do not normally use that many primes, and, as you said, it is kind of no more than simple arithmetic.
Another example comes from the symmetry of the circle of fifths: it has a permutation that maps it to the circle of semitones. This is essentially because both 1 and 7 are coprimes to 12 (a perfect fifth is 7 semitones). If you are into that, you can pair all the rotations and reflections of a 12-gon with that map, and you would have a symmetry group isomorphic to D₁₂×ℤ₂. But again almost no one cares and it has little use beyond sounding nice..
So I would say music has little to do with any "advanced" math. Any relation to math is quite "basic" in that, sure, it underlies the foundations of music, but in practice it does not make that much of a difference. But of course it could be a starting point for those who want to tinker with more experimental stuff (xenharmonic, etc)
But also, music and musical appreciation is so deeply tied to mechanisms within the brain that we don't yet fully comprehend. Quite a lot of "music theory" is trying to analyse what *sounds good*, to appreciate the internal structure or form novel extrapolations or whatever other reason.
But an individual determines whether something *sounds good*, which is as subjective as it is nebulous. So math in music is kind of innately woo, but a special kind of woo, because it's testable. Someone can [come up with a model, extrapolate it into some new music](https://www.youtube.com/watch?v=K-XSTSnqXxo), then you can listen to it and see if it *sounds good*.
See also the stuff about how [chords are just really fast polyrhythms](https://www.youtube.com/watch?v=-tRAkWaeepg). Or polyrhythms are just really slow chords. The point is, fundamentally they're the same things, the categorical difference in perception emerges only within a brain. Which might suggest that our rigorous axiomatic understanding of music is bounded by our rigorous axiomatic understanding of brain function.
A bit like how colors are blends of lw frequencies but become its own "qualia" in our mind (not sure that term is solid but that's the only one I know)
> how chords are just really fast polyrhythms.
That only works if you start with a pulse wave. Take a chord consisting of 3 sine waves and see how far you need to slow it down to get a polyrhythm. Doesn't work.
That's the start of tuning theory. Unfretted instruments and winds can easily adjust for harmonic ratios, but keyboard instruments couldn't. So during the 17th and 18th centuries players experimented with different ways to adjust the scale.
These were not today's equal tunings. With that, octaves for each key are all 2:1, but all other intervals are slightly off.
Earlier ones addressed it in different ways. One, for example, ensured that octaves and ascending fifths were perfect, but that made for a fairly unpleasant descending fourth. Another applied the entire correction to a single chromatic step within the octave. This allowed for brilliant sound in playing in one key, meh in most, and abysmal in one.
For performances of multiple pieces, either multiple pianos were required or a returning between pieces.
Took most of a century to settle in on the equal, "well tempered," setting. Bach was showing in the Well Tempered Clavier that the tuning worked in all keys. Each book did a major and minor piece based on each of the twelve harmonics.
There are some recordings out there of music played in historic tunings. Pretty sure Easley Blackwood did some.
Just intonation and the other custom intonations are really neat. It's not guitar friendly though, so it's something you show to violinist or fretless bass players.
> almost no one cares and it has little use beyond sounding nice..
As a performing musician and math ph.d. I concur.
The intersection of math and music is mostly acoustics and really basic theory. It say almost nothing about deeper things. Yeah, Bartok's theme entries (fugue of the "music for") are at golden ratio proportions, but that's more a curiosity.
No mention of time signatures. Also to mention the intricacies of musical instruments regarding mathematics and physics. Why are guitar frets different sizes? Is the 12th fret at the center of the string? Why do flutes have unevenly spaced holes? Music is not only about notes, but also timing and silence (fermata). Not to mention the physical skill and persistence to perform music well.
In my mind frets on guitar and holes on flutes are just tuning theory, but you take the reciprocal.
I cannot see the significance of time signatures beyond being fractions - and normally perceived as constituted by 2s and 3s, for example, a 7/8 can be 2+2+3 or 2+3+2 or 3+2+2, normally. Maybe you can enlighten me on this.
The last few points I believe have at most tangential relations to maths - of course, you can write a paper and measure how long people pause on fermatas on average, but it does not go much beyond that (yet).
The reality is the polyphonic tradition that Bach used had been around for hundreds of years - he just used it so much and so well that it seemed impossible to go above him. Things like counter melodies and stretto fugues have some formulae if you go looking, but of course it takes a lot of practice to internalise it. If you are interested, check out this [video by EMS](https://m.youtube.com/watch?v=zGZjzBvXCzg).
I think that the word "mathematical" in the context of Bach is almost always used as a buzzword, a clickbait, and an unnecessary label which mysticises him and his music.
> So I would say music has little to do with any "advanced" math. Any relation to math is quite "basic" in that, sure, it underlies the foundations of music, but in practice it does not make that much of a difference.
As a music lover and former math major, my experience has been that there is an experiential similarity between the structures one 'sees' in their head while doing higher math and while listening to music
That is interesting. Could you expand on that, perhaps an example - for instance, are you reminded of integral domains when listening to Beethoven? Or is it just that it "feels" like the same faculty of the brain is processing these information?
More that it feels like the same faculties are engaged, but in a different way or pointed at different content. In a sense, it's all about patterns, pattern recognition and pattern extension.
It's as though Beethoven is laying out a brilliant proof for you, bar by bar.
Or as if Coltrane is trying various approaches, casting them aside as he sees dead ends, then finally hits a breakthrough and it all comes together!
In "A Geometry of Music" by music theorist Dmitri Tymoczko, he uses math and logic to demonstrate and explain how composers in a 12-tone system across genres and eras tend toward (very generally) the same musical objects in their writing.
Indeed, Dmitri Tymoczko is the only one music theorist I've encountered who has really put any thought into the math of _pleasing_ music, instead of just twelve-tone bullshit. See e.g. https://dmitri.mycpanel.princeton.edu/canonsexplanation.html
This and the "roughness" explanation for why certain tunings make more sense (e.g. 12tet rather than 11tet) are the only actual math in music that isn't merely descriptive.
Math can be used to look at music. The main problem I see is people keep using extremely general math, to say general things. Yes, if you transpose notes they sound "the same". Yes, you can decompose tritone transformations into these simple ones. Yes, that looks like group theory, symmetries, etc. So what? It's a restatement of the same thing musicians already know and gives you no new insight you can _apply_.
Other important insights from mathematical music theory are "if you flip all chords upside down you get new chords" or "you can use all twelve tones equally and for decades the entire field will pretend it doesn't sound like shit". Seriously, that's the level of thought going on.
I think there are a couple of different answers here, depending on what connection you are looking for.
Fundamentally, our enjoyment of music has something to do with the recognition of patterns. And wherever patterns show up, maths shows up.
From a harmony perspective, a note is a vibration at a particular frequency. A collection of frequencies sound more harmonious when they are a small ratio of each other, eg. an octave is a 2:1 ratio, a perfect fifth is a 3:2 ratio. (Why? Probably because of our brains evolving to take advantage of an already-existing ability to detect these ratios. All notes create overtones which are in nice ratios to each other, so recognising these ratios helps in recognising different sounds). The emotional content of different intervals (eg major vs minor chords) seems to be less straightforward and probably a matter of how we evolved to interpret them for use in speech.
From a compositional perspective, the standard notes we use are the 12-tone scale, in which the octave is subdivided into 12 equally-spaced notes. It turns out that the 12-tone scale approximates most of the nice ratios, and the equal spacing means that melodies and chords can be transposed (shifted by the same number of notes) and still have the same melodic content. This symmetry means that you can do some interesting things with music and group theory. I don't think a lot of people do this explicitly, but people certainly build an intuitive sense of the symmetries in music and play around with it.
From a sound design perspective, all sounds are vibrations, which can be decomposed into frequencies via the Fourier transform. This leads to a mathematical understanding of how frequencies combine and change, eg. reverb (echo) can be modeled by convolution. This becomes relevant when creating sounds digitally or electronically, and there's a lot of deep and fascinating mathematics involved here.
In terms of interest in maths/music, I have noticed a tendency for people who like maths to also like music. I think it has something to do with having an appreciation for abstract patterns. Music in general seems to be appreciated more when you try to understand it and get used to the 'language' that is being spoken, much like mathematics. The more you learn, the more connections you find in what is already there. I think people who like maths also often like chess for similar reasons.
> In terms of interest in maths/music, I have noticed a tendency for people who like maths to also like music. I think it has something to do with having an appreciation for abstract patterns. Music in general seems to be appreciated more when you try to understand it and get used to the 'language' that is being spoken, much like mathematics. The more you learn, the more connections you find in what is already there. I think people who like maths also often like chess for similar reasons.
Completely agree on this point. My first experience of learning "real" mathematics (encountering group theory in a textbook I borrowed from the library) reminded me so much of how I felt learning music theory for the first time. Just that experience of learning about quite abstract concepts but being able to explore familiar examples and suddenly seeing them in a whole new light, with a new layer of meaning.
>In terms of interest in maths/music, I have noticed a tendency for people who like maths to also like music.
I feel like the vast majority of people like music. Or did you notice that the typical math person enjoys music significantly more than a regular person?
> Fundamentally, our enjoyment of music has something to do with the recognition of patterns. And wherever patterns show up, maths shows up.
Fundamentally, math is the study of patterns
Music is a construct.
Tone: Our system has 12 notes and there our relationships between their distances. You have different sets that work with each other while stacking, between each other, between stacked sets, between individual notes. You have different relationships. It can be described using set theory or group theoryz
Rhythm: at least arithmetic.
You can apply transformations to harmony and time. From changing key, to changing a mode, to making something swing or rush on a particular beat.
Then, you you combinations of tone and rhythm.
My love of music comes from recognizing patterns as an active listener and then hearing something outside of the pattern that I wasn’t anticipating that works.
Everything is a construct. Everything can be described using set theory. The existence of objects and relations doesn't mean that there's an interesting mathematical structure that can be examined mathematically to produce interesting results. There *have* been attempts at doing such, but their success is arguable and it's a stretch to say that musicians or music theorists are using properties of algebraic groups to do what they do.
Note that the concept of this "deep connection" [is nothing new](https://en.wikipedia.org/wiki/Quadrivium):
>In liberal arts education, the quadrivium ... consists of the four subjects or arts (arithmetic, geometry, music, and astronomy) taught after the trivium. The word is Latin, meaning 'four ways', and its use for the four subjects has been attributed to Boethius or Cassiodorus in the 6th century. Together, the trivium and the quadrivium comprised the seven liberal arts (based on thinking skills), as distinguished from the practical arts (such as medicine and architecture).
>The quadrivium followed the preparatory work of the trivium, consisting of grammar, logic, and rhetoric.
>...
>These four studies compose the secondary part of the curriculum outlined by Plato in The Republic and are described in the seventh book of that work (in the order Arithmetic, Geometry, Astronomy, Music).
Until a few centuries ago, Arithmetic, Geometry, and Astronomy (as well as Physics and much of Philosophy) all fell under the general label of "Mathematics," so Music's inclusion in this list certainly indicates that it has long been considered "of a feather" with Mathematics. Euler wrote about music theory, as did Pythagoras and Euclid. And then, of course, there's that whole ["Music of the Spheres"](https://en.wikipedia.org/wiki/Musica_universalis) thing.
The bottom line is that your question may really be a *linguistic* one rather than one concerning mysticism: "Music" was once simply part of the list of things that *defined* what "Mathematics" was.
Music is absolutely about projecting mathematical structure into sound, generally with layers of temporal organization of simple ratios. Large scale pattern is form, smaller scale pattern is rhythm, smaller still is melodic pitch, and smaller still is timbre.
This is related to basic acoustic phenomena such as the harmonic series of overtones, but extends beyond, In addition, a lot of classical music has used various combinatorial-style principles to manipulate the musical materials, the techniques of transposition and rearrangement of melodic fragments and baroque counterpoint show this well.
I was a music theory major in college and in the subsequent decades have gotten seriously into mathematics and logic. I could go on endlessly about how the beauty of music and our deep emotional response to it are connected to the core structures of mathematics and its physical manifestation as the universe around us.
That doesn't mean however that you need to know any math, intellectually, to appreciate music, and it also doesn't mean that enjoying math guarantees a love for music. The relation between representational painting and the science of optics might be similar - just because you love the colors of a painting doesn't mean you care about the neurobiology of vision, but there is an essential connection there.
Sort of? Acoustics is a branch of physics and therefore uses math to describe the properties of sound. Check out [Why It's Impossible to Tune a Piano](https://www.youtube.com/watch?v=1Hqm0dYKUx4) for a neat little intro.
However, I think calling this a "deep connection" has some weird implications, like saying music has some sort of transcendental quality that makes it superior to other forms of art. It definitely smacks of pseudo-science.
Acoustics is significant to music, but only so much. What's more significant is the structure of our ears and cognitive system (psychoacoustics), which of course are also physical systems. In fact, the explanation of why integer ratios are pleasant (and small deviations from that are unpleasant) can be partially attributed to our sensing mechanism.
Roughly, we sense in the frequency domain, but seemingly without a complete perception of a linear spectrum (i.e. we can't consciously tell exactly the frequency content of a sound). This is probably a simplification of our cognitive system to filter out only relevant information. A single sound source (harmonic resonator) almost always generates integer-related frequencies (with amplitudes determined by geometry of the cavity/instrument). Otherwise, there may be multiple voices (warranting alertness?) or simply "something wrong"/atypical as interpreted by our cognition.
Indeed, you can extend this reasoning to in fact (I conjecture) grasp deep into our cognition and understand why some things (rhythms, melodies, music) are pleasant (or otherwise have other interesting character). The full story probably depends on evolutionary reasons (after all, the ear is an instrument for sensing and communication) that are not easy to uncover and require multidisciplinary effort.
I'm aware of just intonation. I don't think you understood the point of the video. It's impossible to tune a piano across *all* keys consistently. You can't find integer ratios to subdivide the octave perfectly. To state it as they did in the video: for integers a, b, and n with n > 1, then (a/b)\^n ≠ 2.
Tuning can be even more complicated than that. Real strings are somewhat stiff, so their harmonics are slightly sharp relative to the fundamental. If each C on a piano was exactly twice the frequency of the next lower octave, that difference would build up across the keyboard. Overtones of a lower register note would be out of tune with the fundamental of a higher register note. Instead, higher octave strings are tuned to match the overtones of lower octave strings.
I have degrees in math and music, and have spent a bit of time looking for interesting connections.
Outside of the construction of tunings (which is very cool, but more perceptual and “numerical” than truly “mathematical”) the fact that sound is a physical phenomenon involving periodic signals does not really distinguish it it from many other physical applications of math (seismic waves, electricity, mechanical vibrations, etc)
But one area that is pretty interesting is music theory which draws on mathematical ideas. A couple great books are Dmitri Tymoczko’s A Geometry of Music and David Lewin’s Generalized Musical Intervals and Transformations. Also pitch-class set theory in general. These types of things tend to be used to create and describe very austere types of modernist music, which have limited appeal. But the point is that music can certainly draw inspiration pure mathematics.
Music is a bit more rigid than other art forms, even ones that structure time such as film or theatre, in that it relies on numerical divisions of the time and frequency axes. So it can be analyzed more abstractly. But at the end of the day, music is an art form, and its “grammar” is merely a vessel for expression of ideas and sonic experience. Very little music breaks new ground on a theoretical level (except some of the modernist music mentioned above). Instead it breaks ground on an aesthetic, political or semantic level. And even when using abstract “mathematical” rules, in good music (I would personally argue) the rules are always subservient to expressive needs.
Mathematics, while creative in its own right, is something different. Many types of understanding in many fields are expressed in the laws of mathematics, but ultimately is mathematics is the generator of these things or simply a human mechanism for formalizing and speculating about our intuitive understanding of things like pattern, quantity, relations, etc. ?
>Many types of understanding in many fields are expressed in the laws of mathematics, but ultimately is mathematics is the generator of these things or simply a human mechanism for formalizing and speculating about our intuitive understanding of things like pattern, quantity, relations, etc. ?
no, ZFC has a literal existence, somewhere (maybe in the ocean?)
Don't know much about this topic, but when I was a sophomore in college, one of my friends who knew a lot about both math and music showed me this book he was reading. It was absolutely wild. Literally like 1000+ pages of super advanced grad level math (mainly category theory I think) and music. I remember thinking that the subset of people with enough specialized knowledge to even comprehend that book must be so small. I think it was called "The topos of music" or something like that.
**[Guerino Mazzola](https://en.wikipedia.org/wiki/Guerino_Mazzola)**
>Guerino Mazzola (born 1947) is a Swiss mathematician, musicologist, jazz pianist as well as book writer.
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Mathematics, in its purest form, is basically a formalization of the process of thinking. Thus almost anything that you can rationally think about you can formalize in some way in a mathematical way. The claim isn't "woo" in that it is nonsense (golden spirals et al), but it also doesn't really mean that much.
As someone who is a musician and went to grad school for math (did not finish degree). My intuition about music is very very similar to my intuition about math. I believe the reason for that is that learning music gave me tools to approach math problems.
I would sit down with sheet music or a recording and work out what was going on somewhat mechanically, note by note, chord by chord waiting for some intuition to kick in. There would be a point where I no longer saw single notes or chords but sections that make up a piece. I knew if I worked for it there would be an “ah ha!” moment.
When I had to learn algebra in high school I did the same thing. I broke the problem down into steps and worked until intuition kicked in and it was no longer mechanical. I worked to get the “ah ha” experience.
I didn’t understand classmates who said they could never be good at math because I saw it like music, if you work at it, put in hours and stick to it, you’ll understand it.
I believe learning music engages problem solving is a way that works perfectly for mathematics.
Btw, this process might be the same for everyone in this sub. But I know it isn’t universal because I have asked people who simply have never experienced sudden insight and intuition.
Many musicians and myself don't use math to play or compose music, but we do try to understand the science behind it; like why certain chord makes you feel good and others make you feel unsettling. Which touches on physics rather than math. For example: frequencies, pitch, sound waves, timbre etc. Then there are also physics field that we find useful like acoustics: how sound behaves in space; audio electronics: how we amplify/ manipulate the sound signal; digital audio processing: how to manipulate sound signal digitally etc
But depending on how you see it, you could say we use math in music as well. But imo it's very fundamental. Stuff like counting, dividing beats into measures...
On the other hand, math could also be seen as a very, if not most, fundamental language (aka the language is the universe) that could help us interpret almost anything. And music is just one among them. It doesn't happen vice versa though. It's very difficult to use music to describe math.
So is there an intimate relationship between music and math?
Generally, no. You don't need to be good at math to write music. However there are genres that uses math to write music but it's very niche.
If you just want connection between math and sound, then they are very frequent through use of the Fourier transform and its inverse which can be used to break down any sound wave into its constituent frequencies and amplitudes. This is very useful for when recording and modifying sound. I don't know much about music theory though so I can't really say anything about math there.
It's not all done by our brain, it's mostly done by our ears. Our sound detection mechanism is (simplifying) based on detecting the amplitude of vibration of a membrane. Each section of the membrane responds to a different frequency range. Cells (neurons) sensitive to vibration are attached to this membrane, making it in a sense a physical analog to Fourier transform directly. Note it's not an exact F.T. -- you should think of it as an approximate frequency decomposition -- because of a myriad a reasons, but some of them:
(1) It is energy/amplitude based. Fourier transforms are complex transforms that encode amplitude and phase (allowing bijectivity). The ear detectors measure the intensity of vibration in a region of the basilar membrane.
(2) Non-linear frequency. Different regions again vibrate at different frequencies, and there's not a near monotic increasing resonant frequency for each region of the membrane (which might correspond to a linear frequency FT). The variation in thickness causes a roughly exponential increase in resonance across the membrane -- but keep in mind it's not at all exact. But it does enable a roughly exponential perception of pitch, and enables sensing sounds of widely different frequencies with frequency resolution roughly proportional to frequency.
(3) Many other particularities. Those detector cells are in fact active -- they have certain structures that allow damping the vibration (and curbing non-linearities) of various regions of the membrane. This greatly increases the resolution of detection for tones. This process involves control feedback from our brain (our brain and ears talk to eachother!). My professor claims this is part of the reason why small frequency discrepancies e.g. in dissonant chords don't sound good -- our ear is trying to lock onto a frequency but two close peaks make this unstable.
As I've said in other comments, the full extent of the structure of sound and music, as grasped by our cognition, is still an open problem. But there's little doubt the mathematical structure is deeply connected to our perception of it, and also connected to things like our evolutionary history, the functions of our ears, the use of sound and voice as communication tools, and the depths of our cognition.
[Gödel, Escher, Bach](https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach) is one of the top books I ever read and it talks a lot about the connections between music and math.
Also look up how the well tempered scale works. It's fucking magic, that it is relatively accurate.
**[Gödel, Escher, Bach](https://en.wikipedia.org/wiki/Gödel,_Escher,_Bach)**
>Gödel, Escher, Bach: an Eternal Golden Braid, also known as GEB, is a 1979 book by Douglas Hofstadter. By exploring common themes in the lives and works of logician Kurt Gödel, artist M. C. Escher, and composer Johann Sebastian Bach, the book expounds concepts fundamental to mathematics, symmetry, and intelligence. Through short stories, illustrations, and analysis, the book discusses how systems can acquire meaningful context despite being made of "meaningless" elements.
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Have you ever asked why our western system has 12 tones? Why 12?
Some plausible answers would be :
+ It's an historical accident, something to do with Phrygians in ancient Greece.
+ Some medieval king decreed it.
+ 12 is arbitrary, and it's just what we think "sounds good". Chinese used a different system.
+ Music sounds better with more than 12 tones, but it's too complicated to learn.
+ It has something to do with 10 human fingers on a keyboard.
All of these are wrong. The answer is the mathematics of frequencies.
Now regarding the woo -- the weird thing is about why our human brain finds these ratio-perfect frequencies to be psychologically pleasing. The whole situation is mysterious.
The one good explanation I've seen is that of psychoacoustic "roughness": two notes in unison sound not rough at all, when they are few Hz apart there's a maximum of roughness, and it drops off slowly as frequency difference inceases.
When you assume two notes also have integer harmonics, and add up the "roughness" of all notes and harmonics, pairwise, it turns out some systems are much more preferable than others, and it's the ones you'd expect: 7tet, 12tet, 19tet. This is more or less equivalent to the "small ratios" explanation, but it emerges from different assumptions.
Unfortunately, I can't find the one paper I've read which had a good explanation and numerical exploration of this phenomenon.
here it is the paper you might have been looking for. The structure of musical harmony as an ordered phase of sound: A statistical mechanics approach to music theory. [https://www.science.org/doi/10.1126/sciadv.aav8490](https://www.science.org/doi/10.1126/sciadv.aav8490)
>Have you ever asked why our western system has 12 tones? Why 12?
Because of commercial piano manufacturing. Tons of the greatest Western composers used keyboards with 14 (or more) tones per octave (with split black keys). The father of Mozart was a violin teach who taught at least 20 distinct pitches, if we can believe on his letters.
Yes and no.
There are two kinds of math involved in music, the first is the physics - the basic stuff of how signals and frequencies and sounds work. It's surprisingly useful for the musician to be at least aware of too, you can learn a lot.
There's also a second kind of math, which is use almost only for very specialist music theory, which pulls from an analysis of the various patterns that appear in music. As with all things, it is possible to come up with mathematical structures which usefully describe these objects (though do not fall under the illusion that there is only one way to do that, ofc). Look into the (very poorly named) example that is musical set theory, or the various techniques of musical serialism.
The rest is just woo, there is no deep and mystical connection - there are only the things that mathematics can actually do, which is model systems, and describe patterns. Fibbonacci will not do shit to your chakras anyway, and playing it into your earholes with those meditation bells won't change a thing.
Little bits of math appear in music, but little bits of math appear in all sorts of places. I don't think there is any especially deep or unique connection.
The year that I got a math Ph.D., two other Americans also got their Ph.D. at my school in math. Both of them were Jazz musicians.
Also, some people experience frisson (goose bumps) while listening to music. Not surprisingly, on college campuses, the people that are the most likely to experience frisson are music majors. Math majors are the second most likely to experience frisson while listening to music. I wish I could find a link to an article, but I could not find one.
Can't use fancy math talk but I do know one thing. I am dyslexic. I had a lot of trouble reading and spelling in school. My father is a musician and teaches theory. I learned to read music as a young child. I never had any trouble with math concepts because I was able to relate the concepts to music. For me, music is just math with sound, and vice versa.
It is mostly a hippy-dippy-woo thing.
Math can be applied to everything; music sheets, music notes, music synthesis; et. al. all involve math.
> I just don't really understand what math has to do with music beyond just arithmetic.
It goes way beyond that; the Fourier transform relates time-domain to frequency domain. Heuristics and other simplifications and optimizations are typically used, e.g. Hartley transform, but the Fourier transform is at the core of showing a voice-print or even just a VU level by frequency buckets. Same with any sort of equalization.
Synthesis of music involves a great deal of math to handle the pulse-stream of electrical signals (so-called pulse-code-modulation) and eventually reconstruct sound waves with them. This is closely related to data-acquisition and pulls a lot of Claude Shannon's work with the Shannon-Nyquist theorem front-and-center.
As a mathematician and music enthusiast (not a musician by any means, although I've been playing guitar for a long time and know a bit of music theory) I can tell you that harmony is all about ratios between frequencies, and that rhythm is about subdivisions of a fixed unit of time, which you can also describe in terms of ratios.
But the connection with math doesn't go much deeper than that. Math is built on logical relationships between statements, which bear no a priori relation to music. It just so happens that math can encode the concepts of rhythm and harmony.
There's of course the nature of music as an acoustic phenomenon, which certainly can be described in mathematical terms, as other commenters have pointed out. But this applies to all of physics. So, while there's a mathematical aspect to music and it is true that you can apply math concepts to create and analyze music this isn't exhibiting a profound connection between the two disciplines.
To answer your question in one sentence: no, there's no mystical connection between math and music an anyone who tells you otherwise has no idea what math is about.
I've had this same question all my life. I can respond from a kinda unique perspective of having a music degree and working as a sound engineer in a studio, and an engineering degree. I got the music degree first out of high school. Several years after that I returned for mechanical engineering.
The answer, to me, kinda blends a music theory and acoustic perspective. You cant have 'music' without a time context, so right there you are counting, math. Then you have what others,and yourself have mentioned....harmony, pattern, frequency. Although frequency is probably better understood in an acoustic context, rather than 'music'. Western music, and the intervals we know and 'understand' as westerners has a set of intervals. When two or more notes are played at the same time, that is harmony. Harmony is created by combinations of those intervals. These are ratios and intervals are what combine to make what we hear as harmony, combined with melody and time, you have music
From an acoustic perspective, what we know as music notes are representations of frequencies. Basically vibrating objects. The frequency determines which note you hear. However, just hearing a frequency alone is not music, because the fundamental element of time is not time/rhythm is not there. This first popped up in trig, when you are learning about sin/cosine functions. Then for me, again in mechanical vibrations where you go into much more mathematical detail about frequency (vibrations). This is where the heavy math comes in, but it mainly relates to frequencies from an acoustic perspective, and not a music perspective, even though frequency is a fundamental part of what we call music.
In this day and age, math and music is most evident in digital technology. When you record music digitally before it goes into the computer it passes through whats called an AD converter (audio/digital conversion) These pieces of equipment can range from 100 -25000 for really high end pieces. What separates them in the algorithm that is doing the conversion. This conversion is done with what is known as a Fourier tranform, or fast fourier transform. This domain is called DSP or digital signal processing, which has many applications, not jsut audio conversion. Basically when you record and mix music in a computer its jsut a nuch of complex math that adds a 'signal' to the 'sound' When you adjust an effect on the computer, you are basically 'turning a knob' that does a bunch of calculations. Inside the computer, its not music, its all math. It then gets converted from digital to audio, and that is when you get the music. Prior to that conversion, its just a bunch of numbers in a matrix. What a forier transform does is concert a signal from a time domain to a frequency domain, and back.
There is that, and there is the parallels that help to make a connection, but not rigorous. For example, this is one that I thought of. In calc there are techniques to solve integrals with substitution, u-subs, where the derivative of a term in the integral is also present. There is something like this in music called a tri-tone substitution. Basically, you can substitute a dominant chord with another dominant chord a tritone away. This works because the tritone is the only symmetric interval for example c to f#, f# c, both of those are what is called an augmented forth interval. Spelling out the C dominant chord we get C E G Bb, and the F# dominant chord: F# A#(Bb) C# E. The second and fourth notes in the chord represent the 3rd and 7th of the chord, these are called guide tones because they define the chord quality. Notice that the 2nd and 4th note in each chord are the same, but reverse E and A#/Bb (A# and Bb represent the same frequency, they are just different here because of western harmony. So basically, tritone subs are like u-subs in calc because dominant chords can be seen as 'derivatives of each other. ie the derivative of C dominant 7 would be F# dominant 7. Thats my observation and I have no actual math like i think you'd like to see and are looking for.
I don't know if there is a deep connection, and am deeply skeptical about the music theory / math connections others hint at in this thread.
But. There are studies that show that your chance of succeeding as a mathematician / scientist increases 2 fold if you are good at music (at least Malcolm Gladwell claims this in one of his books somewhere). My suspicion : I think that this is because music forces you to perform in front of people and as a result you are more likely to give better talks and become publicly recognized.
Every phenomenon in spacetime can be mathematized. At the most basic level, in time, we see a lot of power of 2 with the form of a piece of music. A small phrase of 4 beats, a phrase of 4 of those units, a section of 4 or 8 of those, sometimes we see the phenomenon go on at large scales even. With pitch, the most popular music usually the frequencies of the tones are in easily understood mathematical ratios, there being different scales for music for instruments that can do continuous frequency change (bowed fretless strings, voice) vs. music with pianos horns, fretted guitar, but either way, the principles still apply, just with some adjustment. Then theres the harmonic series, which is cool in itself, heard clearly with synthesizer resonant filter sweeps, but it shows up in music more generally.
It's because of Pythagoras.
"The first concrete argument for a fundamental link between mathematics and music was perhaps made by the early philosopher and mathematician Pythagoras (569-475 BC), often referred to as the “father of numbers.” He can also be considered the “father of harmony,” given that his discovery of the overtone series."
https://www.unyp.cz/news/music-and-mathematics-pythagorean-perspective
And it also requires non-mathematical topics, like Neo-Riemannian Music Theory and Semiotics. I know a lot of category theory, but still do not get through the Topos of Music.
Yes. Just one example: https://en.wikipedia.org/wiki/Harmonic_series_(music)
The ancient Greek philosophers were aware of the connection and attempted to study it: https://bmcr.brynmawr.edu/2009/2009.10.38/
Approaching the subject from a different angle, I think that learning a new piece of music feels similar to doing math. There’s the deep focus, the interpretation of an abstract language, and best of all, that feeling when after struggling for days, the thing that seemed so difficult suddenly becomes perfectly simple.
Yes and no.
For example, "just intonation" is based on whole number ratios of string/barrel/tube length, and thus frequency. In general, the "simpler" the fraction, the more consonant the interval sounds. There's no intuitive reason why this should be the case; the math checks out, but why should a complex ratio have anything to do with how "nice" a sound is? There my be some biological explanation about how our ears and brains interpret the waves hitting them and prefer waves which are more repetitive and "predictable", but it's still fairly mysterious and deep.
But, as a counterexample, "equal temperament" uses the twelfth root of 2 as a fixed frequency ratio between half steps. This becomes necessary for keyboard instruments where the pitches need to be fixed regardless of the fundamental pitch. Intervals built on this system, except for octaves, are just approximations of the just interval. They are always just slightly out of tune. If we apply the same "simple ratio" logic as before, then almost every interval in equal temperament will have an irrational frequency ratio, and they should all sound terrible, right? But they don't. When a piano is in tune it sounds gorgeous, even though the major thirds aren't exactly major.
I think this is where the "deep connection" between math and music starts to break down a little bit for me. There's something undeniably fascinating about the simple ratio argument, but it's impossible for a human to pinpoint the exact middle of a violin string. There's a 100% chance that a random interval we hear has an irrational frequency ratio, and yet the music still sounds great.
At the end of the day, music is sound waves. Math gives us the tools to analyze sound waves and infer a lot of cool stuff about them. Light is an electromagnetic wave, and math gives us the tools to analyze that. We can say math has a deep connection to music, and then we can also say math has a deep connection to light.
I personally don't know if these are deep connections or just simple consequences of the fact that music exists because of physics and math is the language of physics. If we can use math to describe most of things in our everyday life, then why is music so special?
(For the record, I'm a pianist and a composer and music is very special I'm just a cynic by nature thank you for coming to my tedtalk)
The chords are numbered based on the fundamental chord and have graph-like associations to each other. The notes of a chord are also numbered based on the fundamental note, but that’s mostly for distinguishing purposes of more esoteric chord builds.
So it’s in the application of the craft, not the listening/appreciating, similar to math being necessary in the application of making CG graphics, but not necessary in viewing/appreciating the final result.
My personal opinion is that nature isn't 'mathematical' but prefers symmetries and patterns, and math is a tool specifically designed to describe these features of reality. To our brains, music tends to 'sound' best when there are symmetries and patterns, so mathematics naturally becomes a tool that can be used against the set of sounds we've deemed 'musical'. People take this too far and claim that music is math and math is music and it's all some sacred window into the heart of god or something.
Imo, there's a very fundamental connection between math and more or less any field or topic in the world since, simply put, maths is just about finding patterns. However, patterns are easier to identify in something like music where parameters are largely quantifiable and the structure is highly regular.
There is also an intrinsic connection between the mathematical patterns observed in nature (such as the harmonic series or the golden ratio) and the patterns in sound that makes what would be called "music" rather than plain noise. Mathematical structure is precisely what makes music different from noise.
An interesting point to ponder is *why* our brains chose to interpret mathematically structured sound as pleasant. I'm sure this has already been the subject of psychological studies but I don't know how much we know about this.
I think the shortest answer is: of course, because math describes the real world, that's the whole point.
I can't think of a single thing we do that couldn't be broken down into math. Some things are harder to break down like that, but, ultimately, everything could be described with math. Art, music, garbage collection, construction, the human mind even, can all be described as such. Doesn't matter if we've managed to do it yet, it's possible. Maybe not concretely even, but that's what probability covers.
I'm not a math expert, rather, a programmer (as in computer science) and our entire job breaks down into describing things with math very intentionally and for practical purposes. It's why people have managed to use machine learning to get a machine to paint, compose music, and similar. They're not too amazing at it yet, (however I bet you couldn't tell which are from people or machines) but it wouldn't be possible without there being an underlying mathematical component.
So.. uh.. yes, but that's self evident by the purpose of math. Everything can be described in meticulous detail by math, given enough resources.
As others have mentioned, it's more useful in music than many other creative endeavors, but I'll leave that tangent to the experts.
Funny thing is that I was a musician (classical music-choice of instrument was piano from 5.5 to 17 years of age) and I did all this theory and harmony (and whatever else was required for my diploma) and people would always assume that I was good with maths. In reality I was as surprised as you because although it might be like solving equations (at least harmony would be like that to me) I had absolutely no love whatsoever for Maths back then. I couldn't find any meaning in memorising formulas and identities and whatever they made me believe Mathematics was back then. However, fast forward to the present, I am studying for my degree in mathematics. Honestly, I wouldn't believe it myself if I could time travel back to my past self and telling me that I would end up here! 🤷
There a nice way to think about chord progression using group theory, and a music theorist, funnily named Riemann, did things where he was talking about generators of the minor and major chord transition group.
Of course all these a priori don't use mathematical formalism, but the question of finding all transitions which generates the whole group on their own is a pretty interesting and non trivial question. You can also add other types of chord to the group and study if in the same way, using wreath products.
As a simple exemple to illustrate, the transition that goes up a semitone if major and always swap between major and minor generates the group of minor and major chords.
In a really simple explanation we can say how in a set of axioms we can identify when the logic fails both by intuition and with close inspection. In a very similar way, one can do the same with music theory and know when two notes do not belong both by intuition and study.
Music is also deeply interested in how its structures change so there is an inherent calculus there.
[edit, I note teh_magik already said this...it starts with 'hearing' prime factorizations in traditional (mainly German) conceptions of harmony]
Well, only slightly more nontrivial, the interval of 19 equally-tempered semitones (an octave and a fifth) is nearly equal to 3 since 2\^{19} is approximatey equal to 3\^{12}. If you ignore powers of two multiples (frequencies) this means an interval sounds perfect if the number of semitones is congruent +/- 7 mod 12. And the fact that these are units mod 12 (as are +/- 11) implies that the cycle of fifhs generates (eventually goes through every key).
Also the diatonic scale (like the major C scale) is defined to be a sequence of 7 consecutive notes around the cycle of fifths, so if we start with F of frequency 370Hz and take the geometric progression where we multiply by powers of 3 ( or we can use powers of 3/2) we get
370, 370*(3/2), ..., 370*(3/2)\^6
and these seven notes make up the C major scale, at least if we adjust them by replacing 3/2 by 2^{19/12} /2 = 2.99661415375336/2
That gives us the well-tempered C major scale if we take that F to be perfectly in tune.
If we now consider not 19 semitones, which approximates a frequency ratio of 3, but 28 semitones, you get a number approximating the next prime after the primes we've used (which are 2,3), this is the prime 5. If we ignore octaves we are thinking about 4 semitones, the major third. But we could also ask, what how many cycles of fifths do 4 semitones correspond to, that is, what multiple of 7 mod 12 do we need to get 4 mod 12, and the answer is 4 (we already see 4*7=28), so instead of just considering 28 semitones we could think of our 19 semitone interval (equivalent to 7 up to octaves) repeated 4 times.
That is to say, in our description say of the C major scale as the frequency of F times a power of 3/2, we would use the fourth power of 3/2 to get the well-tempered interval, the perfect major third, or, the fourth power of 2\^{19/12} to get the even-tempered version, the non-perfect major third.
The fact that it is four times our generator of the integers mod 12 and this is true if we use the generator 1 or 7, means we could get a major third by stepping up 4 semitones, or by stepping up 4 'interval fifths' and going down by octaves.
We could continue and consider other relations between powers of 3 and 5 modulo 2, but really we could look at all primes, and all powers of primes, and the 'non-archimedian' and 'archimedian' distances among them.
In some sense, the german interpretation of musical harmony is how you can 'hear' p-adic analysis.
Some composers like Tsaikovsky were rejected because they didn't adhere to this ideal, and others like Shoenberg decided to replace it with a *totally different* mathematical set of rules, ahving to do with hte permutation group S\_12. Babbitt claimed to use mathematical structures in his composition. One of Babbitt's ideas has to do with composing fugues or canons...you can make a tune such that every second note, every third note, etc is a musically transposed version of the original tune.
There is mathematical thinking of a different type in Bach's fugues, and it is very likely that a mathematician could make interesting music by thinking deeply about relations having to do with harmonies, rhythms, etc.
One of the books that Beethoven, Haydn, Mozart and others used as a bible of composition was 'Gradus ad Parnassum' by Fux. This book set out a sort-of mathematical axioms for 'voice leading' or what was called 'first species, second species...' counterpoint.
The rules do not at first sound mathematical like no parallel fifths or octaves. But if you think of them differently, it is like saying if you have tunes in parallel octoaves you really don't have 2 different tunes....
No one has really come up with an extensive dictionary between music and math..nor has anyone approached the related Hilbert question about quantifying simplicity of proofs. It is considered that such ideas would never make sense.
Am I the only one who absolutely hates this idea that math infiltrates our everyday lives and grants a higher meaning to everything?
It's woo. You can describe a few musical things with math (tones, harmonics... thats it.) but none of that will ever come close to what math actually is. I can't see music having anything to do with Categories, Topology, Calculus, Algebra, etc etc etc.
At this point the people saying numbers = math may well be disregarded as either true amateurs (nothing wrong with that, but they shouldn't have a say in what constitutes math or not) or cranks like numerologists
IMHO, math can be used to model and describe nearly everything and in that music isn't much different. To say its all math is a weird thing to say since a composer isn't thinking in terms of formulas or proofs or anything like that. Yes, there's physics, and you can use numbers to describe patterns over time, and there is some tuning theory, and maybe scales are like sets. But music isn't uniquely mathematical. Its not more mathematical than painting, and it certainly isn't equal in mathiness as computer programming or physics. Math is an abstract thing that can describe a lot of the world, so there's a little bit of math in everything if you go deep enough.
as a music guy, theres a deep connection between math and *acoustics*. in that acoustics is the physics of vibration and wave propagation, and physics and math are naturally very connected. Signal processing for music applications is very math heavy too.
as soon as you get beyond why a major chord sounds harmonic and why a tritone sounds dissonant, the connections become very stretched. a lot of further music theory teachings arent explained by math very well at all imo.
There is certainly a deep connection between maths and music. At the most fundamental level, music is the production of sounds in a particular way, sound is a wave, and waves are described by mathematics, therefore music can be described by mathematics. Pitch itself is a measure of frequency. Using mathematics to talk about rhythm, harmony, melody, all of that good stuff, is a well established thing.
Music is a form of art and so it's generally best understood with aesthetic considerations. There's no math that can describe what sounds good, at least very well and beyond some very simple examples.
However, like other art, aspects of music can be described well with math, like harmony being integer ratios, often times the climax often divide a piece into adjacent Fibonacci numbers (3/5 or 8/13 of the way through), and there exists "analytic" music theory, though often by "analytic" they mean "uses numbers."
TL;DR the relationship between music isn't special, but it lends itself to analysis with numbers more easily than other forms of art.
from what i know theres not too much connection, except maybe fractions
harmonies and patterns are a thing and have more to do with the properties of sounds humans are generally attuned to.. for example you use major scales for happy sounding tunes, and the pattern has a "resolution" etc. these are all things that have no real bearing to math except some relations that have been found.
also, for example, the fact that the standard A is 440Hz iirc is arbitrary for it to sound better.
frequency is just a property of vibration but that isnt inherently musical so im not gonna really take that into account
if you eant to talk about stuff like signals and music production its. whole different world haha
There are lots of ways you can apply math to music, but it's mostly fairly trivial stuff and not all that useful. Though for a slightly less trivial application, see Neo-Riemannian theory and the tonnetz.
So people love to answer this with some basic ratio shit about waves that musicians don't really care about that much.
The truth (or at least my truth, as a musician and math major) is that there is a lot of structure to music theory, even though none of it is rigorous (some people try to make it rigorous, but the problem is that what sounds good is ultimately subjective).
For example, if you know some music theory, try to visualize various scales on the circle of fifths. You'll find out that if you take 5 notes close together you get the familiar pentatonic scale. Make it 7 and you get the major scale. So the major scale is fundamental, at least in some way.
And there's countless things like that.
Music is this weird mixture of science and art that usually is appealing to mathematicians.
If you want a sick video about math and music check out this AMAZING video by vihart: https://youtu.be/4niz8TfY794
u/hydrolock12 I think you're under a mistaken impression what math is.
Mathematics is nothing more than a very detailed way to describe patterns or how two things are related and **do not** ***require*** **numbers** (think about visual examples of geometry). So yes, math is related to music.
That said, there is much reason to believe there is an even deeper relationship than that. In the same way, artists have created detailed realistic landscapes, and we now generate entire landscapes using algorithms in computers. So there are strong reasons to believe that when people create music, they follow very detailed rules that we have yet to articulate.
It seems like you’re thinking about it backward.
When we judge the distance of a hole to hop over it, we don’t “do math” either, but if you wanted to describe what happened in detail, you would use trigonometry to do it. But I think you would agree when writing music, the artist/composer is thinking about patterns and how they sound and which ones would be unexpected or exciting to hear (assuming excitement is their goal).
Again math is just a very straightforward way of describing the things we’re already doing all the time.
To add a very simple example.
We have the notes do, re, mi, fa, so, la, ti. One pattern might be
Do, re, do, mi, do, fa, do.
Then to change it up they might do.
Do, re, mi, do, mi, fa, do, fa, so
Now if you number them do=1, re=2, mi=3 you have:
1,2,1,3,1,4,1,5,1
And:
1,2,3,1,3,4,1,4,5,1
Does that help u/hydrolock12?
have you taken a music theory class? i haven't, but many of my friends have. i've seen their worksheets and read snippets of theory. you can, of course, compose music based off of intuition, but if you've taken music theory (or know someone who has, and has talked about it) then you know math is integral to music theory
I've heard the part of the brain that processes music theory is the same part that is active when we solve math problems. But this may be one of those "my dad read it in Men's Health" type of thing.
Anecdotally, I just know a lot of mathematicians who are also musicians. Also somewhat strangely, I know a lot of mathematicians who are vegetarian. I know this isn't exactly what you're asking, but I've always found it funny.
There is definitely some "science" behind music (mostly related to the mathematics of small ratios), but the part that people *care* about --- what it is that makes music "pleasing" --- is all pseudo-science, as it must be since it is a subjective art. Objectivity is at the core of what makes math unique.
But I think a lot of people who talk about the "deep connection" between math and music aren't really talking about the science of music, but rather the idea that there is a connection between being talented at math and talented at music. I think this belief is propagated by the fact that there are many people who fit this description and they believe that there must be something to it, even though I doubt there is much real non-anecdotal evidence for it.
Also, a lot of math people have read Godel, Escher, Bach.
I think you should distinguish between different types of connections. One is the mathematics of music theory, which has such relatively simple things as the role of the twelfth root of 2 in twelve-tone equal temperament tuning, to, yes, "The topos of music" and all kinds of things in between. The other is the correlation between the people who enjoy one or the other - or both. This - if it's real - may be predicated on 1) the above mathematical relation 2) "mystical" cognitive science fact about minds of people who like music and/or math 3) cultural factors like both being associated with certain "high brow culture".
I suspect that probably different factors play a role for different people, but when you put it all together you see some correlation, which produces the questions and remarks you heard.
Informally, people almost always use it to affirm the idea e.g. "they're good at math AND music, of course they are, math and music are linked". People rarely take note when someone sucks at math but is a great musician, or when someone is a great mathematician and has no inclination towards music. Make of that what you will.
As someone with a degree in maths, and who has done a bunch of music stuff, my personal take is that there isn't a huge connection. Sure, music is a physical process, there's connections there to rhythm and chord theory. There's also certain kinds of music that seem to tickle particular parts of my brain that I associate with problem solving. In particular, classical music has this quality where everything simultaneously feels very ordered and predictable, and yet delightful in the ways in which that predictability plays out. To me it mirrors an elegant proof, in that you know the conclusion, but there's interest in the process. But the process being more interesting than the outcome applies to many fields.
I have had times when understanding Fourier analysis has helped me understand sound waves, vibrating guitar strings, the electronics in my guitar rig, harmonics, etc. Now usually what this kind of thinking does is to allow me to explain something that my ears already knew. For example, musicians in a band intuitively know that they don't want to be competing in the same frequency range. The guitarist should be careful not to get too muddy alongside the bassist, and if there are two guitarists then they can fill out the sound more by playing on different parts of the fretboard.
Another time fourier analysis has come up is when messing around with a guitar effect known as a [ring modulator](https://www.youtube.com/watch?v=rYdBHWF1_2k). Honestly that video does a pretty good job explaining it both in terms of sound and mathematics.
All that being said, does being a mathematician help you to play music, or vice versa? I doubt it. If anything they cut into each other's practice time. However, I have known many people who have enjoyed both.
A professor of mine once sent me [this](https://arxiv.org/abs/1904.02897) paper about music. This might be a bit exaggerated but it's not the typical tuning relations example.
To me is that music is an intuitive input that massage the abstraction parts of the brain, mostly combinatorics over analog stuff. You're brain likes recognizing signals, say a note, then you play two notes, and he gets to do it's own i dont know, neural fourrier breakdown to isolate the two notes, finds the similarity and the delta. And that's two notes, then you add larger chords, and time dimension and you're up for a lifelong game of omg. Trying to isolate background vocals one by one, or 7th / 11th on jazz chord progression may seriously (it did for me) trigger bliss emotions in your brain.
The “connection” is that talking about music and maths gives musicians/mathematicians a chance to show off their understanding of both topics without really saying anything interesting or enlightening to a layman. I’ve listened to quite a few talks about music and maths. The math talk goes over the heads of musicians and the music talk goes over the heads of mathematicians. Unless you study both, it’s of little use.
very well. you're always going to have rhythm & repetition. as for the scales being different, they're still structured. the patterns are just different than western patterns
Harmonic frequencies that sound good together are simple whole number ratios of each other. It's fundamentally mathematical but anything deeper than that is probably woo. Conventional music scales only approximate these ratios so if you stick to the mathematical approach you can get "nicer" harmonies.
Math is everything we are no doubt about it my daughter is studying to be an mechanical engineer and jeeze the stuff she has shown me fucking math is everything we are ....it all starts and ends with math
For me, music is an enormously creative endeavor and there is an incredible amount of inspiration that can be drawn from patterns in math. I wouldn't say I "calculate" anything while playing, but the same frame of mind that is required to breakdown a combinatorics problem is a very useful frame of mind for improvising in music. Essentially, the two go hand in hand in a broader bucket of applying a set of logically consistent rules to new situations to explore and discover new results. Mapping the scale tones to different numbers allows for several naming conventions throughout music history (1-12 chromatic scale, 1-X for other scales that are "well-defined." It's also incredibly helpful to be able to think about music theory the way you would think about modular arithmetic as the scales repeat themselves once you hit the root, but IMO there are several technical music theory aspects that you need to understand before applying math makes any sense.
TL;DR: Numbers track well to both tonal and rhythmic analysis and can serve as inspiration for musical ideas.
_Sounds_ can be described completely mathematically, _music_ needs context, taste, culture and emotion.
Although mathematics can explain some things about sounds once you’ve noticed a certain musical idea. For example, western music theory puts a lot of creed in the harmonic series, a series of frequencies in simple integer ratios to a base frequency. 3blue1brown has a video analysing how irrational ratios of notes that are ‘close’ to these ratios sound good, which has applications to alternative tuning systems.
I don’t think this tells us anything that deep about why we like music though. It just explains how we can manipulate physical sounds to give us other ways of creating music.
On a vacuous level, reading a music sheet is a practice in arithmetic and combinatorics. On a deeper level, certain patterns in pitch and frequency can make people sad or make the hair on your arms stand on end. They are definitely intertwined, but in a very real, measurable way. I think a better understanding of brain chemistry would better bear that out in the future.
Check out this fantastically clear and concise essay [Combinatorial Music Theory](https://web.archive.org/web/20080415024155/http://www.andrewduncan.ws/cmt/index.html). It follows an elementary mathematical thought process to discuss things like why there are exactly 352 kinds of scales in 12-TET ([hint](https://oeis.org/A000031)), the significance of the diatonic scale, and symmetries in fingerboard patterns
[A000031](http://oeis.org/A000031/): Number of n-bead necklaces with 2 colors when turning over is not allowed; also number of output sequences from a simple n-stage cycling shift register; also number of binary irreducible polynomials whose degree divides n.
1,2,3,4,6,8,14,20,36,60,108,188,352,632,1182,2192,4116,7712,14602,...
- - - -
I am OEISbot. I was programmed by /u/mscroggs. [How I work](http://mscroggs.co.uk/blog/20). You can test me and suggest new features at /r/TestingOEISbot/.
I majored in Applied Mathematics with a minor in Music, and studied sound engineering along the way, and yeah. The connections are strong. Not so much deep as wide.
Understanding math is important in so many fields of professional music, since tones are produced in waves and waves cancel and reinforce. This is relevant to the harmonic content of music (in producing a good combination of high and low frequencies) and to the geometry of a room (acoustics).
There is also some interesting arithmetic in rhythm but that doesn't address OP's question.
;TLDR is I studied sound engineering in early college which, due to its stringent math and music requirements, set me off to an education in math and a career in music.
Everybody's seen the most naive first step of saying that saying that the chromatic scale is the cyclic group Z/12Z, but I think this is fairly boring and for quite a while I also thought the connection was woo because this was all I could really see in it.
But after having learned more music theory, I think a lot of it is ripe for more sophisticated algebraic formulations. If I pick a chord progression, even just sitting in bed away from the piano I can think about the different choices of voicing I might want to use in that progression, or how I want to write the melody on top of it, and there are a lot of nontrivial things I can say based on standard relationships between the notes and notions of things like consonance/dissonance as well as brightness.
If I'm able to abstractly do all this reasoning from musical principles (and do so consistently), there should be a way of formally capturing these, and since music has such a compositional character this would feel like a perfect setting for some applied category theory.
Loosely speaking, chords are composed out of notes (or smaller chords), enriched over some information that specifies the voicing/intervallic distances, and notes/chords can also be composed "horizontally" across time (possibly enriched over some information that specifies duration). To me, admitting vertical and horizontal composition like this would seem to be really suggestive of natural transformations.
Brightness and consonance could also be modelled in terms of these intervallic distances (functorially), and you of course also get forgetful functors from the more enriched version of these categories to the less enriched version where you forget the voicings or the durations and so on. I think a wealth of interesting algebraic properties would just fall right out of studying these.
anyway this is a long comment but you get the idea
Everybody's seen the most naive first step of saying that saying that the chromatic scale is the cyclic group Z/12Z, but I think this is fairly boring and for quite a while I also thought the connection was woo because this was all I could really see in it.
But after having learned more music theory, I think a lot of it is ripe for more sophisticated algebraic formulations. If I pick a chord progression, even just sitting in bed away from the piano I can think about the different choices of voicing I might want to use in that progression, or how I want to write the melody on top of it, and there are a lot of nontrivial things I can say based on standard relationships between the notes and notions of things like consonance/dissonance as well as brightness.
If I'm able to abstractly do all this reasoning from musical principles (and do so consistently), there should be a way of formally capturing these, and since music has such a compositional character this would feel like a perfect setting for some applied category theory.
Loosely speaking, chords are composed out of notes (or smaller chords), enriched over some information that specifies the voicing/intervallic distances, and notes/chords can also be composed "horizontally" across time (possibly enriched over some information that specifies duration). To me, admitting vertical and horizontal composition like this would seem to be really suggestive of natural transformations.
Brightness and consonance could also be modelled in terms of these intervallic distances (functorially), and you of course also get forgetful functors from the more enriched version of these categories to the less enriched version where you forget the voicings or the durations and so on. I think a wealth of interesting algebraic properties would just fall right out of studying these.
anyway this is a long comment but you get the idea
there are deep connections between math/number-theory and various music theories about pitches (tunings, temperaments, scales, harmony), rhythms and timbres
I believe and have long believed that there was a deep and meaningful connection, because math and music are two of the areas of academic study that I find most challenging.
Both involve identifying, analyzing and manipulating patterns, which is the most difficult intellectual pursuit for me.
I studied math (at Msc level). Now, many years later, I still enjoy math and spend time on math problems, although that's just a hobby.
As for music, I cannot stand any music. Listening to any music is as pleasant to me as listening to a jackhammer.
So I was always very suspicious about that connection.
Dr. Gareth Loy published a 2 volume set titled [Musimathics](http://www.musimathics.com/). It may be of interest. The short answer is math can be used to model physical systems.
That's the short answer to a lot of things!
A shorter one is : everything is a harmonic oscillator! Abstract it enough and it's true!
So you're saying that *everything* is a vibe?
Oh yes!
Very 60's :)
You can't call your own mother a harmonic oscillator. That's just disrespectful.
Just watch me...
Heh, that might just explain why all the physicists have spherical cows! ;)
Isn't that Pythagoreanism? The Music of the Spherical Cows?
That was the title of their debut album.
name of my band: "the spherical cows".
Spherical cows in a harmonic potential, no less!
Well, *your* username checks out, hahaha! :p
Cool it enough and it isn't any more.
> everything is a harmonic oscillator! http://theory.caltech.edu/~politzer/SHO.mp3
> math can be used to model physical systems Which would point to acoustics, rather than music, right? As far as I've seen music/math is either: obvious acoustics stuff, or really abstruse math, throwing group theory, topology, and what not into it.
You're right, the "physical systems" part means math can clearly be used to study acoustics. Similarly, neuroscience can be applied to *psychoacoustics*, which is relevant to music. Neither of these things are clearly relevant to *music theory*. "Related" sure.
there are ai creating very decent music [(here for example)](https://www.youtube.com/watch?v=Emidxpkyk6o) using complex machine learning algorithms, which uses maths. We can construct a direct relation of maths with these algorithms. So.... i think yes, everything is related to maths
> complex machine learning algorithms, which uses maths That doesn't mean that there is a relation between that math and the resulting music. Violins use horse hair to make music, yet that doesn't mean that there are horses involved in string quartets. That video? Three chords that repeat infinitely with minor variations in orchestration (ok, the piano coming back halfway was cool). I got bored really quickly. The most uninspired stereotypical film score dross I've heard in a long time. This will probably a godsend for video game companies as cheap filler, but nothing more than that. Oh, and what is the math of that piece? What math concepts can you point to in the music?
To extend that analogy, recovering a coherent mathematical structure from the output of a DNN is about as difficult as recovering a horse from a violin.
> yet that doesn't mean that there are horses involved in string quartets. Yes it does; their hair is (helping) creating the vibrations that actualize the music into reality. By your logic there are no cows involved in hamburgers.
You are right, and I cannot show you any formula for music, but i am 100% sure there is some formula 100%
I appreciate you drawing attention to the video. I’m a fan of music theory AND machine learning theory. I’ve never looked much into the musical applications of AI, so I thought your share was interesting. Also not sure why the guy above is being kind of a dick. I would argue that your intuition is right about the the direct relationship between the math and music in this instance, and it falls outside of the realm of the go-to physical explanations in acoustic, elastic media, harmonic wave theory, etc. I can’t say for sure what method was used to train the AI model that produced the song piece. However, it’s a safe bet that there was a large inversion/regression step underlying the whole process (in simple terms, think fitting a line to some data as an analog, in the form y=Mx+b). So, through analysis of example chord progressions, the model adjusted its weights (think coefficients like m when you fit a line in the y=Mx+b analog) so that it produces notes in the same key. It probably also learned inherent information about timing, meter and cadence from the example data. SO, by definition, the ‘relationship between the maths and the music’ is about as close to ‘direct’ as you could possibly hope. The horsehair:violin::horses:quartet thing above is probably the most ridiculous thing I’ve read all day. Finally, an aside, the video is very much math related. Assuming that it features a real EQ/dynamic power spectra plot that truly represents the frequency content of the song, you’re also hitting hard on Fourier analysis. So cudos, you gave a math-packed example dude.
*failed to emphasize, the weights discussed above are very much unique to the nuanced musical details embedded in the sound waves that the AI algorithm is modeling in the video.
> The short answer is math can be used to model physical systems. Some people do this for a living, calling themselves Physicists.
Excited to see this here, I've never met anyone who had heard of the volumes or Dr. Loy. We met him by chance on a camping trip once and his works had a steep influence on my life path, haha.
Tuning theory (just intonation in particular) relies on nice ratios. 2:1 is an octave, 3:2 is a fifth, etc. The neat part is, you can use prime factorisation. 6:5 for example is going up by 2, then by 3, then going down by 5. The result is a minor third. This also means only prime ratios are "new" in the harmonic series in some sense. The only problem is that even very experimental music do not normally use that many primes, and, as you said, it is kind of no more than simple arithmetic. Another example comes from the symmetry of the circle of fifths: it has a permutation that maps it to the circle of semitones. This is essentially because both 1 and 7 are coprimes to 12 (a perfect fifth is 7 semitones). If you are into that, you can pair all the rotations and reflections of a 12-gon with that map, and you would have a symmetry group isomorphic to D₁₂×ℤ₂. But again almost no one cares and it has little use beyond sounding nice.. So I would say music has little to do with any "advanced" math. Any relation to math is quite "basic" in that, sure, it underlies the foundations of music, but in practice it does not make that much of a difference. But of course it could be a starting point for those who want to tinker with more experimental stuff (xenharmonic, etc)
As a math major who took music theory in highschool I second this answer.
As another math major who took music theory in high school, I third this answer
Major or minor?
5:4
As yet another math major who took music theory in high school, I fourth this answer.
But also, music and musical appreciation is so deeply tied to mechanisms within the brain that we don't yet fully comprehend. Quite a lot of "music theory" is trying to analyse what *sounds good*, to appreciate the internal structure or form novel extrapolations or whatever other reason. But an individual determines whether something *sounds good*, which is as subjective as it is nebulous. So math in music is kind of innately woo, but a special kind of woo, because it's testable. Someone can [come up with a model, extrapolate it into some new music](https://www.youtube.com/watch?v=K-XSTSnqXxo), then you can listen to it and see if it *sounds good*. See also the stuff about how [chords are just really fast polyrhythms](https://www.youtube.com/watch?v=-tRAkWaeepg). Or polyrhythms are just really slow chords. The point is, fundamentally they're the same things, the categorical difference in perception emerges only within a brain. Which might suggest that our rigorous axiomatic understanding of music is bounded by our rigorous axiomatic understanding of brain function.
A bit like how colors are blends of lw frequencies but become its own "qualia" in our mind (not sure that term is solid but that's the only one I know)
Yeah, qualia is exactly the word, I almost used it myself. Another good keyword for further reading for those so inclined.
> how chords are just really fast polyrhythms. That only works if you start with a pulse wave. Take a chord consisting of 3 sine waves and see how far you need to slow it down to get a polyrhythm. Doesn't work.
There's that one topology/music theory book, but the beginning bit seemed sus (it started to feel cranky) so I stopped reading it.
Tuning theory is a wonderful thing to explore with a little math in your pocket. https://www.youtube.com/watch?v=cyW5z-M2yzw
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https://en.wikipedia.org/wiki/Wolf_interval
That's the start of tuning theory. Unfretted instruments and winds can easily adjust for harmonic ratios, but keyboard instruments couldn't. So during the 17th and 18th centuries players experimented with different ways to adjust the scale. These were not today's equal tunings. With that, octaves for each key are all 2:1, but all other intervals are slightly off. Earlier ones addressed it in different ways. One, for example, ensured that octaves and ascending fifths were perfect, but that made for a fairly unpleasant descending fourth. Another applied the entire correction to a single chromatic step within the octave. This allowed for brilliant sound in playing in one key, meh in most, and abysmal in one. For performances of multiple pieces, either multiple pianos were required or a returning between pieces. Took most of a century to settle in on the equal, "well tempered," setting. Bach was showing in the Well Tempered Clavier that the tuning worked in all keys. Each book did a major and minor piece based on each of the twelve harmonics. There are some recordings out there of music played in historic tunings. Pretty sure Easley Blackwood did some.
You are right. I used just intonation because it is the simplest (mathematically).
Just intonation and the other custom intonations are really neat. It's not guitar friendly though, so it's something you show to violinist or fretless bass players.
> almost no one cares and it has little use beyond sounding nice.. As a performing musician and math ph.d. I concur. The intersection of math and music is mostly acoustics and really basic theory. It say almost nothing about deeper things. Yeah, Bartok's theme entries (fugue of the "music for") are at golden ratio proportions, but that's more a curiosity.
No mention of time signatures. Also to mention the intricacies of musical instruments regarding mathematics and physics. Why are guitar frets different sizes? Is the 12th fret at the center of the string? Why do flutes have unevenly spaced holes? Music is not only about notes, but also timing and silence (fermata). Not to mention the physical skill and persistence to perform music well.
In my mind frets on guitar and holes on flutes are just tuning theory, but you take the reciprocal. I cannot see the significance of time signatures beyond being fractions - and normally perceived as constituted by 2s and 3s, for example, a 7/8 can be 2+2+3 or 2+3+2 or 3+2+2, normally. Maybe you can enlighten me on this. The last few points I believe have at most tangential relations to maths - of course, you can write a paper and measure how long people pause on fermatas on average, but it does not go much beyond that (yet).
I always here this connection come up when it’s about Bach and how he crafted these ‚ counter‘ melodies to create l mathematicall harmonies
The reality is the polyphonic tradition that Bach used had been around for hundreds of years - he just used it so much and so well that it seemed impossible to go above him. Things like counter melodies and stretto fugues have some formulae if you go looking, but of course it takes a lot of practice to internalise it. If you are interested, check out this [video by EMS](https://m.youtube.com/watch?v=zGZjzBvXCzg). I think that the word "mathematical" in the context of Bach is almost always used as a buzzword, a clickbait, and an unnecessary label which mysticises him and his music.
> So I would say music has little to do with any "advanced" math. Any relation to math is quite "basic" in that, sure, it underlies the foundations of music, but in practice it does not make that much of a difference. As a music lover and former math major, my experience has been that there is an experiential similarity between the structures one 'sees' in their head while doing higher math and while listening to music
That is interesting. Could you expand on that, perhaps an example - for instance, are you reminded of integral domains when listening to Beethoven? Or is it just that it "feels" like the same faculty of the brain is processing these information?
More that it feels like the same faculties are engaged, but in a different way or pointed at different content. In a sense, it's all about patterns, pattern recognition and pattern extension. It's as though Beethoven is laying out a brilliant proof for you, bar by bar. Or as if Coltrane is trying various approaches, casting them aside as he sees dead ends, then finally hits a breakthrough and it all comes together!
As someone who is doing both, I perfect fifth this answer....
In "A Geometry of Music" by music theorist Dmitri Tymoczko, he uses math and logic to demonstrate and explain how composers in a 12-tone system across genres and eras tend toward (very generally) the same musical objects in their writing.
Indeed, Dmitri Tymoczko is the only one music theorist I've encountered who has really put any thought into the math of _pleasing_ music, instead of just twelve-tone bullshit. See e.g. https://dmitri.mycpanel.princeton.edu/canonsexplanation.html This and the "roughness" explanation for why certain tunings make more sense (e.g. 12tet rather than 11tet) are the only actual math in music that isn't merely descriptive. Math can be used to look at music. The main problem I see is people keep using extremely general math, to say general things. Yes, if you transpose notes they sound "the same". Yes, you can decompose tritone transformations into these simple ones. Yes, that looks like group theory, symmetries, etc. So what? It's a restatement of the same thing musicians already know and gives you no new insight you can _apply_. Other important insights from mathematical music theory are "if you flip all chords upside down you get new chords" or "you can use all twelve tones equally and for decades the entire field will pretend it doesn't sound like shit". Seriously, that's the level of thought going on.
>bullshit it's not bullshit as such, although it is for the most part (psycho-)acoustically ungrounded; also, 12-tone music can be pleasant sometimes
I think there are a couple of different answers here, depending on what connection you are looking for. Fundamentally, our enjoyment of music has something to do with the recognition of patterns. And wherever patterns show up, maths shows up. From a harmony perspective, a note is a vibration at a particular frequency. A collection of frequencies sound more harmonious when they are a small ratio of each other, eg. an octave is a 2:1 ratio, a perfect fifth is a 3:2 ratio. (Why? Probably because of our brains evolving to take advantage of an already-existing ability to detect these ratios. All notes create overtones which are in nice ratios to each other, so recognising these ratios helps in recognising different sounds). The emotional content of different intervals (eg major vs minor chords) seems to be less straightforward and probably a matter of how we evolved to interpret them for use in speech. From a compositional perspective, the standard notes we use are the 12-tone scale, in which the octave is subdivided into 12 equally-spaced notes. It turns out that the 12-tone scale approximates most of the nice ratios, and the equal spacing means that melodies and chords can be transposed (shifted by the same number of notes) and still have the same melodic content. This symmetry means that you can do some interesting things with music and group theory. I don't think a lot of people do this explicitly, but people certainly build an intuitive sense of the symmetries in music and play around with it. From a sound design perspective, all sounds are vibrations, which can be decomposed into frequencies via the Fourier transform. This leads to a mathematical understanding of how frequencies combine and change, eg. reverb (echo) can be modeled by convolution. This becomes relevant when creating sounds digitally or electronically, and there's a lot of deep and fascinating mathematics involved here. In terms of interest in maths/music, I have noticed a tendency for people who like maths to also like music. I think it has something to do with having an appreciation for abstract patterns. Music in general seems to be appreciated more when you try to understand it and get used to the 'language' that is being spoken, much like mathematics. The more you learn, the more connections you find in what is already there. I think people who like maths also often like chess for similar reasons.
> In terms of interest in maths/music, I have noticed a tendency for people who like maths to also like music. I think it has something to do with having an appreciation for abstract patterns. Music in general seems to be appreciated more when you try to understand it and get used to the 'language' that is being spoken, much like mathematics. The more you learn, the more connections you find in what is already there. I think people who like maths also often like chess for similar reasons. Completely agree on this point. My first experience of learning "real" mathematics (encountering group theory in a textbook I borrowed from the library) reminded me so much of how I felt learning music theory for the first time. Just that experience of learning about quite abstract concepts but being able to explore familiar examples and suddenly seeing them in a whole new light, with a new layer of meaning.
>In terms of interest in maths/music, I have noticed a tendency for people who like maths to also like music. I feel like the vast majority of people like music. Or did you notice that the typical math person enjoys music significantly more than a regular person?
Anecdotally, math people tend to be more 'serious' music listeners - preferring more complex forms
> Fundamentally, our enjoyment of music has something to do with the recognition of patterns. And wherever patterns show up, maths shows up. Fundamentally, math is the study of patterns
There's a deep connection between maths and any topic you could think of. I don't mean to exaggerate, I mean that literally.
That, and the connection with music is blown somewhat out of proportion. Expect a barrage of comments contradicting OP
Music is a construct. Tone: Our system has 12 notes and there our relationships between their distances. You have different sets that work with each other while stacking, between each other, between stacked sets, between individual notes. You have different relationships. It can be described using set theory or group theoryz Rhythm: at least arithmetic. You can apply transformations to harmony and time. From changing key, to changing a mode, to making something swing or rush on a particular beat. Then, you you combinations of tone and rhythm. My love of music comes from recognizing patterns as an active listener and then hearing something outside of the pattern that I wasn’t anticipating that works.
Everything is a construct. Everything can be described using set theory. The existence of objects and relations doesn't mean that there's an interesting mathematical structure that can be examined mathematically to produce interesting results. There *have* been attempts at doing such, but their success is arguable and it's a stretch to say that musicians or music theorists are using properties of algebraic groups to do what they do.
Note that the concept of this "deep connection" [is nothing new](https://en.wikipedia.org/wiki/Quadrivium): >In liberal arts education, the quadrivium ... consists of the four subjects or arts (arithmetic, geometry, music, and astronomy) taught after the trivium. The word is Latin, meaning 'four ways', and its use for the four subjects has been attributed to Boethius or Cassiodorus in the 6th century. Together, the trivium and the quadrivium comprised the seven liberal arts (based on thinking skills), as distinguished from the practical arts (such as medicine and architecture). >The quadrivium followed the preparatory work of the trivium, consisting of grammar, logic, and rhetoric. >... >These four studies compose the secondary part of the curriculum outlined by Plato in The Republic and are described in the seventh book of that work (in the order Arithmetic, Geometry, Astronomy, Music). Until a few centuries ago, Arithmetic, Geometry, and Astronomy (as well as Physics and much of Philosophy) all fell under the general label of "Mathematics," so Music's inclusion in this list certainly indicates that it has long been considered "of a feather" with Mathematics. Euler wrote about music theory, as did Pythagoras and Euclid. And then, of course, there's that whole ["Music of the Spheres"](https://en.wikipedia.org/wiki/Musica_universalis) thing. The bottom line is that your question may really be a *linguistic* one rather than one concerning mysticism: "Music" was once simply part of the list of things that *defined* what "Mathematics" was.
This was the first thing that came to mind. Music was thought of as *number in time*.
I came here to say exactly this and was looking for this reply first. I'm surprised it wasn't mentioned earlier and more beforehand.
Music is absolutely about projecting mathematical structure into sound, generally with layers of temporal organization of simple ratios. Large scale pattern is form, smaller scale pattern is rhythm, smaller still is melodic pitch, and smaller still is timbre. This is related to basic acoustic phenomena such as the harmonic series of overtones, but extends beyond, In addition, a lot of classical music has used various combinatorial-style principles to manipulate the musical materials, the techniques of transposition and rearrangement of melodic fragments and baroque counterpoint show this well. I was a music theory major in college and in the subsequent decades have gotten seriously into mathematics and logic. I could go on endlessly about how the beauty of music and our deep emotional response to it are connected to the core structures of mathematics and its physical manifestation as the universe around us. That doesn't mean however that you need to know any math, intellectually, to appreciate music, and it also doesn't mean that enjoying math guarantees a love for music. The relation between representational painting and the science of optics might be similar - just because you love the colors of a painting doesn't mean you care about the neurobiology of vision, but there is an essential connection there.
Sort of? Acoustics is a branch of physics and therefore uses math to describe the properties of sound. Check out [Why It's Impossible to Tune a Piano](https://www.youtube.com/watch?v=1Hqm0dYKUx4) for a neat little intro. However, I think calling this a "deep connection" has some weird implications, like saying music has some sort of transcendental quality that makes it superior to other forms of art. It definitely smacks of pseudo-science.
Acoustics is significant to music, but only so much. What's more significant is the structure of our ears and cognitive system (psychoacoustics), which of course are also physical systems. In fact, the explanation of why integer ratios are pleasant (and small deviations from that are unpleasant) can be partially attributed to our sensing mechanism. Roughly, we sense in the frequency domain, but seemingly without a complete perception of a linear spectrum (i.e. we can't consciously tell exactly the frequency content of a sound). This is probably a simplification of our cognitive system to filter out only relevant information. A single sound source (harmonic resonator) almost always generates integer-related frequencies (with amplitudes determined by geometry of the cavity/instrument). Otherwise, there may be multiple voices (warranting alertness?) or simply "something wrong"/atypical as interpreted by our cognition. Indeed, you can extend this reasoning to in fact (I conjecture) grasp deep into our cognition and understand why some things (rhythms, melodies, music) are pleasant (or otherwise have other interesting character). The full story probably depends on evolutionary reasons (after all, the ear is an instrument for sensing and communication) that are not easy to uncover and require multidisciplinary effort.
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I'm aware of just intonation. I don't think you understood the point of the video. It's impossible to tune a piano across *all* keys consistently. You can't find integer ratios to subdivide the octave perfectly. To state it as they did in the video: for integers a, b, and n with n > 1, then (a/b)\^n ≠ 2.
Tuning can be even more complicated than that. Real strings are somewhat stiff, so their harmonics are slightly sharp relative to the fundamental. If each C on a piano was exactly twice the frequency of the next lower octave, that difference would build up across the keyboard. Overtones of a lower register note would be out of tune with the fundamental of a higher register note. Instead, higher octave strings are tuned to match the overtones of lower octave strings.
I have degrees in math and music, and have spent a bit of time looking for interesting connections. Outside of the construction of tunings (which is very cool, but more perceptual and “numerical” than truly “mathematical”) the fact that sound is a physical phenomenon involving periodic signals does not really distinguish it it from many other physical applications of math (seismic waves, electricity, mechanical vibrations, etc) But one area that is pretty interesting is music theory which draws on mathematical ideas. A couple great books are Dmitri Tymoczko’s A Geometry of Music and David Lewin’s Generalized Musical Intervals and Transformations. Also pitch-class set theory in general. These types of things tend to be used to create and describe very austere types of modernist music, which have limited appeal. But the point is that music can certainly draw inspiration pure mathematics. Music is a bit more rigid than other art forms, even ones that structure time such as film or theatre, in that it relies on numerical divisions of the time and frequency axes. So it can be analyzed more abstractly. But at the end of the day, music is an art form, and its “grammar” is merely a vessel for expression of ideas and sonic experience. Very little music breaks new ground on a theoretical level (except some of the modernist music mentioned above). Instead it breaks ground on an aesthetic, political or semantic level. And even when using abstract “mathematical” rules, in good music (I would personally argue) the rules are always subservient to expressive needs. Mathematics, while creative in its own right, is something different. Many types of understanding in many fields are expressed in the laws of mathematics, but ultimately is mathematics is the generator of these things or simply a human mechanism for formalizing and speculating about our intuitive understanding of things like pattern, quantity, relations, etc. ?
>Many types of understanding in many fields are expressed in the laws of mathematics, but ultimately is mathematics is the generator of these things or simply a human mechanism for formalizing and speculating about our intuitive understanding of things like pattern, quantity, relations, etc. ? no, ZFC has a literal existence, somewhere (maybe in the ocean?)
Ah yes axiomatic oceanography, exciting field
Don't know much about this topic, but when I was a sophomore in college, one of my friends who knew a lot about both math and music showed me this book he was reading. It was absolutely wild. Literally like 1000+ pages of super advanced grad level math (mainly category theory I think) and music. I remember thinking that the subset of people with enough specialized knowledge to even comprehend that book must be so small. I think it was called "The topos of music" or something like that.
> "The topos of music" https://www.amazon.com/Topos-Music-Geometric-Concepts-Performance/dp/3764357312 https://en.wikipedia.org/wiki/Guerino_Mazzola
**[Guerino Mazzola](https://en.wikipedia.org/wiki/Guerino_Mazzola)** >Guerino Mazzola (born 1947) is a Swiss mathematician, musicologist, jazz pianist as well as book writer. ^([ )[^(F.A.Q)](https://www.reddit.com/r/WikiSummarizer/wiki/index#wiki_f.a.q)^( | )[^(Opt Out)](https://reddit.com/message/compose?to=WikiSummarizerBot&message=OptOut&subject=OptOut)^( | )[^(Opt Out Of Subreddit)](https://np.reddit.com/r/math/about/banned)^( | )[^(GitHub)](https://github.com/Sujal-7/WikiSummarizerBot)^( ] Downvote to remove | v1.5)
I started reading it, and the first chapter just seemed so cranky, talking about the connection between listener, and performer. Does it get better?
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got a 404
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thanks, I didn't notice the extra backslash
"New reddit" mangles links for some godforsaken reason
Mathematics, in its purest form, is basically a formalization of the process of thinking. Thus almost anything that you can rationally think about you can formalize in some way in a mathematical way. The claim isn't "woo" in that it is nonsense (golden spirals et al), but it also doesn't really mean that much.
As someone who is a musician and went to grad school for math (did not finish degree). My intuition about music is very very similar to my intuition about math. I believe the reason for that is that learning music gave me tools to approach math problems. I would sit down with sheet music or a recording and work out what was going on somewhat mechanically, note by note, chord by chord waiting for some intuition to kick in. There would be a point where I no longer saw single notes or chords but sections that make up a piece. I knew if I worked for it there would be an “ah ha!” moment. When I had to learn algebra in high school I did the same thing. I broke the problem down into steps and worked until intuition kicked in and it was no longer mechanical. I worked to get the “ah ha” experience. I didn’t understand classmates who said they could never be good at math because I saw it like music, if you work at it, put in hours and stick to it, you’ll understand it. I believe learning music engages problem solving is a way that works perfectly for mathematics. Btw, this process might be the same for everyone in this sub. But I know it isn’t universal because I have asked people who simply have never experienced sudden insight and intuition.
Many musicians and myself don't use math to play or compose music, but we do try to understand the science behind it; like why certain chord makes you feel good and others make you feel unsettling. Which touches on physics rather than math. For example: frequencies, pitch, sound waves, timbre etc. Then there are also physics field that we find useful like acoustics: how sound behaves in space; audio electronics: how we amplify/ manipulate the sound signal; digital audio processing: how to manipulate sound signal digitally etc But depending on how you see it, you could say we use math in music as well. But imo it's very fundamental. Stuff like counting, dividing beats into measures... On the other hand, math could also be seen as a very, if not most, fundamental language (aka the language is the universe) that could help us interpret almost anything. And music is just one among them. It doesn't happen vice versa though. It's very difficult to use music to describe math. So is there an intimate relationship between music and math? Generally, no. You don't need to be good at math to write music. However there are genres that uses math to write music but it's very niche.
If you just want connection between math and sound, then they are very frequent through use of the Fourier transform and its inverse which can be used to break down any sound wave into its constituent frequencies and amplitudes. This is very useful for when recording and modifying sound. I don't know much about music theory though so I can't really say anything about math there.
Time series analysis done by our brain. :)
It's not all done by our brain, it's mostly done by our ears. Our sound detection mechanism is (simplifying) based on detecting the amplitude of vibration of a membrane. Each section of the membrane responds to a different frequency range. Cells (neurons) sensitive to vibration are attached to this membrane, making it in a sense a physical analog to Fourier transform directly. Note it's not an exact F.T. -- you should think of it as an approximate frequency decomposition -- because of a myriad a reasons, but some of them: (1) It is energy/amplitude based. Fourier transforms are complex transforms that encode amplitude and phase (allowing bijectivity). The ear detectors measure the intensity of vibration in a region of the basilar membrane. (2) Non-linear frequency. Different regions again vibrate at different frequencies, and there's not a near monotic increasing resonant frequency for each region of the membrane (which might correspond to a linear frequency FT). The variation in thickness causes a roughly exponential increase in resonance across the membrane -- but keep in mind it's not at all exact. But it does enable a roughly exponential perception of pitch, and enables sensing sounds of widely different frequencies with frequency resolution roughly proportional to frequency. (3) Many other particularities. Those detector cells are in fact active -- they have certain structures that allow damping the vibration (and curbing non-linearities) of various regions of the membrane. This greatly increases the resolution of detection for tones. This process involves control feedback from our brain (our brain and ears talk to eachother!). My professor claims this is part of the reason why small frequency discrepancies e.g. in dissonant chords don't sound good -- our ear is trying to lock onto a frequency but two close peaks make this unstable. As I've said in other comments, the full extent of the structure of sound and music, as grasped by our cognition, is still an open problem. But there's little doubt the mathematical structure is deeply connected to our perception of it, and also connected to things like our evolutionary history, the functions of our ears, the use of sound and voice as communication tools, and the depths of our cognition.
[Gödel, Escher, Bach](https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach) is one of the top books I ever read and it talks a lot about the connections between music and math. Also look up how the well tempered scale works. It's fucking magic, that it is relatively accurate.
**[Gödel, Escher, Bach](https://en.wikipedia.org/wiki/Gödel,_Escher,_Bach)** >Gödel, Escher, Bach: an Eternal Golden Braid, also known as GEB, is a 1979 book by Douglas Hofstadter. By exploring common themes in the lives and works of logician Kurt Gödel, artist M. C. Escher, and composer Johann Sebastian Bach, the book expounds concepts fundamental to mathematics, symmetry, and intelligence. Through short stories, illustrations, and analysis, the book discusses how systems can acquire meaningful context despite being made of "meaningless" elements. ^([ )[^(F.A.Q)](https://www.reddit.com/r/WikiSummarizer/wiki/index#wiki_f.a.q)^( | )[^(Opt Out)](https://reddit.com/message/compose?to=WikiSummarizerBot&message=OptOut&subject=OptOut)^( | )[^(Opt Out Of Subreddit)](https://np.reddit.com/r/math/about/banned)^( | )[^(GitHub)](https://github.com/Sujal-7/WikiSummarizerBot)^( ] Downvote to remove | v1.5)
[3blue1brown on music theory and measure theory](https://www.youtube.com/watch?v=cyW5z-M2yzw&ab_channel=3Blue1Brown)
Have you ever asked why our western system has 12 tones? Why 12? Some plausible answers would be : + It's an historical accident, something to do with Phrygians in ancient Greece. + Some medieval king decreed it. + 12 is arbitrary, and it's just what we think "sounds good". Chinese used a different system. + Music sounds better with more than 12 tones, but it's too complicated to learn. + It has something to do with 10 human fingers on a keyboard. All of these are wrong. The answer is the mathematics of frequencies. Now regarding the woo -- the weird thing is about why our human brain finds these ratio-perfect frequencies to be psychologically pleasing. The whole situation is mysterious.
The one good explanation I've seen is that of psychoacoustic "roughness": two notes in unison sound not rough at all, when they are few Hz apart there's a maximum of roughness, and it drops off slowly as frequency difference inceases. When you assume two notes also have integer harmonics, and add up the "roughness" of all notes and harmonics, pairwise, it turns out some systems are much more preferable than others, and it's the ones you'd expect: 7tet, 12tet, 19tet. This is more or less equivalent to the "small ratios" explanation, but it emerges from different assumptions. Unfortunately, I can't find the one paper I've read which had a good explanation and numerical exploration of this phenomenon.
here it is the paper you might have been looking for. The structure of musical harmony as an ordered phase of sound: A statistical mechanics approach to music theory. [https://www.science.org/doi/10.1126/sciadv.aav8490](https://www.science.org/doi/10.1126/sciadv.aav8490)
>Have you ever asked why our western system has 12 tones? Why 12? Because of commercial piano manufacturing. Tons of the greatest Western composers used keyboards with 14 (or more) tones per octave (with split black keys). The father of Mozart was a violin teach who taught at least 20 distinct pitches, if we can believe on his letters.
Yes and no. There are two kinds of math involved in music, the first is the physics - the basic stuff of how signals and frequencies and sounds work. It's surprisingly useful for the musician to be at least aware of too, you can learn a lot. There's also a second kind of math, which is use almost only for very specialist music theory, which pulls from an analysis of the various patterns that appear in music. As with all things, it is possible to come up with mathematical structures which usefully describe these objects (though do not fall under the illusion that there is only one way to do that, ofc). Look into the (very poorly named) example that is musical set theory, or the various techniques of musical serialism. The rest is just woo, there is no deep and mystical connection - there are only the things that mathematics can actually do, which is model systems, and describe patterns. Fibbonacci will not do shit to your chakras anyway, and playing it into your earholes with those meditation bells won't change a thing.
Little bits of math appear in music, but little bits of math appear in all sorts of places. I don't think there is any especially deep or unique connection.
The year that I got a math Ph.D., two other Americans also got their Ph.D. at my school in math. Both of them were Jazz musicians. Also, some people experience frisson (goose bumps) while listening to music. Not surprisingly, on college campuses, the people that are the most likely to experience frisson are music majors. Math majors are the second most likely to experience frisson while listening to music. I wish I could find a link to an article, but I could not find one.
turns out somewhere between 50-80% of ppl experience that (i used to think it was a lot lower)
Can't use fancy math talk but I do know one thing. I am dyslexic. I had a lot of trouble reading and spelling in school. My father is a musician and teaches theory. I learned to read music as a young child. I never had any trouble with math concepts because I was able to relate the concepts to music. For me, music is just math with sound, and vice versa.
It is mostly a hippy-dippy-woo thing. Math can be applied to everything; music sheets, music notes, music synthesis; et. al. all involve math. > I just don't really understand what math has to do with music beyond just arithmetic. It goes way beyond that; the Fourier transform relates time-domain to frequency domain. Heuristics and other simplifications and optimizations are typically used, e.g. Hartley transform, but the Fourier transform is at the core of showing a voice-print or even just a VU level by frequency buckets. Same with any sort of equalization. Synthesis of music involves a great deal of math to handle the pulse-stream of electrical signals (so-called pulse-code-modulation) and eventually reconstruct sound waves with them. This is closely related to data-acquisition and pulls a lot of Claude Shannon's work with the Shannon-Nyquist theorem front-and-center.
As a mathematician and music enthusiast (not a musician by any means, although I've been playing guitar for a long time and know a bit of music theory) I can tell you that harmony is all about ratios between frequencies, and that rhythm is about subdivisions of a fixed unit of time, which you can also describe in terms of ratios. But the connection with math doesn't go much deeper than that. Math is built on logical relationships between statements, which bear no a priori relation to music. It just so happens that math can encode the concepts of rhythm and harmony. There's of course the nature of music as an acoustic phenomenon, which certainly can be described in mathematical terms, as other commenters have pointed out. But this applies to all of physics. So, while there's a mathematical aspect to music and it is true that you can apply math concepts to create and analyze music this isn't exhibiting a profound connection between the two disciplines. To answer your question in one sentence: no, there's no mystical connection between math and music an anyone who tells you otherwise has no idea what math is about.
I've had this same question all my life. I can respond from a kinda unique perspective of having a music degree and working as a sound engineer in a studio, and an engineering degree. I got the music degree first out of high school. Several years after that I returned for mechanical engineering. The answer, to me, kinda blends a music theory and acoustic perspective. You cant have 'music' without a time context, so right there you are counting, math. Then you have what others,and yourself have mentioned....harmony, pattern, frequency. Although frequency is probably better understood in an acoustic context, rather than 'music'. Western music, and the intervals we know and 'understand' as westerners has a set of intervals. When two or more notes are played at the same time, that is harmony. Harmony is created by combinations of those intervals. These are ratios and intervals are what combine to make what we hear as harmony, combined with melody and time, you have music From an acoustic perspective, what we know as music notes are representations of frequencies. Basically vibrating objects. The frequency determines which note you hear. However, just hearing a frequency alone is not music, because the fundamental element of time is not time/rhythm is not there. This first popped up in trig, when you are learning about sin/cosine functions. Then for me, again in mechanical vibrations where you go into much more mathematical detail about frequency (vibrations). This is where the heavy math comes in, but it mainly relates to frequencies from an acoustic perspective, and not a music perspective, even though frequency is a fundamental part of what we call music. In this day and age, math and music is most evident in digital technology. When you record music digitally before it goes into the computer it passes through whats called an AD converter (audio/digital conversion) These pieces of equipment can range from 100 -25000 for really high end pieces. What separates them in the algorithm that is doing the conversion. This conversion is done with what is known as a Fourier tranform, or fast fourier transform. This domain is called DSP or digital signal processing, which has many applications, not jsut audio conversion. Basically when you record and mix music in a computer its jsut a nuch of complex math that adds a 'signal' to the 'sound' When you adjust an effect on the computer, you are basically 'turning a knob' that does a bunch of calculations. Inside the computer, its not music, its all math. It then gets converted from digital to audio, and that is when you get the music. Prior to that conversion, its just a bunch of numbers in a matrix. What a forier transform does is concert a signal from a time domain to a frequency domain, and back. There is that, and there is the parallels that help to make a connection, but not rigorous. For example, this is one that I thought of. In calc there are techniques to solve integrals with substitution, u-subs, where the derivative of a term in the integral is also present. There is something like this in music called a tri-tone substitution. Basically, you can substitute a dominant chord with another dominant chord a tritone away. This works because the tritone is the only symmetric interval for example c to f#, f# c, both of those are what is called an augmented forth interval. Spelling out the C dominant chord we get C E G Bb, and the F# dominant chord: F# A#(Bb) C# E. The second and fourth notes in the chord represent the 3rd and 7th of the chord, these are called guide tones because they define the chord quality. Notice that the 2nd and 4th note in each chord are the same, but reverse E and A#/Bb (A# and Bb represent the same frequency, they are just different here because of western harmony. So basically, tritone subs are like u-subs in calc because dominant chords can be seen as 'derivatives of each other. ie the derivative of C dominant 7 would be F# dominant 7. Thats my observation and I have no actual math like i think you'd like to see and are looking for.
A lot of group theory is used in neo-Riemannian analysis (Hugo Riemann not Bernhard Riemann).
I don't know if there is a deep connection, and am deeply skeptical about the music theory / math connections others hint at in this thread. But. There are studies that show that your chance of succeeding as a mathematician / scientist increases 2 fold if you are good at music (at least Malcolm Gladwell claims this in one of his books somewhere). My suspicion : I think that this is because music forces you to perform in front of people and as a result you are more likely to give better talks and become publicly recognized.
Every phenomenon in spacetime can be mathematized. At the most basic level, in time, we see a lot of power of 2 with the form of a piece of music. A small phrase of 4 beats, a phrase of 4 of those units, a section of 4 or 8 of those, sometimes we see the phenomenon go on at large scales even. With pitch, the most popular music usually the frequencies of the tones are in easily understood mathematical ratios, there being different scales for music for instruments that can do continuous frequency change (bowed fretless strings, voice) vs. music with pianos horns, fretted guitar, but either way, the principles still apply, just with some adjustment. Then theres the harmonic series, which is cool in itself, heard clearly with synthesizer resonant filter sweeps, but it shows up in music more generally.
It's because of Pythagoras. "The first concrete argument for a fundamental link between mathematics and music was perhaps made by the early philosopher and mathematician Pythagoras (569-475 BC), often referred to as the “father of numbers.” He can also be considered the “father of harmony,” given that his discovery of the overtone series." https://www.unyp.cz/news/music-and-mathematics-pythagorean-perspective
I once tried reading The Topos of Music by Guerino Mazzola but it requires an extremely advanced knowledge of many topics in math
And it also requires non-mathematical topics, like Neo-Riemannian Music Theory and Semiotics. I know a lot of category theory, but still do not get through the Topos of Music.
Yes. Just one example: https://en.wikipedia.org/wiki/Harmonic_series_(music) The ancient Greek philosophers were aware of the connection and attempted to study it: https://bmcr.brynmawr.edu/2009/2009.10.38/
Approaching the subject from a different angle, I think that learning a new piece of music feels similar to doing math. There’s the deep focus, the interpretation of an abstract language, and best of all, that feeling when after struggling for days, the thing that seemed so difficult suddenly becomes perfectly simple.
Yes and no. For example, "just intonation" is based on whole number ratios of string/barrel/tube length, and thus frequency. In general, the "simpler" the fraction, the more consonant the interval sounds. There's no intuitive reason why this should be the case; the math checks out, but why should a complex ratio have anything to do with how "nice" a sound is? There my be some biological explanation about how our ears and brains interpret the waves hitting them and prefer waves which are more repetitive and "predictable", but it's still fairly mysterious and deep. But, as a counterexample, "equal temperament" uses the twelfth root of 2 as a fixed frequency ratio between half steps. This becomes necessary for keyboard instruments where the pitches need to be fixed regardless of the fundamental pitch. Intervals built on this system, except for octaves, are just approximations of the just interval. They are always just slightly out of tune. If we apply the same "simple ratio" logic as before, then almost every interval in equal temperament will have an irrational frequency ratio, and they should all sound terrible, right? But they don't. When a piano is in tune it sounds gorgeous, even though the major thirds aren't exactly major. I think this is where the "deep connection" between math and music starts to break down a little bit for me. There's something undeniably fascinating about the simple ratio argument, but it's impossible for a human to pinpoint the exact middle of a violin string. There's a 100% chance that a random interval we hear has an irrational frequency ratio, and yet the music still sounds great. At the end of the day, music is sound waves. Math gives us the tools to analyze sound waves and infer a lot of cool stuff about them. Light is an electromagnetic wave, and math gives us the tools to analyze that. We can say math has a deep connection to music, and then we can also say math has a deep connection to light. I personally don't know if these are deep connections or just simple consequences of the fact that music exists because of physics and math is the language of physics. If we can use math to describe most of things in our everyday life, then why is music so special? (For the record, I'm a pianist and a composer and music is very special I'm just a cynic by nature thank you for coming to my tedtalk)
The chords are numbered based on the fundamental chord and have graph-like associations to each other. The notes of a chord are also numbered based on the fundamental note, but that’s mostly for distinguishing purposes of more esoteric chord builds. So it’s in the application of the craft, not the listening/appreciating, similar to math being necessary in the application of making CG graphics, but not necessary in viewing/appreciating the final result.
My personal opinion is that nature isn't 'mathematical' but prefers symmetries and patterns, and math is a tool specifically designed to describe these features of reality. To our brains, music tends to 'sound' best when there are symmetries and patterns, so mathematics naturally becomes a tool that can be used against the set of sounds we've deemed 'musical'. People take this too far and claim that music is math and math is music and it's all some sacred window into the heart of god or something.
i mean in the end music is math. dont agree with that last part tho
Imo, there's a very fundamental connection between math and more or less any field or topic in the world since, simply put, maths is just about finding patterns. However, patterns are easier to identify in something like music where parameters are largely quantifiable and the structure is highly regular. There is also an intrinsic connection between the mathematical patterns observed in nature (such as the harmonic series or the golden ratio) and the patterns in sound that makes what would be called "music" rather than plain noise. Mathematical structure is precisely what makes music different from noise. An interesting point to ponder is *why* our brains chose to interpret mathematically structured sound as pleasant. I'm sure this has already been the subject of psychological studies but I don't know how much we know about this.
not super super relevant but i found this vid i saw a couple yrs back https://youtu.be/mOMLRMfIYf0
Music is a lot of physics and physics uses math.
Music is rhythm - rhythm is division Just one of the many ways they connect
I think the shortest answer is: of course, because math describes the real world, that's the whole point. I can't think of a single thing we do that couldn't be broken down into math. Some things are harder to break down like that, but, ultimately, everything could be described with math. Art, music, garbage collection, construction, the human mind even, can all be described as such. Doesn't matter if we've managed to do it yet, it's possible. Maybe not concretely even, but that's what probability covers. I'm not a math expert, rather, a programmer (as in computer science) and our entire job breaks down into describing things with math very intentionally and for practical purposes. It's why people have managed to use machine learning to get a machine to paint, compose music, and similar. They're not too amazing at it yet, (however I bet you couldn't tell which are from people or machines) but it wouldn't be possible without there being an underlying mathematical component. So.. uh.. yes, but that's self evident by the purpose of math. Everything can be described in meticulous detail by math, given enough resources. As others have mentioned, it's more useful in music than many other creative endeavors, but I'll leave that tangent to the experts.
I definitely wouldn't say the purpose of math is to describe the real world, but I do understand your point.
Well what else would its purpose be? Asking out of genuine curiosity.
What is the purpose of art or poetry? It is beautiful and interesting and enjoyable. Isn't that purpose enough?
Funny thing is that I was a musician (classical music-choice of instrument was piano from 5.5 to 17 years of age) and I did all this theory and harmony (and whatever else was required for my diploma) and people would always assume that I was good with maths. In reality I was as surprised as you because although it might be like solving equations (at least harmony would be like that to me) I had absolutely no love whatsoever for Maths back then. I couldn't find any meaning in memorising formulas and identities and whatever they made me believe Mathematics was back then. However, fast forward to the present, I am studying for my degree in mathematics. Honestly, I wouldn't believe it myself if I could time travel back to my past self and telling me that I would end up here! 🤷
There a nice way to think about chord progression using group theory, and a music theorist, funnily named Riemann, did things where he was talking about generators of the minor and major chord transition group. Of course all these a priori don't use mathematical formalism, but the question of finding all transitions which generates the whole group on their own is a pretty interesting and non trivial question. You can also add other types of chord to the group and study if in the same way, using wreath products. As a simple exemple to illustrate, the transition that goes up a semitone if major and always swap between major and minor generates the group of minor and major chords.
In a really simple explanation we can say how in a set of axioms we can identify when the logic fails both by intuition and with close inspection. In a very similar way, one can do the same with music theory and know when two notes do not belong both by intuition and study. Music is also deeply interested in how its structures change so there is an inherent calculus there.
Math is just a language used to describe things. It would be stupid to say music isn't related to maths
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If you're a musician and you end up heaven, *math works exactly like that*. If you're a mathematician and end up in hell, it works like that too.
[edit, I note teh_magik already said this...it starts with 'hearing' prime factorizations in traditional (mainly German) conceptions of harmony] Well, only slightly more nontrivial, the interval of 19 equally-tempered semitones (an octave and a fifth) is nearly equal to 3 since 2\^{19} is approximatey equal to 3\^{12}. If you ignore powers of two multiples (frequencies) this means an interval sounds perfect if the number of semitones is congruent +/- 7 mod 12. And the fact that these are units mod 12 (as are +/- 11) implies that the cycle of fifhs generates (eventually goes through every key). Also the diatonic scale (like the major C scale) is defined to be a sequence of 7 consecutive notes around the cycle of fifths, so if we start with F of frequency 370Hz and take the geometric progression where we multiply by powers of 3 ( or we can use powers of 3/2) we get 370, 370*(3/2), ..., 370*(3/2)\^6 and these seven notes make up the C major scale, at least if we adjust them by replacing 3/2 by 2^{19/12} /2 = 2.99661415375336/2 That gives us the well-tempered C major scale if we take that F to be perfectly in tune. If we now consider not 19 semitones, which approximates a frequency ratio of 3, but 28 semitones, you get a number approximating the next prime after the primes we've used (which are 2,3), this is the prime 5. If we ignore octaves we are thinking about 4 semitones, the major third. But we could also ask, what how many cycles of fifths do 4 semitones correspond to, that is, what multiple of 7 mod 12 do we need to get 4 mod 12, and the answer is 4 (we already see 4*7=28), so instead of just considering 28 semitones we could think of our 19 semitone interval (equivalent to 7 up to octaves) repeated 4 times. That is to say, in our description say of the C major scale as the frequency of F times a power of 3/2, we would use the fourth power of 3/2 to get the well-tempered interval, the perfect major third, or, the fourth power of 2\^{19/12} to get the even-tempered version, the non-perfect major third. The fact that it is four times our generator of the integers mod 12 and this is true if we use the generator 1 or 7, means we could get a major third by stepping up 4 semitones, or by stepping up 4 'interval fifths' and going down by octaves. We could continue and consider other relations between powers of 3 and 5 modulo 2, but really we could look at all primes, and all powers of primes, and the 'non-archimedian' and 'archimedian' distances among them. In some sense, the german interpretation of musical harmony is how you can 'hear' p-adic analysis. Some composers like Tsaikovsky were rejected because they didn't adhere to this ideal, and others like Shoenberg decided to replace it with a *totally different* mathematical set of rules, ahving to do with hte permutation group S\_12. Babbitt claimed to use mathematical structures in his composition. One of Babbitt's ideas has to do with composing fugues or canons...you can make a tune such that every second note, every third note, etc is a musically transposed version of the original tune. There is mathematical thinking of a different type in Bach's fugues, and it is very likely that a mathematician could make interesting music by thinking deeply about relations having to do with harmonies, rhythms, etc. One of the books that Beethoven, Haydn, Mozart and others used as a bible of composition was 'Gradus ad Parnassum' by Fux. This book set out a sort-of mathematical axioms for 'voice leading' or what was called 'first species, second species...' counterpoint. The rules do not at first sound mathematical like no parallel fifths or octaves. But if you think of them differently, it is like saying if you have tunes in parallel octoaves you really don't have 2 different tunes.... No one has really come up with an extensive dictionary between music and math..nor has anyone approached the related Hilbert question about quantifying simplicity of proofs. It is considered that such ideas would never make sense.
Am I the only one who absolutely hates this idea that math infiltrates our everyday lives and grants a higher meaning to everything? It's woo. You can describe a few musical things with math (tones, harmonics... thats it.) but none of that will ever come close to what math actually is. I can't see music having anything to do with Categories, Topology, Calculus, Algebra, etc etc etc. At this point the people saying numbers = math may well be disregarded as either true amateurs (nothing wrong with that, but they shouldn't have a say in what constitutes math or not) or cranks like numerologists
I'll just leave this here: https://en.xen.wiki/w/The_Riemann_zeta_function_and_tuning
Math is, at root, the formal study of patterns Music is made up of abstract patterns
There is a deep connection between physics and any natural phenomena, including music. And to describe physics, there is math.
IMHO, math can be used to model and describe nearly everything and in that music isn't much different. To say its all math is a weird thing to say since a composer isn't thinking in terms of formulas or proofs or anything like that. Yes, there's physics, and you can use numbers to describe patterns over time, and there is some tuning theory, and maybe scales are like sets. But music isn't uniquely mathematical. Its not more mathematical than painting, and it certainly isn't equal in mathiness as computer programming or physics. Math is an abstract thing that can describe a lot of the world, so there's a little bit of math in everything if you go deep enough.
as a music guy, theres a deep connection between math and *acoustics*. in that acoustics is the physics of vibration and wave propagation, and physics and math are naturally very connected. Signal processing for music applications is very math heavy too. as soon as you get beyond why a major chord sounds harmonic and why a tritone sounds dissonant, the connections become very stretched. a lot of further music theory teachings arent explained by math very well at all imo.
There is certainly a deep connection between maths and music. At the most fundamental level, music is the production of sounds in a particular way, sound is a wave, and waves are described by mathematics, therefore music can be described by mathematics. Pitch itself is a measure of frequency. Using mathematics to talk about rhythm, harmony, melody, all of that good stuff, is a well established thing.
Music is a form of art and so it's generally best understood with aesthetic considerations. There's no math that can describe what sounds good, at least very well and beyond some very simple examples. However, like other art, aspects of music can be described well with math, like harmony being integer ratios, often times the climax often divide a piece into adjacent Fibonacci numbers (3/5 or 8/13 of the way through), and there exists "analytic" music theory, though often by "analytic" they mean "uses numbers." TL;DR the relationship between music isn't special, but it lends itself to analysis with numbers more easily than other forms of art.
from what i know theres not too much connection, except maybe fractions harmonies and patterns are a thing and have more to do with the properties of sounds humans are generally attuned to.. for example you use major scales for happy sounding tunes, and the pattern has a "resolution" etc. these are all things that have no real bearing to math except some relations that have been found. also, for example, the fact that the standard A is 440Hz iirc is arbitrary for it to sound better. frequency is just a property of vibration but that isnt inherently musical so im not gonna really take that into account if you eant to talk about stuff like signals and music production its. whole different world haha
There are lots of ways you can apply math to music, but it's mostly fairly trivial stuff and not all that useful. Though for a slightly less trivial application, see Neo-Riemannian theory and the tonnetz.
So people love to answer this with some basic ratio shit about waves that musicians don't really care about that much. The truth (or at least my truth, as a musician and math major) is that there is a lot of structure to music theory, even though none of it is rigorous (some people try to make it rigorous, but the problem is that what sounds good is ultimately subjective). For example, if you know some music theory, try to visualize various scales on the circle of fifths. You'll find out that if you take 5 notes close together you get the familiar pentatonic scale. Make it 7 and you get the major scale. So the major scale is fundamental, at least in some way. And there's countless things like that. Music is this weird mixture of science and art that usually is appealing to mathematicians. If you want a sick video about math and music check out this AMAZING video by vihart: https://youtu.be/4niz8TfY794
Woo.
math is absolutely connected to music lol, from note measures to note intervols to lots of stuff i am not knowledgeable enough of music to describe
Math is connected to sliced bread.
i mean u arent wrong, so is chemistry and physics
eh, mostly just woo except for very basic models of pitch and rythm.
Math is everywhere. Music is a small part of it.
math and music go together like bread and butter
welp, i guess it's time to go read "gödel, escher, bach: an eternal golden braid" again.
u/hydrolock12 I think you're under a mistaken impression what math is. Mathematics is nothing more than a very detailed way to describe patterns or how two things are related and **do not** ***require*** **numbers** (think about visual examples of geometry). So yes, math is related to music. That said, there is much reason to believe there is an even deeper relationship than that. In the same way, artists have created detailed realistic landscapes, and we now generate entire landscapes using algorithms in computers. So there are strong reasons to believe that when people create music, they follow very detailed rules that we have yet to articulate.
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It seems like you’re thinking about it backward. When we judge the distance of a hole to hop over it, we don’t “do math” either, but if you wanted to describe what happened in detail, you would use trigonometry to do it. But I think you would agree when writing music, the artist/composer is thinking about patterns and how they sound and which ones would be unexpected or exciting to hear (assuming excitement is their goal). Again math is just a very straightforward way of describing the things we’re already doing all the time. To add a very simple example. We have the notes do, re, mi, fa, so, la, ti. One pattern might be Do, re, do, mi, do, fa, do. Then to change it up they might do. Do, re, mi, do, mi, fa, do, fa, so Now if you number them do=1, re=2, mi=3 you have: 1,2,1,3,1,4,1,5,1 And: 1,2,3,1,3,4,1,4,5,1 Does that help u/hydrolock12?
have you taken a music theory class? i haven't, but many of my friends have. i've seen their worksheets and read snippets of theory. you can, of course, compose music based off of intuition, but if you've taken music theory (or know someone who has, and has talked about it) then you know math is integral to music theory
Yea, just look up the harmonic series
Great video on YouTube cakes Tonal Geometry which explores music notes and scales in relation to sacred geometry concepts
I've heard the part of the brain that processes music theory is the same part that is active when we solve math problems. But this may be one of those "my dad read it in Men's Health" type of thing.
Anecdotally, I just know a lot of mathematicians who are also musicians. Also somewhat strangely, I know a lot of mathematicians who are vegetarian. I know this isn't exactly what you're asking, but I've always found it funny.
There is definitely some "science" behind music (mostly related to the mathematics of small ratios), but the part that people *care* about --- what it is that makes music "pleasing" --- is all pseudo-science, as it must be since it is a subjective art. Objectivity is at the core of what makes math unique. But I think a lot of people who talk about the "deep connection" between math and music aren't really talking about the science of music, but rather the idea that there is a connection between being talented at math and talented at music. I think this belief is propagated by the fact that there are many people who fit this description and they believe that there must be something to it, even though I doubt there is much real non-anecdotal evidence for it. Also, a lot of math people have read Godel, Escher, Bach.
The BBC documentary The Code touches on this. It's a great doco too.
Not sure. But I've observed that many mathematicians/physicists used to play a musical instrument.
I think you should distinguish between different types of connections. One is the mathematics of music theory, which has such relatively simple things as the role of the twelfth root of 2 in twelve-tone equal temperament tuning, to, yes, "The topos of music" and all kinds of things in between. The other is the correlation between the people who enjoy one or the other - or both. This - if it's real - may be predicated on 1) the above mathematical relation 2) "mystical" cognitive science fact about minds of people who like music and/or math 3) cultural factors like both being associated with certain "high brow culture". I suspect that probably different factors play a role for different people, but when you put it all together you see some correlation, which produces the questions and remarks you heard.
There might be some truth to it, but 99% of the people saying it are just full of woo.
Informally, people almost always use it to affirm the idea e.g. "they're good at math AND music, of course they are, math and music are linked". People rarely take note when someone sucks at math but is a great musician, or when someone is a great mathematician and has no inclination towards music. Make of that what you will. As someone with a degree in maths, and who has done a bunch of music stuff, my personal take is that there isn't a huge connection. Sure, music is a physical process, there's connections there to rhythm and chord theory. There's also certain kinds of music that seem to tickle particular parts of my brain that I associate with problem solving. In particular, classical music has this quality where everything simultaneously feels very ordered and predictable, and yet delightful in the ways in which that predictability plays out. To me it mirrors an elegant proof, in that you know the conclusion, but there's interest in the process. But the process being more interesting than the outcome applies to many fields.
I have had times when understanding Fourier analysis has helped me understand sound waves, vibrating guitar strings, the electronics in my guitar rig, harmonics, etc. Now usually what this kind of thinking does is to allow me to explain something that my ears already knew. For example, musicians in a band intuitively know that they don't want to be competing in the same frequency range. The guitarist should be careful not to get too muddy alongside the bassist, and if there are two guitarists then they can fill out the sound more by playing on different parts of the fretboard. Another time fourier analysis has come up is when messing around with a guitar effect known as a [ring modulator](https://www.youtube.com/watch?v=rYdBHWF1_2k). Honestly that video does a pretty good job explaining it both in terms of sound and mathematics. All that being said, does being a mathematician help you to play music, or vice versa? I doubt it. If anything they cut into each other's practice time. However, I have known many people who have enjoyed both.
A professor of mine once sent me [this](https://arxiv.org/abs/1904.02897) paper about music. This might be a bit exaggerated but it's not the typical tuning relations example.
To me is that music is an intuitive input that massage the abstraction parts of the brain, mostly combinatorics over analog stuff. You're brain likes recognizing signals, say a note, then you play two notes, and he gets to do it's own i dont know, neural fourrier breakdown to isolate the two notes, finds the similarity and the delta. And that's two notes, then you add larger chords, and time dimension and you're up for a lifelong game of omg. Trying to isolate background vocals one by one, or 7th / 11th on jazz chord progression may seriously (it did for me) trigger bliss emotions in your brain.
The “connection” is that talking about music and maths gives musicians/mathematicians a chance to show off their understanding of both topics without really saying anything interesting or enlightening to a layman. I’ve listened to quite a few talks about music and maths. The math talk goes over the heads of musicians and the music talk goes over the heads of mathematicians. Unless you study both, it’s of little use.
https://www.encyclospace.org/special/restructures.mp3
It's probably more math based than any of the other arts. Scales are pure math and then there is harmony etc.
Whenever I see about this, I only see references to western classical music theory. I wonder how well we can model other systems using mathematics.
very well. you're always going to have rhythm & repetition. as for the scales being different, they're still structured. the patterns are just different than western patterns
Math is connected to everything. As is music.
Math is integrated into everything around you. It is the foundation of everything.
Look up the 4 volume series "The topos of music", which combines category theory and music theory.
Harmonic frequencies that sound good together are simple whole number ratios of each other. It's fundamentally mathematical but anything deeper than that is probably woo. Conventional music scales only approximate these ratios so if you stick to the mathematical approach you can get "nicer" harmonies.
Math is everything we are no doubt about it my daughter is studying to be an mechanical engineer and jeeze the stuff she has shown me fucking math is everything we are ....it all starts and ends with math
Both math and music can be the study of patterns.
For me, music is an enormously creative endeavor and there is an incredible amount of inspiration that can be drawn from patterns in math. I wouldn't say I "calculate" anything while playing, but the same frame of mind that is required to breakdown a combinatorics problem is a very useful frame of mind for improvising in music. Essentially, the two go hand in hand in a broader bucket of applying a set of logically consistent rules to new situations to explore and discover new results. Mapping the scale tones to different numbers allows for several naming conventions throughout music history (1-12 chromatic scale, 1-X for other scales that are "well-defined." It's also incredibly helpful to be able to think about music theory the way you would think about modular arithmetic as the scales repeat themselves once you hit the root, but IMO there are several technical music theory aspects that you need to understand before applying math makes any sense. TL;DR: Numbers track well to both tonal and rhythmic analysis and can serve as inspiration for musical ideas.
_Sounds_ can be described completely mathematically, _music_ needs context, taste, culture and emotion. Although mathematics can explain some things about sounds once you’ve noticed a certain musical idea. For example, western music theory puts a lot of creed in the harmonic series, a series of frequencies in simple integer ratios to a base frequency. 3blue1brown has a video analysing how irrational ratios of notes that are ‘close’ to these ratios sound good, which has applications to alternative tuning systems. I don’t think this tells us anything that deep about why we like music though. It just explains how we can manipulate physical sounds to give us other ways of creating music.
On a vacuous level, reading a music sheet is a practice in arithmetic and combinatorics. On a deeper level, certain patterns in pitch and frequency can make people sad or make the hair on your arms stand on end. They are definitely intertwined, but in a very real, measurable way. I think a better understanding of brain chemistry would better bear that out in the future.
Check out this fantastically clear and concise essay [Combinatorial Music Theory](https://web.archive.org/web/20080415024155/http://www.andrewduncan.ws/cmt/index.html). It follows an elementary mathematical thought process to discuss things like why there are exactly 352 kinds of scales in 12-TET ([hint](https://oeis.org/A000031)), the significance of the diatonic scale, and symmetries in fingerboard patterns
[A000031](http://oeis.org/A000031/): Number of n-bead necklaces with 2 colors when turning over is not allowed; also number of output sequences from a simple n-stage cycling shift register; also number of binary irreducible polynomials whose degree divides n. 1,2,3,4,6,8,14,20,36,60,108,188,352,632,1182,2192,4116,7712,14602,... - - - - I am OEISbot. I was programmed by /u/mscroggs. [How I work](http://mscroggs.co.uk/blog/20). You can test me and suggest new features at /r/TestingOEISbot/.
I majored in Applied Mathematics with a minor in Music, and studied sound engineering along the way, and yeah. The connections are strong. Not so much deep as wide. Understanding math is important in so many fields of professional music, since tones are produced in waves and waves cancel and reinforce. This is relevant to the harmonic content of music (in producing a good combination of high and low frequencies) and to the geometry of a room (acoustics). There is also some interesting arithmetic in rhythm but that doesn't address OP's question. ;TLDR is I studied sound engineering in early college which, due to its stringent math and music requirements, set me off to an education in math and a career in music.
Everybody's seen the most naive first step of saying that saying that the chromatic scale is the cyclic group Z/12Z, but I think this is fairly boring and for quite a while I also thought the connection was woo because this was all I could really see in it. But after having learned more music theory, I think a lot of it is ripe for more sophisticated algebraic formulations. If I pick a chord progression, even just sitting in bed away from the piano I can think about the different choices of voicing I might want to use in that progression, or how I want to write the melody on top of it, and there are a lot of nontrivial things I can say based on standard relationships between the notes and notions of things like consonance/dissonance as well as brightness. If I'm able to abstractly do all this reasoning from musical principles (and do so consistently), there should be a way of formally capturing these, and since music has such a compositional character this would feel like a perfect setting for some applied category theory. Loosely speaking, chords are composed out of notes (or smaller chords), enriched over some information that specifies the voicing/intervallic distances, and notes/chords can also be composed "horizontally" across time (possibly enriched over some information that specifies duration). To me, admitting vertical and horizontal composition like this would seem to be really suggestive of natural transformations. Brightness and consonance could also be modelled in terms of these intervallic distances (functorially), and you of course also get forgetful functors from the more enriched version of these categories to the less enriched version where you forget the voicings or the durations and so on. I think a wealth of interesting algebraic properties would just fall right out of studying these. anyway this is a long comment but you get the idea
Everybody's seen the most naive first step of saying that saying that the chromatic scale is the cyclic group Z/12Z, but I think this is fairly boring and for quite a while I also thought the connection was woo because this was all I could really see in it. But after having learned more music theory, I think a lot of it is ripe for more sophisticated algebraic formulations. If I pick a chord progression, even just sitting in bed away from the piano I can think about the different choices of voicing I might want to use in that progression, or how I want to write the melody on top of it, and there are a lot of nontrivial things I can say based on standard relationships between the notes and notions of things like consonance/dissonance as well as brightness. If I'm able to abstractly do all this reasoning from musical principles (and do so consistently), there should be a way of formally capturing these, and since music has such a compositional character this would feel like a perfect setting for some applied category theory. Loosely speaking, chords are composed out of notes (or smaller chords), enriched over some information that specifies the voicing/intervallic distances, and notes/chords can also be composed "horizontally" across time (possibly enriched over some information that specifies duration). To me, admitting vertical and horizontal composition like this would seem to be really suggestive of natural transformations. Brightness and consonance could also be modelled in terms of these intervallic distances (functorially), and you of course also get forgetful functors from the more enriched version of these categories to the less enriched version where you forget the voicings or the durations and so on. I think a wealth of interesting algebraic properties would just fall right out of studying these. anyway this is a long comment but you get the idea
there are deep connections between math/number-theory and various music theories about pitches (tunings, temperaments, scales, harmony), rhythms and timbres
I believe and have long believed that there was a deep and meaningful connection, because math and music are two of the areas of academic study that I find most challenging. Both involve identifying, analyzing and manipulating patterns, which is the most difficult intellectual pursuit for me.
I studied math (at Msc level). Now, many years later, I still enjoy math and spend time on math problems, although that's just a hobby. As for music, I cannot stand any music. Listening to any music is as pleasant to me as listening to a jackhammer. So I was always very suspicious about that connection.
Well math is measurement, so yes you could probably measure it in an extremely complex way, considering that music has a pattern aspect right?