I often see people frame music as mathematical manipulation or try to approach music making from a “first principles” approach, where those principles are mathematics and physics. But watching musicians talk about making music, I seldom see any discussion of the underlying math, and instead see discussions of timbres, instruments, and stylistic/historical influences; musicians who make good music seems to believe “first principles” involves historical knowledge and a well-listened ear, and nothing involving math. My question is: Is thinking about music as applied mathematics a good way to create good music? Or is it just the most easily digestible model of music for the crowd on this site?
> Is thinking about music as applied mathematics a good way to create good music?
As an instruction, I think clearly not, the fact that lots of musicians aren't mathematical at all but create great music seems to prove it to me.
But it is interesting to think about musicians who do seem to think about music this way. Bach is definitely a good example where the system of counterpoint is very complex. I'm not sure if she'd describe herself in these terns, but I've always got the impression Laurie Speigel thinks about music a little like that too. Then there's stuff like Coltrane's Giant Steps, where the whole piece is based around a sort of music theory "trick".
So maybe not generally, but there's definitely some awesome music out of that kind of relationship.
Maths and physics are a terrible way to learn the artistic side of music, but if you are interested in "why does a fifth chord sound nice" or "why are the black and white keys on a piano in that particular pattern" you can get interesting (partial) answers by looking into the maths of frequency ratios and the physics of overtones and how they affect the cilia of the inner ear. Music differs between cultures but there are some universals such as the Octave (edit: by which I mean doubling of frequency, not how its divided up) and nearly all cultures have some form of music ... There is something universally human about it, and so its a doorway to studying how our minds work.
> but there are some universals such as the Octave
Universal in the sense that a number of rocks or a number of sheep can be doubled just as a frequency can?
The notion that there are 8 sub divisions to a doubled frequency interval isn't universal. Balinese Gamelan doesn't even neccessarily have an agreed number of "notes" in an "Octave" from one village to the next.
There aren't 8 subdivisions in an octave in western music either. Well, there are in any given scale, but there are also many scales. "Octave" is a misleading term. Given that it's just a doubling of frequency, the term is sort of as good as any other, and that douibling exists in pretty much all cultures that have developed string, pipe or other resonant body based music (including hitting hollow logs and plucking vibrating reeds / sticks / tines).
It's pretty much the foundational idea of any modality. No matter how you divide it up, the purest harmony is doubling or halving.
The commenter presumably was talking about octave equivalence, which is reportedly present across all or nearly all historical musical cultures that we know about. It’s also supposedly present in some other mammals.
reportedly present, yes .. but the debate is still hot on universal.
I was asking to tease out some PoV perspective, again Gamelan doesn't neccessarily have powers of two, or 12, etc divisions of a doubling (or Octave, if we're using that term); it's a non western style of percussion that has a suprising number of local variations (it's essentially near unique to Balinese culture) in divisions and tunings.
The Octave wikipedia entry includes:
Octave equivalence is a part of most musical cultures, but is far from universal in "primitive" and early music
Yeah, that sentence on Wikipedia is a bit unclear though. It might be merely claiming that, for some musical cultures, we don’t have a written record of an explicit notion of octave equivalence or tone name circularity.
But I suspect there’s a clear biological mechanism which makes it easy to mistake one octave for another from any source of roughly harmonic sound. This is due to the similarity in the overtones of two harmonic sounds that differ by an octave. I would be surprised if this mechanism isn’t universal, although its on various musical systems can obviously vary a lot.
Universal in the sense that a number of rocks or a number of sheep can be doubled just as a frequency can?
Yes thats what I meant, the doubling of frequency. It might seem trivial but the fact that doubling frequency sounds "right" to humans is actually quite interesting. Why does it sound "right"?
Interference is most of the answer. With frequencies f and 2f you get the smoothest interference patterns, even if the tones have a lot of harmonics. This applies reducingly to increasingly fractional ratios.
> So yes, the 12-tone scale is a universal thing -
I don't follow the logic here though. It's certainly true that a 12-tone / Chromatic scale is ubiquitous within the Western Music tradition .. but the universe is reportedly a little larger.
Even Western Music includes exceptions like the 9-note augmented scale, though the argument can be made that it's a 12-scale with 3 bits "missing" - not a case that can be made about a non-western 7 note percussive scale.
All scales in all cultures are based on octaves and fifths. (E.g., the ancient Chinese musical scale also has 12 tones.)
Also the so-called "Western music" standardized on 12 tones very late in the process, long after the Chinese figured it out.
> a 12-scale with 3 bits "missing"
That's all scales, even the "non-Western" ones. Microtonality is added to the standard 12 tone to add tone effects. (Synthesizers in pop music do the same trick.)
To confirm the claim that "all scales in all cultures are based on octaves and fifths" one might study the scales.zip scale files and find those that do not contain octaves and fifths, which should naturally be zero if the claim is true.
Note also that certain musical traditions were suppressed or eradicated due to their unfortunate habit of using dissonant notes such as minor seconds, as opposed to the consonant traids favored by a particular group recently in power around the world. Happy Easter!
Thank you, I am somewhat aware of the knobs present on a synth, though fail to see the relevance given that various other instruments do not have dynamic retuning options. Which 12 tone scale did you have in mind (for there are many) and why do you think 12 (for there are many other numbers, some of which are used by various scale systems) is a natural property of the world? Perhaps with a more cogent argument you could make a better case for your opinion.
129.74 is not really close to a power of two. 31-tet scales have a better approximation of a 5th (and an impressively better approximation of a minor 7th).
The obvious exception in the western system would be the blues scale, which arguably has 9 tones (7 equal tempered notes, plus a just tempered 3rd and 7th).
And Indian ragas break all of these rules. They have scales that don't have 8 notes, scales that don't use equal temperament, and even a few scales that don't repeat on octaves.
As another commenter below has said, "mathematics might be a useful way to understand music", but it's not how compelling music is made.
Mathematics are fundamental to scales and the harmonic series, and knowing about them will help you refine certain choices, but it's not going to help you write a dramatic melody or an emotionally resonant chord progression, or play an energizing rhythm, even if there are mathematical explanations sometimes.
Good music comes from being a good listener, having a strong sense of what's possible, where it could go, and then delivering something surprising. Telling a story with your melody and supporting the arc of that gesture with harmony that accentuates or contrasts it.
Again, there's a mathematical explanation for harmony and dissonance, but players aren't thinking that granular. They're operating at a higher level of abstraction one, two, or three levels above that: They're thinking about telling a story, evoking an emotion, and exciting an audience in the moment.
Music is more like language. (This is profoundly true). There is very little that's mathematical about music; and even the bit that is, isn't really (since nobody actually uses just temperament).
Digital sound production, however. Yes. There's all kinds of thoroughly unpleasant mathematics, none of which you actually need to know unless you're writing computer music software.
(I write computer music software, and I am also a jazz musician).
It sounds pedantic, but I think it's important: maths and physics are often used to describe sounds, their relationships and emergent properties through combination. Maths and physics aren't ever really used to describe music.
It's like telling someone they can paint a masterpiece because they understand Fe4[Fe(CN)6]3 makes an aesthetically pleasant blue pigment.
> Is thinking about music as applied mathematics a good way to create good music? Or is it just the most easily digestible model of music for the crowd on this site?
It's a great way to analyse music (e.g. to categorise, understand, and communicate detail), but that does not mean it's a good way to create it. There's a lot of beauty in finding those abstractions and I think that representation appeals to a lot of people here.
Discussions about timbre, instrumentation, and stylistic influence are often symmetric to those about math. When you have 90 minutes to spare, highly recommend strapping in for a listen to https://malwebb.com/notnoi.html.
There's a lot of really incredible musicians, composers, producers, and educators that go deep on the math. There's also plenty that don't. People build mental models in different ways. That's a good thing and a big part of what makes most art interesting.
Wonderful question. I suspect it's partially the culture issue you point to, but also a practical issue of composition. If we decompose sound into the basic waveforms, similar to the subject pdf on page 18, we then have parts that we can reassemble. We can take the defense-funded DSP math of the likes of a John Cooley or a John Tukey and build an engine for assembling the parts of sound.
All this being said, I think that's a process of convenience and a historical path not a absolute constraint. We have some more flexible means of communicating with the machines today. And I strongly encourage someone to work on a new UI for computer music. "Jazz trio piano, upright bass, and drums. start drummer laid-back, piano blowing over the changes, then piano on top."
Indeed it described an output and was also a UI. I meant that describing the output could be the UI. I pictured a textbox or a whisper-style text to speech session. Basically an ai chatbot specialized for generating music.
I couldn't figure out precisely what that video showed, but it was fascinating.
Somehow it reminded me of the orca music programming environment.
It doesn't take that long to learn to read sheet music (or tabs especially) and you could treat it like just playing a sequence of notes but you're never going to get far that way. You need to understand why certain notes go together. Some people have done that without theory but you're going to get much further with even some basic theory.
Think of it this way: if you first saw the word "HELLO". You could deconstruct that and remember that there are 11 lines and 1 circle but that's not how you learn to read or write. You learn letters, which are collections of lines. So you learn the concept of "H" and it having a sound and that it is 3 lines. You then learn to put them together and how you can sound out something thats's written and with varying degree (depending on language) take something said and write it down.
Music theory is like that. Sheet music may be a bunch of circles and lines on a sheet but really it's describing keys and usually a chord-progression. Some sheet music will explicitly just list the chords at the top like A, Em, Asus4, etc.
The 12 notes are constructed from harmonics, specifically 2:1 and 3:2. This part is maths. But the frequencies are adjusted slightly in a system called "equal temperament" where the ratio of 2 adjacent notes is the 12th root of 2.
From there you generally play a subset of those notes (often 7). That's called a scale (eg major or minor scale). The chords in that scale can then be identified by a Roman numeral within a key. So the I chord in the C major scale is a C. The IV chord is the F. Depending on the starting note of the scale you'll get sharps (#) and flats (♭) to denote that they are a different pitch. An easy way to remember this is that the white keys represent those whole and half steps with just the white keys (starting from C). As an aside, so does the A minor scale.
Why do I say all this? Because a huge amount of modern music is simply a I-IV-V chord progression within whatever scale you're using. So if you know a little theory, you can choose a key and a chord progression that will inherently sound nice together. There's more to it of course but understanding what a key is, what chords are and what a chord progression is is a pretty good start.
Upvoted because it’s a thoughtful question, but honestly I think it’s just that this book and many others like it are addressed primarily to people who are going to use tools like SuperCollider or CSound or raw dsp to create their own entirely original technology stacks for creative work, and an understanding of the physics/math of sound is pretty key to that kind of work, regardless of the musicality of their later creative production.
> My question is: Is thinking about music as applied mathematics a good way to create good music? Or is it just the most easily digestible model of music for the crowd on this site?
You are probably aware that there are these things called synthesizers, which exist both hardware and software, complex pieces of technology that can shape sound. There are people who are specialized in creating them (with code and/or electronics), people who are specialized in programming them (creating presets) and people who excel in using them to make music. And many more different profiles who are in between. Each will care about different aspects, they all contribute to making music.
Life is not black and white, and music neither. What is even "good music"? What is your mental model for "the crowd on this site"? In your questions, aren't you reducing the possibilities of learning by putting these into boxes?
The world is big, life is rich and people are much more diverse than what one typically perceives.
Both of these things. The timbres can be explained through evolutionary biology. The same brain centers used for processing movement in 3d space also start firing up to start predicting where any given piece of music will go.
In my opinion, good music typically comes from the flow state. In other words, less thinking and more practice/exploration leads to the best results.
From personal experience, pattern-recognition is the most useful "applied math" skill when making music. I use it when identifying intervals between notes, and chord progressions, which you need when you're trying to get the idea out of your head and onto the instrument you're playing or song you're writing.
Theory and practice are both important. A few analogues:
- fundamentals on an instrument vs performance
- low-level graphics programming vs using a game engine
- comp sci vs software engineering
The theory side here gets to the root of things, is valid for any sort of DAW/DSP software, and has the benefit of being easier to teach. Practice is obviously more important though, especially in the arts. It's better to grope in the dark than do nothing.
I'm no musician, but it sounds like trying to understand human psychology by studying the brain, or cells by studying physics, or the colony by isolating a single ant. Answer: no. It's the idea of emergent phenomena not being reducible to its parts.
Musicians are already experiencing this. The likes of Suno are churning out high quality songs with only a minimal amount of prompting material.
One can roughly prototype a song, giving it the structure, melody, harmony, rhythm, lyrics that a finished song might have, upload it and request a cover in a particular style. The output will often resemble a highly competent human performance.
I'm a lifelong musician, went to music school to study jazz and orchestration, was a professional film composer for 15 years prior to pivoting to programming. I've read quite a few books on the intersection of math and music.
And not once have I ever felt that these so-called intersections were anything other than contrived.
Of course we can interface with music from a mathematical perspective, but that doesn't mean that we should or that there's anything particularly illuminating to gleen from doing so.
Beyond the very basic math (honestly even that's perhaps too strong a word -- just because something is expressed in numbers doesn't make it _math_) of time signatures and some harmonic concepts up to maybe some of Slonimsky's work, doing so is IMO a fool's errand that exists only to fill space on a TEDx stage.
Microtonal music throws all of that out the window.
Logic only works in the context of definite ontologies. But audio frequencies are continuous, not discrete. It really is all vibes at the end of the day.
Plug for Angine de Poitrine for a contemporary example of music that breaks the rules that define traditional music.
Understanding the soul of music and creativity at a mathematical is something that not that many people are trying to do. But there is an entire world of technology that underpins modern music and sound that is built soundly on math, like digital recording digital signal processing, synthesis, physical modeling, and plenty of other stuff, and this seems to be what the book's focus is.
Sure, there have been plenty of attempts to distill music to a mathematical essence. Certainly the ancient Greeks tried this, and traditional counterpoint resembles math in a number of ways. But at the end of the day, mathematical descriptions of math and music theory more generally are more useful as descriptive tools to help give language to what people are doing musically and to understand why we perceive some things as sounding better than others.
Starting with numbers can be good in some respects, like understanding the circle of fifths or how scales are built out of intervals, how chord progressions and harmony work and how to reharmonize, all of which can be augmented with a solid conceptual understanding. But at the end of the day, your ear and creative spirit are your primary asset when it comes to creating good music. This is why computer-generated music has been so bad up until AI took over. Great for building arpeggiators or backing tracks, but good luck creating a beautiful melody in a purely numerical rule-based system.
> I seldom see any discussion of the underlying math, and instead see discussions of timbres, instruments, and stylistic/historical influences
Music today is utter crap at all levels, this is a verifiable scientific fact.
This is probably why.
Music "theory" was invented as a critical tool (i.e., basically to enable reviewers to describe and evaluate the music of the time), not as a composition tool.
Basically, we're holding it wrong and it's doing us harm.
> Music today is utter crap at all levels, this is a verifiable scientific fact.
No it's not, and it's not a verifiable fact. Unless you have a source?
Rick Beato knocks the sami-ness of 'the charts', but there's more to music than that...take a look at who he interviews.
Bach is considered the greatest musical genius of all time, but he was part of an industry and composing was his day job. Each of those BWV's was written in a couple days. Bach's performers at the time didn't study for years for a single recital, they read the sheet music in an afternoon and then performed the BWV next day.
Beethoven improvised his pieces on the fly and performed them himself. This wasn't considered as something out of the ordinary at the time.
Can you imagine the average conservatory graduate imporovising anything today? Even a pentatonic blues riff?
Clearly we went off the rails bigtime somewhere along the way. The framework we're using to teach and compose music is actively hindering us.
I don't agree. I grew up with piano and had music friends all through my life. Classical music requires a certain level of math ability. Modern musicians scorn this and frankly it shows.
Many classical musicians have only a cursory understanding of mathematics. Many modern musicians are pushing the boundaries. There's a reason there's a genre named "math rock". Also, Jazz probably pushed the maths of music beyond classical music. As a last example, listen to some Meshuggah :-)
Why so rude? You really think Meshuggah is some profound group? Do you have any background in classical music? European screamo bands are just the worst. And personally people I've met from that scene rarely understand even the basics of rhythm. So your comment does not match my experience at all.
I've been playing piano for over 40 years and other instruments almost as long. I'm a massive fan of classical music. If you think metal bands don't understand rhythm you obviously haven't listened to much.
I'm not claiming any band to be profound, but many have complex interplay between multiple rhythms, very non standard chord progressions, often non standard tunings, and even microtonal instrumentation.
You seem to be stuck in a world where everyone from Stockhausen suddenly stopped learning music theory. I'm telling you that you're wrong, and obviously not listening to enough modern music.
Stop being a music snob. You are free to like whatever you want but making glaringly incorrect statements about an entire time period encompassing many genres is going to get push back.
Why so rude? This site is ridiculous. You made statements about two entire periods, the current period and the past. Additionally, I am not a snob and I have listened to a lot. I won't even bother to get into this. My college roommate had thousands of records and forced me to listen to all of them.
Amusing to see how attitudes toward AI change over time. On page 6, part of the original text has a footnote apologizing to readers far in the future for outdated speculations, then mentions that future readers "may even be an artificial intelligence rather than a human, how wonderful!"
But just a bit before that in the foreword written in the present day, bars AI scrapers from reading or referencing the materials under any circumstances!
Anyway, this seems fantastic and I'll definitely be spending some time diving in.
> then mentions that future readers "may even be an artificial intelligence rather than a human, how wonderful!"
My first thought seeing this post was, I need to find more literature like this, fine-tune a model with that + Logic Pro documentation, then give it an MCP to control Logic Pro and see if it can be my music production assistant.
Note: Nick Collins (the author of this book) and Alex McLean created Algorave. The time I spent learning from the Algorave community was crucial to my later work on Glicol (https://glicol.org/).
Btw, I have a feeling that if you want to learn about computer music, you can send the PDF to LLM and ask what the chapter is about and how to represent it using csound or supercollider.
My experience is that with computer music, you have to keep experimenting and listening in order to truly understand and innovate.
This appears to be mercifully shorter and less intimidating than the must-have bible, "Curtis Roads. The Computer Music Tutorial. MIT Press, Cambs, MA, 1996".
It says it was originally published by Wiley in 2009, and the rights reverted to the author in 2025, whereupon the author released it on the net for free.
If someone wanted to start making computer music I'm not sure I'd recommend this or Curtis Roads' book as a starting point.
These aren't resources for getting started. They're more like encyclopedias for learning about DSP and tech once you've established the fundamentals of music and sequencing.
If a beginner wants practical knowledge for making records with electronic instruments I'd give them a DAW, teach them to record and sequence, teach them basic music theory, and then point them to something like Ableton's synthesis tutorials that will teach them about oscillators, envelopes, filters, LFOs, and basic sample manipulation.
I'm so happy to see Nick Collins taking this on. If you haven't seen his book, Handmade Music, it's an excellent book for music projects. This contribution looks exceptional as well.
I was reading up on the author and saw this interesting bit[0]:
> An algorave (from an algorithm and rave) is an event where people dance to music generated from algorithms, often using live coding techniques. Alex McLean of Slub and Nick Collins coined the word "algorave" in 2011, and the first event under such a name was organised in London, England. It has since become a movement, with algoraves taking place around the world.
I love the world of music production. I started with Ableton Live six years or so ago and it's been a wonderful hobby. It has such a vibrant cottage industry of plugins (sampled instruments, synthesizers, effects, etc) thanks to the VST standard.
I work professionally with music, including using ableton. I do create but don't sell/adverise, I'm strictly 'backstage'. I love everything about creating music, less so for reading about music (reviews, critiques, dissecting) though there are occasional exceptions. Are you putting your creativity online publicly?
Do you put yours online? I enjoy listening to other musicians. People give soundcloud a lot of grief but I love the service and the musicians I have met there.
Thank you. Your music is enjoyable, I listened to a couple (twice, first track and the fast one) and skimmed (four times) through others (my listening time is packed). Each piece has a filmic quality. The most impressive part is that they are (more of less) finished, which is a feat in itself. Very good.
I am very backstage, I don't wish to be known professionally, except by clients, all of which came via word of mouth - I wouldn't know how to advertise and I'd probably mess it up/give the wrong impression/attract the 'wrong' clients.
I won't link here but I do use soundcloud, I would say it's just ok. Tend to go Faircamp, bandcamp, geocities. MySpace lost 12 tunes.
I'll make a note of our exchange and if I change my mind I'll let you know. Some of my music is not dissimilar to yours, which is often the case, and not a bad thing - like you, it's the act of creation that brings immense pleasure. Thanks again for sharing I enjoyed listening.
Thanks for listening. Feel free to reach out if you ever want to share. I create music strictly for my own enjoyment so I understand where you are coming from. Take care!
Loved this book when I was a student! This is for DSP enclined audience with a focus on musical applications. Not very in-depth if you want to develop your own plugins or DAW but still very informative
Juan García Castillejo published 'La telegrafía rápida, el triteclado y la música eléctrica' (High-speed telegraphy, the three-keyboard system, and electric music) in 1944.
Reminds me of starting college and photocopying digitized instrument wave forms (e.g. guitar string pluck) and then feeding them to a 6502 subroutine that output the waveform to the 8-bit companding DAC in my OSI C2-4P computer from a Basic driver that would play different notes through for songs.
Wow, this book has been published in 2025, and it has zero mention of AI generated music. Not saying it's a bad thing - from the table of content it covers a lot of important fundamentals, but ignoring the elephant in the room is... weird.
I often see people frame music as mathematical manipulation or try to approach music making from a “first principles” approach, where those principles are mathematics and physics. But watching musicians talk about making music, I seldom see any discussion of the underlying math, and instead see discussions of timbres, instruments, and stylistic/historical influences; musicians who make good music seems to believe “first principles” involves historical knowledge and a well-listened ear, and nothing involving math. My question is: Is thinking about music as applied mathematics a good way to create good music? Or is it just the most easily digestible model of music for the crowd on this site?
> Is thinking about music as applied mathematics a good way to create good music?
As an instruction, I think clearly not, the fact that lots of musicians aren't mathematical at all but create great music seems to prove it to me.
But it is interesting to think about musicians who do seem to think about music this way. Bach is definitely a good example where the system of counterpoint is very complex. I'm not sure if she'd describe herself in these terns, but I've always got the impression Laurie Speigel thinks about music a little like that too. Then there's stuff like Coltrane's Giant Steps, where the whole piece is based around a sort of music theory "trick".
So maybe not generally, but there's definitely some awesome music out of that kind of relationship.
Maths and physics are a terrible way to learn the artistic side of music, but if you are interested in "why does a fifth chord sound nice" or "why are the black and white keys on a piano in that particular pattern" you can get interesting (partial) answers by looking into the maths of frequency ratios and the physics of overtones and how they affect the cilia of the inner ear. Music differs between cultures but there are some universals such as the Octave (edit: by which I mean doubling of frequency, not how its divided up) and nearly all cultures have some form of music ... There is something universally human about it, and so its a doorway to studying how our minds work.
> but there are some universals such as the Octave
Universal in the sense that a number of rocks or a number of sheep can be doubled just as a frequency can?
The notion that there are 8 sub divisions to a doubled frequency interval isn't universal. Balinese Gamelan doesn't even neccessarily have an agreed number of "notes" in an "Octave" from one village to the next.
There aren't 8 subdivisions in an octave in western music either. Well, there are in any given scale, but there are also many scales. "Octave" is a misleading term. Given that it's just a doubling of frequency, the term is sort of as good as any other, and that douibling exists in pretty much all cultures that have developed string, pipe or other resonant body based music (including hitting hollow logs and plucking vibrating reeds / sticks / tines).
It's pretty much the foundational idea of any modality. No matter how you divide it up, the purest harmony is doubling or halving.
The commenter presumably was talking about octave equivalence, which is reportedly present across all or nearly all historical musical cultures that we know about. It’s also supposedly present in some other mammals.
reportedly present, yes .. but the debate is still hot on universal.
I was asking to tease out some PoV perspective, again Gamelan doesn't neccessarily have powers of two, or 12, etc divisions of a doubling (or Octave, if we're using that term); it's a non western style of percussion that has a suprising number of local variations (it's essentially near unique to Balinese culture) in divisions and tunings.
The Octave wikipedia entry includes:
but gets woolly on examples.Cheers for the response, appreciated.
Yeah, that sentence on Wikipedia is a bit unclear though. It might be merely claiming that, for some musical cultures, we don’t have a written record of an explicit notion of octave equivalence or tone name circularity.
But I suspect there’s a clear biological mechanism which makes it easy to mistake one octave for another from any source of roughly harmonic sound. This is due to the similarity in the overtones of two harmonic sounds that differ by an octave. I would be surprised if this mechanism isn’t universal, although its on various musical systems can obviously vary a lot.
Universal in the sense that a number of rocks or a number of sheep can be doubled just as a frequency can?
Yes thats what I meant, the doubling of frequency. It might seem trivial but the fact that doubling frequency sounds "right" to humans is actually quite interesting. Why does it sound "right"?
Interference is most of the answer. With frequencies f and 2f you get the smoothest interference patterns, even if the tones have a lot of harmonics. This applies reducingly to increasingly fractional ratios.
1.5**12 is about 129.74, which is as close as you can reasonably get to a power of two.
So yes, the 12-tone scale is a universal thing - you want both octaves and fifths in your scale.
(12 is actually too much, so usually that's pared down to something like 4 or 5 or 7 tones, this is where you get cultural variation.)
> 1.5*12 is about 129.74,
Math checks out.
> So yes, the 12-tone scale is a universal thing -
I don't follow the logic here though. It's certainly true that a 12-tone / Chromatic scale is ubiquitous within the Western Music tradition .. but the universe is reportedly a little larger.
Even Western Music includes exceptions like the 9-note augmented scale, though the argument can be made that it's a 12-scale with 3 bits "missing" - not a case that can be made about a non-western 7 note percussive scale.
All scales in all cultures are based on octaves and fifths. (E.g., the ancient Chinese musical scale also has 12 tones.)
Also the so-called "Western music" standardized on 12 tones very late in the process, long after the Chinese figured it out.
> a 12-scale with 3 bits "missing"
That's all scales, even the "non-Western" ones. Microtonality is added to the standard 12 tone to add tone effects. (Synthesizers in pop music do the same trick.)
To confirm the claim that "all scales in all cultures are based on octaves and fifths" one might study the scales.zip scale files and find those that do not contain octaves and fifths, which should naturally be zero if the claim is true.
https://www.huygens-fokker.org/scala/
Note also that certain musical traditions were suppressed or eradicated due to their unfortunate habit of using dissonant notes such as minor seconds, as opposed to the consonant traids favored by a particular group recently in power around the world. Happy Easter!
The 12 tone scale is a natural property of the world that has been independently discovered by many different cultures across the world.
The microtonalities or strange scales (like gamelan) are evolutions of the natural octave/fifths grid.
Every synthesizer has a "detune" knob.
Thank you, I am somewhat aware of the knobs present on a synth, though fail to see the relevance given that various other instruments do not have dynamic retuning options. Which 12 tone scale did you have in mind (for there are many) and why do you think 12 (for there are many other numbers, some of which are used by various scale systems) is a natural property of the world? Perhaps with a more cogent argument you could make a better case for your opinion.
> microtonalities or strange scales (like gamelan) are evolutions of the natural octave/fifths grid.
What is your source for this opinion?
~ https://en.wikipedia.org/wiki/GamelanHow do you know anything that comes prior to recorded history?
129.74 is not really close to a power of two. 31-tet scales have a better approximation of a 5th (and an impressively better approximation of a minor 7th).
The obvious exception in the western system would be the blues scale, which arguably has 9 tones (7 equal tempered notes, plus a just tempered 3rd and 7th).
And Indian ragas break all of these rules. They have scales that don't have 8 notes, scales that don't use equal temperament, and even a few scales that don't repeat on octaves.
powers of 2 seem to work well in many things. in rhythm too. so dont be so quick to dismiss.
In short: Not really.
As another commenter below has said, "mathematics might be a useful way to understand music", but it's not how compelling music is made.
Mathematics are fundamental to scales and the harmonic series, and knowing about them will help you refine certain choices, but it's not going to help you write a dramatic melody or an emotionally resonant chord progression, or play an energizing rhythm, even if there are mathematical explanations sometimes.
Good music comes from being a good listener, having a strong sense of what's possible, where it could go, and then delivering something surprising. Telling a story with your melody and supporting the arc of that gesture with harmony that accentuates or contrasts it.
Again, there's a mathematical explanation for harmony and dissonance, but players aren't thinking that granular. They're operating at a higher level of abstraction one, two, or three levels above that: They're thinking about telling a story, evoking an emotion, and exciting an audience in the moment.
Music is more like language. (This is profoundly true). There is very little that's mathematical about music; and even the bit that is, isn't really (since nobody actually uses just temperament).
Digital sound production, however. Yes. There's all kinds of thoroughly unpleasant mathematics, none of which you actually need to know unless you're writing computer music software.
(I write computer music software, and I am also a jazz musician).
It sounds pedantic, but I think it's important: maths and physics are often used to describe sounds, their relationships and emergent properties through combination. Maths and physics aren't ever really used to describe music.
It's like telling someone they can paint a masterpiece because they understand Fe4[Fe(CN)6]3 makes an aesthetically pleasant blue pigment.
That's a very nice analogy, thank you.
> Is thinking about music as applied mathematics a good way to create good music? Or is it just the most easily digestible model of music for the crowd on this site?
It's a great way to analyse music (e.g. to categorise, understand, and communicate detail), but that does not mean it's a good way to create it. There's a lot of beauty in finding those abstractions and I think that representation appeals to a lot of people here.
Discussions about timbre, instrumentation, and stylistic influence are often symmetric to those about math. When you have 90 minutes to spare, highly recommend strapping in for a listen to https://malwebb.com/notnoi.html.
There's a lot of really incredible musicians, composers, producers, and educators that go deep on the math. There's also plenty that don't. People build mental models in different ways. That's a good thing and a big part of what makes most art interesting.
Good musicians care about music theory / “first principles” as much as good writers care about language theory / grammar.
Wonderful question. I suspect it's partially the culture issue you point to, but also a practical issue of composition. If we decompose sound into the basic waveforms, similar to the subject pdf on page 18, we then have parts that we can reassemble. We can take the defense-funded DSP math of the likes of a John Cooley or a John Tukey and build an engine for assembling the parts of sound.
All this being said, I think that's a process of convenience and a historical path not a absolute constraint. We have some more flexible means of communicating with the machines today. And I strongly encourage someone to work on a new UI for computer music. "Jazz trio piano, upright bass, and drums. start drummer laid-back, piano blowing over the changes, then piano on top."
This is the kind of UI people should be building.
https://youtu.be/3poN6FDyB28?is=QjDzlmRQCMMbP_lS
What you wrote described an output, not a UI.
Indeed it described an output and was also a UI. I meant that describing the output could be the UI. I pictured a textbox or a whisper-style text to speech session. Basically an ai chatbot specialized for generating music.
I couldn't figure out precisely what that video showed, but it was fascinating. Somehow it reminded me of the orca music programming environment.
https://www.youtube.com/watch?v=r28Xy-1_F8I
https://www.youtube.com/watch?v=-tAAsolMG-M
It doesn't take that long to learn to read sheet music (or tabs especially) and you could treat it like just playing a sequence of notes but you're never going to get far that way. You need to understand why certain notes go together. Some people have done that without theory but you're going to get much further with even some basic theory.
Think of it this way: if you first saw the word "HELLO". You could deconstruct that and remember that there are 11 lines and 1 circle but that's not how you learn to read or write. You learn letters, which are collections of lines. So you learn the concept of "H" and it having a sound and that it is 3 lines. You then learn to put them together and how you can sound out something thats's written and with varying degree (depending on language) take something said and write it down.
Music theory is like that. Sheet music may be a bunch of circles and lines on a sheet but really it's describing keys and usually a chord-progression. Some sheet music will explicitly just list the chords at the top like A, Em, Asus4, etc.
The 12 notes are constructed from harmonics, specifically 2:1 and 3:2. This part is maths. But the frequencies are adjusted slightly in a system called "equal temperament" where the ratio of 2 adjacent notes is the 12th root of 2.
From there you generally play a subset of those notes (often 7). That's called a scale (eg major or minor scale). The chords in that scale can then be identified by a Roman numeral within a key. So the I chord in the C major scale is a C. The IV chord is the F. Depending on the starting note of the scale you'll get sharps (#) and flats (♭) to denote that they are a different pitch. An easy way to remember this is that the white keys represent those whole and half steps with just the white keys (starting from C). As an aside, so does the A minor scale.
Why do I say all this? Because a huge amount of modern music is simply a I-IV-V chord progression within whatever scale you're using. So if you know a little theory, you can choose a key and a chord progression that will inherently sound nice together. There's more to it of course but understanding what a key is, what chords are and what a chord progression is is a pretty good start.
Upvoted because it’s a thoughtful question, but honestly I think it’s just that this book and many others like it are addressed primarily to people who are going to use tools like SuperCollider or CSound or raw dsp to create their own entirely original technology stacks for creative work, and an understanding of the physics/math of sound is pretty key to that kind of work, regardless of the musicality of their later creative production.
Thinking about music mathematically is at least a good way to understand music
> My question is: Is thinking about music as applied mathematics a good way to create good music? Or is it just the most easily digestible model of music for the crowd on this site?
You are probably aware that there are these things called synthesizers, which exist both hardware and software, complex pieces of technology that can shape sound. There are people who are specialized in creating them (with code and/or electronics), people who are specialized in programming them (creating presets) and people who excel in using them to make music. And many more different profiles who are in between. Each will care about different aspects, they all contribute to making music.
Life is not black and white, and music neither. What is even "good music"? What is your mental model for "the crowd on this site"? In your questions, aren't you reducing the possibilities of learning by putting these into boxes?
The world is big, life is rich and people are much more diverse than what one typically perceives.
I am the least musical person I know, but I can help you out here. Math? John Coltrane has you covered. https://www.americanjazzmusicsociety.com/blog/john-coltrane-...
Both of these things. The timbres can be explained through evolutionary biology. The same brain centers used for processing movement in 3d space also start firing up to start predicting where any given piece of music will go.
An interesting note has a fundamental and harmonics and allows analogies to be drawn in RF engineering and quantum mechanics: https://www.google.com/search?q=any+good+parallels+between+i...
In my opinion, good music typically comes from the flow state. In other words, less thinking and more practice/exploration leads to the best results.
From personal experience, pattern-recognition is the most useful "applied math" skill when making music. I use it when identifying intervals between notes, and chord progressions, which you need when you're trying to get the idea out of your head and onto the instrument you're playing or song you're writing.
Theory and practice are both important. A few analogues:
- fundamentals on an instrument vs performance
- low-level graphics programming vs using a game engine
- comp sci vs software engineering
The theory side here gets to the root of things, is valid for any sort of DAW/DSP software, and has the benefit of being easier to teach. Practice is obviously more important though, especially in the arts. It's better to grope in the dark than do nothing.
> nothing involving math.
It's like Escher; he didn't have any clue that his intricate work would excite mathematicians and crystallographers.
Mandatory reference to GEB
Why not both approaches? Creativity is not just making the most use of what you have but also the most of what you are.
I'm no musician, but it sounds like trying to understand human psychology by studying the brain, or cells by studying physics, or the colony by isolating a single ant. Answer: no. It's the idea of emergent phenomena not being reducible to its parts.
Likely historically true, but not anymore.
As a software developer I see that LLMs are better at the "craft" of making software.
Software developers training are overwhelmingly analytical.
Musicians will experience the same. That the quality of Ai generated music is superior. But it will come more as a chock for the reasons you explain.
Musicians are already experiencing this. The likes of Suno are churning out high quality songs with only a minimal amount of prompting material.
One can roughly prototype a song, giving it the structure, melody, harmony, rhythm, lyrics that a finished song might have, upload it and request a cover in a particular style. The output will often resemble a highly competent human performance.
I'm a lifelong musician, went to music school to study jazz and orchestration, was a professional film composer for 15 years prior to pivoting to programming. I've read quite a few books on the intersection of math and music.
And not once have I ever felt that these so-called intersections were anything other than contrived.
Of course we can interface with music from a mathematical perspective, but that doesn't mean that we should or that there's anything particularly illuminating to gleen from doing so.
Beyond the very basic math (honestly even that's perhaps too strong a word -- just because something is expressed in numbers doesn't make it _math_) of time signatures and some harmonic concepts up to maybe some of Slonimsky's work, doing so is IMO a fool's errand that exists only to fill space on a TEDx stage.
No. Create good music from the principles of creating good music, then, a few years down the line, add maths if you really need it.
tbf the essay is clearly titled 'Introduction to Computer Music', and not 'Introduction to Music'.
Microtonal music throws all of that out the window.
Logic only works in the context of definite ontologies. But audio frequencies are continuous, not discrete. It really is all vibes at the end of the day.
Plug for Angine de Poitrine for a contemporary example of music that breaks the rules that define traditional music.
Understanding the soul of music and creativity at a mathematical is something that not that many people are trying to do. But there is an entire world of technology that underpins modern music and sound that is built soundly on math, like digital recording digital signal processing, synthesis, physical modeling, and plenty of other stuff, and this seems to be what the book's focus is.
Sure, there have been plenty of attempts to distill music to a mathematical essence. Certainly the ancient Greeks tried this, and traditional counterpoint resembles math in a number of ways. But at the end of the day, mathematical descriptions of math and music theory more generally are more useful as descriptive tools to help give language to what people are doing musically and to understand why we perceive some things as sounding better than others.
Starting with numbers can be good in some respects, like understanding the circle of fifths or how scales are built out of intervals, how chord progressions and harmony work and how to reharmonize, all of which can be augmented with a solid conceptual understanding. But at the end of the day, your ear and creative spirit are your primary asset when it comes to creating good music. This is why computer-generated music has been so bad up until AI took over. Great for building arpeggiators or backing tracks, but good luck creating a beautiful melody in a purely numerical rule-based system.
> I seldom see any discussion of the underlying math, and instead see discussions of timbres, instruments, and stylistic/historical influences
Music today is utter crap at all levels, this is a verifiable scientific fact.
This is probably why.
Music "theory" was invented as a critical tool (i.e., basically to enable reviewers to describe and evaluate the music of the time), not as a composition tool.
Basically, we're holding it wrong and it's doing us harm.
> Music today is utter crap at all levels, this is a verifiable scientific fact.
No it's not, and it's not a verifiable fact. Unless you have a source? Rick Beato knocks the sami-ness of 'the charts', but there's more to music than that...take a look at who he interviews.
Bach is considered the greatest musical genius of all time, but he was part of an industry and composing was his day job. Each of those BWV's was written in a couple days. Bach's performers at the time didn't study for years for a single recital, they read the sheet music in an afternoon and then performed the BWV next day.
Beethoven improvised his pieces on the fly and performed them himself. This wasn't considered as something out of the ordinary at the time.
Can you imagine the average conservatory graduate imporovising anything today? Even a pentatonic blues riff?
Clearly we went off the rails bigtime somewhere along the way. The framework we're using to teach and compose music is actively hindering us.
I don't agree. I grew up with piano and had music friends all through my life. Classical music requires a certain level of math ability. Modern musicians scorn this and frankly it shows.
Many classical musicians have only a cursory understanding of mathematics. Many modern musicians are pushing the boundaries. There's a reason there's a genre named "math rock". Also, Jazz probably pushed the maths of music beyond classical music. As a last example, listen to some Meshuggah :-)
I don't agree at all.
You don't have to agree. That's fine.
Why so rude? You really think Meshuggah is some profound group? Do you have any background in classical music? European screamo bands are just the worst. And personally people I've met from that scene rarely understand even the basics of rhythm. So your comment does not match my experience at all.
I've been playing piano for over 40 years and other instruments almost as long. I'm a massive fan of classical music. If you think metal bands don't understand rhythm you obviously haven't listened to much.
I'm not claiming any band to be profound, but many have complex interplay between multiple rhythms, very non standard chord progressions, often non standard tunings, and even microtonal instrumentation.
You seem to be stuck in a world where everyone from Stockhausen suddenly stopped learning music theory. I'm telling you that you're wrong, and obviously not listening to enough modern music.
Stop being a music snob. You are free to like whatever you want but making glaringly incorrect statements about an entire time period encompassing many genres is going to get push back.
Why so rude? This site is ridiculous. You made statements about two entire periods, the current period and the past. Additionally, I am not a snob and I have listened to a lot. I won't even bother to get into this. My college roommate had thousands of records and forced me to listen to all of them.
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Amusing to see how attitudes toward AI change over time. On page 6, part of the original text has a footnote apologizing to readers far in the future for outdated speculations, then mentions that future readers "may even be an artificial intelligence rather than a human, how wonderful!"
But just a bit before that in the foreword written in the present day, bars AI scrapers from reading or referencing the materials under any circumstances!
Anyway, this seems fantastic and I'll definitely be spending some time diving in.
> then mentions that future readers "may even be an artificial intelligence rather than a human, how wonderful!"
My first thought seeing this post was, I need to find more literature like this, fine-tune a model with that + Logic Pro documentation, then give it an MCP to control Logic Pro and see if it can be my music production assistant.
https://github.com/ahujasid/ableton-mcp
Note: Nick Collins (the author of this book) and Alex McLean created Algorave. The time I spent learning from the Algorave community was crucial to my later work on Glicol (https://glicol.org/).
Btw, I have a feeling that if you want to learn about computer music, you can send the PDF to LLM and ask what the chapter is about and how to represent it using csound or supercollider.
My experience is that with computer music, you have to keep experimenting and listening in order to truly understand and innovate.
This appears to be mercifully shorter and less intimidating than the must-have bible, "Curtis Roads. The Computer Music Tutorial. MIT Press, Cambs, MA, 1996".
It says it was originally published by Wiley in 2009, and the rights reverted to the author in 2025, whereupon the author released it on the net for free.
If someone wanted to start making computer music I'm not sure I'd recommend this or Curtis Roads' book as a starting point.
These aren't resources for getting started. They're more like encyclopedias for learning about DSP and tech once you've established the fundamentals of music and sequencing.
If a beginner wants practical knowledge for making records with electronic instruments I'd give them a DAW, teach them to record and sequence, teach them basic music theory, and then point them to something like Ableton's synthesis tutorials that will teach them about oscillators, envelopes, filters, LFOs, and basic sample manipulation.
That's 80% of the necessary skills right there.
I'm so happy to see Nick Collins taking this on. If you haven't seen his book, Handmade Music, it's an excellent book for music projects. This contribution looks exceptional as well.
I was reading up on the author and saw this interesting bit[0]:
> An algorave (from an algorithm and rave) is an event where people dance to music generated from algorithms, often using live coding techniques. Alex McLean of Slub and Nick Collins coined the word "algorave" in 2011, and the first event under such a name was organised in London, England. It has since become a movement, with algoraves taking place around the world.
[0] https://en.wikipedia.org/wiki/Algorave
Nicolas Collins is actually a different person: https://www.nicolascollins.com/handmade.htm
I love the world of music production. I started with Ableton Live six years or so ago and it's been a wonderful hobby. It has such a vibrant cottage industry of plugins (sampled instruments, synthesizers, effects, etc) thanks to the VST standard.
colkassad
I work professionally with music, including using ableton. I do create but don't sell/adverise, I'm strictly 'backstage'. I love everything about creating music, less so for reading about music (reviews, critiques, dissecting) though there are occasional exceptions. Are you putting your creativity online publicly?
Yes, I have a soundcloud account: https://soundcloud.com/emmets-music
I like to mix electronic and orchestral sounds. Some examples:
https://soundcloud.com/emmets-music/the-seven-hills-of-rome
https://soundcloud.com/emmets-music/dark-matter
I also like piano a lot (I only know how to play guitar badly, I just draw in the notes and work velocity until it sounds good to me):
https://soundcloud.com/emmets-music/life-is-delicate-remixed
https://soundcloud.com/emmets-music/soledad
Do you put yours online? I enjoy listening to other musicians. People give soundcloud a lot of grief but I love the service and the musicians I have met there.
Thank you. Your music is enjoyable, I listened to a couple (twice, first track and the fast one) and skimmed (four times) through others (my listening time is packed). Each piece has a filmic quality. The most impressive part is that they are (more of less) finished, which is a feat in itself. Very good.
I am very backstage, I don't wish to be known professionally, except by clients, all of which came via word of mouth - I wouldn't know how to advertise and I'd probably mess it up/give the wrong impression/attract the 'wrong' clients.
I won't link here but I do use soundcloud, I would say it's just ok. Tend to go Faircamp, bandcamp, geocities. MySpace lost 12 tunes.
I'll make a note of our exchange and if I change my mind I'll let you know. Some of my music is not dissimilar to yours, which is often the case, and not a bad thing - like you, it's the act of creation that brings immense pleasure. Thanks again for sharing I enjoyed listening.
Thanks for listening. Feel free to reach out if you ever want to share. I create music strictly for my own enjoyment so I understand where you are coming from. Take care!
The info page, perhaps a better url for submission instead of the large/hugged PDF
https://composerprogrammer.com/introcompmusic.html
I enjoyed Miller Puckette's Theory and Techniques of Electronic Music, using Pd. https://msp.ucsd.edu/techniques.htm
That's a classic :) The cool thing is that all examples are written in Pd (naturally) and can be explored interactively by the reader.
For what it's worth there's a new book on The Science of Music by Mark Newman who also the author of the popular book on Computational Physics [1].
[1] Mark Newman's new book: The Science of Music (2023):
https://lsa.umich.edu/cscs/news-events/all-news/search-news/...
A similar endeavour,
Introduction to Computer Music, by Prof. Jeffrey Hass
https://news.ycombinator.com/item?id=44744578
Loved this book when I was a student! This is for DSP enclined audience with a focus on musical applications. Not very in-depth if you want to develop your own plugins or DAW but still very informative
https://youtu.be/0Ssi-9wS1so?is=2tc8mEWK9ndA3G-4
Math rock, and microtonality.
Oh this is just awesome, thank you!
Juan García Castillejo published 'La telegrafía rápida, el triteclado y la música eléctrica' (High-speed telegraphy, the three-keyboard system, and electric music) in 1944.
https://archive.org/details/latelegrafiarapida.eltritecladoy...
Reminds me of starting college and photocopying digitized instrument wave forms (e.g. guitar string pluck) and then feeding them to a 6502 subroutine that output the waveform to the 8-bit companding DAC in my OSI C2-4P computer from a Basic driver that would play different notes through for songs.
I like that Programming for Music and Music for Programming are both on the front page together
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Wow, this book has been published in 2025, and it has zero mention of AI generated music. Not saying it's a bad thing - from the table of content it covers a lot of important fundamentals, but ignoring the elephant in the room is... weird.
It wasn't published in 2025. It was published in 2009 and the rights reverted to the author in 2025, who released it for free.
Oh ok, makes sense then