The Mathematics of Music [pdf](imaginary.org)
imaginary.org
The Mathematics of Music [pdf]
https://imaginary.org/sites/default/files/20190911-lala-booklet-v0.4-web-text.pdf
34 comments
C, E-flat, and G go into a bar. The bartender says, "Sorry, but we don't serve minors." So E-flat leaves, and C and G have an open fifth between them. After a few drinks, the fifth is diminished, and G is out flat. F comes in and tries to augment the situation, but is not sharp enough. D comes in and heads for the bathroom, saying, "Excuse me; I'll just be a second." Then A comes in, but the bartender is not convinced that this relative of C is not a minor. Then the bartender notices B-flat hiding at the end of the bar and says, "Get out! You're the seventh minor I've found in this bar tonight." E-flat comes back the next night in a three-piece suit with nicely shined shoes. The bartender says, "You're looking sharp tonight. Come on in, this could be a major development." Sure enough, E-flat soon takes off his suit and everything else, and is au natural. Eventually C sobers up and realizes in horror that he's under a rest. C is brought to trial, found guilty of contributing to the diminution of a minor, and is sentenced to 10 years of D.S. without Coda at an upscale correctional facility.
[deleted]
Made my day!
Another good book in the same vein is Dave Benson's 'Music: a Mathematical Offering'.
https://homepages.abdn.ac.uk/d.j.benson/pages/html/maths-mus...
https://homepages.abdn.ac.uk/d.j.benson/pages/html/maths-mus...
That's an incredible book! The observation that harmony depends on the timbre of instruments was mind blowing (e.g., if we used different musical instruments, the consonant and dissonant intervals would be different).
William Sethares has a good webpage about this (https://sethares.engr.wisc.edu/consemi.html). He talks about how you can create a synthesised instrument to fit a given scale. He even has example pieces you can listen to that are written in scales that would sound terrible with our usual instruments.
Notice that the "scalelab" tool of TFA allows you to also synthesize dissonant octaves and whatnot: https://github.com/IMAGINARY/ScaleLab
Musimathics is great [1].
If you want a more interactive source you can play with python notebooks from the music information retrieval site [2]. I've found it helpful, as you try some of the described music theory concepts in a programming environment.
Shameless plug, I've also written about how audio fingerprinting works [3], which touches on some of the topics regarding music theory.
[1] http://www.musimathics.com
[2] https://musicinformationretrieval.com
[3] https://emysound.com/blog/open-source/2020/06/12/how-audio-f...
If you want a more interactive source you can play with python notebooks from the music information retrieval site [2]. I've found it helpful, as you try some of the described music theory concepts in a programming environment.
Shameless plug, I've also written about how audio fingerprinting works [3], which touches on some of the topics regarding music theory.
[1] http://www.musimathics.com
[2] https://musicinformationretrieval.com
[3] https://emysound.com/blog/open-source/2020/06/12/how-audio-f...
Recommended:
A Geometry of Music by Dmitri Tymoczko - http://dmitri.mycpanel.princeton.edu/geometry-of-music.html
The Geometry of Musical Rhythm by Godfried Toussaint
- Original paper - http://cgm.cs.mcgill.ca/~godfried/publications/geometry-of-r... (PDF)
- Book - Review https://mtosmt.org/issues/mto.13.19.2/mto.13.19.2.gotham.php
A Geometry of Music by Dmitri Tymoczko - http://dmitri.mycpanel.princeton.edu/geometry-of-music.html
The Geometry of Musical Rhythm by Godfried Toussaint
- Original paper - http://cgm.cs.mcgill.ca/~godfried/publications/geometry-of-r... (PDF)
- Book - Review https://mtosmt.org/issues/mto.13.19.2/mto.13.19.2.gotham.php
I picked that book up on a whim a couple years ago and was surprised how engaged I was. It introduces you to some very interesting concepts.
I admit much of it's over my head, but that's exactly the kind of books I like. I keep coming back to chew on different parts.
In a similar vein, there's the Topos of Music book[0], which, according to Wikipedia, has been somewhat controversial among mathematicians and musicians alike.
[0] https://www.amazon.com/Topos-Music-Geometric-Concepts-Perfor...
[0] https://www.amazon.com/Topos-Music-Geometric-Concepts-Perfor...
Thanks for the book recommendation. I was curious about the controversy, so dug into it a bit.
From the Wikipedia page of the author:
> Dmitri Tymoczko..said of Mazzola: "If you can't learn algebraic geometry, he sometimes seems to be saying, then you have no business trying to understand Mozart."
https://en.wikipedia.org/wiki/Guerino_Mazzola
The critic, Tymoczko, is the author of A Geometry of Music. Oh, his review of Topos of Music is online.
http://dmitri.mycpanel.princeton.edu/files/publications/mazz... (PDF)
Abstract:
This paper critiques Guerino Mazzola’s derivation of traditional counterpoint rules, arguing that those rules are not well-modeled by pitch-class intervals; that Mazzola’s differential treatment of fifths and octaves is not supported musically or by traditional counterpoint texts; that Mazzola’s specific calculations are not reproducible; that there are a number of intuitive considerations weighing against Mazzola’s explanation; that the fit between theory and evidence is not good; and that Mazzola’s statistical arguments are flawed. This leads to some general methodological reflections on different approaches to mathematical music theory, as well as to an alternative model of first-species counterpoint featuring the orbifold T2/S2.
From the Wikipedia page of the author:
> Dmitri Tymoczko..said of Mazzola: "If you can't learn algebraic geometry, he sometimes seems to be saying, then you have no business trying to understand Mozart."
https://en.wikipedia.org/wiki/Guerino_Mazzola
The critic, Tymoczko, is the author of A Geometry of Music. Oh, his review of Topos of Music is online.
http://dmitri.mycpanel.princeton.edu/files/publications/mazz... (PDF)
Abstract:
This paper critiques Guerino Mazzola’s derivation of traditional counterpoint rules, arguing that those rules are not well-modeled by pitch-class intervals; that Mazzola’s differential treatment of fifths and octaves is not supported musically or by traditional counterpoint texts; that Mazzola’s specific calculations are not reproducible; that there are a number of intuitive considerations weighing against Mazzola’s explanation; that the fit between theory and evidence is not good; and that Mazzola’s statistical arguments are flawed. This leads to some general methodological reflections on different approaches to mathematical music theory, as well as to an alternative model of first-species counterpoint featuring the orbifold T2/S2.
Have posted this link before, but so relevant to the discussion: https://softwareengineering.stackexchange.com/questions/1360...
If music were Turing complete, it could imply we've been developing the ability to derive theorems in it as a species forever, which has some speculatively interesting philosophical implications as well. :)
If music were Turing complete, it could imply we've been developing the ability to derive theorems in it as a species forever, which has some speculatively interesting philosophical implications as well. :)
What would some of those philosophic implications be?
Also, this link asks if music notation is Turing complete, not music itself. Is out method of writing musc turing complete or is the music itself turing complete? What about something like maqam or other microtonal systems? Or temperament beyond the twelve-tone temperament we know and love?
Good find though, that's a very interesting question...
Also, this link asks if music notation is Turing complete, not music itself. Is out method of writing musc turing complete or is the music itself turing complete? What about something like maqam or other microtonal systems? Or temperament beyond the twelve-tone temperament we know and love?
Good find though, that's a very interesting question...
IANA-Philosopher, but if music, geometry, and logic could each encode all the same things, and we have a musical "sense," for its rules and symmetry, it's reasonable to speculate that music is an artifact of patterns and theorems that appear in these other areas, and where we don't have an matching rule in one area, it implies one should exist in the other.
It's stuff like the machine learning kit that produced a new AC/DC song (https://www.youtube.com/watch?v=vpEVsDN84Hc), meaning that machine learning has discovered an algorithm for producing outputs from rules effectively hidden in the band members heads.
The function the ML model discovered that produces theorems in the domain of plausible AC/DC songs isn't as useful to us as the one that produces plausible fluid dynamics models, but the possibility that the properties (or category) of that function might contain other analogies and isomorphisms to geometric figures, graphs, and other objects may imply that when we listen to music or compose, it our minds could just be in effect, "doing math," - and whether we in fact do anything that is "not math," either.
High level, I suspect this may have been what Hofstadter, Dennett, and Searle were on about.
It's stuff like the machine learning kit that produced a new AC/DC song (https://www.youtube.com/watch?v=vpEVsDN84Hc), meaning that machine learning has discovered an algorithm for producing outputs from rules effectively hidden in the band members heads.
The function the ML model discovered that produces theorems in the domain of plausible AC/DC songs isn't as useful to us as the one that produces plausible fluid dynamics models, but the possibility that the properties (or category) of that function might contain other analogies and isomorphisms to geometric figures, graphs, and other objects may imply that when we listen to music or compose, it our minds could just be in effect, "doing math," - and whether we in fact do anything that is "not math," either.
High level, I suspect this may have been what Hofstadter, Dennett, and Searle were on about.
the software can be found here, https://imaginary.org/programs
I started learning music recently, and another type that I would like to see, is mathematics for music genres, mathematics for music effects and mathematics for music theory
> mathematics for music theory
Oh boy now that's a fun one
There's a great book called "The Oxford Handbook of Algorithmic Music"[0] that has many insights into that! Some philosophical, some really mathy.
[0] https://global.oup.com/academic/product/the-oxford-handbook-...
Oh boy now that's a fun one
There's a great book called "The Oxford Handbook of Algorithmic Music"[0] that has many insights into that! Some philosophical, some really mathy.
[0] https://global.oup.com/academic/product/the-oxford-handbook-...
Mathematics for music effects is called digital signal processing. Proakis's book is good if you can handle analysis.
Also, a word of warning/advice: Music theory is not mathematics - it's just a way of storing patterns that sound nice on western instruments. There are some musical genres that you can basically generate algorithmically, but if you want to actually learn musicianship stay well clear.
Mark Levine's books are a good "textbook style" music theory book with lots of examples of actual jazz music.
Also, a word of warning/advice: Music theory is not mathematics - it's just a way of storing patterns that sound nice on western instruments. There are some musical genres that you can basically generate algorithmically, but if you want to actually learn musicianship stay well clear.
Mark Levine's books are a good "textbook style" music theory book with lots of examples of actual jazz music.
Actually, there is lots of nice maths in music theory.
Yes, music theory is descriptive rather than prescriptive. But messing around (mathematically) with its ideas can help one find nice musical possibilities one might not otherwise have considered.
For example, the major scale, the (ascending) melodic minor scale, and the harmonic minor scale are all seven-note scales such that all scale-wise thirds are major or minor.
This is important, because people like using harmony built on stacking major and minor thirds.
But are there any other seven-note scales whose thirds are all major or minor, beside the aforementioned (and their modes)?
Yes, the harmonic major scale (and its modes). Not so well known, it presents lots of different, but still ultimately convential, possibilities.
Yes, music theory is descriptive rather than prescriptive. But messing around (mathematically) with its ideas can help one find nice musical possibilities one might not otherwise have considered.
For example, the major scale, the (ascending) melodic minor scale, and the harmonic minor scale are all seven-note scales such that all scale-wise thirds are major or minor.
This is important, because people like using harmony built on stacking major and minor thirds.
But are there any other seven-note scales whose thirds are all major or minor, beside the aforementioned (and their modes)?
Yes, the harmonic major scale (and its modes). Not so well known, it presents lots of different, but still ultimately convential, possibilities.
After digging a bit into math approach to music, I'd agree with mhh__. Music is the business of making vibrations sound good. Definition of "good" is cultural and changes with times (see Devil's interval). The standard mathematical approach to music starts from axioms that are only applicable to western music and imply no evolution.
Maybe linguistic methods would serve composers better, as they deal in similar problems: how to to create meaningful whole from meaningless parts that relate to each other in complex way. Just as linguistics, music involves not only the mechanics of ear sensing, but also pattern recognition machine that likes to have just the right amount of repetition, and just the right amount of new stuff.
Maybe linguistic methods would serve composers better, as they deal in similar problems: how to to create meaningful whole from meaningless parts that relate to each other in complex way. Just as linguistics, music involves not only the mechanics of ear sensing, but also pattern recognition machine that likes to have just the right amount of repetition, and just the right amount of new stuff.
Yes, ultimately taste, or `what sounds good' decides. But mathematical approaches to all the building blocks of music can bring new materials to the ear which might sound good.
For example, I love the third mode of harmonic major (Phrygian flat 4) because I like the way it sounds. But I wouldn't have discovered it if I hadn't been thinking about scales in a systematic, mathematical kind of way.
For example, I love the third mode of harmonic major (Phrygian flat 4) because I like the way it sounds. But I wouldn't have discovered it if I hadn't been thinking about scales in a systematic, mathematical kind of way.
"... but if you want to actually learn musicianship stay well clear."
Species counterpoint is entirely mathematical, as is (to a great extent) common-practice partwriting. Yet, I guarantee you that most people who listen to that music would think it is entirely musical, when in fact very little is more than rule-following. (And yes, this is for a lot of genres! Sonata form, pop structure, etc., etc.)
Also, involving the work of David Cope, his rule-based composition systems repeatedly passed the turing test.
(I'm hoping, once I get to grad school, for my master's thesis to be around this exact idea: the computational aspect of creativity, and how the idea of "musicianship" is really just a human attribute we hold on to.)
I agree that music theory is mainly descriptive rather than prescriptive, but the fact that many works of genius center around patterns, well... it's hard for me to believe that we aren't heavily dependent, on some level, on math.
Species counterpoint is entirely mathematical, as is (to a great extent) common-practice partwriting. Yet, I guarantee you that most people who listen to that music would think it is entirely musical, when in fact very little is more than rule-following. (And yes, this is for a lot of genres! Sonata form, pop structure, etc., etc.)
Also, involving the work of David Cope, his rule-based composition systems repeatedly passed the turing test.
(I'm hoping, once I get to grad school, for my master's thesis to be around this exact idea: the computational aspect of creativity, and how the idea of "musicianship" is really just a human attribute we hold on to.)
I agree that music theory is mainly descriptive rather than prescriptive, but the fact that many works of genius center around patterns, well... it's hard for me to believe that we aren't heavily dependent, on some level, on math.
> Species counterpoint is entirely mathematical, as is (to a great extent) common-practice partwriting.
It really isn't. You can try to translate the guidelines of species counterpoint into "hard" constraints and generate pieces that precisely fit these constraints - but this does not result in good or interesting music. And this is even more true for actual composition, which is developed to a level that far exceeds the guidelines of species counterpoint itself. Thinking of music patterns as being entirely "mathematical" was historically common - there is a lot of old music theory that simply involves a mathematical/numerological exploration of musically-viable ratios between frequencies - but it's not really helpful if you're trying to figure out what makes music tick.
It really isn't. You can try to translate the guidelines of species counterpoint into "hard" constraints and generate pieces that precisely fit these constraints - but this does not result in good or interesting music. And this is even more true for actual composition, which is developed to a level that far exceeds the guidelines of species counterpoint itself. Thinking of music patterns as being entirely "mathematical" was historically common - there is a lot of old music theory that simply involves a mathematical/numerological exploration of musically-viable ratios between frequencies - but it's not really helpful if you're trying to figure out what makes music tick.
As a composer I agree to some extent. Like, I don't sit down and say "oh lets use some augmented sixths today", but I definitely refer back to the "rules" of music theory a lot when I get stuck or I'm looking for a way to communicate a certain mood in a piece.
And this kinda device-reference approach, I believe, is mathematical. And I also think that a lot of what makes music tick is that inherent usage of the musical language we study or grow up with. Truly original music imo doesn't really exist[0]; the intentional or unintentional borrowing of ideas is why music theory is a field of study: to describe the ways in which music works. I think even modern techniques (set theory, 12-tone rows, and atonal music overall) has to look at the existing rules and say "what can we do differently" -- even as Romantic-era composers got more adventerous in their use of texture, harmony, tonality, etc.
(The story of Charles Ives, who spent most of his younger life being "untrained" by his father to sing duets in overlapping keys and the like, is a good example of this.)
> ... actual composition, which is developed to a level that far exceeds the guidelines of species counterpoint itself
While true, I still think it's more of a spectrum than a binary "all or nothing" mathematical approach, which is why I believe that musicianship is more mathematical than not. I agree that it far exceeds the "use a forth and you're beheaded" approach but I'd be interested to hear music that is entirely free of some sort of process.
[0] note: the discussion of originality in music is probably an entire essay on its own, lol
And this kinda device-reference approach, I believe, is mathematical. And I also think that a lot of what makes music tick is that inherent usage of the musical language we study or grow up with. Truly original music imo doesn't really exist[0]; the intentional or unintentional borrowing of ideas is why music theory is a field of study: to describe the ways in which music works. I think even modern techniques (set theory, 12-tone rows, and atonal music overall) has to look at the existing rules and say "what can we do differently" -- even as Romantic-era composers got more adventerous in their use of texture, harmony, tonality, etc.
(The story of Charles Ives, who spent most of his younger life being "untrained" by his father to sing duets in overlapping keys and the like, is a good example of this.)
> ... actual composition, which is developed to a level that far exceeds the guidelines of species counterpoint itself
While true, I still think it's more of a spectrum than a binary "all or nothing" mathematical approach, which is why I believe that musicianship is more mathematical than not. I agree that it far exceeds the "use a forth and you're beheaded" approach but I'd be interested to hear music that is entirely free of some sort of process.
[0] note: the discussion of originality in music is probably an entire essay on its own, lol
Absolutely fascinating section on Lissajous figures. I had only been exposed to the concept through orbits.[1]
[1] - https://en.wikipedia.org/wiki/Lissajous_orbit
[1] - https://en.wikipedia.org/wiki/Lissajous_orbit
I'm not sure whom this is aimed at. The name "La La Lab" and the interactivity suggests kids, but there's a Fourier integral on Page 7 which immediately suggests undergraduates.
I have the same confusion about the target audience but there are some interesting tidbits. Seems more like a brain dump.
Why would the interactivity be limited to kids? I think the name is a pun on "La La Land", a musical film.
Neither... I suggested the name in a brainstorming session. We wanted to transmit the exhibition was like a Lab and I joined with the “La La La” you would use to warm up your voice, or the onomatopoeia of singing.
We were concerned of the movie association, but this exhibition was initially targeted at Germany so it wouldn’t be as strong as in the US.
Edit: the exhibition IS kid-friendly though... we have lots of families and schools visiting.
Surprised and happy to find a mention to LA LA LAB here ️ We just launched a digital exhibition on AI at www.i-am.ai
We were concerned of the movie association, but this exhibition was initially targeted at Germany so it wouldn’t be as strong as in the US.
Edit: the exhibition IS kid-friendly though... we have lots of families and schools visiting.
Surprised and happy to find a mention to LA LA LAB here ️ We just launched a digital exhibition on AI at www.i-am.ai
the expression predates the film by many years, fwiw.
I highly recommend Drew Nobile's book or papers on functional circuits as an adjunct to this on how to recursively generate songs from an initial seed progression.