The Mathematics of Music [pdf]
imaginary.org
imaginary.org
https://homepages.abdn.ac.uk/d.j.benson/pages/html/maths-mus...
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...
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
[0] https://www.amazon.com/Topos-Music-Geometric-Concepts-Perfor...
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.
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. :)
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...
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.
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-...
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.
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.
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.
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.
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.
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.
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
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