The Mathematics of Music [pdf]

https://imaginary.org/sites/default/files/20190911-lala-booklet-v0.4-web-text.pdf

by kondor • 6 years ago

210 32 6 years ago

brudgers• 6 years ago

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

peterkos3• 6 years ago

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-...

mhh__• 6 years ago

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.

parenthesis2• 6 years ago

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.

peterkos3• 6 years ago

"... 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.

siraben• 6 years ago

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...

lioeters3• 6 years ago

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.

OscarCunningham2• 6 years ago

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...

enriquto2• 6 years ago

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).

OscarCunningham2• 6 years ago

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.

TheOtherHobbes• 6 years ago

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.

nathell• 6 years ago

Why would the interactivity be limited to kids? I think the name is a pun on "La La Land", a musical film.

elondaits• 6 years ago

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

empath75• 6 years ago

the expression predates the film by many years, fwiw.

bluetwo• 6 years ago

I have the same confusion about the target audience but there are some interesting tidbits. Seems more like a brain dump.

082349872349872• 6 years ago

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.

arc_of_descent• 6 years ago

Made my day!

lioeters3• 6 years ago

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

RogueBurger• 6 years ago

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.

lioeters3• 6 years ago

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.

addictedcs• 6 years ago

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...

motohagiography2• 6 years ago

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. :)

totemandtoken• 6 years ago

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...

motohagiography2• 6 years ago

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.

aaron-santos• 6 years ago

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

kian• 6 years ago

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.

The Mathematics of Music [pdf]

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