
The Mathematics of Music [pdf] - kondor
https://imaginary.org/sites/default/files/20190911-lala-booklet-v0.4-web-text.pdf
======
082349872349872
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
Made my day!

------
OscarCunningham
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-music.html)

~~~
enriquto
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).

~~~
OscarCunningham
William Sethares has a good webpage about this
([https://sethares.engr.wisc.edu/consemi.html](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.

~~~
enriquto
Notice that the "scalelab" tool of TFA allows you to also synthesize dissonant
octaves and whatnot:
[https://github.com/IMAGINARY/ScaleLab](https://github.com/IMAGINARY/ScaleLab)

------
addictedcs
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](http://www.musimathics.com)

[2]
[https://musicinformationretrieval.com](https://musicinformationretrieval.com)

[3] [https://emysound.com/blog/open-source/2020/06/12/how-
audio-f...](https://emysound.com/blog/open-source/2020/06/12/how-audio-
fingerprinting-works.html)

------
lioeters
Recommended:

A Geometry of Music by Dmitri Tymoczko -
[http://dmitri.mycpanel.princeton.edu/geometry-of-
music.html](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...](http://cgm.cs.mcgill.ca/~godfried/publications/geometry-of-
rhythm.pdf) (PDF)

\- Book - Review
[https://mtosmt.org/issues/mto.13.19.2/mto.13.19.2.gotham.php](https://mtosmt.org/issues/mto.13.19.2/mto.13.19.2.gotham.php)

~~~
RogueBurger
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.

~~~
lioeters
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.

------
siraben
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...](https://www.amazon.com/Topos-Music-Geometric-Concepts-
Performance/dp/3764357312)

~~~
lioeters
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](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...](http://dmitri.mycpanel.princeton.edu/files/publications/mazzola.pdf)
(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.

------
motohagiography
Have posted this link before, but so relevant to the discussion:
[https://softwareengineering.stackexchange.com/questions/1360...](https://softwareengineering.stackexchange.com/questions/136085/is-
musical-notation-turing-complete)

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
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...

~~~
motohagiography
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](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.

------
brudgers
the software can be found here,
[https://imaginary.org/programs](https://imaginary.org/programs)

------
kevindeasis
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

~~~
mhh__
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.

~~~
parenthesis
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.

~~~
jmiskovic
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.

~~~
parenthesis
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.

------
aaron-santos
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](https://en.wikipedia.org/wiki/Lissajous_orbit)

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

~~~
elondaits
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

------
kian
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.

