Math and Music
 

Have you ever wondered where the rules of music theory come from? Musical structure, i.e. songs, scales, chords, etc. are some of the most beautiful mathematical structures one can see. In fact on of the most interesting groups to look at in math is the dihedral groups. Dihedrals are the most symmetrical shapes in the world (triangle, square, pentagon, and so forth). Well there is actually an isomorphism between the set of all the major and minor triad chords and the D12 dihedral group, the group generated by the translations and inversions of the 12 sided dihedral. This is such an amazing outcome, these are both very complicated unrelated objects, but written as a mathematical structure, you cannot tell the difference between the two.

What other kinds of musical structures do you think could be represented using mathematical structures?

free improv == chaos theory.

Hahaha great!

lol.

I am taking a Pre-Calc honors course but that made my head hurt...wtf are some of those words? :wavey:

My Dad's friend Gareth wrote a book on this called 'Musimathics' or something of that ilk.

I imagine the maths surrounding the timbre of different musical instruments would be quite interesting. I've been thinking about this lately because I've been working on coding an app which uses voice recognition.

Also, I've heard of Euclid's Algorithm being used to tune instruments.


Then there's Tool - Lateralus. :D

i know there are a bunch of rock and metal bands that use math quite a bit when creating their rhythms, but autechre is the only group i can think of that creates their music as a result of mathematic formulas they create for their pieces.

I didn't know Autechre did that. :eek:

Chiastic Slide is a great album.

Wow, theres so much I don't know. It would be great if you could let me know the names of some of the songs that Autechre uses math to create. Also, proggyman, if you could get me the name of that book, that would be awesome.
I love the Fibonacci sequence, it's another one of those things that is seen everywhere in the natural world and is an excellent example of how deeply everything is connected.
Physics is used widely with the timbre of instruments, there are many physics equations that deal with that, I however do not know much about the physical equations, I look more abstractly at the structures underneath.
And yes, Euclids algorithm to find the highest common factor of two numbers can also be used to find the largest measure of string segment that would fit into both. This is would have been extremely useful in tuning instruments in Euclids time since they talked about lengths of string rather then pitch and frequency.
This is awesome, keep throwing the ideas out there, what more does everyone know? Have you ever thought about what happens when you invert the notes of a chord mod 12? It's actually quite interesting. If you refer to the notes as numbers in the chromatic scale rather than by the note name, then all you have to do is negate each number of the chord and mod it by 12 and you will actually get the chord which is rooted on the minor third of the chord, and it will switch from major to minor, or vice versa.
E.g. Take C major (C,E,G) or in terms of the numbers on the scale (1,5,8). Now negate those (-1,-5,-8) and mod by 12 (11,7,4) This is an Eb minor chord. It works like this for any triad chord.

Amazon.com: Musimathics, Volume 1: The Mathematical Foundations of Music: Gareth Loy: Books
Here it is, I know the author personally, he's into Traditional Irish Folk mostly but I think he minored in music theory in college. Maybe you can find a bio.

Care to back that up?

I'm not sure if you are saying "I dont think so" to the inversion method, but it really does work for any major or minor three note chord, meaning I am excluding all seventh, diminished, or any other more complicated chord. It's actually quite easy to show that this is true. There are 12 notes in the chromatic scale, C D E F G A B, plus all the accidentals. So if we just write them as numbers rather than letters, then we get 0,1,2,3,4,5,6,7,8,9,11 (I just realized a mistake I made in my previous post; I should have started C on 0 rather than 1). There are 24 three note chords using these notes and the rules of constructing chords. 12 major and 12 minor. I can just write it in list form to show that what I said earlier is true.

C Major - (0,4,7) - (0,8,5) - F minor (another mistake I made, it is not rooted on the minor third)
C# Major - (1,5,8) - (11,7,4) - E minor
D Major - (2,6,9) - (10,6,3) - D# minor
D# Major - (3,7,10) - (9,5,2) - D minor
E Major - (4,8,11) - (8,4,1) - C# minor
F Major - (5,8,0 - (7,2,0) - C minor

I'm going to leave it at that because there is actually a really nice little pattern that is coming out of this. Once the pattern is found it is very easy to figure out the rest.
Also, if you inverted the minor chords, you would get the same major chords out as above.
So F minor inverted is C Major and so on.

as far as i know autechre's entire catalog was created using formulas. haven't really paid attention to them in a while though. i've only got their 'confield' disc, had a bunch of mp3s too a while back.

a friend of mine was telling me that there was a correlation between the frequencies of the different colours of refracted light and the major tones. any truth to that?

I actually read an article that was a study about how all music can be plotted to make geometric surface structues that are really quit stunning. They went on to explain that our love for music is based on our understanding of mathematics. They also showed that some of the great past musicians were very talented in geometry specifically. Here is another interesting article, unrelated to the article I was just talking about Music Has Its Own Geometry, Researchers Find