Very nice web chord app, BTW. I use GuitarePro which also has excellent facilities for chord exploration particularly when paired to scale alternatives. A pet wish of mine is something similar for the raga systems.

I went to an organ concert while I was in college. I forget who was performing, but he was amazing. As part of the concert, he improvised fugues on themes handed to him on stage that had been prepared earlier. As I was leaving the concert, I told a friend who had not been there about the improvisations. I mentioned that one of the themes sounded like a Schoenberg tone row. (I didn’t know for certain that it was a tone row, only that it sounded weird.) I thought that was a dirty trick, but the organist took it in stride and improvised a fugue to the theme. A music professor overheard our conversation and approached me. He was impressed that I recognized the tone row but disappointed when I told him I was not a music major.

Are you familiar with the composer Arnold Schoenberg? He decided to create scales (”tone rows”) by more irregular permutations of all twelve tones and compose music from these new musical beasts. The math behind it may be of interest:
http://en.wikipedia.org/wiki/Twelve-tone_technique

Last, I’ll leave everyone with a deceptively simple puzzle to test their knowledge of basic harmony and the circle of fifths. What are the proper note names (not just the enharmonics–too easy!) for the chord D# major and in what key will you find it? What is its diatonic functionality in this key?

It turns out that learning how to answer that question properly (for me) resulted in a month of study where I learned basic music theory (basic harmony). Of course to test how much I really understood I taught my knowledge to a computer:

http://www.updike.org/uchord4/

This guitar/piano chord finder turns this simple list of 75 guitar chord forms:

http://spreadsheets.google.com/pub?key=p31a0PLmUWUiwYILN4F-_MQ&single=true&gid=0&output=html

into 928 guitar chords. (It also has the answer to the puzzle if you poke around enough.)

You could think of an infinite range of notes of the form k rⁿ where r is the 12th root of 2 and k is your reference pitch, say 440 for A 440. Then map each note k rⁿ to the remainder when n is divided by 12.

In group theory terminology, 7 is a generator of the group Z₁₂ and so you can solve 7x = b (mod 12) for any b. The generators are 1, 5, 7, and 11 and so these correspond to intervals that will generate all the notes in the chromatic scale. There’s no circle of major thirds, for example, because 4 is not a generator of Z₁₂.

I’m trying to visualize the cycles in terms of powers, like in modular arithmetic. For example, for the circle of 5ths, I compute 2^(7/12) mod 2 (loosely speaking) :

1. 1 * 2^(7/12) = 2^(7/12)
2. 2^(7/12) * 2^(7/12) = 2^(14/12)
3. 2^(2/12) * 2^(7/12) = 2^(9/12)
4. 2^(9/12) * 2^(7/12) = 2^(16/12)
5. 2^(4/12) * 2^(7/12) = 2^(11/12)
6. 2^(11/12) * 2^(7/12) = 2^(18/12)
7. 2^(6/12) * 2^(7/12) = 2^(13/12)
8. 2^(1/12) * 2^(7/12) = 2^(8/12)
9. 2^(8/12) * 2^(7/12) = 2^(15/12)
10. 2^(3/12) * 2^(7/12) = 2^(10/12)
11. 2^(10/12) * 2^(7/12) = 2^(17/12)
12. 2^(5/12) * 2^(7/12) = 2^(12/12)
13. 1 * 2^(7/12) = 2^(7/12)
…

Is that how you think of it?