The piece linked to in the above post is worth hearing. Check it out.
This post, however, is just to document one of "life's little victories" -- like that cartoon by ?Keith Knight in Funny Times. He's good.
The victory is that I solved a technical problem in no time at all, without knowing what I was doing. Pure intuition, use of tech, muddling through, dumb luck.
The problem is this: Given a long, perfect self-similar sequence, how do you derive shorter sequences from it that A) most resemble it and B) have the closest thing to perfect self-similarity that is possible? There's no formula that I'm aware of.
Here's my perfect self-similar sequence for starters, from the indices of 3 ^ n
mod 17: (I'm using dots for spacers so that the tables line up, because I'm too lazy to make tables the right way):
16..14..17..12..5...15..11..10..2...3...7...13..4. ..9...6...8
http://www.google.com/search?q=%2216...5%2C15%2C11%22
Take every second number from the second number, every third from the third, every nth from the nth, and it's perfectly self-identical. That's math.
Now, problem is, to get a 14 or 12 or 10-number version, you can't just lop off a few numbers and expect it will work.
But somehow, I managed to get the results below by looking at the desired columns and just plowing through the problem. If I'd really tried to understand what I'm doing, it would take me a long time, which I don't have. I'm a composer, dammit.
These are, at least, results that are exactly what I was looking for. I just can't entirely explain how I did it or why it works. Took me about an hour. (If I didn't do this by muddling through, it might have hung me up for days or weeks.)
16..14..17..12..5...15..11..10..2...3...7...13..4. ..9...6...8
15..13..16..11..4...14..10..9...1...2...6...12..3. ..8...5...7
14..12..15..10..3...13..9...8...16..1...5...11..2. ..7...4...6
13..11..14..9...2...12..8...7...15..16..4...10..1. ..6...3...5
12..10..13..8...1...11..7...6...14..15..3...9...16 ..5...2...4
11..9...12..7...16..10..6...5...13..14..2...8...15 ..4...1...3
10..8...11..6...15..9...5...4...12..13..1...7...14 ..3...16..2
9...7...10..5...14..8...4...3...11..12..16..6...13 ..2...15..1
8...6...9...4...13..7...3...2...10..11..15..5...12 ..1...14..16
7...5...8...3...12..6...2...1...9...10..14..4...11 ..16..13..15
6...4...7...2...11..5...1...16..8...9...13..3...10 ..15..12..14
5...3...6...1...10..4...16..15..7...8...12..2...9. ..14..11..13
4...2...5...16..9...3...15..14..6...7...11..1...8. ..13..10..12
3...1...4...15..8...2...14..13..5...6...10..16..7. ..12..9...11
2...0...3...14..7...1...13..12..4...5...9...15..6. ..11..8...10
1...15..2...13..6...16..12..11..3...4...8...14..5. ..10..7...9
best fit 14!
13..11..14..9...4...12..8...7...1...2...5...10..3. ..6
11..9...12..7...2...10..5...6...13..14..3...8...1. ..4
9...7...10..6...14..8...3...4...11..12..1...5...13 ..2
7...6...8...4...12..5...1...2...9...10..13..3...11 ..14
6...4...5...2...10..3...13..14..7...8...11..1...9. ..12
4...2...3...14..8...1...11..12..6...5...9...13..7. ..10
2...14..1...12..5...13..9...10..4...3...7...11..6. ..8
14..12..13..10..3...11..7...8...2...1...6...9...4. ..5
12..10..11..8...1...9...6...5...14..13..4...7...2. ..3
10..8...9...5...13..7...4...3...12..11..2...6...14 ..1
8...5...7...3...11..6...2...1...10..9...14..4...12 ..13
5...3...6...1...9...4...14..13..8...7...12..2...10 ..11
3...1...4...13..7...2...12..11..5...6...10..14..8. ..9
1...13..2...11..6...14..10..9...3...4...8...12..5. ..7
best fit 12:
11..9...12..7...3...10..1...5...4...2...6...8
9...7...10..5...2...8...11..12..3...1...4...6
7...5...8...12..1...6...9...10..2...11..3...4
5...12..6...10..11..4...7...8...1...9...2...3
12..10..4...8...9...3...5...6...11..7...1...2
10..8...3...6...7...2...12..4...9...5...11..1
8...6...2...4...5...1...10..3...7...12..9...11
6...4...1...3...12..11..8...2...5...10..7...9
4...3...11..2...10..9...6...1...12..8...5...7
3...2...9...1...8...7...4...11..10..6...12..5
2...1...7...11..6...5...3...9...8...4...10..12
1...11..5...9...4...12..2...7...6...3...8...10
best fit 10:
8...7...10..6...3...9...5...4...1...2
7...6...9...4...2...8...10..3...5...1
6...4...8...3...1...7...9...2...10..5
4...3...7...2...5...6...8...1...9...10
3...2...6...1...10..4...7...5...8...9
2...1...4...5...9...3...6...10..7...8
1...5...3...10..8...2...4...9...6...7
5...10..2...9...7...1...3...8...4...6
10..9...1...8...6...5...2...7...3...4
9...8...5...7...4...10..1...6...2...3
These satisfy the musical goal -- which is to have a pre-ordained series of pitches, within which lines move at slower speeds, strongly resemble each other, and add up -- as a composite texture -- to exactly that pre-ordained series. It's a trick, but it sounds good, and it's not possible to achieve, exactly, by writing counterpoint the old-fashioned way.