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Old 26th March 2013, 08:13 AM   #93
calebprime
Penultimate Amazing
 
Join Date: Jul 2006
Posts: 13,001
I just had a little encounter with 69Dodge, who has made me a handful of applications that I use in my work.

I was trying to use one of them to nail down the best-fit 14-element self-similar series. This turns out to be a little hard, because 14+1 -- unlike 16+1, 12+1, and 10+1 -- is not prime. (15 is not prime.) So some of my tricks don't work as easily.

The goal of having a self-similar series that gets shorter but remains self-similar, and still sounds like itself, is slightly trickier than I first thought.

This is because in practice, the rhythm, the series, and the mapping to actual pitches are all different things.

Turns out the best way to think of a 14-element self-similar series in this context is: 7+7 mapping to two octaves of any 7-note scale. But rhythmically, it's 14 elements with 3 rests, so that it's a 17-beat (or if more silence is desired 19-beat, 23-beat, etc.) cycle. That allows all the possible polyrhythms to work as they should.

I got a little confused trying to mentally juggle all this, while still keeping my musical goals in mind.

Looking for the answer, I found what seemed to be a little flaw in one of the apps 69Dodge wrote for me.

Our conversation went like this:

Calebprime: lots of

69Dodge, moments later:
Quote:
Ok, I think I understand what you want.

The attached new program works the same as the old one, except for one extra step: After it calculates each deviation d, it replaces d with modulus - d, if the latter is smaller.
I have to live with the way my brain works, when it works. Wordy and messy, but basically grounded in musical reality. 69Dodge always gets his end done without a lot of verbiage. He's efficient.

The app that 69Dodge revised turned out to not be necessary. Once I got clear in my head what I wanted to do with a 14-element series, I realized I had to treat it differently than the 12 and 10-element series.

Here are the finalfinal-until-I-revise-them-again tables, from which I can derive all my pitch material. The 16-element table is good for two octaves of octotonic (diminished) scales, 8+8. Or another approach I need to think about is whether mapping it to a pitch grid that doesn't match -- at the octave -- the number series, will work also. I have to try it.


Original 3 ^ n mod 17..16 elements
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

14..revised again

13..11..0...9...4...12..8...7...1...2...5...10..3. ..6...--..--..--
12..10..13..8...3...11..7...6...0...1...4...9...2. ..5...--..--..--
11..9...12..7...2...10..6...5...13..0...3...8...1. ..4...--..--..--
10..8...11..6...1...9...5...4...12..13..2...7...0. ..3...--..--..--
9...7...10..5...0...8...4...3...11..12..1...6...13 ..2...--..--..--
8...6...9...4...13..7...3...2...10..11..0...5...12 ..1...--..--..--
7...5...8...3...12..6...2...1...9...10..13..4...11 ..0...--..--..--
6...4...7...2...11..5...1...0...8...9...12..3...10 ..13..--..--..--
5...3...6...1...10..4...0...13..7...8...11..2...9. ..12..--..--..--
4...2...5...0...9...3...13..12..6...7...10..1...8. ..11..--..--..--
3...1...4...13..8...2...12..11..5...6...9...0...7. ..10..--..--..--
2...0...3...12..7...1...11..10..4...5...8...13..6. ..9...--..--..--
1...13..2...11..6...0...10..9...3...4...7...12..5. ..8...--..--..--
0...12..1...10..5...13..9...8...2...3...6...11..4. ..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..4..9..1..5..2..3
7..6..9..5..3..8..0..4..1..2
6..5..8..4..2..7..9..3..0..1
5..4..7..3..1..6..8..2..9..0
4..3..6..2..0..5..7..1..8..9
3..2..5..1..9..4..6..0..7..8
2..1..4..0..8..3..5..9..6..7
1..0..3..9..7..2..4..8..5..6
0..9..2..8..6..1..3..7..4..5
9..8..1..7..5..0..2..6..3..4

Last edited by calebprime; 26th March 2013 at 08:21 AM.
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