caleb, morgan (2006) mallalieu & self-sim series

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mallalieu & self-sim series

Posted by Caleb Morgan on 9/25/2006, 1:11 pm68.166.234.51

What I'm searching for are new simple formulas that will generate self-similar12-tone series.

I've looked at quadratic residues, binomial coefficients, Stirling numbers, Fibonacciseries, Golomb rulers, Sidon Sets; Collatz-type sequences, & Costas arrays.

Discussion of 2 techniques of generating self-similarity:

There is a kind of 12-tone row that is sort of fractal. When you skip some notes inthe sequence and take every other note, or every third note, or "pick and skip" insome irregular pattern, you get a sequence that resembles the original sequence.

The vast majority of self-similarity of this kind is limited and imperfect. There areexactly four and only four 12-tone rows that exhibit complete, "perfect"self-similarity. Here they are:

C C# E D A F B Eb Ab Bb G F# the so-called "Mallalieu" rowC F Ab Bb A C# G D# E D B F# "Mallalieu" intervals multiplied by 5C G E D Eb B F A G# Bb C# F# "Mallalieu" intervals multiplied by 7C B Ab Bb Eb G C# A E D F F# "Mallalieu" intervals multiplied by 11

Theorists, including Milton Babbitt, have observed that the Mallalieu series can bederived from the series of numbers that come from the following formula: i ^ n modp. (2 ^ n mod 13 in this case.)

I was fascinated to learn that we can take the formula i ^ n mod p, and plug indifferent numbers to get other series that exhibit this "self-similar" property, only toa lesser degree.

Now, these series have certain limitations. The pattern of self-similarity is alwaysthe same--every other note starting on the second note, every third note starting onthe third note, and so on. And there are other limitations.

So this leads to the need for an alternative, "hand algorithm" approach. I haveencountered at least four in the music-theory literature--by Tom Johnson, AndrewMead, Phillip Batstone, and Robert Morris. I've come up with my own approach.

HAND ALGORITHM OR RECIPE FOR SELF-SIMILAR TWELVE-TONE ROWS

The recipe has four to six steps:

1) Grid

2) "Pick or Skip" pattern(often from limited-interval rows. see table of 62 rows below

Music Theory Forum: mallalieu & self-sim series http://members.boardhost.com/mtr1/msg/1159215088.html

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3) Cyclical Permutation series

4) X Swap (or interpreting Cyclical Permutation series as order numbers rather thanpitches)

5) Optional: Further Transformation of Series (including, primarily, intervallicexpansions)

6) Optional: Test and Tweak

1) Grid--This is simply the 12 tones of the chromatic scale in order.

grids:C C# D Eb E F F# G G# A Bb B

B Bb A Ab G F# F E Eb D Db C

or: the same thing in numbers:0 1 2 3 4 5 6 7 8 9 10 11

2) "Pick or Skip" pattern--starting on some note or number other than the first orlast, we create a pattern, at will, of numbers picked, in sequence. Having chosen anoverall direction, either to the left or to the right (ascending or descending) we stickwith it.

I've obtained the best results so far by limiting the number of different intervals inmy pattern. For this purpose, one can use a table of rows which are limited to only 3intervals. (See Table below) these can be used in all transformations & rotations

Note that the Mallalieu row is made by the simplest possible pattern. (This patterncorresponds to row/map # 14 in the Table)1,3,5,7,9,11,0,2,4,6,8,10

Next we align this pattern of picks, for convenience, underneath the "grid".

0 1 2 3 4 5 6 7 8 9 10 11 (grid)1 3 5 7 9 11 0 2 4 6 8 10 (sequence of values picked)

3) Cyclical Permutation series

The next step is to represent the relationship between our "grid" and our "pick orskip" series as if it were a cyclical permutation.

0 goes to 1 underneath. 1 above goes to 3 underneath. Now we find the 3 in theupper row and note the 7 beneath it. So 3 goes to 7 in the row underneath. Find the7 in the upper row. 7 goes to 2 in the row underneath. And so on.

Continuing this way, we have: (0,1,3,7,2,5,11,10, 8, 4, 9, 6) This is the same thingas writing "0 goes to 1, 1 goes to 3, 3 goes to 7", and so on. 6 goes to 0, or thebeginning of the cycle again.

cyclical permutation notation:(0,1,3,7,2,5,11,10, 8, 4, 9, 6)

4) X Swap (or interpreting Cyclical Permutation series as position numbers ratherthan pitches)


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Now, we could interpret these numbers as representing the pitches of a 12-toneseries. In fact, in this case it would make a rather pleasant all-interval series, a sortof "growth" series. But instead, we interpret these numbers as representing theposition numbers of each of the 12 notes of the chromatic scale. An easy way to dothis is to align the cyclical permutation series with the numbers 0 through 11underneath it:

(0, 1, 3, 7, 2, 5, 11, 10, 8, 4, 9, 6)0 1 2 3 4 5 6 7 8 9 10 11

We start on the lower left with the lower 0. We find the matching 0 above it. Welook underneath it and write down what we find, the 0. Same thing with the nextvalue, the 1. With the 2, we find the 2 in the upper row, and write down what wefind underneath it, the 4. With the next value--the 3--we find the 3 in the upper rowand write down the 2 underneath it. And so on. Doing this for all twelve values, weget:

0 1 4 2 9 5 11 3 8 10 7 6 (voila: our Mallalieu series)

This is what we get when we interpret the cyclical permutation series as if itrepresented the positions (or the "wheres" instead of "whats" of each of the 12tones. That is, taking 0, 1, 3, 7, 2, 5, 11, 10, 8, 4, 9, 6,instead of thinking of this as, for example, 11, or B in the 6th position, we areinstead thinking of it as 6, or F# in the 11th position. We are swapping pitch andposition, in effect.

5) Optional: Further Transformation of Series (including, primarily, intervallicexpansions)

By the end of step 4 in this recipe, we have a series with a pattern of self-similaritythat we chose in step 2. However, we might not like the intervallic pattern of theseries that resulted, and so we can change it. Here are 3 ways:

1) multiply the intervals of the series by 5 or 7.2) multiply the intervals of the series by 2 through 10, and take mod 13. Replacethe one duplicate pitch.3) "Map" the pitches using one of many possible mappings. Again, we can use ourtable of series having only three intervals. (See Table) Different transpositions of asingle series will produce different new series under mapping. Choose which of theresulting series to keep. Some of these mappings have the same effect asmultiplying mod 13.

6) Hand Tweak series

(needs no explanation)

Conclusion: Two methods for producing self-similar 12-tone series have beenpresented. The first uses the formula i ^ n mod p. The second uses a "handalgorithm" worked out by calebprime.The series derived from i ^ n mod p have a consistent pattern of self-similarity butthe self-similarity is "fuzzy". The "hand algorithm" creates series with variedpatterns of self-similarity, in which the self-similarity is exact.

TABLE: LIMITED INTERVALS-ROWS

1.12B C D D# E F F# G G# A A# B C#2.129 C D E F F# G G# A A# B C# D#


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3.139 C C# E F F# G G# A A# B D D#4.127 C D E F# G G# A A# B C# D# F5.125 C D E F# G# A A# B C# D# F G6.135 C C# E F G# A A# B D D# F# G7.123 C D E F G G# A# B C# D# F# A8.123 C D D# F G G# A# B C# E F# A9.123 C C# D# F G G# A# B D E F# A10.123 C C# D# E G G# A# B D F F# A11.123 C D D# F F# G# A# B C# E G A12.123 C C# D# F F# G# A# B D E G A13.123 C C# D# E F# G# A# B D F G A14.123 C D E F# G# A# B C# D# F G A15.123 C C# D# F# G# A# B D E F G A16.123 C D D# F G# A# B C# E F# G A17.123 C C# D# F G# A# B D E F# G A18.123 C C# D# E F# G A# B D F G# A19.123 C C# D# F# G A# B D E F G# A20.123 C C# D# F G A# B D E F# G# A21.23b C D# F G A B D E F# G# A# C#22.13b C D# E G G# B D F F# A A# C#23.137 C D# F# A A# C# E G G# B D F24.237 C D# F# G# A# C# E G A B D F25.235 C D# F# A B D F G# A# C# E G26.234 C D E G A B D# F# A# C# F G#27.234 C D E F# A B D# G A# C# F G#28.234 C D E G A# C# F A B D# F# G#29.234 C D E F# G# B D# G A# C# F A30.234 C D E G B D# F# G# A# C# F A31.234 C D F G# A# C# E G B D# F# A32.234 C D E G# A# C# F G B D# F# A33.234 C D F G A# C# E G# B D# F# A34.234 C D E G A# C# F G# B D# F# A35.234 C D E F# A# C# F G# B D# G A36.134 C D# F# A C# E G A# D F G# B37.34b C E G A# D F G# B D# F# A C#38.35b C D# F# A D F G# B E G A# C#39.349 C E G A# D F# A C# F G# B D#40.347 C E G# B D# G A# D F# A C# F41.357 C D# G# B E G A# C# F# A D F42.345 C D# F# A# C# F G# B E A D G43.345 C D# F# A# D G B E A C# F G#44.345 C D# G A# D F A C# F# B E G#45.345 C D# F# A# C# F A D G B E G#46.345 C D# F# A C# F A# D G B E G#47.135 C F G# C# E A D G A# D# F# B48.145 C E G# C# F A D F# A# D# G B49.45b C F A D F# A# D# G B E G# C#50.459 C F A D G B E G# C# F# A# D#51.359 C F G# C# F# B E A D G A# D#52.456 C E A D G B F A# D# G# C# F#53.456 C E G# D G B F A# D# A C# F#54.456 C E A C# G B F A# D# G# D F#55.456 C E G# C# G B F A# D# A D F#56.456 C E G# D F# B F A# D# A C# G


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57.456 C E A C# F# B F A# D# G# D G58.456 C E G# C# F# B F A# D# A D G59.456 C E A D# G# C# F# B F A# D G60.356 C F A# D# G# D G C# F# B E A61.156 C F A# E A D# G# D G C# F# B62.56b C F# B F A# E A D# G# D G C#

Message Thread

mallalieu & self-sim series - Caleb Morgan 9/25/2006, 1:11 pmRe: mallalieu & self-sim series - Caleb Morgan 9/27/2006, 6:07 amRe: mallalieu & self-sim series - caleb "quixotic" morgan 9/27/2006, 6:54am

Re: mallalieu & self-sim series - PSM113 10/3/2006, 12:39 pmRe: mallalieu & self-sim series - caleb "middlebrow" morgan10/4/2006, 3:28 am

Re: mallalieu & self-sim series - mark Q 10/3/2006, 12:29 pmRe: mallalieu & self-sim series - caleb 10/3/2006, 5:58 pmRe: mallalieu & self-sim series - Caleb "Overman" Morgan10/4/2006, 7:44 am

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caleb, morgan (2006) mallalieu & self-sim series

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