Music Theory ---> 12-tone theory
This is a study of some 12-tone rows. The basic idea is to take a big list of series that were collected for having a property of loose self-similarity (because of how they were generated) and to test those series when rotated for a different definition of self-similarity. The best finds will be series with "dual citizenship".
Series are generated as truncated power-residue index series. As such, they will all possess loose self-similarity of the first 6 notes.
An example with 3 ^ n mod 17:
Base: 3 Mod: 17 Period: 16
-----------------------------------------------------------
Power: 1 2 3 4 5 6 7 8 9 10 11 12
-----------------------------------------------------------
Raw: 16 14 1 12 5 15 11 10 2 3 7 13
Rank: 11 9 0 7 3 10 6 5 1 2 4 8
-----------------------------------------------------------
Mod 11: 5 3 1 1 5 4 0 10 2 3 7 2
Mod 12: 4 2 1 0 5 3 11 10 2 3 7 1
Mod 13: 3 1 1 12 5 2 11 10 2 3 7 0
When truncated and ranked, the 12-tone series becomes B,A,C,G,Eb,Bb,F#,F,C#,D,E,G#.
displaying in an "every other" permutation square:
The loose partial self-similarity can be quantified by noting the intervals of the colums: 10,10,10,7,10,10,8,9,2,11,4,5.
A series derived from taking every other note from some positions of this series will resemble the original series, so this can be used musically and perceived -- it's not hard to hear.
Every truncated ranked power-residue index series will have some of this self-similarity. A list of these series can be built up.
These power-residue series tend to be relatively "musical" because they usually have nice mixtures of intervals and contours.
Then series made from such a list can be tested from starting points other than the first note, to see if any of them also possess self-similarity as defined by the parameters of the Microtonal Scales program. These parameters can be adjusted to only select the series with the most extensive self-similiarty.
The study as I intend it will just be individual dual series, with analysis.
A first example. Unfortunately, I don't know what power-residue series generated it.
(0 10 2 3 6 7 4 1 9 11 5 8)
The first six notes C,Bb,D,Eb,F#,G are the ones passing the second test, and the segment beginning on 4 is the power-res portion: E,C#,A,B,F,Ab
The purpose of the thread is to share these special-property rows with anyone else who might be interested in using them. Music is a big tent that allows for all kinds of activities aside from the main ones of composing and performing.
This is a study of some 12-tone rows. The basic idea is to take a big list of series that were collected for having a property of loose self-similarity (because of how they were generated) and to test those series when rotated for a different definition of self-similarity. The best finds will be series with "dual citizenship".
Series are generated as truncated power-residue index series. As such, they will all possess loose self-similarity of the first 6 notes.
An example with 3 ^ n mod 17:
Base: 3 Mod: 17 Period: 16
-----------------------------------------------------------
Power: 1 2 3 4 5 6 7 8 9 10 11 12
-----------------------------------------------------------
Raw: 16 14 1 12 5 15 11 10 2 3 7 13
Rank: 11 9 0 7 3 10 6 5 1 2 4 8
-----------------------------------------------------------
Mod 11: 5 3 1 1 5 4 0 10 2 3 7 2
Mod 12: 4 2 1 0 5 3 11 10 2 3 7 1
Mod 13: 3 1 1 12 5 2 11 10 2 3 7 0
When truncated and ranked, the 12-tone series becomes B,A,C,G,Eb,Bb,F#,F,C#,D,E,G#.
displaying in an "every other" permutation square:
Code:
B, A, C, G, Eb, Bb, Gb, F, Db, D, E, Ab
A, G, Bb, F, D, Ab, B, C, Eb, Gb, Db, E
G, F, Ab, C, Gb, E, A, Bb, D, B, Eb, Db
F, C, E, Bb, B, Db, G, Ab, Gb, A, D, Eb
C, Bb, Db, Ab, A, Eb, F, E, B, G, Gb, D
Bb, Ab, Eb, E, G, D, C, Db, A, F, B, Gb
Ab, E, D, Db, F, Gb, Bb, Eb, G, C, A, B
E, Db, Gb, Eb, C, B, Ab, D, F, Bb, G, A
Db, Eb, B, D, Bb, A, E, Gb, C, Ab, F, G
Eb, D, A, Gb, Ab, G, Db, B, Bb, E, C, F
D, Gb, G, B, E, F, Eb, A, Ab, Db, Bb, C
Gb, B, F, A, Db, C, D, G, E, Eb, Ab, Bb
A series derived from taking every other note from some positions of this series will resemble the original series, so this can be used musically and perceived -- it's not hard to hear.
Every truncated ranked power-residue index series will have some of this self-similarity. A list of these series can be built up.
These power-residue series tend to be relatively "musical" because they usually have nice mixtures of intervals and contours.
Then series made from such a list can be tested from starting points other than the first note, to see if any of them also possess self-similarity as defined by the parameters of the Microtonal Scales program. These parameters can be adjusted to only select the series with the most extensive self-similiarty.
The study as I intend it will just be individual dual series, with analysis.
A first example. Unfortunately, I don't know what power-residue series generated it.
(0 10 2 3 6 7 4 1 9 11 5 8)
The first six notes C,Bb,D,Eb,F#,G are the ones passing the second test, and the segment beginning on 4 is the power-res portion: E,C#,A,B,F,Ab
Code:
C,Bb,D,Eb,Gb,G,E,Db,A,B,F,Ab
|
v
C, Bb, D, Eb, Gb, G, E, Db, A, B, F, Ab
D, C, E, F, Ab, A, Gb, Eb, B, Db, G, Bb
Bb, Ab, C, Db, E, F, D, B, G, A, Eb, Gb
A, G, B, C, Eb, E, Db, Bb, Gb, Ab, D, F ------>
Gb, E, Ab, A, C, Db, Bb, G, Eb, F, B, D
F, Eb, G, Ab, B, C, A, Gb, D, E, Bb, Db
Ab, Gb, Bb, B, D, Eb, C, A, F, G, Db, E
B, A, Db, D, F, Gb, Eb, C, Ab, Bb, E, G
Eb, Db, F, Gb, A, Bb, G, E, C, D, Ab, B
Db, B, Eb, E, G, Ab, F, D, Bb, C, Gb, A
G, F, A, Bb, Db, D, B, Ab, E, Gb, C, Eb
E, D, Gb, G, Bb, B, Ab, F, Db, Eb, A, C
^
|
Starting from the seventh note above, tranposed to C,
and displayed in every-other square:
C, A, F, G, Db, E, Ab, Gb, Bb, B, D, Eb
A, G, E, Gb, B, Eb, C, F, Db, Ab, Bb, D
G, Gb, Eb, F, Ab, D, A, E, B, C, Db, Bb
Gb, F, D, E, C, Bb, G, Eb, Ab, A, B, Db
F, E, Bb, Eb, A, Db, Gb, D, C, G, Ab, B
E, Eb, Db, D, G, B, F, Bb, A, Gb, C, Ab
Eb, D, B, Bb, Gb, Ab, E, Db, G, F, A, C
D, Bb, Ab, Db, F, C, Eb, B, Gb, E, G, A
Bb, Db, C, B, E, A, D, Ab, F, Eb, Gb, G
Db, B, A, Ab, Eb, G, Bb, C, E, D, F, Gb
B, Ab, G, C, D, Gb, Db, A, Eb, Bb, E, F
Ab, C, Gb, A, Bb, F, B, G, D, Db, Eb, E
9,10,11,11,11,11,11,8,3,10,9, intervals of columns
via International Skeptics Forum http://ift.tt/2CqOCcs
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