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Old 2nd January 2018, 07:15 AM   #1
calebprime
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Twelve Tone Rows with Dual Properties

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:

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
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

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
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.

Last edited by calebprime; 2nd January 2018 at 07:30 AM.
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Old 3rd January 2018, 05:49 AM   #2
calebprime
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yesterday's series: C,Bb,D,Eb,Gb,G,E,Db,A,B,F,Ab 10,4,1,3,1,9,9,8,2,6,3,4


Row of the Day: 90 ^ n mod 101, or "Compromising Adult".
Code:
======================================================================
 Base:  90       Mod:  101       Period:  100
-----------------------------------------------------------------------
 Power:    1    2    3    4    5    6    7    8    9    10   11   12   
-----------------------------------------------------------------------
 Raw:      100  27   63   54   48   90   43   81   26   75   51   17   
 Rank:     11   2    7    6    4    10   3    9    1    8    5    0    
-----------------------------------------------------------------------
 Mod 11:   1    5    8    10   4    2    10   4    4    9    7    6    
 Mod 12:   4    3    3    6    0    6    7    9    2    3    3    5    
 Mod 13:   9    1    11   2    9    12   4    3    0    10   12   4

If we shift (rotate) this back three positions, we get (0 9 4 3 6 11 10 8 2 7 1 5)
which tests well in Microtonal Scales.


Creative Process: Chakras stimulated, glands expressed, hard jets of water sprayed, libations consumed, stimulants titrated, visualization of unrequited lovers consummated, explanations proffered, tender offered, letters mailed, finally one felt that one's whole life had been leading up to this materially, spiritually, and in whatever narrative could be conceived. One saw -- in one's mind's eye -- a cool green glade.

In the cool green glade, Sahib, was a tiger all jeweled. Three were his gleaming eyes and mouth, if you get my drift. He spoke, saying, in his like tigery voice, "Ninety Mod 101 is the bit of cleverness that you seek O Seeker".


Creative Process: Visualization. "I like older men" she said, "especially them wit' th' mad bad math skilz", twirling her hair in her left hand.


Code:
(0 9 4 3 6 11 10 8 2 7 1 5)    Compromising Adult

C,A,E,Eb,F#,B,Bb,Ab,D,G,Db,F  9,7,11,3,5,11,10,6,5,6,4,7




Compromising Adult

            |
C,  A,  E,  Eb, F#, B,  Bb, Ab, D,  G,  Db, F
Eb, C,  G,  F#, A,  D,  Db, B,  F,  Bb, E,  Ab   --->
Ab, F,  C,  B,  D,  G,  F#, E,  Bb, Eb, A,  Db
A,  F#, Db, C,  Eb, Ab, G,  F,  B,  E,  Bb, D    <---
F#, Eb, Bb, A,  C,  F,  E,  D,  Ab, Db, G,  B
Db, Bb, F,  E,  G,  C,  B,  A,  Eb, Ab, D,  F#   ---->
D,  B,  F#, F,  Ab, Db, C,  Bb, E,  A,  Eb, G
E,  Db, Ab, G,  Bb, Eb, D,  C,  F#, B,  F,  A     ---->
Bb, G,  D,  Db, E,  A,  Ab, F#, C,  F,  B,  Eb
F,  D,  A,  Ab, B,  E,  Eb, Db, G,  C,  F#, Bb    ---->
B,  Ab, Eb, D,  F,  Bb, A,  G,  Db, F#, C,  E
G,  E,  B,  Bb, Db, F#, F,  Eb, A,  D,  Ab, C

p/0/0:...............C  A  E  Eb F# B  Bb Ab D  G  C# F  



p/3/1:...............C  G  F# A  D  C# B  F  Bb E  Ab Eb 
.....................c........a.................e.....eb....
...........................f#..........b.....bb....ab.......
........................g........d..c#....f                  slight cheat




r/9/8:...............C  C# F# A  D  Bb E  B  F  G  Ab Eb 
.....................c........a........e..............eb...
...........................f#.......bb....b........ab......  slight cheat
.................................d..............g..........
........................c#...................f.............




p/0/0:...............C  A  E  Eb F# B  Bb Ab D  G  C# F  



p/4/7:...............C  F# B  F  A  E  C# Ab G  Bb Eb D    
.....................c...........a..e..............eb....
........................f#.b..............ab....bb....d..
..............................f........c#....g...........    massive fraud at end



p/5/9:...............C  F# Bb F  D  A  Ab B  E  Eb C# G 
.....................c..............a........e..eb.......
........................f#................b..............
...........................bb..........ab................
.................................d....................g..
..............................f....................c#....   slight cheat


p/0/0:...............C  A  E  Eb F# B  Bb Ab D  G  C# F  





i/0/0:...............C  Eb Ab A  F# C# D  E  Bb F  B  G  
.....................c........a...........e...............
........................eb.......f#..........bb....b......   slight cheat
...........................ab..........d..............g...
....................................c#..........f.........



i/10/6:..............C  D  Ab Eb A  F  Bb C# F# G  E  B  
.....................c...........a.................e......
..............................eb.............f#.......b...
...........................ab..........bb.................     slight cheat
........................d...........f.....c#....g.........     massive shameless fraud


 
ri/7/2:..............C  F  B  A  Ab C# E  Eb Bb G  D  F# 
.....................c........a........e..eb..........f#..
...........................b.....ab..........bb....d......
........................f...........c#....................     minor hanky-panky


 
ri/6/7:..............C  Eb D  A  F# C# F  B  E  Bb Ab G  
.....................c........a..............e............
........................eb.......f#.......b.....bb.ab.g...     naughty!
...........................d........c#.f.................. 







5x/i/3/3:............C  A  Ab C# B  F  E  Bb D  Eb F# G  
5x/i/10/8:...........C  B  F  A  Bb C# D  G  E  Eb Ab F# 
5x5p/p/0/0:..........C  G  F  Eb Bb A  D  C# F# B  Ab E  
5x5p/i/3/3:..........C  F  F# C# D  A  E  G  B  Eb Ab Bb 
 


90 m101	


=======================================================================
 Base:  90       Mod:  101       Period:  100  Compromising Adult
-----------------------------------------------------------------------
 Power:    1    2    3    4    5    6    7    8    9    10   11   12   
-----------------------------------------------------------------------
 Raw:      100  27   63   54   48   90   43   81   26   75   51   17   
 Rank:     11   2    7    6    4    10   3    9    1    8    5    0    
-----------------------------------------------------------------------
 Mod 11:   1    5    8    10   4    2    10   4    4    9    7    6    
 Mod 12:   4    3    3    6    0    6    7    9    2    3    3    5    
 Mod 13:   9    1    11   2    9    12   4    3    0    10   12   4    

B,D,G,F#,E,Bb,Eb,A,C#,G#,F,C

Compromising Adult

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


Compromising Adult

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

3,4,3,10,3,10,5,8,3,4,7,8
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Old 3rd January 2018, 01:25 PM   #3
calebprime
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Originally Posted by calebprime View Post
...

Row of the Day: 90 ^ n mod 101, or "Compromising Adult".
Code:
...

3,4,3,10,3,10,5,8,3,4,7,8
inverted one interval. the actual self-sim is slightly better:

3,4,3,10,3,2,5,8,3,4,7
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Old 4th January 2018, 04:17 AM   #4
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so far: C,Bb,D,Eb,Gb,G,E,Db,A,B,F,Ab

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



Row of the Day: Base 4, Mod 5,148,359: C,D,Eb,A,Bb,F#,G,C#,F,Ab,E,B
or 0,2,3,9,10,6,7,1,5,8,4,11

================================================== ================================================== ===================
Base: 4 Mod: 5148359 Period: 2574179
-----------------------------------------------------------------------------------------------------------------------
Power: 1 2 3 4 5 6 7 8 9 10 11 12
-----------------------------------------------------------------------------------------------------------------------
Raw: 5148358 1287090 2121447 1 1718776 834358 891153 1287091 1668715 431687 501751 2121448
Rank: 11 5 9 0 8 3 4 6 7 1 2 10
-----------------------------------------------------------------------------------------------------------------------
Mod 11: 6 2 9 1 4 8 10 3 4 3 8 10
Mod 12: 10 6 3 1 4 10 9 7 7 11 7 4
Mod 13: 7 12 3 1 7 5 3 0 9 9 3 4



Here is that series starting on G in a standard inversion square and then an "every other" square


Code:
Standard inversion square

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





every-other square

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

6,7,6,3,6,7,4,5,1,5,3,7 column intervals; center of self-sim is 6.  This is relatively good for this method.

For resemblance of this kind, the "every other" square is the one to look at, and in particular the columns, as the intervals of the columns give a clear picture of how tight the self-similarity is.

Then the series is rotated 6 notes to show the version that passes tests in Microtonal Scales restrictive enough that only around 12 series were found out of thousands. Below that are some embedding diagrams.

Then harmonic analysis and some chorales or arrays using this series.


Rotation 6 places, every-other square:

Code:
C,  D,  Eb, A,  Bb, F#, G,  C#, F,  Ab, E,  B
D,  A,  F#, C#, Ab, B,  C,  Eb, Bb, G,  F,  E
A,  C#, B,  Eb, G,  E,  D,  F#, Ab, C,  Bb, F
C#, Eb, E,  F#, C,  F,  A,  B,  G,  D,  Ab, Bb
Eb, F#, F,  B,  D,  Bb, C#, E,  C,  A,  G,  Ab
F#, B,  Bb, E,  A,  Ab, Eb, F,  D,  C#, C,  G
B,  E,  Ab, F,  C#, G,  F#, Bb, A,  Eb, D,  C
E,  F,  G,  Bb, Eb, C,  B,  Ab, C#, F#, A,  D
F,  Bb, C,  Ab, F#, D,  E,  G,  Eb, B,  C#, A
Bb, Ab, D,  G,  B,  A,  F,  C,  F#, E,  Eb, C#
Ab, G,  A,  C,  E,  C#, Bb, D,  B,  F,  F#, Eb
G,  C,  C#, D,  F,  Eb, Ab, A,  E,  Bb, B,  F#


        |           p
        V           
C,  D,  Eb, A,  Bb, F#, G,  C#, F,  Ab, E,  B
Bb, C,  C#, G,  Ab, E,  F,  B,  Eb, F#, D,  A
A,  B,  C,  F#, G,  Eb, E,  Bb, D,  F,  C#, Ab      <-- part
Eb, F,  F#, C,  C#, A,  Bb, E,  Ab, B,  G,  D
D,  E,  F,  B,  C,  Ab, A,  Eb, G,  Bb, F#, C#
F#, Ab, A,  Eb, E,  C,  C#, G,  B,  D,  Bb, F
F,  G,  Ab, D,  Eb, B,  C,  F#, Bb, C#, A,  E
B,  C#, D,  Ab, A,  F,  F#, C,  E,  G,  Eb, Bb    <--- part
G,  A,  Bb, E,  F,  C#, D,  Ab, C,  Eb, B,  F#
E,  F#, G,  C#, D,  Bb, B,  F,  A,  C,  Ab, Eb    <-- part
Ab, Bb, B,  F,  F#, D,  Eb, A,  C#, E,  C,  G
C#, Eb, E,  Bb, B,  G,  Ab, D,  F#, A,  F,  C     <-- most


c,  f,  a,  f#, d,  ab, g,  b,  bb, e,  eb, c#

c...............d.......................eb......
........a.......................bb..............
............f#..........g...................c#..
....f...............ab......b.......e...........   e and b reversed

Last edited by calebprime; 4th January 2018 at 04:57 AM.
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Old 4th January 2018, 04:49 AM   #5
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Some quick analysis of:

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

lots of 0147.

If broken into 2 groups of 6, you have
C,D,Eb,F#,A,Bb set
and
B,C#,E,F,G,G# set


C,D,Eb,F#,A,Bb allows for either C,D,Eb,F,Gb,A,Bb interp., or C,D,Eb,F#,G,A,Bb interp. -- that is, b5 or #4.

the second set has no such ambiguity -- the only superset is octotonic: E,F,G,G#,A#,B,C#,D to pick one starting-point.


a couple of nice 4-part combinations:

Code:
5x/i/6/1:............Ab Eb A  E  C  G  C# F  D  Bb B  F# 
5x/r/7/1:............Eb B  Ab C  F# C# A  E  Bb F  G  D  
ri/1/2:..............F  Ab C  F# G  Eb E  Bb B  C# D  A  
p/0/0:...............C  D  Eb A  Bb F# G  C# F  Ab E  B  

5x/i/6/1:............Ab Eb A  E  C  G  C# F  D  Bb B  F# 
5x/r/11/9:...........D  A  B  F# G  Eb C  E  Bb F  C# Ab 
ri/5/10:.............Eb F  F# C# A  C  E  Bb B  G  Ab D  
p/0/0:...............C  D  Eb A  Bb F# G  C# F  Ab E  B
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Old 4th January 2018, 08:27 AM   #6
calebprime
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Originally Posted by calebprime View Post
Some quick analysis of:

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

lots of 0147.

If broken into 2 groups of 6, you have
C,D,Eb,F#,A,Bb set
and
B,C#,E,F,G,G# set


C,D,Eb,F#,A,Bb allows for either C,D,Eb,F,Gb,A,Bb interp., or C,D,Eb,F#,G,A,Bb interp. -- that is, b5 or #4.

the second set has no such ambiguity -- the only superset is octotonic: E,F,G,G#,A#,B,C#,D to pick one starting-point.


a couple of nice 4-part combinations:

Code:
p/0/0:...............C  D  Eb A  Bb F# G  C# F  Ab E  B  
ri/1/2:..............F  Ab C  F# G  Eb E  Bb B  C# D  A
5x/r/7/1:............Eb B  Ab C  F# C# A  E  Bb F  G  D 
5x/i/6/1:............Ab Eb A  E  C  G  C# F  D  Bb B  F# 
 
  
p/0/0:...............C  D  Eb A  Bb F# G  C# F  Ab E  B  
ri/5/10:.............Eb F  F# C# A  C  E  Bb B  G  Ab D  
5x/r/11/9:...........D  A  B  F# G  Eb C  E  Bb F  C# Ab 
5x/i/6/1:............Ab Eb A  E  C  G  C# F  D  Bb B  F#
4-part combo happens to sound much better voiced like this, I think. (Upside down.) This is designed so that each chord pairs with the one before it and the one after it, but not 3 in a row. So any 2 adjacent chords will sound nice mushed together. There will also be many opportunities to suspend voices or anticipate because of this design.
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Old 4th January 2018, 08:37 AM   #7
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(0 10 2 3 6 7 4 1 9 11 5 8)
C,Bb,D,Eb,Gb,G,E,Db,A,B,F,Ab First Example

intervals: 10,4,1,3,1,9,9,8,2,6,3,4


--------------------------------------------------------

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

C, A, E, Eb, F#, B, Bb, Ab, D, G, Db, F
intervals: 9,7,11,3,5,11,10,6,5,6,4,7

--------------------------------------------------------
C,D,Eb,A,Bb,F#,G,C#,F,Ab,E,B Five Large

0,2,3,9,10,6,7,1,5,8,4,11
or rotated
0,6,10,1,9,4,5,7,8,2,3,11


intervals: 2,1,6,1,8,1,6,4,3,8,7,1
--------------------------------------------------------



Here's what we have so far, the 3 series. In order to find out how close one is to another, I'll test them with another program I have.

It's helpful to make up nicknames, no matter how stupid or offensive or delightful or hilarious, for remembering them and keeping them straight. We have First Example, Compromising Adult, and Five Large.
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Old 4th January 2018, 08:42 AM   #8
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First Example is not like Compromising Adult:


Code:
?add row:c,a,e,eb,f#,b,bb,ab,d,g,db,f
?voc #1:1
?voc #2:1
?inc har:0,0
?har errs: 0 to 8
?run
working...
r/4/3:...............F# C  D  Eb Bb G  Ab C# E  A  F  B  
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab 

i/6/0:...............F# A  D  Eb C  G  Ab Bb E  B  F  C# 
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab 

5p/ri/4/1:...........C  A  Bb B  F# G  D  C# Eb F  E  Ab 
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab 

5x/p/0/0:............C  A  Ab Eb F# G  D  E  Bb B  F  C# 
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab 

5x/r/6/7:............C  A  D  Eb F# G  B  F  E  Bb Ab C# 
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab 

5x/i/3/9:............E  Bb D  Eb F# G  C  A  Ab C# B  F  
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab 

5x/ri/10/3:..........C  F# Ab Eb E  G  D  C# Bb A  F  B  
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab 

5x5p/p/5/3:..........Ab Eb D  G  F# B  E  C# A  F  C  Bb 
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab 

5x5p/r/4/1:..........C  Eb Bb F  F# C# D  G  A  B  E  Ab 
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab 

5x5p/r/1/2:..........C  G  D  Eb Bb B  E  F# Ab C# F  A  
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab 

5x5p/i/2/6:..........C  C# Ab Eb F# Bb D  G  A  B  E  F  
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab 

5x5p/ri/3/8:.........C  Bb Ab Eb B  G  E  A  D  C# F# F  
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab

by the same process, Five Large is not like Compromising Adult.

and finally, First Example is not like Five Large:


Code:
?new row #1:c,bb,d,eb,gb,g,e,db,a,b,f,ab
?add row:c,d,eb,a,bb,f#,g,c#,f,ab,e,b
?voc #1:1
?voc #2:1
?inc har:0,0
?har errs: 0 to 8
?run
working...
p/3/9:...............B  G  D  Eb F  F# C  C# A  Bb E  Ab 
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab 

p/4/9:...............C  Ab Eb E  F# G  C# D  Bb B  F  A  
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab 

ri/1/4:..............C  F# G  Eb E  Bb B  C# D  A  F  Ab 
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab 

5p/p/3/5:............F  Bb D  C# B  F# E  Eb A  G  C  Ab 
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab 

5p/r/3/8:............C  G  A  Eb E  F# B  C# D  Bb F  Ab 
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab 

5x/i/5/2:............D  Ab Eb B  F# C  E  C# A  Bb F  G  
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab 

5x5p/ri/5/1:.........D  C# Eb Bb F# G  E  Ab A  B  F  C  
p/0/0:...............C  Bb D  Eb F# G  E  C# A  B  F  Ab

Last edited by calebprime; 4th January 2018 at 08:51 AM.
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Old 4th January 2018, 08:45 AM   #9
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I really, really wish I understood this.
__________________
Up the River!
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Old 4th January 2018, 09:14 AM   #10
calebprime
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Hey, it's easy, but I'm being sketchy, so just ask me about a given step, and I'll explain that.
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Old 4th January 2018, 09:22 AM   #11
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Maybe a key concept is this: The so called extended row class.

That is, we're implicitly thinking of a series as being part of a group that is considered the same row -- all the transpositions, rotations, inversions, backwards, and multiplying-intervals-by-five, and possibly even taking-every-5th-note-of-12. We can define this, but it almost always includes the first stuff on the list.

Then I want to know if the 3 rows I just generated and found and tested are really different enough from each other to deserve to be called different.

So I have a program that will accept two series as inputs, and vary the second one according to its extended row class, and see if it can line them up so that they are the same. It couldn't, in the case of these 3, so I'm comfortable with giving them 3 different names.

The names sound like horses, which is good.
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Old 4th January 2018, 12:27 PM   #12
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Here's a good list to have, all the MOFs of Mallalieu type starting with narrow spacing and gradually widening in a fairly systematic fashion, ending in self-similarity with spacing of 6 and 7 notes.


C# E A Ab G C Eb F# B D F Bb intervals: 3 5 11 11 5 3 3 5 3 3 5 3
C# C B E F F# A D G Bb Eb Ab intervals: 11 11 5 1 1 3 5 5 3 5 5 5
C# Eb Bb Ab F# D F E B C A G intervals: 2 7 10 10 8 3 11 7 1 9 10 6
C# Bb B Ab F C A G E F# Eb D intervals: 9 1 9 9 7 9 10 9 2 9 11 11
C# Eb Bb A F F# G C D Ab E B intervals: 2 7 11 8 1 1 5 2 6 8 7 2
C# F# Bb E D A Ab G B C F Eb intervals: 5 4 6 10 7 11 11 4 1 5 10 10
C# D F Eb F# Ab B E G Bb A C intervals: 1 3 10 3 2 3 5 3 3 11 3 1
C# E D A F B Eb Ab Bb G F# C intervals: 3 10 7 8 6 4 5 2 9 11 6 1
C# Bb B F G C A E F# Ab Eb D intervals: 9 1 6 2 5 9 7 2 2 7 11 11
C# F# A E D Ab G Bb B C F Eb intervals: 5 3 7 10 6 11 3 1 1 5 10 10
C# Bb C A Ab E F F# D B G Eb intervals: 9 2 9 11 8 1 1 8 9 8 8 10
C# F F# A G Bb C E Ab B Eb D intervals: 4 1 3 10 3 2 4 4 3 4 11 11
C# D Bb Eb F B A E Ab G F# C intervals: 1 8 5 2 6 10 7 4 11 11 6 1
C# D B G Eb Bb A C F# Ab F E intervals: 1 9 8 8 7 11 3 6 2 9 11 9
C# F# Bb D F E G Ab B A Eb C intervals: 5 4 4 3 11 3 1 3 10 6 9 1
C# D Bb G Eb A B C F# Ab F E intervals: 1 8 9 8 6 2 1 6 2 9 11 9
C# F E G Ab B A Eb D C F# Bb intervals: 4 11 3 1 3 10 6 11 10 6 4 3
C# D F C E Eb A Bb F# B Ab G intervals: 1 3 7 4 11 6 1 8 5 9 11 6
C# Bb F Eb D Ab B A G F# C E intervals: 9 7 10 11 6 3 10 10 11 6 4 9
C# D F# A Ab Eb E C F G B Bb intervals: 1 4 3 11 7 1 8 5 2 4 11 3
C# Ab G B Eb C F# D F E A Bb intervals: 7 11 4 4 9 6 8 3 11 5 1 3
C# Bb D Ab B G Eb E A F# F C intervals: 9 4 6 3 8 8 1 5 9 11 7 1
C# D F C E Eb G Ab B F# Bb A intervals: 1 3 7 4 11 4 1 3 7 4 11 4
C# A Ab B E C G Eb D F Bb F# intervals: 8 11 3 5 8 7 8 11 3 5 8 7
C# Ab F F# Bb Eb G D B C E A intervals: 7 9 1 4 5 4 7 9 1 4 5 4
C# A D B Bb F# G Eb Ab F E C intervals: 8 5 9 11 8 1 8 5 9 11 8 1
C# Bb B Eb F# D G E F A C Ab intervals: 9 1 4 3 8 5 9 1 4 3 8 5
C# E B Eb C Ab G Bb F A F# D intervals: 3 7 4 9 8 11 3 7 4 9 8 11
C# Bb A G E C Ab Eb D F F# B intervals: 9 11 10 9 8 8 7 11 3 1 5 2
C# E F# G Bb Ab Eb D B C F A intervals: 3 2 1 3 10 7 11 9 1 5 4 4
C# E B F Bb G Ab C Eb A F# D intervals: 3 7 6 5 9 1 4 3 6 9 8 11
C# Bb B F E G D F# Eb A C Ab intervals: 9 1 6 11 3 7 4 9 6 3 8 5
C# Ab C B F F# D G Bb A Eb E intervals: 7 4 11 6 1 8 5 3 11 6 1 9
C# F# Ab D C G E A F B Eb Bb intervals: 5 2 6 10 7 9 5 8 6 4 7 3
C# F# C B E Ab F Eb D A Bb G intervals: 5 6 11 5 4 9 10 11 7 1 9 6
C# E D F# C F B A Eb Bb Ab G intervals: 3 10 4 6 5 6 10 6 7 10 11 6
C# Bb B G Eb C F# F Ab A D E intervals: 9 1 8 8 9 6 11 3 1 5 2 9
C# B F# F D Eb A C E Ab G Bb intervals: 10 7 11 9 1 6 3 4 4 11 3 3
C# D G F# Bb Eb A B C F Ab E intervals: 1 5 11 4 5 6 2 1 5 3 8 9
C# Ab G F B F# D Eb Bb A C E intervals: 7 11 10 6 7 8 1 7 11 3 4 9
C# F# Eb D Ab B G A F Bb E C intervals: 5 9 11 6 3 8 2 8 5 6 8 1
C# C A F# B Bb Eb Ab G E F D intervals: 11 9 9 5 11 5 5 11 9 1 9 11
C# E A Bb G C B D Eb F# F Ab intervals: 3 5 1 9 5 11 3 1 3 11 3 5
C# Bb B Ab G E F C Eb D A F# intervals: 9 1 9 11 9 1 7 3 11 7 9 7
C# E Eb F# G D A Bb F Ab B C intervals: 3 11 3 1 7 7 1 7 3 3 1 1
C# A F F# Eb C D G Ab B Bb E intervals: 8 8 1 9 9 2 5 1 3 11 6 9
C# C F D B F# G Bb Eb E A Ab intervals: 11 5 9 9 7 1 3 5 1 5 11 5
C# E B C G Ab Eb D A F# F Bb intervals: 3 7 1 7 1 7 11 7 9 11 5 3
C# Bb F E B Ab Eb C G F# A D intervals: 9 7 11 7 9 7 9 7 11 3 5 11
C# D G Bb B F# Eb E A C F Ab intervals: 1 5 3 1 7 9 1 5 3 5 3 5
C# F# Ab Bb B G E F Eb C A D intervals: 5 2 2 1 8 9 1 10 9 9 5 11
C# Ab B D E Eb F# Bb A G F C intervals: 7 3 3 2 11 3 4 11 10 10 7 1
C# D C Ab A B Eb E Bb F# G F intervals: 1 10 8 1 2 4 1 6 8 1 10 8
C# G F# D C B Eb F E Ab Bb A intervals: 6 11 8 10 11 4 2 11 4 2 11 4
C# E F# Bb A G Eb D C Ab B F intervals: 3 2 4 11 10 8 11 10 8 3 6 8
C# Eb E Ab Bb B G F D F# C A intervals: 2 1 4 2 1 8 10 9 4 6 9 4
C# F# C B D Bb G A Eb F Ab E intervals: 5 6 11 3 8 9 2 6 2 3 8 9
C# Ab B Eb C Bb E D F A F# G intervals: 7 3 4 9 10 6 10 3 4 9 1 6
C# F A Ab F# E B C G Bb D Eb intervals: 4 4 11 10 10 7 1 7 3 4 1 10
C# F# Ab Bb B G Eb F E C A D intervals: 5 2 2 1 8 8 2 11 8 9 5 11
C# B A Ab E C Eb F G F# Bb D intervals: 10 10 11 8 8 3 2 2 11 4 4 11
C# Eb F F# D Bb B A G E Ab C intervals: 2 2 1 8 8 1 10 10 9 4 4 1
C# F A Bb Ab D E Eb B G F# C intervals: 4 4 1 10 6 2 11 8 8 11 6 1
C# D Bb F F# C A Eb E B G Ab intervals: 1 8 7 1 6 9 6 1 7 8 1 5
C# Ab G B E Eb A C F# F Bb D intervals: 7 11 4 5 11 6 3 6 11 5 4 11
C# D G C E F Eb A B Bb F# Ab intervals: 1 5 5 4 1 10 6 2 11 8 2 5
C# B Eb E D Ab Bb A F C G F# intervals: 10 4 1 10 6 2 11 8 7 7 11 7
C# G Eb E D B F# C F Ab Bb A intervals: 6 8 1 10 9 7 6 5 3 2 11 4
C# A F B Bb E G Ab C F# D Eb intervals: 8 8 6 11 6 3 1 4 6 8 1 10
C# C A E Ab Bb G D F# B F Eb intervals: 11 9 7 4 2 9 7 4 5 6 10 10
C# B G C Eb E F# Ab D A F Bb intervals: 10 8 5 3 1 2 2 6 7 8 5 3
C# F# Bb F D E B C Eb Ab A G intervals: 5 4 7 9 2 7 1 3 5 1 10 6
C# B D G Eb Bb E F# F C A Ab intervals: 10 3 5 8 7 6 2 11 7 9 11 5
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Old 5th January 2018, 04:16 AM   #13
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Two words of explanation before the row of the day.

> the relation between the base/mod combination and the actual series in this formula is mysterious, so it's more or less useless, as far as I know. The base number determines where the number that is b2, or a half-step or minor second above the first note, which is always the lowest. Choosing the base determines where 0 and 1 fall, or 1 and b2, if you will.

> this generation technique therefore might be compared to a sort of concentrated jet -- all the results are confined to a narrow range of possibility, but the specifics seem random. All the drops/series that hit within a range form the potential database to be tested for other properties when rotated.

> Musically, these series are like freeze-dried themes. Just add "water". "Water" is rhythm, register, rhetoric, roulades and other ridiculous things. Soon, oatmeal is served! rut row!

Last edited by calebprime; 5th January 2018 at 04:18 AM.
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Old 5th January 2018, 05:05 AM   #14
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Row of the Day: Gathering Blandness, or (0 5 3 1 7 2 9 10 6 4 8 11), when
rotated, the power residue series is 0,4,7,8,1,11,9,3,10,5,6,2.

The other little trick I've been doing is finding a good series first, then seeing if I can find it in the database of series. This is because with the current hodgpodge of programs, the rows I'm finding aren't identifed by generator.

The only thing I can tell is that it will 5 ^ n mod something, if I can find it.

In its current form (0 5 3 1 7 2 9 10 6 4 8 11) C, F, Eb, Db, G, D, A, Bb, Gb, E, Ab, B, I thought that it tended to sort of modulate in a brighter and therefore blander direction, as if it were "skating backwards". It reminded me of someone with sad memories in the act of trying to blot them out with a narcotic that brings gathering warm numbness. I could have called it Oxy Overdose, but that would be too topical and on-the-nose. Mothers Little Helper? DrinkeePoo? These, too could be future names.

This one in its power res position has a classic pattern of self-similarity that makes it special.

Gathering Blandness (rotated to power res position start)

Code:
Rotation of Gathering Blandness and Display in Every-Other square:

C,  E,  G,  Ab, Db, B,  A,  Eb, Bb, F,  Gb, D
E,  Ab, B,  Eb, F,  D,  C,  G,  Db, A,  Bb, Gb
Ab, Eb, D,  G,  A,  Gb, E,  B,  F,  C,  Db, Bb
Eb, G,  Gb, B,  C,  Bb, Ab, D,  A,  E,  F,  Db
G,  B,  Bb, D,  E,  Db, Eb, Gb, C,  Ab, A,  F
B,  D,  Db, Gb, Ab, F,  G,  Bb, E,  Eb, C,  A
D,  Gb, F,  Bb, Eb, A,  B,  Db, Ab, G,  E,  C
Gb, Bb, A,  Db, G,  C,  D,  F,  Eb, B,  Ab, E
Bb, Db, C,  F,  B,  E,  Gb, A,  G,  D,  Eb, Ab
Db, F,  E,  A,  D,  Ab, Bb, C,  B,  Gb, G,  Eb
F,  A,  Ab, C,  Gb, Eb, Db, E,  D,  Bb, B,  G
A,  C,  Eb, E,  Bb, G,  F,  Ab, Gb, Db, D,  B

The column intervals: 4,4,7,4,4,3,4,4,3,4,4,3
which is as near-perfect as one will find, so this
is a special series

Last edited by calebprime; 5th January 2018 at 05:12 AM.
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Old 5th January 2018, 05:09 AM   #15
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Yes, this is a special series, and not one generated with the power-res technique, but rather a related one that someone named G Systems, someone named Ladma.

Oh, here he is:

http://mathforum.org/library/view/18422.html

Library Home || Full Table of Contents || Library Help
_____________________________________
Visit this site: http://www.sweb.cz/vladimir_ladma/en...les/dide99.htm
Author: Vladimir Ladma
Description: A paper presenting an algebraic theory of musical chord structures that seeks to provide a simple algorithm for generating these structures.
Levels: College, Research
Languages: English
Resource Types: Articles
Math Topics: Modern Algebra, Music


http://cyclesresearchinstitute.org/s...tronomy/ladma/

etc.

Last edited by calebprime; 5th January 2018 at 05:11 AM.
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Old 5th January 2018, 05:28 AM   #16
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Gathering Blandness (Ladma) is distinct from: First Example, Compromising Adult,

and here's as close as it gets to Five Large:

Code:
?add row:c,d,eb,a,bb,f#,g,c#,f,ab,e,b
?voc #1:1
?voc #2:1
?inc har:0,0
?har errs: 0 to 8
?run
working...
5x/ri/3/9:...........C  F  Eb Ab G  B  D  Bb E  A  C# F# 
p/0/0:...............C  F  Eb C# G  D  A  Bb F# E  Ab B  

5x5p/r/0/11:.........C  F  Eb E  D  G  B  Bb C# A  Ab F# 
p/0/0:...............C  F  Eb C# G  D  A  Bb F# E  Ab B

Last edited by calebprime; 5th January 2018 at 05:41 AM.
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Old 6th January 2018, 03:22 AM   #17
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Previously:

First Example C,Bb,D,Eb,Gb,G,E,Db,A,B,F,Ab (0 10 2 3 6 7 4 1 9 11 5 8)
Intervals: 10,4,1,3,1,9,9,8,2,6,3,4


Compromising Adult 90 ^ n mod 101
11 2 7 6 4 10 3 9 1 8 5 0
(0 9 4 3 6 11 10 8 2 7 1 5) C,A,E,Eb,F#,B,Bb,Ab,D,G,Db,F
Intervals: 9,7,11,3,5,11,10,6,5,6,4,7


Five Large ... C,D,Eb,A,Bb,F#,G,C#,F,Ab,E,B ... 0,2,3,9,10,6,7,1,5,8,4,11
or rotated 0,6,10,1,9,4,5,7,8,2,3,11
Intervals 2,1,6,1,8,1,6,4,3,8,7,1


Gathering Ladma
(0 5 3 1 7 2 9 10 6 4 8 11) ... C, F, Eb, Db, G, D, A, Bb, Gb, E, Ab, B,
best self-sim rotation: C, E, G, Ab, Db, B, A, Eb, Bb, F, Gb, D
Intervals: 4,3,1,5,10,10,6,7,7,1,8,10



=======================

Today's Row of the Day:

The Ni of '73


================================================== =====================
Base: 73 Mod: 233 Period: 232
-----------------------------------------------------------------------
Power: 1 2 3 4 5 6 7 8 9 10 11 12
-----------------------------------------------------------------------
Raw: 232 24 97 48 229 121 190 72 194 21 85 145
Rank: 11 1 5 2 10 6 8 3 9 0 4 7
-----------------------------------------------------------------------
Mod 11: 1 2 9 4 9 0 3 6 7 10 8 2
Mod 12: 4 0 1 0 1 1 10 0 2 9 1 1
Mod 13: 11 11 6 9 8 4 8 7 12 8 7 2



The series that tests well in Microtonal Scales is the rotation of this by 1 note:

(0 4 6 10 7 3 11 1 8 2 5 9) ... C,E,F#,Bb,G,Eb,B,C#,G#,D,F,A

intervals: 4,2,4,9,8,8,2,7,6,3,4,3

But the best power-res rotation is this, as above:

Code:
The Ni of '73


73 m233


Every-other square:

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

column intervals: 2,1,1,2,1,1,9,5,1,2,8,5



Standard inversion square:

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








The Ni of '73 is distinct from the other 4 rows:

?add row:c,bb,d,eb,f#,g,e,c#,a,b,f,g#
?voc #1:1
?voc #2:1
?inc har:0,0
?har errs: 0 to 6
?run
working...

?add row:c,a,e,eb,f#,b,bb,ab,d,g,c#,f
?voc #1:1
?voc #2:1
?inc har:0,0
?har errs: 0 to 6
?run
working...

?add row:c,d,eb,a,bb,f#,g,c#,f,ab,e,b
?voc #1:1
?voc #2:1
?inc har:0,0
?har errs: 0 to 6
?run
working...

?add row:c,f,eb,db,g,d,a,bb,gb,e,ab,b
?voc #1:1
?voc #2:1
?inc har:0,0
?har errs: 0 to 6
?run
working...

Last edited by calebprime; 6th January 2018 at 03:38 AM.
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Old 6th January 2018, 07:32 AM   #18
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You're certainly doing a lot of work on this.
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Old 6th January 2018, 09:38 AM   #19
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I love this kind of work, I wish I could do pre-composition research on rows for other composers. (That will never happen on this planet.)

I've lost the heart for composition, so this is a hobby with a sort of applied math angle. It should keep me out of trouble.

Over the years, I've developed quite a bag of techniques for working with these materials, so I might as well use them.

To me, the series are often beautiful and I still enjoy jamming with them a little.

And there is another motivation -- a small area of mastery. You can do one thing right, might as well do it.

Last edited by calebprime; 6th January 2018 at 09:49 AM.
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Old 6th January 2018, 10:53 AM   #20
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Another thing: I'm writing what I'd be delighted to stumble on and read myself.


I'd certainly welcome ideas such as how to generate good self-similar series. But there the devil is in the details, so I'm not that interested in defining it for others, except, heck, give it one go.

The kind of self-similarity I'm interested in is "growth" or "Multiple Order Function" or Mallalieu type. Growth because a series can be thought of as embedded in another, so the the series grows as notes are interpolated, added between existing notes. Whole pieces can grow from this simple process, if one is so inclined.

These series have some recognizable version or variation of themselves embedded in the series at some definable spacing (that makes some justifiable sense musically.)

Series that are self-similar at every 2cnd note starting on the second, and simultaneously every 3rd note starting on the 3rd, every 4th on the 4th, etc. are the power-residue type series, of which Mallalieu (2 ^ n mod 13 index series [or log series*]) is the prime example. The regularity of the self-similarity can be exploited to make polyrhythms in a way that irregular self-similarity cannot do. (They can make ideosyncratic irregular polyrhythms.)





*pardon my absymal math education and ignorance of many things.

Last edited by calebprime; 6th January 2018 at 10:58 AM.
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Old 6th January 2018, 10:57 AM   #21
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alfaniner's right, you've done a lot of work, why don't you give it a rest, stop awhile, maybe go to a bar, you don't want to lose your tone.
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Old 6th January 2018, 11:04 AM   #22
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I'm going to clean the back bathroom, vacuum, take out the recycling, clean the fridge, and also the trash.

eta: I'll try to keep my pace here to a post or two a day, until I can't think of any new rows, or whatever else stops me. It's an important project to me in that nothing really interests me any more, so this is important to fill the time. This is the last thing I'm a little interested in that I'm aware of, besides bestial pleasures and animal comforts.

Tired of living, scared of dying. That Ol' Man River.

Last edited by calebprime; 6th January 2018 at 12:38 PM.
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Old 7th January 2018, 03:50 AM   #23
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Today's Row: CovFEfe C,F,E,Bb,C#,A,G,Eb,Ab,F#,B,D
intervals: 5,11,6,3,8,10,8,5,10,5,3,10

Base: 5 Mod: 709453
Mod 12: 0 5 4 10 1 9 7 3 8 6 11 2

This is derived a little differently -- this time taking the mod12 values of 5 ^ n mod 709,453. When this method yields enough distinct values for a series, the series are usually slightly better as self-similar series than the truncated rank method. Variants on this method include doing the same but taking the series as mod11 or mod13 values. The results are self-similar series with flaws, usually less consistent than mod12.


Code:
CovFEfe

Every-other square:

C,  F,  E,  Bb, C#, A,  G,  Eb, Ab, Gb, B,  D       5,11,6,3,8,10,8,5,10,5,3,10
F,  Bb, A,  Eb, Gb, D,  C,  E,  C#, G,  Ab, B
Bb, Eb, D,  E,  G,  B,  F,  A,  Gb, C,  C#, Ab
Eb, E,  B,  A,  C,  Ab, Bb, D,  G,  F,  Gb, C#
E,  A,  Ab, D,  F,  C#, Eb, B,  C,  Bb, G,  Gb
A,  D,  C#, B,  Bb, Gb, E,  Ab, F,  Eb, C,  G
D,  B,  Gb, Ab, Eb, G,  A,  C#, Bb, E,  F,  C
B,  Ab, G,  C#, E,  C,  D,  Gb, Eb, A,  Bb, F
Ab, C#, C,  Gb, A,  F,  B,  G,  E,  D,  Eb, Bb
C#, Gb, F,  G,  D,  Bb, Ab, C,  A,  B,  E,  Eb
Gb, G,  Bb, C,  B,  Eb, C#, F,  D,  Ab, A,  E
G,  C,  Eb, F,  Ab, E,  Gb, Bb, B,  C#, D,  A

5,5,5,1,5,5,9,9,5,5,1,5

Standard inversion matrix:


C,  F,  E,  Bb, C#, A,  G,  Eb, Ab, Gb, B,  D
G,  C,  B,  F,  Ab, E,  D,  Bb, Eb, C#, Gb, A
Ab, C#, C,  Gb, A,  F,  Eb, B,  E,  D,  G,  Bb
D,  G,  Gb, C,  Eb, B,  A,  F,  Bb, Ab, C#, E
B,  E,  Eb, A,  C,  Ab, Gb, D,  G,  F,  Bb, C#
Eb, Ab, G,  C#, E,  C,  Bb, Gb, B,  A,  D,  F
F,  Bb, A,  Eb, Gb, D,  C,  Ab, C#, B,  E,  G
A,  D,  C#, G,  Bb, Gb, E,  C,  F,  Eb, Ab, B
E,  A,  Ab, D,  F,  C#, B,  G,  C,  Bb, Eb, Gb
Gb, B,  Bb, E,  G,  Eb, C#, A,  D,  C,  F,  Ab
C#, Gb, F,  B,  D,  Bb, Ab, E,  A,  G,  C,  Eb
Bb, Eb, D,  Ab, B,  G,  F,  C#, Gb, E,  A,  C


A nice 4-part, though it has the esoteric "5p" in the tenor:

5x/i/4/4:............B  G  F  C# C  Bb A  F# E  Eb Ab D  
5x/r/0/8:............D  Ab C# C  Bb G  F# E  Eb B  A  F  
5p/p/1/7:............Eb D  G  F  E  C# Bb C  B  A  F# Ab 
p/0/0:...............C  F  E  Bb C# A  G  Eb Ab F# B  D
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Old 7th January 2018, 04:11 AM   #24
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CovFEfe is self-identical to 8 places out of 12 in the 5x and 5x5p squares:


Code:
?inc har:0,0
?har errs: 0 to 4
?run
working...
5x/r/10/0:...........Ab F  E  D  C# A  G  Eb C  F# B  Bb 
p/0/0:...............C  F  E  Bb C# A  G  Eb Ab F# B  D  

5x5p/r/10/8:.........C  F  G  Bb C# F# E  Eb Ab A  B  D  
p/0/0:...............C  F  E  Bb C# A  G  Eb Ab F# B  D
CovFEfe passes all tests for distinctness from the other series so far.
Last time, however, I was using the wrong app to test, so there might be some uncertainty about The Ni of '73. I'll do again. Probably ok. eta: ran tests again, ok.

Last edited by calebprime; 7th January 2018 at 04:17 AM.
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Old 8th January 2018, 04:02 AM   #25
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Monday's Series: Say I'm Sick Backwards

Base: 3 Mod: 592157 Period: 592156

4 9 1 6 11 10 5 3 2 8 0 7
substitute: 4 and 0:

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

Code:
C,A,C#,F#,B,Bb,F,Eb,D,Ab,E,G

  every-other square
C,  A,  C#, F#, B,  Bb, F,  Eb, D,  Ab, E,  G  9,4,5,5,11,7,10,11,6,8,3,5
A,  F#, Bb, Eb, Ab, G,  C,  C#, B,  F,  D,  E
F#, Eb, G,  C#, F,  E,  A,  Bb, Ab, C,  B,  D
Eb, C#, E,  Bb, C,  D,  F#, G,  F,  A,  Ab, B
C#, Bb, D,  G,  A,  B,  Eb, E,  C,  F#, F,  Ab
Bb, G,  B,  E,  F#, Ab, C#, D,  A,  Eb, C,  F
G,  E,  Ab, D,  Eb, F,  Bb, B,  F#, C#, A,  C
E,  D,  F,  B,  C#, C,  G,  Ab, Eb, Bb, F#, A
D,  B,  C,  Ab, Bb, A,  E,  F,  C#, G,  Eb, F#
B,  Ab, A,  F,  G,  F#, D,  C,  Bb, E,  C#, Eb
Ab, F,  F#, C,  E,  Eb, B,  A,  G,  D,  Bb, C#
F,  C,  Eb, A,  D,  C#, Ab, F#, E,  B,  G,  Bb


9,9,9,10,9,9,9,10,9,9,9,7
at first glance, the series has an 012 per 4: Eb, D, Ab, E, but no 012 per contiguous 3 segment, wrapping around.

Sort of Mozart and Rakowski stuff: http://home.earthlink.net/~ziodavino/album1_001.htm

Last edited by calebprime; 8th January 2018 at 04:05 AM.
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Old 8th January 2018, 04:25 AM   #26
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A better way to check that each row is unique enough. Simply create a file with each row, in Microtonal Scales format, and check for dupes using the remove near-duplicates feature.





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

number of rows found: 7

removing near-duplicates...done

number of rows remaining: 7

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

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




This test says that the rows are unique to 5 places, and then 1 pair is the same to 6 places.

I'd say that anything that is identical to 8 places out of 12 should still be included but given a name related to the near-identical entry, so we can keep track of clumps of these. Anything identical to 9 or more places should be considered too similar and therefore trivial.

Last edited by calebprime; 8th January 2018 at 04:27 AM.
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Old 8th January 2018, 04:52 AM   #27
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Originally Posted by calebprime View Post

every-other square
C, A, C#, F#, B, Bb, F, Eb, D, Ab, E, G 9,4,5,5,11,7,10,11,6,8,3,5
A, F#, Bb, Eb, Ab, G, C, C#, B, F, D, E
F#, Eb, G, C#, F, E, A, Bb, Ab, C, B, D
Eb, C#, E, Bb, C, D, F#, G, F, A, Ab, B
C#, Bb, D, G, A, B, Eb, E, C, F#, F, Ab
Bb, G, B, E, F#, Ab, C#, D, A, Eb, C, F
G, E, Ab, D, Eb, F, Bb, B, F#, C#, A, C
E, D, F, B, C#, C, G, Ab, Eb, Bb, F#, A
D, B, C, Ab, Bb, A, E, F, C#, G, Eb, F#
B, Ab, A, F, G, F#, D, C, Bb, E, C#, Eb
Ab, F, F#, C, E, Eb, B, A, G, D, Bb, C#
F, C, Eb, A, D, C#, Ab, F#, E, B, G, Bb


9,9,9,10,9,9,9,10,9,9,9,7



[/code]

at first glance, the series has an 012 per 4: Eb, D, Ab, E, but no 012 per contiguous 3 segment, wrapping around.

Sort of Mozart and Rakowski stuff: http://home.earthlink.net/~ziodavino/album1_001.htm

the embedding is really pretty:

Code:
C,  A,  C#, F#, B,  Bb, F,  Eb, D,  Ab, E,  G  9,4,5,5,11,7,10,11,6,8,3,5
A,  F#, Bb, Eb, Ab, G,  C,  C#, B,  F,  D,  E
F#, Eb, G,  C#, F,  E,  A,  Bb, Ab, C,  B,  D




F#, Eb, G,  C#, F,  E,  A,  Bb, Ab, C,  B,  D
........g...........e...........ab..........d...
....eb..........f...........bb..........b.......
f#..........c#..........a...........c
What's unusally good here is the resemblance between the row and the embedded row, not that the every-fourth series when taken backwards is every-three. That's trivially true.

Last edited by calebprime; 8th January 2018 at 05:20 AM.
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Old 8th January 2018, 05:54 AM   #28
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One last post on Say I'm Sick. There will be a Mallalieu Exclusion rule. Any row that is found to be essentially the same as the same row-class as Mallalieu series (c,c#,e,d,a,f,b,eb,ab,bb,g,f#) or (0 1 4 2 9 5 11 3 8 10 7 6) will be excluded.

On further examination, we'll also have to eliminate series that are members of the same every-other square. We would have to account for those permutations being considered part of the same class. Currently, they're not.


Say I'm Sick is distinct from Mallalieu.


Code:
?new row #1:c,c#,e,d,a,f,b,eb,ab,bb,g,f#
?add row:c,a,c#,f#,b,bb,f,eb,d,ab,e,g
?voc #1:1
?voc #2:1
?inc har:0,0
?har errs: 0 to 6
?run
working...

?add row:c,a,c#,f#,b,bb,f,eb,d,ab,e,g
?voc #1:1
?voc #2:1
?inc har:0,0
?har errs: 0 to 8
?run
working...
p/4/9:...............C  Ab B  E  C# F  Bb Eb D  A  G  F# 
p/0/0:...............C  C# E  D  A  F  B  Eb Ab Bb G  F# 

r/10/3:..............C  C# Eb Ab A  E  B  G  Bb F  D  F# 
p/0/0:...............C  C# E  D  A  F  B  Eb Ab Bb G  F# 

i/7/3:...............C# Ab A  D  E  F  B  Eb C  G  Bb F# 
p/0/0:...............C  C# E  D  A  F  B  Eb Ab Bb G  F# 

i/9/8:...............G  C# F  D  A  C  Ab Eb Bb B  E  F# 
p/0/0:...............C  C# E  D  A  F  B  Eb Ab Bb G  F# 

ri/3/4:..............C  Bb F  E  A  D  F# Eb Ab B  G  C# 
p/0/0:...............C  C# E  D  A  F  B  Eb Ab Bb G  F# 

ri/1/9:..............C  E  C# F# A  F  B  Bb Ab Eb D  G  
p/0/0:...............C  C# E  D  A  F  B  Eb Ab Bb G  F# 

5p/p/7/2:............B  C# A  E  C  D  F# Eb Ab Bb G  F  
p/0/0:...............C  C# E  D  A  F  B  Eb Ab Bb G  F# 

5p/r/1/10:...........B  C# E  D  A  C  Ab F# Bb Eb G  F  
p/0/0:...............C  C# E  D  A  F  B  Eb Ab Bb G  F# 

5x/i/6/1:............A  C# C  B  E  F  Eb Ab D  Bb G  F# 
p/0/0:...............C  C# E  D  A  F  B  Eb Ab Bb G  F# 

5x/ri/0/11:..........C  C# E  Ab D  A  B  Bb F  F# G  Eb 
p/0/0:...............C  C# E  D  A  F  B  Eb Ab Bb G  F# 

5x5p/p/10/3:.........E  Ab G  B  A  F  D  Eb C# Bb C  F# 
p/0/0:...............C  C# E  D  A  F  B  Eb Ab Bb G  F# 

5x5p/r/4/9:..........C  F# E  G  A  Ab B  Eb F  C# D  Bb 
p/0/0:...............C  C# E  D  A  F  B  Eb Ab Bb G  F#

always pleasing when of the millions of 4-part combinations, it only finds 6:

Code:
ri/6/1:..............D  Bb E  Eb C# Ab G  C  F  A  F# B  
i/3/5:...............F  Bb C  C# G  B  Ab Eb F# D  A  E  
r/3/8:...............A  E  C  Eb Bb G  B  F  F# Ab C# D  
p/0/0:...............C  A  C# F# B  Bb F  Eb D  Ab E  G  

ri/8/8:..............D  G  B  Ab C# E  C  F# F  Eb Bb A  
i/2/0:...............D  F  C# Ab Eb E  A  B  C  F# Bb G  
r/6/8:...............C  G  Eb F# C# Bb D  Ab A  B  E  F  
p/0/0:...............C  A  C# F# B  Bb F  Eb D  Ab E  G  

ri/0/2:..............E  Bb A  G  D  C# F# B  Eb C  F  Ab 
i/6/6:...............C# Eb E  Bb D  B  F# A  F  C  G  Ab 
r/6/8:...............C  G  Eb F# C# Bb D  Ab A  B  E  F  
p/0/0:...............C  A  C# F# B  Bb F  Eb D  Ab E  G  

ri/3/2:..............G  C# C  Bb F  E  A  D  F# Eb Ab B  
i/9/6:...............E  F# G  C# F  D  A  C  Ab Eb Bb B  
r/6/8:...............C  G  Eb F# C# Bb D  Ab A  B  E  F  
p/0/0:...............C  A  C# F# B  Bb F  Eb D  Ab E  G  

ri/0/3:..............Bb A  G  D  C# F# B  Eb C  F  Ab E  
i/3/7:...............C  C# G  B  Ab Eb F# D  A  E  F  Bb 
r/9/8:...............Eb Bb F# A  E  C# F  B  C  D  G  Ab 
p/0/0:...............C  A  C# F# B  Bb F  Eb D  Ab E  G  

ri/0/3:..............Bb A  G  D  C# F# B  Eb C  F  Ab E  
i/6/6:...............C# Eb E  Bb D  B  F# A  F  C  G  Ab 
r/6/9:...............G  Eb F# C# Bb D  Ab A  B  E  F  C  
p/0/0:...............C  A  C# F# B  Bb F  Eb D  Ab E  G

Last edited by calebprime; 8th January 2018 at 06:10 AM.
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Old 9th January 2018, 03:17 AM   #29
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"It's long gone buddy now, run and go after it". Aimee Mann -- Til Tuesday

I can't in good conscience even ask you to watch this YouTube slide montage that someone assembled for this song -- it features finches, strewn flowers, thongs, maidens, more strewn flowers, and some cold trucks with icicles. Brr!

For me, this line compares the foolish dog who will always chase after the stick to the foolish human heart.

https://www.youtube.com/watch?v=ikUZ6YZTLJY

Those with good listening setup can hear how they used a little breathing sound in the rhythm track. Buddy panting.

Ah the glories of the Roland D50.

Buddy is a lab. Me, I'm an English Setter.

"We're making...such a mess!"



candidates for Too Tuesday, today's row:
C,Bb,Db,Ab,F,B,E,F#,D,Eb,G,A
C,Bb,Db,Ab,F,G,B,F#,D,Eb,E,A
E,F#,A,G#,F,B,C,Bb,D,G,Eb,Db

or


0 2 8 4 5 10 1 6 3 7 9 11
C,D,Ab,E,F,Bb,Db,Gb,Eb,G,A,B heck, this one for Too Tuesday

Code:
Too Tuesday:


Every other square


C,  D,  Ab, E,  F,  Bb, Db, Gb, Eb, G,  A,  B  2,6,8,1,5,3,5,9,4,2,2,1
D,  E,  Bb, Gb, G,  B,  C,  Ab, F,  Db, Eb, A
E,  Gb, B,  Ab, Db, A,  D,  Bb, G,  C,  F,  Eb
Gb, Ab, A,  Bb, C,  Eb, E,  B,  Db, D,  G,  F
Ab, Bb, Eb, B,  D,  F,  Gb, A,  C,  E,  Db, G
Bb, B,  F,  A,  E,  G,  Ab, Eb, D,  Gb, C,  Db
B,  A,  G,  Eb, Gb, Db, Bb, F,  E,  Ab, D,  C
A,  Eb, Db, F,  Ab, C,  B,  G,  Gb, Bb, E,  D
Eb, F,  C,  G,  Bb, D,  A,  Db, Ab, B,  Gb, E
F,  G,  D,  Db, B,  E,  Eb, C,  Bb, A,  Ab, Gb
G,  Db, E,  C,  A,  Gb, F,  D,  B,  Eb, Bb, Ab
Db, C,  Gb, D,  Eb, Ab, G,  E,  A,  F,  B,  Bb   

column intervals: 2,2,2,2,2,1,10,6,2,2,6,11


standard inversion square

C,  D,  Ab, E,  F,  Bb, Db, Gb, Eb, G,  A,  B
Bb, C,  Gb, D,  Eb, Ab, B,  E,  Db, F,  G,  A
E,  Gb, C,  Ab, A,  D,  F,  Bb, G,  B,  Db, Eb
Ab, Bb, E,  C,  Db, Gb, A,  D,  B,  Eb, F,  G
G,  A,  Eb, B,  C,  F,  Ab, Db, Bb, D,  E,  Gb
D,  E,  Bb, Gb, G,  C,  Eb, Ab, F,  A,  B,  Db
B,  Db, G,  Eb, E,  A,  C,  F,  D,  Gb, Ab, Bb
Gb, Ab, D,  Bb, B,  E,  G,  C,  A,  Db, Eb, F
A,  B,  F,  Db, D,  G,  Bb, Eb, C,  E,  Gb, Ab
F,  G,  Db, A,  Bb, Eb, Gb, B,  Ab, C,  D,  E
Eb, F,  B,  G,  Ab, Db, E,  A,  Gb, Bb, C,  D
Db, Eb, A,  F,  Gb, B,  D,  G,  E,  Ab, Bb, C




"It's not that I'm frightened of being alone. It's just that I know what a burden this grave can be." chucka-kah.



"That love is... SO UNKIND!"


"I must confess, I don't halve." (What don't you halve, Aimee? Aren't you just a nice Catholic girl from around here?)

Last edited by calebprime; 9th January 2018 at 03:41 AM.
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Old 9th January 2018, 03:33 AM   #30
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Too Tuesday is distinct in this set:


removing near-duplicates...done

number of rows remaining: 8

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

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

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



nice 4-part structure by making an every-other square with a transposed version of the row as column,
then selecting 4 lines from the square:

Code:
C,  D,  Ab, E,  F,  Bb, Db, Gb, Eb, G,  A,  B
D,  E,  C,  F,  G,  Eb, A,  B,  Gb, Db, Bb, Ab
E,  F,  D,  G,  Db, Gb, Bb, Ab, B,  A,  Eb, C
F,  G,  E,  Db, A,  B,  Eb, C,  Ab, Bb, Gb, D

Last edited by calebprime; 9th January 2018 at 04:50 AM.
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Old 9th January 2018, 03:19 PM   #31
sir drinks-a-lot
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I second the bar idea.
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Old 10th January 2018, 03:00 AM   #32
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I'll lose the ghastly whimsy, keep to the project. The interested reader should find a new nugget or two every post in addition to the row of the day. The reason to keep this in Arts as opposed to Community is that people who are interested in 12-tone theory can Google it.

Today's row: Beggar Man


Code:

Every-other square


E,  G,  B,  Bb, Ab, D,  C,  C#, F#, Eb, A,  F
G,  Bb, D,  C#, Eb, F,  E,  B,  Ab, C,  F#, A
Bb, C#, F,  B,  C,  A,  G,  D,  Eb, E,  Ab, F#
C#, B,  A,  D,  E,  F#, Bb, F,  C,  G,  Eb, Ab
B,  D,  F#, F,  G,  Ab, C#, A,  E,  Bb, C,  Eb
D,  F,  Ab, A,  Bb, Eb, B,  F#, G,  C#, E,  C
F,  A,  Eb, F#, C#, C,  D,  Ab, Bb, B,  G,  E
A,  F#, C,  Ab, B,  E,  F,  Eb, C#, D,  Bb, G
F#, Ab, E,  Eb, D,  G,  A,  C,  B,  F,  C#, Bb
Ab, Eb, G,  C,  F,  Bb, F#, E,  D,  A,  B,  C#
Eb, C,  Bb, E,  A,  C#, Ab, G,  F,  F#, D,  B
C,  E,  C#, G,  F#, B,  Eb, Bb, A,  Ab, F,  D
column intervals: 3,3,3,10,3,3,4,9,2,7,9,4


Standard square:

E,  G,  B,  Bb, Ab, D,  C,  C#, F#, Eb, A,  F
C#, E,  Ab, G,  F,  B,  A,  Bb, Eb, C,  F#, D
A,  C,  E,  Eb, C#, G,  F,  F#, B,  Ab, D,  Bb
Bb, C#, F,  E,  D,  Ab, F#, G,  C,  A,  Eb, B
C,  Eb, G,  F#, E,  Bb, Ab, A,  D,  B,  F,  C#
F#, A,  C#, C,  Bb, E,  D,  Eb, Ab, F,  B,  G
Ab, B,  Eb, D,  C,  F#, E,  F,  Bb, G,  C#, A
G,  Bb, D,  C#, B,  F,  Eb, E,  A,  F#, C,  Ab
D,  F,  A,  Ab, F#, C,  Bb, B,  E,  C#, G,  Eb
F,  Ab, C,  B,  A,  Eb, C#, D,  G,  E,  Bb, F#
B,  D,  F#, F,  Eb, A,  G,  Ab, C#, Bb, E,  C
Eb, F#, Bb, A,  G,  C#, B,  C,  F,  D,  Ab, E
Beggar Man tests as distinct from the other 8 including Mallalieu:

finding rows....done

number of rows found: 10

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

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

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

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

Last edited by calebprime; 10th January 2018 at 04:10 AM.
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Old 10th January 2018, 04:46 AM   #33
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Originally Posted by calebprime View Post
Here's a good list to have, all the MOFs of Mallalieu type starting with narrow spacing and gradually widening in a fairly systematic fashion, ending in self-similarity with spacing of 6 and 7 notes.

...
taking a series from that list as an example:


C# D F# A Ab Eb E C F G B Bb intervals: 1 4 3 11 7 1 8 5 2 4 11 3

Code:
C# D F# A Ab Eb E C F G B Bb 
...d.........eb.......g...bb...
........a.......e...f..........
c#...f#...ab......c.....b......
Here's an example of how the series from that big list work.

The self-similarity is always exact, at 1 semitone interval. The spacing of the self-similarity is the one thing that the user can define in advance, without knowing the outcome or the particular sound of the row. If that spacing corresponds to a possible series, this procedure will find it.

The procedure is this:
You define a "spacing" series, that corresponds to the spacing pattern of self-similarity. (The spacing series C,D,E,F#,G#,A#,B,C#,D#,F,G,A is the one that corresponds to the "perfect" series, Mallalalieu: (0 1 4 2 9 5 11 3 8 10 7 6)).


Trial-and-error test, if it's possible to make a complete permutation orbit with another series, 0,1,2,3,4,5,6,7,8,9,10,11 and its rotations.

If a series is found, perform a 90-degree reflection, or swapping position and pitch values in pairs, and the resulting series will have self-similarity at the semitone, in the pattern you chose.

For self-similarity at 5 and 7, multiply your series by 5, and 7 will be the inversion.

For self-similarity with one position of accuracy lost at 2 and 10, multiply the series by a function like [for 0,1,2,3,4,5,6,7,8,9,10,11 map to 0,2,4,6,8,10,1,3,5,7,9,11]. For 10, take the inversion.

We now have 1,2,5,7,10,11. We want 3 and 4 and 6, which will give us 8 and 9 by inversion.

My understanding is rudimentary here. I know of only one permutation pattern that will be perfectly self-similar at 4.

It corresponds to the series I call Mof3, so I'll refer to it. A,C#,D#,C,F,G#,G,D,B,E,Bb,F#

Code:
A..C#.D#.C..F..Ab.G..D..B..E..Bb.F#.
...C#.......f.....g........e..........
a........c..............b........f#...
......d#.......g#....d........bb......

Here by computer search are all the rest of the series up to inversion that also have perfect self-sim with that pattern at 4:


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



Last edited by calebprime; 10th January 2018 at 04:50 AM.
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Old 10th January 2018, 05:34 AM   #34
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I concur with 3point14. I have no idea what you're talking about.


I vaguely suspect it is about music, but there it ends for me.
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Old 10th January 2018, 05:38 AM   #35
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Well, let me know if you have specific questions. It's about the design of 12-tone rows for music composition.

This is essentially applied math, except that composers are interested in different questions from mathematicians.

A live issue for me is: How to have series that are self-similar at every possible interval. Minor third, major third, and what degree possible tritone -- these are of particular interest.

Last edited by calebprime; 10th January 2018 at 05:41 AM.
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Old 10th January 2018, 08:38 AM   #36
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Originally Posted by calebprime View Post
A live issue for me is: How to have series that are self-similar at every possible interval. Minor third, major third, and what degree possible tritone -- these are of particular interest.
I think that depends on whether you can have 12 tempered scales, which I don't think is practical.

Friend, take a break and listen to (but don't analyze) a couple symphonies.
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Old 10th January 2018, 08:51 AM   #37
Cheetah
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Originally Posted by erwinl View Post
I concur with 3point14. I have no idea what you're talking about.
He sais cofveve, I say covfefe lets call the whole thing off.
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Old 10th January 2018, 08:56 AM   #38
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I just posted one at interval 4. The question is how to extend it.

Listen, I'm not oblivious and I'm aware that this thread is not of interest.


However, there is a shrinkage of membership and a lack of posts.

Might as well post about what I'm interested in, see if that draws someone in from outside who is interested in twelve-tone theory.

I'm quite serious that it's my only interest right now, so it has life-sustaining value for me.

If I could reframe this as a math question, it might interest some people, but then I would no longer understand my own thread.

You yourself, alfa, are a depressive type with music theory interests. I'd hardly see why you'd discourage such a thread.

People can contribute by coming up with appropriate rows or generating techniques.

Otherwise, what can I do besides responding to questions?

Last edited by calebprime; 10th January 2018 at 09:00 AM.
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Old 10th January 2018, 09:05 AM   #39
3point14
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Originally Posted by erwinl View Post
I concur with 3point14. I have no idea what you're talking about.


I vaguely suspect it is about music, but there it ends for me.

It's definitely music.

Give that my new years resolution is to actually write some songs again (one a fortnight, done, finished, arranged, all year. Yes, I'm being optimistic ) I'm going to use this thread as inspiration for messing about with chord progressions and see what comes up.


Originally Posted by calebprime View Post
I just posted one at interval 4. The question is how to extend it.

Listen, I'm not oblivious and I'm aware that this thread is not of interest.


However, there is a shrinkage of membership and a lack of posts.

Might as well post about what I'm interested in, see if that draws someone in from outside who is interested in twelve-tone theory.

I didn't mean to criticise. I would dearly love to have a deep understanding of what you're saying and, perhaps just a small disagreement, so we could argue in a good natured fashion.
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Old 10th January 2018, 09:08 AM   #40
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it's not a debate thread

it's a how-to-generate, perhaps how-to-assess thread

anyone could start a serial music sucks thread if they're so inclined
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