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While working on a project to make a MOS rhythm generator audio plugin (to be released in 2019), I found out that I liked the sound of mirror symmetrical or palindromic rhythms, rhythms that are the same pattern both forwards and backwards and so I wanted to find a method to wrestle out ALL the palindromic sequences of a single MOS rhythm.
While working on a project to make a MOS rhythm generator audio plugin (to be released in 2019), I found out that I liked the sound of mirror symmetrical or palindromic rhythms, rhythms that are the same pattern both forwards and backwards and so I wanted to find a method to wrestle out ALL the palindromic patterns of a single MOS rhythm.
 
I knew how to make the whole MOS scale palindromic, by using the same amount of positive and negative generators. Added together along with the starting note, it turned out that I all of them odd.
I knew how to make the whole MOS scale palindromic, by using the same amount of positive and negative generators. And that only all the palindromic subsets of MOS scales of ODD cardinality could be found in this way. So for MOS scales of even cardinality, the process described here won't work because there is no palindromic mode for the whole scale.  
 
And since this palindromic mode is mirror symmetric through and through, all you need to do to find the smaller palindromic sequences was to delete two notes from it, one from either end.
Since this primary palindromic mode is of course mirror symmetric through and through, all you need to do to find smaller palindromic subsets was to delete two notes from it at a time, one from either end.
 
A trivial example of this would be the diatonic scale. By using the same amount of positive and negative generators, we end up with the following pattern, which we call the dorian mode:
A trivial example of this would be the diatonic scale. By using the same amount of positive and negative generators, we end up with the following pattern, which we call the dorian mode:
 
7 notes: DEFGABC, 7 steps: 2122212
7 notes: DEFGABC, 7 steps: 2122212
 
And by chopping of both the end notes repeatedly, we end up with the following patterns that are also palindromic:
And by chopping of both the end notes repeatedly, we end up with the following patterns that are also palindromic:
 
DEFGABCD
DEFGABCD
EFGABC
  EFGABC
  FGAB
  FGAB
  GA
    GA
 
2122212
2122212
12221
  12221
  222
  222
  2
    2
 
Paul Erlich then showed me that by removing the middle step of this mode while leaving the notes untouched, would give you another scale which was also palindromic:
Paul Erlich then showed me that by removing the middle step of the primary palindromic mode while leaving the notes untouched, would give you another secondary scale which was also palindromic:
 
7 notes: ABCDEFG, 6 steps 212212
7 notes: ABCDEFG, 6 steps 212212
 
And by chopping of both the end notes repeatedly, we end up with the additional patterns that are also palindromic:
And by chopping of both the end notes repeatedly, we end up with the additional patterns that are also palindromic:
 
ABCDEFG
ABCDEFG
BCDEF
  BCDEF
  CDE
  CDE
 
212212
212212
1221
  1221
  22
  22
 
So now we had a neat way of finding the palindromic mode of a scale and it's counterpart on the other end of the scale (when the scale is viewed as a complete circle anyway), that would let us repeatedly chop of both ends to locate ALL the palindromic patterns.
So now we had a neat way of finding the palindromic mode of a scale and it's counterpart on the other end of the scale (when the scale is viewed as a complete circle anyway), that would let us repeatedly chop of both ends to locate ALL the palindromic patterns.
 
Still, I could find palindromic patterns that were not part of any of these "parent palindrome" rhythms, like 212 and so the quest was on to find the missing parent that contained the rest of them.
Still, we could find palindromic patterns that were not part of any of these "parent palindrome" rhythms, like 212 and so the quest was on to find a third parent scale that maybe contained the rest of them.
 
I can't remember how I discovered it, but I know that the rhythm I was working on was 34.21.13, where the palindromic mode is:
I can't remember how I discovered it, but I know that the rhythm I was working on was 34.21.13, where the palindromic mode is:
 
13 notes and 13 steps: 3232332332323
13 notes and 13 steps: 3232332332323
 
By chopping of both the end notes repeatedly, we end up with the additional patterns that are also palindromic:
By chopping of both the end notes repeatedly, we end up with the additional patterns that are also palindromic:
 
3232332332323
3232332332323
23233233232
  23233233232
  323323323
  323323323
  2332332
    2332332
    33233
    33233
    323
      323
      2
      2
 
By removing the middle step of it we get the other parent palindromic scale:
By removing the middle step of it we get the other parent palindromic scale:
13 notes and 12 steps: 332323323233
 
13 notes and 12 steps: 332323323233
By chopping of both the end notes repeatedly, we end up with the additional patterns that are also palindromic:
 
By chopping of both the end notes repeatedly, we end up with the additional patterns that are also palindromic:
332323323233
3232332323
332323323233
  23233232
  3232332323
  323323
  23233232
    2332
    323323
    33
    2332
 
      33
It was while I was working with this scale I discovered that if you delete either the first or the last note of the symmetrical chain of generators that was used to make the first or primary palindromic mode, that you hit upon a new palindromic parent scale that contained the remaining palindromic patterns.
 
Because I was working with this scale visualized as a circle I discovered that if you delete either the first or the last note of the symmetrical chain of generators that was used to make the primary palindromic mode, you would hit upon a third palindromic parent scale that contained the remaining palindromic patterns.
32332323323
233232332
32332323323
  3323233
  233232332
  32323
  3323233
    232
    32323
    3
    232
 
      3
It may be a bit hard to picture this with such an unfamiliar scale, but if we go back to our first example with the diatonic scale, we see that while we already have DEFGABCD and ABCDEFG, we can chop of either F or B from the chain FCGDAEB that was used to generate them both, we end up with either FCGDAE or CGDAEB as the generator chain for our new parent palindromic scale:
 
It may be a bit hard to picture this with such an unfamiliar scale, but if we go back to our first example with the diatonic scale, we see that if we chop of either F or B from the chain FCGDAEB that was used to generate both the two primary palindromic modes DEFGABCD and ABCDEFG, we end up with either FCGDAE or CGDAEB as the generator chain for our new parent palindromic scale:
CDEFGA
DEFG
CDEFGA
  EF
  DEFG
 
  EF
22122
212
22122
  1
  212
 
  1
or...
 
or...
GABCDE
ABCD
GABCDE
  BC
  ABCD
 
  BC
22122
212
22122
  1
  212
 
  1
So, depending on your definition of palindromic scale (is a scale consisting of two notes with one step between them a "scale"?), the diatonic scale has the following palindromic step sequences:
 
So, depending on your definition of palindromic SCALES (is a palindromic scale consisting of only two notes with one step between them a "scale"?), the diatonic scale at least can be said to have the following palindromic SUBSETS:
2122212, 212212, 12221, 22122, 1221, 222, 212, 22, 2, 1
 
2122212, 212212, 12221, 22122, 1221, 222, 212, 22, 2, 1
10 or 8 if you discount the one step patterns.
 
A total of 10 (or 8 if you discount the one step subsets).
And for the 34.21.13 scale:
 
And for the 34.21.13 scale:
3232332332323, 332323323233, 23233233232, 32332323323, 3232332323, 323323323, 233232332, 23233232, 3323233, 2332332, 323323, 33233, 32323, 2332, 323, 232, 33, 3, 2
 
3232332332323, 332323323233, 23233233232, 32332323323, 3232332323, 323323323, 233232332, 23233232, 3323233, 2332332, 323323, 33233, 32323, 2332, 323, 232, 33, 3, 2
A total of 19 or 17 if you discount the one step patterns.
 
A total of 19 (or 17 if you discount the one step subsets).
I have no way of verifying if these indeed are all the palindromic step sequences of these MOS scales, but I think that's all of them and I have verified that the method finds them all in two more MOS scales.
 
I know of no present mathematical way of verifying if these are indeed ALL the palindromic subsets of these MOS scales, but I think that's all of them and can verify that the method has worked for all the other MOS scales I have tried it on.
It would be fun if someone good with maths could verify if my theory of the three parent palindromic patterns is true.
 
-Joakim Bang Larsen (February 2019)
You might think, what is the point of all of this? But when it comes to rhythm it can now be proven that these three parent palindromic rhythms contain all of the smaller palindromic rhythms and it makes a cool feature of the rhythm generator to be able to rotate the rhythm to the three modes that together contain all of them.  
 
-Joakim Bang Larsen (February 2019)
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