Theory of palindromic MOS scales or rhythms

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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 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.  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. 

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:

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:

DEFGABCD
 EFGABC
  FGAB
   GA

2122212
 12221
  222
   2

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

And by chopping of both the end notes repeatedly, we end up with the additional patterns that are also palindromic:

ABCDEFG
 BCDEF
  CDE

212212
 1221
  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.

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:

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:

3232332332323
 23233233232
  323323323
   2332332
    33233
     323
      2

By removing the middle step of it we get the other parent palindromic scale:

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:

332323323233
 3232332323
  23233232
   323323
    2332
     33

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
  3323233
   32323
    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 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
  EF

22122
 212
  1

or...

GABCDE
 ABCD
  BC

22122
 212
  1

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

A total of 10 (or 8 if you discount the one step subsets).

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

A total of 19 (or 17 if you discount the one step subsets).

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.

-Joakim Bang Larsen (February 2019)