Theory of palindromic MOS scales or rhythms

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.

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.

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.

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 this mode while leaving the notes untouched, would give you another 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, 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.

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

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.

32332323323

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:

CDEFGA

DEFG
 EF

22122

212
 1

or...

GABCDE

ABCD
 BC

22122

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:

2122212, 212212, 12221, 22122, 1221, 222, 212, 22, 2, 1

10 or 8 if you discount the one step patterns.

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 patterns.

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.

It would be fun if someone good with maths could verify if my theory of the three parent palindromic patterns is true.

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)