Rank-3 tools: MOS substitution

[frostburn]

inthar — Yesterday at 8:50 PM

Guide frames are my idea and guide frames are a kind of product of one GS with another GS. You also make sure that the second interleaves the first. Generator sequences are a degenerate version of guide frames.

You can associate aLbMcs with some temperament subgroup[a & a+b & a+b+c]. If gcd(a, b, c) = 1 then MOS substitution can be a way to get a generator sequence using two distinct generators which depends on the temperament. It works for diamech at least.

https://en.xen.wiki/w/User:Inthar/MOS_substitution

Hemifamity is 7-lim[5 & 7 & 12], which means you get 5L2m5s. Assuming 5L(2m5s) (with template MOS 5L7X) you get the following MOS substitution scales:

LsLsLsmLsLsm (5sL diachrome)
LmLsLssLmLss
LsLmLssLsLms
LsLsLmsLsLsm (5sC diachrome)
LmLsLsmLsLss
LsLmLssLmLss
LsLsLmsLsLms (5sR diachrome)

Because of the Hemifamity interpretation you get L = 10/9, m = 256/243, s = 81/80~64/63.
The generator is 7 steps, you can work out the chain of fifths from that. Perfect fifths are 3L + m + 3s in hemifamity edos such as 41, 46, and 53. Wait, this might not be a good example for demonstrating the nice GS. Let me work out something with Marvel.

Marvel = 7-lim[10 & 12 & 19] -> 10L2M7s
Let's assume 7s(10L2M) so that we can make the filler MOS 10L2M a 2-period one, simplifying the scale. The template MOS is 7s12X:

                    XXsXXsXXsXsXXsXXsXs template
filler       LLLLLM LLsLLsLMsLsLLsLLsMs
             LLLLML LLsLLsMLsLsLLsLMsLs
             LLLMLL LLsLMsLLsLsLLsMLsLs  
             LLMLLL LLsMLsLLsLsLMsLLsLs          
...

Wow, I just serendipitously found this. This is a very nice Marvel detempering of Meantone[19], though I don't really know why it's so nice:
https://sw3.lumipakkanen.com/scale/2MjzjusQ5

https://sw3.lumipakkanen.com/scale/2Mjt4IB9S
This is a simplification that happens when two of the numbers of occurrences of steps are the same. This is 5L7M7s, so you can build a scale that has a length-2 generator sequence. I based it off Pentacircle, which is 2.3.7.11[5&12&19].

You can do the same thing for 7L5M5s from Parapyth = 2.3.11.13[7, 12[-11], 17]:
https://sw3.lumipakkanen.com/scale/3MjBxjeGV