Just intonation subgroup/Mike's tips

This page contains an archived discussion between Chris Vaisvil and Mike Battaglia about just intonation subgroups and regular temperaments.

On Thu, Nov 1, 2012 at 9:10 PM, Chris Vaisvil <[email protected]> wrote:

> For certain, no. I could only guess that subgroups are actually harmonic series prime limits.

Subgroups expand the concept of a prime limit. For instance, say you want the 7-limit, but you don't care about prime 5; you just want primes 2, 3, and 7. So that's the 2.3.7 subgroup. Or say that you want the 7-limit, but you don't care about 3/1 but you do care about 9/1. Then that's the 2.5.7.9 subgroup. Or, say that you want primes 2 and 3, and the composite interval 7/5; that's the 2.3.7/5 subgroup. The rule for any subgroup is that if you multiply or divide intervals, that's also in the subgroup. So for the 2.3.7 subgroup, 7*3 = 21/1 is in the subgroup, as is 7/(2*3) = 7/6. And for the 2.3.7/5 subgroup, 2/(7/5 * 7/5) = 50/49 is in the subgroup, and so on. They're infinite lattices of intervals.

11-EDO happens to be a decent temperament for the 2.7.9.11 subgroup.

-Mike

Anyway, you asked about figuring out what steps in 11-EDO approximate what intervals. So if 11-EDO supports the 2.7.9.11 subgroup, I can just give you the mappings for 2/1, 7/1, 9/1, and 11/1, and you can mix and match them to get what intervals you want, right. So for instance, 2/1 is 11 steps, 7/1 is 31 steps, 9/1 is 35 steps, and 11/1 is 38 steps. So then 11/9 is 38-35 = 3 steps. That should be enough information for you to get all the intervals.

-Mike

OK, so rather than write all of that out in English, though, we can just use a simple notation. So if 2/1 is 11 steps, 7/1 is 31 steps, 9/1 is 35 steps, and 11/1 is 38 steps, then we can just condense all that as follows:

<11 31 35 38|

where it's understood in this particular case that the coefficients represent how many steps map to 2/1, 7/1, 9/1 and 11/1, respectively. This is called a val, and this is why we use them; so we can figure out how many steps every interval maps to. So 9/7 in the above case is 35-31 = 4 steps.

-Mike

See also

• Equal-step tuning/Jake's tips (contains another archived discussion with Chris Vaisil)