The tertiaseptal family

If we do a search for {2,5,7}-commas of size less than 50 cents and
epimericity less than 0.6, the complete list seems to be 128/125,
50/49, 3136/3125, 2100875/2097152; all of which except for the last
are quite familiar. I've mentioned it on occasion, but this is may be
the first close look anyone has given.

The {2,5,7} linear temperament has generator an approximate 256/245,
three of which make up a septimal tone of 8/7, this relationship
defining the comma. Among the DE scales are 15, 16, and 31 notes.

The 7-limit temperament with TM basis 2401/2400 and 65625/65536 has
essentially the same TOP generator of 77.2 cents, and I propose naming
both the {2,5,7} temperament and its extension to the 7-limit by the
name "tertiaseptal", meaning the generator is 1/3 of a septimal tone
of 8/7. Now would be a good time to suggest a better name.
Tertiaseptal is an excellent high-complexity temperament, especially
if we stick to the 7 odd limit and forgo 9, which 171-et does a good
job for.

If we have a {2,5,7} linear temperament, we can do something like the
nexial business, except that we use <<a2 0 0 -a5 -a7 0|| where
<a2 a5 a7| is a {2,5,7}-val belonging to the comma. Using this
definition of family relationship, we find valentine belongs to the
same family:

{2,5,7}-comma tertiaseptal 2100875/2097152
[1200.073, 77.200]

7-limit family

0: <<22 -5 3 -59 -57 21|| tertiaseptal {2401/2400, 65625/65536}
[1200.074, 77.199]

-16: <<6 -5 3 -22 -12 21|| {49/48, 3584/3375}
[1198.126, 80.798]

-31: <<9 5 -3 -13 -30 -21|| valentine {126/125, 1029/1024}
[1199.793, 77.833]

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> If we do a search for {2,5,7}-commas of size less than 50 cents and
> epimericity less than 0.6, the complete list seems to be 128/125,
> 50/49, 3136/3125, 2100875/2097152; all of which except for the last
> are quite familiar.

A similar search on {2,3,7}-commas gives the following list of commas
less than 50 cents in size with epimericity less than 0.6:

49/48, 64/63, 65536/64827, 1029/1024, 118098/117649

I'll take a look at these.