An additional comma in the Wurschmidt family?

This seems like a niggling point, but while building a scale that tempers
out the Wurschmidt comma (393216/390625, or |17 1 -8>), it seemed to me
that it would be beneficial to temper out 29360128/29296875 = |22 -1 -10 1>
as well. Tempering out Wurschmidt makes eight 5/4s = 3/2
(octave-equivalent); tempering out |22 -1 -10 1> makes ten 5/4s = 7/6
(octave-equivalent). (There's probably reason to temper out another comma
for some number of 5/4s to make an 11/9, but I haven't bothered to check
what it is yet. I will, if anyone's interested.)

This comma also shows up in Vishnu and a Hemifamity temperament called
Alphaquarter -- though I really don't know why, since the documentation
doesn't talk about 5/4 as a generator in either.

I also can't find a name for this comma.

I would add something to the Wurschmidt page if I had confidence that I
knew why various commas were added to make any given temperament; since I
don't, I'm tossing the idea out to the group, in case anyone else is
interested and knowledgeable enough to point the way.

Regards,
Jake

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
>
> This seems like a niggling point, but while building a scale that tempers
> out the Wurschmidt comma (393216/390625, or |17 1 -8>), it seemed to me
> that it would be beneficial to temper out 29360128/29296875 = |22 -1 -10 1>
> as well. Tempering out Wurschmidt makes eight 5/4s = 3/2
> (octave-equivalent); tempering out |22 -1 -10 1> makes ten 5/4s = 7/6
> (octave-equivalent).

What you describe here is exactly 7-limit würschmidt. :)

If adding two 5/4s to 3/2 leads to 7/3, it essentially means that two 5/4s represent (7/3) / (3/2) = 14/9, which in turn means that the marvel comma |-5 2 2 -1> = 225/224 is tempered out.

|-5 2 2 -1> is the exact difference between |17 1 -8> and |22 -1 -10 1>, and tempering out two of those commas always implies tempering out the other comma as well.

If you add 3*|-5 2 2 -1> and |17 1 -8>, you get |2 7 -2 -3> = 8748/8575, so tempering out 225/224 and 8748/8575 (the commas listed for 7-limit würschmidt) leads to the commas you mentioned.

> This comma also shows up in Vishnu and a Hemifamity temperament called
> Alphaquarter -- though I really don't know why, since the documentation
> doesn't talk about 5/4 as a generator in either.

If two temperaments temper out the same comma, it doesn't mean that they have the same generator. For example, tempering out the marvel comma 225/224 means that two 5/4s and a 9/7 add up to an octave, or that two secors (representing 16/15 or 15/14) add up to a 8/7.
However, if you take a look at the marvel temperament page, you see lots of temperaments with different generators:

http://xenharmonic.wikispaces.com/Marvel+temperaments

The same applies to the comma you mentioned; while tempering out that comma implies that 10 5/4s make a 7/6, it doesn't mean that all temperaments that temper out this comma have a 5/4 generator.

Best
- Gedankenwelt

Thank you very much, this was an excellent explanation. It's the kind of
thing that I want to know for every comma, and every comma combination.
It's hard to see what the relationships are, or the differences between
temperaments, sometimes.

Thanks,
Jake

In case you're interested, I took a look at those other temperaments that temper out 29360128/29296875 = |22 -1 -10 1>:

* The reduced mapping for alphaquarter is [<1 2 2 0|, <0 -9 7 61|], which means that -9 generators make a 3/2, 7 generators a 5/4, and 61 generators a 7/4 (octaves ignored).
That means that 7/6 consists of 61 - (-9) = 70 generators, or 10 5/4s, since a 5/4 consists of 7 generators.

* Vishnu is a little more complicated, since it has a half octave-period. The reduced mapping is [<2 4 5 10|, <0 -7 -3 -37|], so -7 25/24 generators make a 3/2, 1\2 ("a half octave") minus 3 generators a 5/4, and -37 generators a 7/4 (again, octaves ignored).
Hence, 7/6 consists of -37 - (-7) = -30 generators, or 10 5/4s, since 10 * (1\2 - 3 generators) = -30 generators and some octaves. ;)

Btw., are you using Graham's temperament finder? It takes some time to get familiar with, but once you are it's extremely useful when working with temperaments.

- Gedankenwelt

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
>
> Thank you very much, this was an excellent explanation. It's the kind of
> thing that I want to know for every comma, and every comma combination.
> It's hard to see what the relationships are, or the differences between
> temperaments, sometimes.
>
> Thanks,
> Jake