Planar temperaments -- 126/125

Looking at 7-limit commas, 81/80 is meantone, and 64/63 encompasses
12,22, and 27. The 225/224 temperament is the one I discussed in
connection with the Miracle-Magic Square. I drew 126/125 out of the
bag of 7-limit commas and took a look at it.

While there are always many ways to put coordinates on a planar
tempermant, I think generators consisting of a flat fifth and a sharp
major sixth work as well as anything. The following 9 note scale is
typical:

15/8
|
3/2 -- 5/4 -- 21/20 -- 7/4
| | | |
1 -- 5/3 -- 7/5 -- 7/6

The least squares 7-limit values for the generators f (the flat
fifth) and s (the sharp 5/3) are f = 699.984 cents, and s = 889.285
cents, the first 1.97 cents flat and the second 4.93 cents sharp. We
have a 5 ~ 2fs which is 2.96 cents sharp, and a 7 ~ fs^3 which is
0.99 cents flat. Clearly we are getting quite practical values for
the 7-limit. Ets with 126/125 in the kernel are
12,15,19,27,31,34,43,46,50,58 and 77.

--- In tuning@y..., genewardsmith@j... wrote:
> Looking at 7-limit commas, 81/80 is meantone, and 64/63 encompasses
> 12,22, and 27. The 225/224 temperament is the one I discussed in
> connection with the Miracle-Magic Square. I drew 126/125 out of the
> bag of 7-limit commas and took a look at it.
>
> While there are always many ways to put coordinates on a planar
> tempermant, I think generators consisting of a flat fifth and a
sharp
> major sixth work as well as anything. The following 9 note scale is
> typical:
>
> 15/8
> |
> 3/2 -- 5/4 -- 21/20 -- 7/4
> | | | |
> 1 -- 5/3 -- 7/5 -- 7/6
>
> The least squares 7-limit values for the generators f (the flat
> fifth) and s (the sharp 5/3) are f = 699.984 cents, and s = 889.285
> cents, the first 1.97 cents flat and the second 4.93 cents sharp.
We
> have a 5 ~ 2fs which is 2.96 cents sharp, and a 7 ~ fs^3 which is
> 0.99 cents flat.

I think I've seen this before. Take a look at this:

/tuning/topicId_6313.html#6313

That was almost two years ago! Thank you, memory!

On Thu, 08 Nov 2001 21:40:16 -0000, "Paul Erlich" <paul@stretch-music.com>
wrote:

>> The least squares 7-limit values for the generators f (the flat
>> fifth) and s (the sharp 5/3) are f = 699.984 cents, and s = 889.285
>> cents, the first 1.97 cents flat and the second 4.93 cents sharp.
>We
>> have a 5 ~ 2fs which is 2.96 cents sharp, and a 7 ~ fs^3 which is
>> 0.99 cents flat.
>
>I think I've seen this before. Take a look at this:
>
>/tuning/topicId_6313.html#6313
>
>That was almost two years ago! Thank you, memory!

I didn't recognize this at first (of course, the 126/125 should've given me
a clue), but as Paul Erlich pointed out to me, this is similar to what I
call Starling temperament (built from 388-cent major thirds and 312-cent
minor thirds, which results in a fifth of 700 cents and a major sixth of
888 cents). Here are a couple of relevant links:

http://www.io.com/~hmiller/music/marrgarrel.html
http://www.uq.net.au/~zzdkeena/Music/DistributingCommas.htm

In addition to the regular Starling temperament, I have a version with a
tempered octave (2 cents sharp). This improves the 7/4 approximation at the
expense of some of the other intervals (the optimum distribution mentioned
by Dave Keenan in his article referred to above actually has slighly narrow
octaves).

/tuning/topicId_6385.html#6385

--
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--- In tuning@y..., Herman Miller <hmiller@I...> wrote:

> I didn't recognize this at first (of course, the 126/125 should've
given me
> a clue), but as Paul Erlich pointed out to me, this is similar to
what I
> call Starling temperament (built from 388-cent major thirds and 312-
cent
> minor thirds, which results in a fifth of 700 cents and a major
sixth of
> 888 cents). Here are a couple of relevant links:

I was just now cooking up an example on the math group for Paul,
trying to explain lattice basis reduction and what it might be good
for, and another example plopped out, this one with a sharp major
third and something which is approximately 375/256 as generators,
good for 11-limit musics. The planar temperament possibilities are a
little dizzying.