325/324, the twin comma to 225/224

We've talked a lot about 676/675 lately, which is a 2.3.5.13 comma. Two other such commas are 625/624 and 325/324. 325/324 can be added to the 11-limit version of marvel, tempering out 225/224 and 385/384 to get 13-limit marvel. But it's also interesting to leave 11 out of it. From 225/224 we get that a 5-limit approximation for 7 is 225/224 * 7 = 225/32. Similarly from 325/324 we get a 5-limit approximation of 13 from 324/325 * 13 = 324/25. If we define the major/minor transformation of the 5-limit as the result of fixing 2 and 3 and replacing 5 by 24/5, then major/minor applied to 225/32 is 162/25, which is (324/25)/2. Similarly, major/minor applied to 324/25 is 225/16 = 2 * (225/32). 225/224 tells us that two 16/15 in a row are an approximate 8/7, and 325/324 tells us two 10/9 in a row are an approximate 16/13. Needless to say, major/minor applied to 16/15 is 10/9, and applied to 10/9 is 16/15.

325/324 seems like a great comma. If you take 10/9 as the generator,
and two of them gets you to 16/3, then three of them gets you to a
flat 11/8, and four gets you to something looking like 32/21. If
you're mixing it with 225/224, then 105/104 also ends up being a good
fit, which Graham's temperament finder calls "Supernatural". It's also
got (16/13)^(1/2) as a generator, which suggests continuing the
pattern to see what we might find. There were some really interesting
higher-accuracy results, and some interesting lower-accuracy results.

(16/13)^(1/3) is kind of between negri and miracle and may imply 65/64
being tempered out; not so great for you high accuracy people, can be
interesting for the pun-happy folks out there. I didn't bother past
having an exponent of 1/3 for (16/13), as I don't care much for scales
with generators that are that small. Repeating the process for 13/8
gives some interesting results:

(13/8)^(1/2) gives you an interesting sensi-ish sort of scale, which
isn't really too good for sensi at all - the generator can be mapped
to either 9/7 or 14/11 (or both to get you to rank 4), two of which
get you to 13/8.

(13/8)^(1/3) could be taken to give you a sharp 7/6, at which point
you get a strange orwell variant (becomes a 13-limit extension of
orwell if 65/64 vanishes), or it could also be 13/11, which is a bit
higher in accuracy. Five of these generators gets you to an almost
perfect 9/4, and six gets you to a pretty good 5/2, and seven gets you
to an almost perfect 28/9. It's 2 AM and I'm way too tired to work out
the commas for now, but that looks like a good temperament.

(13/8)^(1/4) gets you a slightly sharp major 2nd, which suggests
subdividing the generator further into a slightly sharp fifth; this
means that the pythagorean augmented fifth now becomes 13/8. This
looks like it'd be a pleasant tuning and seems to be supported by
17-equal.

(13/8)^(1/5) gives you a really interesting choice with a generator
somewhere between porcupine and tetracot; its generator is about 3
cents off from 11/10, which makes for a really neat tuning! 1
generator is an almost perfect 11/10, two generators is about 336
cents, which is 0.08 cents off from 17/14 should you want to get the
17-limit involved, 3 generators is 504 cents, which suggests 81/80
might go well with this; 5 generators is 13/8, and 6 generators is a
decent 9/5, further suggesting that 81/80 vanish. Not bad at all. 5
generators gives you a 672 cent mavila sized fifth, which makes for a
good reason to start exploring there being two different mappings for
3, and having 135b/128 vanish.

As a random thing I tried to try, (32/13)^(1/4) gives you a 389 cent
major third, which gives a perfect 32/13 after 4 generators, thus
tempering out 8192/8125. It then goes on to give an almost perfect
20/13 after 5 generators, as you'd expect. Might be worth something.

Thus ends my report, for now.

-Mike

On Mon, Jun 20, 2011 at 9:47 PM, genewardsmith
<genewardsmith@...> wrote:
>
>
>
> We've talked a lot about 676/675 lately, which is a 2.3.5.13 comma. Two other such commas are 625/624 and 325/324. 325/324 can be added to the 11-limit version of marvel, tempering out 225/224 and 385/384 to get 13-limit marvel. But it's also interesting to leave 11 out of it. From 225/224 we get that a 5-limit approximation for 7 is 225/224 * 7 = 225/32. Similarly from 325/324 we get a 5-limit approximation of 13 from 324/325 * 13 = 324/25. If we define the major/minor transformation of the 5-limit as the result of fixing 2 and 3 and replacing 5 by 24/5, then major/minor applied to 225/32 is 162/25, which is (324/25)/2. Similarly, major/minor applied to 324/25 is 225/16 = 2 * (225/32). 225/224 tells us that two 16/15 in a row are an approximate 8/7, and 325/324 tells us two 10/9 in a row are an approximate 16/13. Needless to say, major/minor applied to 16/15 is 10/9, and applied to 10/9 is 16/15.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> 325/324 seems like a great comma.

It is, but i'm going to need to wait until tomorrow to respond to your comments. Meanwhile, I'd strongly recommend that people interested in it take a look at cata temperament here:

http://xenharmonic.wikispaces.com/Chromatic+pairs#Cata

Here is a chain of 6/5's reduced by 325/325 and 625/625 (which means also by (325/324)/(625/624) = 676/675.) It's therefore a transversal for the Cata[19] MOS.

! precata19.scl
Cata[19] transversal
19
!
25/24
13/12
10/9
52/45
6/5
5/4
13/10
4/3
18/13
13/9
3/2
20/13
8/5
5/3
26/15
9/5
24/13
25/13
2/1

Among its step sizes are 25/24, 26/25 and 27/26; (25/24)/(26/25) = 625/624 and (26/25)/(27/26) = 676/675. Gotta love those square numerators!

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> 325/324 seems like a great comma. If you take 10/9 as the generator,
> and two of them gets you to 16/3, then three of them gets you to a
> flat 11/8, and four gets you to something looking like 32/21.

(11/8)/(10/9)^3 = 8019/8000, which like 325/324 is a comma of 72et. (32/21)/(10/9)^4 = 4374/4375, also a comma of 72et. Put them all together and you get the 7&19&46 temperament with 72 a good tuning. If by "take 10/9 as the generator" you mean to find a linear temperament, then you end up with unidec, the 46&72 temperament, with period 1/2 octave and a 10/9 generator.

If
> you're mixing it with 225/224, then 105/104 also ends up being a good
> fit, which Graham's temperament finder calls "Supernatural".

That's a sort of slightly detempered magic.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> (11/8)/(10/9)^3 = 8019/8000, which like 325/324 is a comma of 72et. (32/21)/(10/9)^4 = 4374/4375, also a comma of 72et. Put them all together and you get the 7&19&46 temperament with 72 a good tuning. If by "take 10/9 as the generator" you mean to find a linear temperament, then you end up with unidec, the 46&72 temperament, with period 1/2 octave and a 10/9 generator.

Sorry, I didn't make my point. You end up with unidec, because the alternative is 13&72 and that's not much good.