2.9.5.7

Since Mike has been talking up the 2.9.7.11 subgroup with its 1-11/8-7/4-9/4 chord, I thought I'd suggest we take a look at the 2.9.5.7 subgroup with its 1-5/4-7/4-9/4 chord. Instead of tempering out 64/63 and 99/98 like Mike's machine temperament, we would temper out 81/80 and 126/125, just like meantone. This is already sort of familiar, the 6 note MOS being the whole tone scale retuned for better harmony. Aside from 6edo, which takes all the fun out of the idea, there's 19, 25 and 31 to tune it in. 25edo removes the evil temptations of meantone, so I say go for that one. This temperament not much more complex than Mike's "machine" and in much better tune. But is it xenharmonic?

On Sun, Feb 13, 2011 at 3:57 AM, genewardsmith
<genewardsmith@...> wrote:
>
> Since Mike has been talking up the 2.9.7.11 subgroup with its 1-11/8-7/4-9/4 chord, I thought I'd suggest we take a look at the 2.9.5.7 subgroup with its 1-5/4-7/4-9/4 chord. Instead of tempering out 64/63 and 99/98 like Mike's machine temperament, we would temper out 81/80 and 126/125, just like meantone. This is already sort of familiar, the 6 note MOS being the whole tone scale retuned for better harmony. Aside from 6edo, which takes all the fun out of the idea, there's 19, 25 and 31 to tune it in. 25edo removes the evil temptations of meantone, so I say go for that one. This temperament not much more complex than Mike's "machine" and in much better tune. But is it xenharmonic?

Another interesting thing about 25edo is that it makes, using the
notation I had laid out on tuning-math a while ago, for a good
2.3.7.9'.11 temperament, where you have a 9' that is mapped
differently than 3^2. If you do that, you have the following
Blackwood-ish scale to play around with:

! C:\Program Files\Scala22\scl\15-out-of-25.scl
!
15-out-of-25
!
144.00000
192.00000
240.00000
384.00000
432.00000
480.00000
624.00000
672.00000
720.00000
864.00000
912.00000
960.00000
1104.00000
1152.00000
2/1

The 3/2's are so much worse than the other intervals mapped that it
might be worth just using them as intervals to modulate around with,
rather than as part of the chords themselves.

You kind of have to think differently about a tuning like this, as
there's this new "9/9'" interval now, which corresponds to one step of
the scale. One step of the scale is also the difference between 5/4
and 9/7 where 9/7 is derived from the 3-based "non-prime" 9, so you
can say that 9/9' and 36/35 are equated.

This notation might make more sense if you just think of it in terms
of smonzos in which there are two different parameters for 3 and 9.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Another interesting thing about 25edo is that it makes, using the
> notation I had laid out on tuning-math a while ago, for a good
> 2.3.7.9'.11 temperament, where you have a 9' that is mapped
> differently than 3^2.

I had in mind a more obvious extension: add 11 to the subgroup, and 99/98 to the commas. Now as well as 25 and 31, you have 37 to consider as a possible tuning. The POTE generator, by the way, is almost exactly 16\99.

Since not much milage is normally gotten out of the rather complex 11 in either version of 11-limit meantone, this looks like a way to make the whole thing definitely xenharmonic. The 11 now has a Graham complexity of 9, not 18, and the 13 note MOS would do nicely. Familiar, and yet unfamiliar, a bit like mohajira.

Since you gave a scale I think I'll give one too:

! tutone13.scl
!
2.9.5.7.11 with {81/80, 126/125, 99/98} temperament in 16\99 tuning
13
!
157.57576
193.93939
351.51515
387.87879
545.45455
581.81818
739.39394
775.75758
933.33333
969.69697
1127.27273
1163.63636
2/1

On Sun, Feb 13, 2011 at 1:25 PM, genewardsmith
<genewardsmith@...> wrote:
>
> I had in mind a more obvious extension: add 11 to the subgroup, and 99/98 to the commas. Now as well as 25 and 31, you have 37 to consider as a possible tuning. The POTE generator, by the way, is almost exactly 16\99.
>
> Since not much milage is normally gotten out of the rather complex 11 in either version of 11-limit meantone, this looks like a way to make the whole thing definitely xenharmonic. The 11 now has a Graham complexity of 9, not 18, and the 13 note MOS would do nicely. Familiar, and yet unfamiliar, a bit like mohajira.
>
> Since you gave a scale I think I'll give one too:
>
> ! tutone13.scl
> !
> 2.9.5.7.11 with {81/80, 126/125, 99/98} temperament in 16\99 tuning
> 13
> !
> 157.57576
> 193.93939
> 351.51515
> 387.87879
> 545.45455
> 581.81818
> 739.39394
> 775.75758
> 933.33333
> 969.69697
> 1127.27273
> 1163.63636
> 2/1

Very nice! Has a different feel to it than machine. More whole
tone-ish, in a sense.

-Mike