Linear temperaments compatible with HTT

I took a look at 13-limit linear temperaments which killed the commas
352/351 and 364/363, and which had a TOP error less than 2.0. Below I
list some of these; the funny order is because they are listed in
order of a badness figure for the 11-limit reduction. As I expected,
mystery, Graham's discovery, put in an appearance, but the 13-limit
version of hemififths looks like a good choice also, allowing for a 41
note DE as well as a 58 note one (or 99, if you really get worked up.)
Magic, shrutar and supersuper also show up. "Triton" is really better
in the 11-limit, an 896/891 temperament.

Triton {325/324, 352/351, 364/363, 540/539}
[7, 26, 25, -3, -24, 25, 20, -29, -64, -15, -97, -152, -95, -160, -72]
[[1, 5, 15, 15, 2, -8], [0, -7, -26, -25, 3, 24]]
50 .647154 439.161631

Mystery {196/195, 352/351, 364/363, 676/675}
[0, 29, 29, 29, 29, 46, 46, 46, 46, -14, -33, -40, -19, -26, -7] [[29,
46, 67, 81, 100, 107], [0, 0, 1, 1, 1, 1]]
29 .651558 178.354113

Hemififths {144/143, 196/195, 243/242, 352/351}
[2, 25, 13, 5, -1, 35, 15, 1, -9, -40, -75, -95, -31, -51, -22] [[1,
1, -5, -1, 2, 4], [0, 2, 25, 13, 5, -1]]
26 1.056165 241.002003

"Number 80"
[6, -12, 10, -14, -32, -33, -1, -43, -73, 57, 9, -30, -74, -127, -59]
[[2, 4, 3, 7, 5, 3], [0, -3, 6, -5, 7, 16]]
44 .974054 534.159090

Shrutar
[4, -8, 14, -2, -14, -22, 11, -17, -37, 55, 23, -3, -54, -91, -41]
[[2, 3, 5, 5, 7, 8], [0, 2, -4, 7, -1, -7]]
28 1.845166 476.393162

Magic
[5, 1, 12, -8, -23, -10, 5, -30, -55, 25, -22, -57, -64, -109, -50]
[[1, 0, 2, -1, 6, 11], [0, 5, 1, 12, -8, -23]]
35 1.688049 632.167104

[1, 21, 15, 11, 8, 31, 21, 14, 9, -24, -47, -59, -21, -33, -13] [[1,
2, 11, 9, 8, 7], [0, -1, -21, -15, -11, -8]]
21 1.589486 254.071241

Supersupermajor
[3, 17, -1, -13, -22, 20, -10, -31, -46, -50, -89, -114, -33, -58,
-28] [[1, 1, -1, 3, 6, 8], [0, 3, 17, -1, -13, -22]]
39 .958363 429.837463

[2, -4, 30, 22, 16, -11, 42, 28, 18, 81, 65, 52, -42, -66, -26] [[2,
3, 5, 3, 5, 6], [0, 1, -2, 15, 11, 8]]
34 1.076337 384.072477

[2, -4, -16, -24, -30, -11, -31, -45, -55, -26, -42, -55, -12, -25,
-15] [[2, 3, 5, 7, 9, 10], [0, 1, -2, -8, -12, -15]]
34 1.267597 452.320425

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

> Hemififths {144/143, 196/195, 243/242, 352/351}
> [2, 25, 13, 5, -1, 35, 15, 1, -9, -40, -75, -95, -31, -51, -22] [[1,
> 1, -5, -1, 2, 4], [0, 2, 25, 13, 5, -1]]
> 26 1.056165 241.002003

The five mapping is the most complex, and 352/351 and 364/363 are
no-fives commas. We can get a no-fives linear temperament which is the
no-fives reduction of 13-limit hemififths from the commas 144/143,
243/242, 364/363. Margo might possibly find this of interest--the
fifths are of low complexity (two generator steps), with 512/507 a
comma of 13-limit hemififths, 16/13 being the generator and
(16/13)^2/(3/2) = 512/507. It's a no-fives system with a sharp fifth,
which is what she seems to favor. The Graham complexity is 14, which
allows a lot of no-fives 13-limit chords in the 24-note DE, and even
the 17 note DE isn't bad.

Gene Ward Smith wrote:
> I took a look at 13-limit linear temperaments which killed the commas
> 352/351 and 364/363, and which had a TOP error less than 2.0. Below I
> list some of these; the funny order is because they are listed in
> order of a badness figure for the 11-limit reduction. As I expected,
> mystery, Graham's discovery, put in an appearance, but the 13-limit
> version of hemififths looks like a good choice also, allowing for a 41
> note DE as well as a 58 note one (or 99, if you really get worked up.)
> Magic, shrutar and supersuper also show up. "Triton" is really better
> in the 11-limit, an 896/891 temperament.

The 13-limit hemififths looks pretty nice: the 41-note version is comparable to Erv Wilson's Cassandra 1 ( <<1, -8, -14, 23, 20, -15, -25, 33, 28, -10, 81, 76, 113, 108, -16]] ), but has a greater range of 13-limit chords available (15 complete hexads compared to only 4 for Cassandra 1[41]) Supersupermajor also has a good 41-note version, but is too complex to work very well with only 41. With only a few more notes (46), <<2, -4, 30, 22, 16, -11, 42, 28, 18, 81, 65, 52, -42, -66, -26]] has a better mapping.

What really looks interesting is <<1, 21, 15, 11, 8, 31, 21, 14, 9, -24, -47, -59, -21, -33, -13]]. It actually looks a little better than nonkleismic, and the 29-note MOS could potentially use Graham Breed's fourth-based keyboard mapping.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> What really looks interesting is <<1, 21, 15, 11, 8, 31, 21, 14, 9,
-24,
> -47, -59, -21, -33, -13]]. It actually looks a little better than
> nonkleismic, and the 29-note MOS could potentially use Graham Breed's
> fourth-based keyboard mapping.

It's also got a fifth as a generator, a big plus for Margo and George
I suspect.

Here's another possibility: 352/351 is a {2,3,11,13} comma; if you put
it together with another such, you get a {2,3,11,13} temperament, the
mapping of which you can then extend. 144/143 is one choice for this,
leading to hemififths. It's a nice comma but it lowers the accuracy
352/351 gives some. Another possibility is 2197/2187 = 13^3/3^7. This
starts the mapping game off with a generator which is approximately
13/9, about two cents narrow, and we now have

3 ~ (13/9)^3
11 ~ 2^(-5) (13/9)^16
13 ~ (13/9)^7

This is a {2,3,11,13} linear temperament with a Graham complexity of
16. If we now add 364/363 we will get a no-fives linear temperament
compatible with HTT, and mapping

7 ~ 2^(-12) (13/9)^28

The 36 note MOS would seem like a good choice with this. If you
absolutely must, you can add 5 by including 245/243 among the commas,
giving

5 ~ 2^24 (13/9)^(-41)

At this point you should probably just switch to 87-equal and call it
a day.

George's preferred 3 approximation for HTT is 2 (504/13)^(1/9) =
(258048/13)^(1/9), so one possible tuning would have a generator of
size (258048/13)^(1/27), which works fine, giving something whose
tuning works better as an HTT extension (much more accurate for 11 and
13 in particular) than 13-limit hemififths.