Some 11-limit temperaments supported by 152

Below I give the wedige, the mapping, a badness figure computed using
Graham complexity and TOP error, and then the complexity and error.

<<23 -1 13 42 -55 -44 -13 33 101 73|| has a generator of a major
third, so we might call it voodoo, as a relative of magic, muggles,
and wizard.

<<1 33 27 -18 50 40 -32 -30 -156 -144|| has a generator of a fifth,
like meantone and schismic, with TM basis {540/539, 1375/1372,
5120/5103}. The poptimal range runs from about 0.01 to 0.67 cents
sharp. It can be gotten by wedging the standard vals for any two of
41, 111, or 152 together, and so can be taken to be 41&152, with the
41 note DE an obvious possibility. None of this gives me a strong clue
about a good name. Anyone want to take another trip to the zoo?

Octoid
[24, 32, 40, 24, -5, -4, -45, 3, -55, -71] [[8, 13, 19, 23, 28], [0,
-3, -4, -5, -3]]
171.094038 48 .269877

Voodoo
[23, -1, 13, 42, -55, -44, -13, 33, 101, 73] [[1, 9, 2, 7, 17], [0,
-23, 1, -13, -42]]
201.010324 47 .328389

[1, 33, 27, -18, 50, 40, -32, -30, -156, -144] [[1, 2, 16, 14, -4],
[0, -1, -33, -27, 18]]
224.092825 51 .319505

[25, 65, 67, 6, 45, 36, -77, -27, -211, -215] [[1, 8, 19, 20, 5], [0,
-25, -65, -67, -6]]
331.030402 67 .299509

[9, -7, -61, -10, -32, -122, -47, -122, 1, 183] [[1, 2, 2, 0, 3], [0,
-9, 7, 61, 10]]
365.559169 79 .251331

[3, -53, -71, -54, -91, -121, -96, -16, 58, 94] [[1, 2, -5, -7, -4],
[0, -3, 53, 71, 54]]
370.926981 77 .266157

[13, -27, 47, 70, -73, 38, 66, 185, 256, 34] [[1, 8, -11, 26, 38], [0,
-13, 27, -47, -70]]
490.345468 97 .239451

[15, 39, -51, 34, 27, -123, 2, -228, -56, 272] [[1, 8, 19, -19, 18],
[0, -15, -39, 51, -34]]
499.053026 90 .276106

Amity
[5, 13, -17, 62, 9, -41, 81, -76, 99, 233] [[1, 3, 6, -2, 21], [0, -5,
-13, 17, -62]]
500.754537 79 .344281

[47, 31, 53, 66, -60, -48, -58, 36, 46, 2] [[1, -8, -4, -8, -10], [0,
47, 31, 53, 66]]
506.523230 94 .260647

Enneadecal
[19, 19, 57, -38, -14, 37, -126, 79, -154, -304] [[19, 30, 44, 53,
66], [0, 1, 1, 3, -2]]
518.966378 95 .262382

[27, -21, -31, -30, -96, -125, -141, -13, 3, 23] [[1, 11, -5, -8, -7],
[0, -27, 21, 31, 30]]
533.178660 85 .324470

[8, -40, 64, 8, -82, 79, -15, 261, 157, -199] [[8, 13, 17, 25, 28],
[0, -1, 5, -8, -1]]
541.551816 104 .235460

[41, -15, 43, 22, -119, -47, -107, 142, 103, -87] [[1, -3, 4, -2, 1],
[0, 41, -15, 43, 22]]
553.207897 97 .270148

[17, -47, 3, -2, -114, -43, -62, 139, 158, -16] [[1, -3, 15, 2, 4],
[0, 17, -47, 3, -2]]
579.688915 81 .382284

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

> <<1 33 27 -18 50 40 -32 -30 -156 -144|| has a generator of a fifth,
> like meantone and schismic, with TM basis {540/539, 1375/1372,
> 5120/5103}. The poptimal range runs from about 0.01 to 0.67 cents
> sharp.

This range stuck me as way too big, so I recalculated and got
0.658 to 0.670, much more reasonable for a temperament of this
accuracy. The 152 fifth, 0.6766 cents sharp, is just beyond this
range, but works well enough; 291/497, 0.6607 cents sharp, is poptimal.

This thing has an interesting {7,11}-limit comma, 65536/65219; which
is one of the five {7,11} commas of size less than 50 cents and
epimericity less than 0.6:

352/343 44.840223 .376383
14641/14336 36.445863 .597701
65536/65219 8.394360 .519498
117649/117128 7.683669 .536007
5767168/5764801 .710691 .499084

Here I've listed the comma, size in cents, and epimericity; the most
striking is clearly the last one, 2^18 7^(-8) 11, which could be a
gold mine for people seeking the exotic harmonies of 7 and 11 in
conjunction.

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

> Here I've listed the comma, size in cents, and epimericity; the most
> striking is clearly the last one, 2^18 7^(-8) 11, which could be a
> gold mine for people seeking the exotic harmonies of 7 and 11 in
> conjunction.

If we take this comma and add to it 43923/43904, a {2,3,7,11} comma,
we get what we can consider to be either an 11-limit planar
temperament (with 2, 5, and 7 as generators) or {2,3,7,11} linear
temperament; ie, a no-fives temperament. This has period an octave and
generator a very slightly narrow 8/7. (If we use (48)^(1/29) as a
generator, 0.0722 cents flat; if we use 2^(161/836), 0.0736 cents
flat.) In terms of this generator, the mapping is

3 ~ 2^(-4) (8/7)^29
11 ~ 2^5 (8/7)^(-8)

This is a strong no-fives microtemperament but trying to add five to
it doesn't work so well. The Graham complexity is 66, or 37 if you are
willing to forgo 9 as well as 5; we have DE of size 57 and 83. A bit
too much for most of us. Of course if we are willing to settle for a
{7, 11} temperament, the Graham complexity is now 9, and it's
reminiscent of schismic, but of course more accurate. DE of size 11,
16, 21, 26 and 31 become relevant.