Some 13-limit temperaments

Here are a few temperaments found by the brute-force approach (try all possible generators and periods until something interesting comes up).

No. 1: Coendou? (a South American porcupine)
7L + 15s (period=1201.0c, generator=166.2c)
[[1, 2, 3, 1, 4, 3], [0, -3, -5, 13, -4, 5]]
<<3, 5, -13, 4, -5, 1, -29, -4, -19, -44, -8, -30, 56, 34, -32]]

No. 2: Roman? (26 letters in the Roman alphabet)
3L + 23s (period=1200.6c, generator=414.5c)
[[1, 4, 3, -1, 0, 3], [0, -7, -2, 11, 10, 2]]
<<7, 2, -11, -10, -2, -13, -37, -40, -29, -31, -30, -12, 10, 35, 30]]

No. 3: Leapday? (Feb. 29)
17L + 12s (period=1200.0c, generator=495.7c)
[[1, 2, 11, 9, 8, 7], [0, -1, -21, -15, -11, -8]]
<<1, 21, 15, 11, 8, 31, 21, 14, 9, -24, -47, -59, -21, -33, -13]]

No. 4: Myna / Pangolin (formerly Nonkleismic)
27L + 4s (period=1198.5c, generator=310.0c)
[[1, -1, 0, 1, -3, 5], [0, 10, 9, 7, 25, -5]]
<<10, 9, 7, 25, -5, -9, -17, 5, -45, -9, 27, -45, 46, -40, -110]]

No. 5:
26L + 6s (period=599.6c, generator=185.8c)
[[2, 1, 0, 5, 6, 4], [0, 7, 15, 2, 3, 11]]
<<14, 30, 4, 6, 22, 15, -33, -39, -17, -75, -90, -60, 3, 47, 54]]

No. 6:
29L + 5s (period=1199.9c, generator=247.8c)
[[1, 2, 5, 9, 8, 7], [0, -2, -13, -30, -22, -16]]
<<2, 13, 30, 22, 16, 16, 42, 28, 18, 33, 6, -11, -42, -66, -26]]

No. 7: Hitchcock? (39 steps)
7L + 32s (period=1201.1c, generator=339.7c)
[[1, 3, 6, -2, 6, 2], [0, -5, -13, 17, -9, 6]]
<<5, 13, -17, 9, -6, 9, -41, -3, -28, -76, -24, -62, 84, 46, -54]]

No. 8: Supersupermajor
5L + 36s (period=1200.1c, generator=234.5c)
[[1, 1, -1, 3, 6, 8], [0, 3, 17, -1, -13, -22]]
<<3, 17, -1, -13, -22, 20, -10, -31, -46, -50, -89, -114, -33, -58, -28]]

No. 9: Cassandra 1
12L + 29s (period=1200.2c, generator=498.0c)
[[1, 2, -1, -3, 13, 12], [0, -1, 8, 14, -23, -20]]
<<1, -8, -14, 23, 20, -15, -25, 33, 28, -10, 81, 76, 113, 108, -16]]

No. 10: Hemififths
17L + 24s (period=1198.9c, generator=351.2c)
[[1, 1, -5, -1, 2, 4], [0, 2, 25, 13, 5, -1]]
<<2, 25, 13, 5, -1, 35, 15, 1, -9, -40, -75, -95, -31, -51, -22]]

No. 11:
17L + 26s (period=1201.0c, generator=140.3c)
[[1, 1, -2, 0, 1, 3], [0, 5, 37, 24, 21, 6]]
<<5, 37, 24, 21, 6, 47, 24, 16, -9, -48, -79, -123, -24, -72, -57]]

No. 12:
4L + 42s (period=600.5c, generator=287.8c)
[[2, 7, 8, 8, 5, 5], [0, -8, -7, -5, 4, 5]]
<<16, 14, 10, -8, -10, -15, -29, -68, -75, -16, -67, -75, -57, -65, -5]]

No. 13: Diaschismic
12L + 34s (period=599.6c, generator=103.6c)
[[2, 3, 5, 7, 9, 10], [0, 1, -2, -8, -12, -15]]
<<2, -4, -16, -24, -30, -11, -31, -45, -55, -26, -42, -55, -12, -25, -15]]

No. 14: (Number 33 from the 7-limit list)
26L + 20s (period=600.5c, generator=183.3c)
[[2, 5, 8, 5, 6, 8], [0, -6, -11, 2, 3, -2]]
<<12, 22, -4, -6, 4, 7, -40, -51, -38, -71, -90, -72, -3, 26, 36]]

No. 15:
34L + 12s (period=599.7c, generator=104.9c)
[[2, 3, 5, 3, 5, 6], [0, 1, -2, 15, 11, 8]]
<<2, -4, 30, 22, 16, -11, 42, 28, 18, 81, 65, 52, -42, -66, -26]]

No. 16: Vulture
5L + 43s (period=1199.3c, generator=475.4c)
[[1, 0, -6, 4, -12, -7], [0, 4, 21, -3, 39, 27]]
<<4, 21, -3, 39, 27, 24, -16, 48, 28, -66, 18, -15, 120, 87, -51]]

No: 17: (Number 76 from the 7-limit list)
3L + 47s (period=1200.8c, generator=408.0c)
[[1, 6, 3, 13, -3, 2], [0, -13, -2, -30, 19, 5]]
<<13, 2, 30, -19, -5, -27, 11, -75, -56, 64, -51, -19, -157, -125, 53]]

No. 18:
26L + 25s (period=1201.8c, generator=46.7c)
[[1, 1, 1, 3, 4, 3], [0, 15, 34, -5, -14, 18]]
<<15, 34, -5, -14, 18, 19, -50, -74, -27, -107, -150, -84, -22, 69, 114]]

No. 19:
29L + 22s (period=1199.9c, generator=165.0c)
[[1, 2, 7, 9, 8, 7], [0, -3, -34, -45, -33, -24]]
<<3, 34, 45, 33, 24, 47, 63, 42, 27, 9, -41, -70, -63, -99, -39]]

No. 20:
5L + 48s (period=1199.4c, generator=475.8c)
[[1, 0, 17, 4, 11, 16], [0, 4, -37, -3, -19, -31]]
<<4, -37, -3, -19, -31, -68, -16, -44, -64, 97, 84, 65, -43, -76, -37]]

No. 21: Catakleismic (Hanson)
19L + 34s (period=1200.9c, generator=317.0c)
[[1, 0, 1, -3, 9, 0], [0, 6, 5, 22, -21, 14]]
<<6, 5, 22, -21, 14, -6, 18, -54, 0, 37, -66, 14, -135, -42, 126]]

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> Here are a few temperaments found by the brute-force approach (try all
> possible generators and periods until something interesting comes up).
>
> No. 1: Coendou? (a South American porcupine)
{55/54, 65/64, 100/99, 105/104}
> 7L + 15s (period=1201.0c, generator=166.2c)
> [[1, 2, 3, 1, 4, 3], [0, -3, -5, 13, -4, 5]]
> <<3, 5, -13, 4, -5, 1, -29, -4, -19, -44, -8, -30, 56, 34, -32]]
>
> No. 2: Roman? (26 letters in the Roman alphabet)
> 3L + 23s (period=1200.6c, generator=414.5c)
> [[1, 4, 3, -1, 0, 3], [0, -7, -2, 11, 10, 2]]
> <<7, 2, -11, -10, -2, -13, -37, -40, -29, -31, -30, -12, 10, 35, 30]]
>
> No. 3: Leapday? (Feb. 29) Eh?
{91/90, 121/120, 169/168, 352/351}
> 17L + 12s (period=1200.0c, generator=495.7c)
> [[1, 2, 11, 9, 8, 7], [0, -1, -21, -15, -11, -8]]
> <<1, 21, 15, 11, 8, 31, 21, 14, 9, -24, -47, -59, -21, -33, -13]]
>
> No. 4: Myna / Pangolin (formerly Nonkleismic)
> 27L + 4s (period=1198.5c, generator=310.0c)
> [[1, -1, 0, 1, -3, 5], [0, 10, 9, 7, 25, -5]]
> <<10, 9, 7, 25, -5, -9, -17, 5, -45, -9, 27, -45, 46, -40, -110]]
>
> No. 5:
{144/143, 351/350, 364/363, 441/440}
Another extension of HTT to a linear temperament.
> 26L + 6s (period=599.6c, generator=185.8c)
> [[2, 1, 0, 5, 6, 4], [0, 7, 15, 2, 3, 11]]
> <<14, 30, 4, 6, 22, 15, -33, -39, -17, -75, -90, -60, 3, 47, 54]]
>
> No. 6:
> 29L + 5s (period=1199.9c, generator=247.8c)
> [[1, 2, 5, 9, 8, 7], [0, -2, -13, -30, -22, -16]]
> <<2, 13, 30, 22, 16, 16, 42, 28, 18, 33, 6, -11, -42, -66, -26]]
>
> No. 7: Hitchcock? (39 steps) This is in the amity family; it has the
same TOP tuning as the 11-limit reduction, but there are other
11-limit versions of amity which accord better with 5 and 7 limit
amity (adjustments of the 11-limit reduction by +53 or -46.)

> 7L + 32s (period=1201.1c, generator=339.7c)
> [[1, 3, 6, -2, 6, 2], [0, -5, -13, 17, -9, 6]]
> <<5, 13, -17, 9, -6, 9, -41, -3, -28, -76, -24, -62, 84, 46, -54]]
>
> No. 8: Supersupermajor
> 5L + 36s (period=1200.1c, generator=234.5c)
> [[1, 1, -1, 3, 6, 8], [0, 3, 17, -1, -13, -22]]
> <<3, 17, -1, -13, -22, 20, -10, -31, -46, -50, -89, -114, -33, -58,
-28]]
>
> No. 9: Cassandra 1
I've always hated this name. It's either cassandra or it isn't, but no
subscripts, please! My suggestion of "septishis" for this is pretty
lousy, but what about just calling it cassandra? The 11 and 13
versions have the same TOP tuning, and the 7-limit is close. Schismic
would be a different deal, what I once called counterschismic.
> 12L + 29s (period=1200.2c, generator=498.0c)
> [[1, 2, -1, -3, 13, 12], [0, -1, 8, 14, -23, -20]]
> <<1, -8, -14, 23, 20, -15, -25, 33, 28, -10, 81, 76, 113, 108, -16]]
>
> No. 10: Hemififths
> 17L + 24s (period=1198.9c, generator=351.2c)
> [[1, 1, -5, -1, 2, 4], [0, 2, 25, 13, 5, -1]]
> <<2, 25, 13, 5, -1, 35, 15, 1, -9, -40, -75, -95, -31, -51, -22]]
>
> No. 11:
> 17L + 26s (period=1201.0c, generator=140.3c)
> [[1, 1, -2, 0, 1, 3], [0, 5, 37, 24, 21, 6]]
> <<5, 37, 24, 21, 6, 47, 24, 16, -9, -48, -79, -123, -24, -72, -57]]
>
> No. 12:
> 4L + 42s (period=600.5c, generator=287.8c)
> [[2, 7, 8, 8, 5, 5], [0, -8, -7, -5, 4, 5]]
> <<16, 14, 10, -8, -10, -15, -29, -68, -75, -16, -67, -75, -57, -65, -5]]
>
> No. 13: Diaschismic
Number 15 is closer in tuning to 5-limit diaschismic
> 12L + 34s (period=599.6c, generator=103.6c)
> [[2, 3, 5, 7, 9, 10], [0, 1, -2, -8, -12, -15]]
> <<2, -4, -16, -24, -30, -11, -31, -45, -55, -26, -42, -55, -12, -25,
-15]]

I tagged the name unidec onto the 11 and 13 limit temperament a long
time ago, so the 7-limit version should have been unidec also.
> No. 14: (Number 33 from the 7-limit list)
> 26L + 20s (period=600.5c, generator=183.3c)
> [[2, 5, 8, 5, 6, 8], [0, -6, -11, 2, 3, -2]]
> <<12, 22, -4, -6, 4, 7, -40, -51, -38, -71, -90, -72, -3, 26, 36]]
>
> No. 15:
Closer to 5-limit diaschismic than number 13
> 34L + 12s (period=599.7c, generator=104.9c)
> [[2, 3, 5, 3, 5, 6], [0, 1, -2, 15, 11, 8]]
> <<2, -4, 30, 22, 16, -11, 42, 28, 18, 81, 65, 52, -42, -66, -26]]
>
> No. 16: Vulture
{176/175, 351/350, 540/539, 640/637}
> 5L + 43s (period=1199.3c, generator=475.4c)
> [[1, 0, -6, 4, -12, -7], [0, 4, 21, -3, 39, 27]]
> <<4, 21, -3, 39, 27, 24, -16, 48, 28, -66, 18, -15, 120, 87, -51]]
>
> No: 17: (Number 76 from the 7-limit list)
> 3L + 47s (period=1200.8c, generator=408.0c)
> [[1, 6, 3, 13, -3, 2], [0, -13, -2, -30, 19, 5]]
> <<13, 2, 30, -19, -5, -27, 11, -75, -56, 64, -51, -19, -157, -125, 53]]
>
> No. 18:
> 26L + 25s (period=1201.8c, generator=46.7c)
> [[1, 1, 1, 3, 4, 3], [0, 15, 34, -5, -14, 18]]
> <<15, 34, -5, -14, 18, 19, -50, -74, -27, -107, -150, -84, -22, 69,
114]]
>
> No. 19:
> 29L + 22s (period=1199.9c, generator=165.0c)
> [[1, 2, 7, 9, 8, 7], [0, -3, -34, -45, -33, -24]]
> <<3, 34, 45, 33, 24, 47, 63, 42, 27, 9, -41, -70, -63, -99, -39]]
>
> No. 20:
> 5L + 48s (period=1199.4c, generator=475.8c)
> [[1, 0, 17, 4, 11, 16], [0, 4, -37, -3, -19, -31]]
> <<4, -37, -3, -19, -31, -68, -16, -44, -64, 97, 84, 65, -43, -76, -37]]
>
> No. 21: Catakleismic (Hanson)
> 19L + 34s (period=1200.9c, generator=317.0c)
> [[1, 0, 1, -3, 9, 0], [0, 6, 5, 22, -21, 14]]
> <<6, 5, 22, -21, 14, -6, 18, -54, 0, 37, -66, 14, -135, -42, 126]]