A symmetric-based 7-limit temperament list

This is not a proposed list for a paper, nor even a starting point for
such a list, so I used a complexity bound (set high) and a badness
bound. The starting point was my big list of 32000-odd wedgies; the
complexity bound was a symmetric complexity squared of 15000 and the
badness bound was a symmetric badness of 2000. The results are sorted
by rms error, and no error bound was set, so you might want to skip
down to about number 15 to get an idea of how things worked. I list
rms error, squared symmetric complexity, and symmetric badness. Since
there seemed no point in taking the square root just to square it
again, the badness is just the rms error times the squared symmetric
complexity, which is an integer. This complexity measure, or else
whatever we would get as the dual to Hahn taxicab distance, seem to be
the logical ones to use when we are using symmetric, octave equivalent
rms error. Since that has a history going back to Woolhouse and 7/26
comma meantone, it seems to me to be of interest.

1: [1, 1, 0, -1, -3, -3] [[1, 2, 3, 3], [0, -1, -1, 0]]
rms: 225.884103 symcom: 4.000000 symbad: 903.536412

2: [1, 2, 1, 1, -1, -3] [[1, 2, 3, 3], [0, -1, -2, -1]]
rms: 157.889659 symcom: 8.000000 symbad: 1263.117274

3: [1, -1, 1, -4, -1, 5] [[1, 2, 2, 3], [0, -1, 1, -1]]
rms: 154.263172 symcom: 11.000000 symbad: 1696.894891

4: [1, -1, 0, -4, -3, 3] [[1, 2, 2, 3], [0, -1, 1, 0]]
rms: 142.097096 symcom: 8.000000 symbad: 1136.776766

5: [1, -1, -2, -4, -6, -2] [[1, 2, 2, 2], [0, -1, 1, 2]]
rms: 65.953083 symcom: 20.000000 symbad: 1319.061657

6: [0, 0, 3, 0, 5, 7] [[3, 5, 7, 9], [0, 0, 0, -1]]
rms: 61.312549 symcom: 27.000000 symbad: 1655.438815

7: [2, -1, 1, -6, -4, 5] [[1, 2, 2, 3], [0, -2, 1, -1]]
rms: 59.930923 symcom: 20.000000 symbad: 1198.618460

8: [0, 2, 2, 3, 3, -1] [[2, 3, 5, 6], [0, 0, -1, -1]]
rms: 59.723378 symcom: 16.000000 symbad: 955.574045

9: [2, 1, -1, -3, -7, -5] [[1, 1, 2, 3], [0, 2, 1, -1]]
rms: 53.747748 symcom: 20.000000 symbad: 1074.954969

10: [2, 1, 3, -3, -1, 4] [[1, 1, 2, 2], [0, 2, 1, 3]]
rms: 48.926006 symcom: 20.000000 symbad: 978.520120

11: [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]]
rms: 43.659491 symcom: 35.000000 symbad: 1528.082200

12: [2, -2, -2, -8, -9, 1] [[2, 3, 5, 6], [0, 1, -1, -1]]
rms: 43.142169 symcom: 44.000000 symbad: 1898.255432

13: [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]]
rms: 41.524693 symcom: 35.000000 symbad: 1453.364254

14: [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
rms: 34.566097 symcom: 20.000000 symbad: 691.321943

15: [4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]]
rms: 23.945252 symcom: 32.000000 symbad: 766.248055

16: [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
rms: 20.163282 symcom: 75.000000 symbad: 1512.246136

17: [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]]
rms: 19.136993 symcom: 48.000000 symbad: 918.575644

18: [6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]]
rms: 18.042924 symcom: 108.000000 symbad: 1948.635783

19: [6, 0, 3, -14, -12, 7] [[3, 4, 7, 8], [0, 2, 0, 1]]
rms: 16.786584 symcom: 99.000000 symbad: 1661.871769

20: [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
rms: 16.598678 symcom: 99.000000 symbad: 1643.269152

21: [0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]]
rms: 15.815352 symcom: 75.000000 symbad: 1186.151431

22: [2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]]
rms: 12.690078 symcom: 155.000000 symbad: 1966.962143

23: [6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]]
rms: 12.273810 symcom: 84.000000 symbad: 1031.000003

24: [4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]]
rms: 12.188571 symcom: 107.000000 symbad: 1304.177049

25: [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
rms: 10.903177 symcom: 108.000000 symbad: 1177.543168

26: [8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]]
rms: 10.132266 symcom: 144.000000 symbad: 1459.046340

27: [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
rms: 8.100679 symcom: 171.000000 symbad: 1385.216092

28: [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]]
rms: 6.808962 symcom: 276.000000 symbad: 1879.273474

29: [1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]]
rms: 6.410458 symcom: 280.000000 symbad: 1794.928214

30: [2, -9, -4, -19, -12, 16] [[1, 1, 5, 4], [0, 2, -9, -4]]
rms: 6.245316 symcom: 283.000000 symbad: 1767.424344

31: [7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]]
rms: 5.052932 symcom: 355.000000 symbad: 1793.790776

32: [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
rms: 4.139051 symcom: 356.000000 symbad: 1473.502082

33: [14, 11, 9, -15, -25, -10] [[1, 5, 5, 5], [0, -14, -11, -9]]
rms: 4.006991 symcom: 436.000000 symbad: 1747.048215

34: [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
rms: 3.665035 symcom: 243.000000 symbad: 890.603432

35: [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]]
rms: 3.579262 symcom: 420.000000 symbad: 1503.290125

36: [4, 16, 9, 16, 3, -24] [[1, 3, 8, 6], [0, -4, -16, -9]]
rms: 3.443812 symcom: 571.000000 symbad: 1966.416662

37: [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]]
rms: 3.320167 symcom: 244.000000 symbad: 810.120816

38: [9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]]
rms: 3.065962 symcom: 339.000000 symbad: 1039.361092

39: [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
rms: 2.859338 symcom: 603.000000 symbad: 1724.180520

40: [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
rms: 2.589237 symcom: 344.000000 symbad: 890.697699

41: [17, 6, 15, -30, -24, 18] [[1, -5, 0, -3], [0, 17, 6, 15]]
rms: 2.469727 symcom: 756.000000 symbad: 1867.113518

42: [8, 18, 11, 10, -5, -25] [[1, 1, 1, 2], [0, 8, 18, 11]]
rms: 2.064340 symcom: 667.000000 symbad: 1376.914655

43: [15, -2, -5, -38, -50, -6] [[1, 4, 2, 2], [0, -15, 2, 5]]
rms: 1.731230 symcom: 952.000000 symbad: 1648.130712

44: [13, -10, 6, -46, -27, 42] [[1, 2, 2, 3], [0, -13, 10, -6]]
rms: 1.678518 symcom: 1139.000000 symbad: 1911.832046

45: [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
rms: 1.637405 symcom: 347.000000 symbad: 568.179603

46: [6, 5, 22, -6, 18, 37] [[1, 0, 1, -3], [0, 6, 5, 22]]
rms: 1.610555 symcom: 1091.000000 symbad: 1757.115994

47: [23, -1, 13, -55, -44, 33] [[1, 9, 2, 7], [0, -23, 1, -13]]
rms: 1.226222 symcom: 1571.000000 symbad: 1926.394008

48: [24, 20, 16, -24, -42, -19] [[4, 6, 9, 11], [0, 6, 5, 4]]
rms: .881659 symcom: 1328.000000 symbad: 1170.842682

49: [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]]
rms: .875363 symcom: 611.000000 symbad: 534.846775

50: [5, 13, -17, 9, -41, -76] [[1, 3, 6, -2], [0, -5, -13, 17]]
rms: .845880 symcom: 1931.000000 symbad: 1633.393513

51: [12, 34, 20, 26, -2, -49] [[2, 4, 7, 7], [0, -6, -17, -10]]
rms: .600319 symcom: 2444.000000 symbad: 1467.178486

52: [2, 25, 13, 35, 15, -40] [[1, 1, -5, -1], [0, 2, 25, 13]]
rms: .585156 symcom: 1592.000000 symbad: 931.569106

53: [34, 29, 23, -33, -59, -28] [[1, -7, -5, -3], [0, 34, 29, 23]]
rms: .404751 symcom: 2708.000000 symbad: 1096.066600

54: [20, 52, 31, 36, -7, -74] [[1, 3, 6, 5], [0, -20, -52, -31]]
rms: .345464 symcom: 5651.000000 symbad: 1952.215876

55: [17, 35, -21, 16, -81, -147] [[1, -1, -3, 6], [0, 17, 35, -21]]
rms: .255750 symcom: 6859.000000 symbad: 1754.190470

56: [22, -5, 3, -59, -57, 21] [[1, 3, 2, 3], [0, -22, 5, -3]]
rms: .253343 symcom: 1672.000000 symbad: 423.589817

57: [52, 56, 41, -32, -81, -62] [[1, -21, -22, -15], [0, 52, 56, 41]]
rms: .244554 symcom: 7883.000000 symbad: 1927.817350

58: [23, -13, 42, -74, 2, 134] [[1, 11, -3, 20], [0, -23, 13, -42]]
rms: .239309 symcom: 7144.000000 symbad: 1709.625905

59: [20, -30, -10, -94, -72, 61] [[10, 16, 23, 28], [0, -2, 3, 1]]
rms: .228948 symcom: 5200.000000 symbad: 1190.529406

60: [38, -3, 8, -93, -94, 27] [[1, -7, 3, 1], [0, 38, -3, 8]]
rms: .228693 symcom: 4219.000000 symbad: 964.856656

61: [1, -8, 39, -15, 59, 113] [[1, 2, -1, 19], [0, -1, 8, -39]]
rms: .223412 symcom: 5320.000000 symbad: 1188.553383

62: [40, 22, 21, -58, -79, -13] [[1, 21, 13, 13], [0, -40, -22, -21]]
rms: .222189 symcom: 3211.000000 symbad: 713.449285

63: [26, -37, -12, -119, -92, 76] [[1, -1, 6, 4], [0, 26, -37, -12]]
rms: .221987 symcom: 8227.000000 symbad: 1826.286511

64: [21, 3, -36, -44, -116, -92] [[3, 5, 7, 8], [0, -7, -1, 12]]
rms: .221824 symcom: 6840.000000 symbad: 1517.273890

65: [2, -57, -28, -95, -50, 95] [[1, 1, 19, 11], [0, 2, -57, -28]]
rms: .201747 symcom: 9259.000000 symbad: 1867.972277

66: [56, 24, 26, -92, -116, -7] [[2, 4, 5, 6], [0, -28, -12, -13]]
rms: .187109 symcom: 6316.000000 symbad: 1181.780562

67: [41, 14, 60, -73, -20, 100] [[1, -14, -3, -20], [0, 41, 14, 60]]
rms: .186938 symcom: 8683.000000 symbad: 1623.186237

68: [4, -32, -15, -60, -35, 55] [[1, 1, 7, 5], [0, 4, -32, -15]]
rms: .183810 symcom: 3211.000000 symbad: 590.213786

69: [58, 49, 39, -57, -101, -47] [[1, -13, -10, -7], [0, 58, 49, 39]]
rms: .182983 symcom: 7828.000000 symbad: 1432.388742

70: [74, 51, 44, -91, -138, -41] [[1, -25, -16, -13], [0, 74, 51, 44]]
rms: .154407 symcom: 11491.000000 symbad: 1774.294552

71: [3, -24, -54, -45, -94, -58] [[3, 5, 5, 4], [0, -1, 8, 18]]
rms: .146908 symcom: 8379.000000 symbad: 1230.939666

72: [14, 59, 33, 61, 13, -89] [[1, -3, -17, -8], [0, 14, 59, 33]]
rms: .143876 symcom: 7828.000000 symbad: 1126.265088

73: [19, 19, 57, -14, 37, 79] [[19, 30, 44, 53], [0, 1, 1, 3]]
rms: .140199 symcom: 6859.000000 symbad: 961.625009

74: [59, 41, 78, -72, -42, 66] [[1, 4, 4, 6], [0, -59, -41, -78]]
rms: .137131 symcom: 13300.000000 symbad: 1823.841297

75: [42, -35, -7, -153, -129, 82] [[7, 9, 18, 20], [0, 6, -5, -1]]
rms: .132906 symcom: 12152.000000 symbad: 1615.071114

76: [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
rms: .130449 symcom: 1539.000000 symbad: 200.760896

77: [60, -8, 11, -152, -151, 48] [[1, 19, 0, 6], [0, -60, 8, -11]]
rms: .129643 symcom: 11171.000000 symbad: 1448.244216

78: [78, 19, 29, -151, -173, 14] [[1, 29, 9, 13], [0, -78, -19, -29]]
rms: .126772 symcom: 13268.000000 symbad: 1682.017076

79: [15, 51, 72, 46, 72, 24] [[3, 3, 1, 0], [0, 5, 17, 24]]
rms: .077212 symcom: 12996.000000 symbad: 1003.453443

80: [37, 46, 75, -13, 15, 45] [[1, 15, 19, 30], [0, -37, -46, -75]]
rms: .070153 symcom: 11476.000000 symbad: 805.075750

>squared symmetric
>complexity, which is an integer. This complexity measure, or else
>whatever we would get as the dual to Hahn taxicab distance,

What is the difference between symmetric complexity and Hahn
taxicab distance?

-Carl

Here is an abridged version of the aame list, where the badness is
less than 900. This gets rid of everything with higher error than beep
without setting an error bound. The three temperaments between
hemiwuerschmidt and ennealimmal have come up before, but because of
the widespread disdain for high complexity I've not named them. They
are all well-convered by 171; ennealimmal is also but unlike with
them, a continued fraction finds 441-et instead when using the rms
generators.

Beep
14: [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
rms: 34.566097 symcom: 20.000000 symbad: 691.321943

Decimal
15: [4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]]
rms: 23.945252 symcom: 32.000000 symbad: 766.248055

Meantone
34: [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
rms: 3.665035 symcom: 243.000000 symbad: 890.603432

Nonkleismic
37: [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]]
rms: 3.320167 symcom: 244.000000 symbad: 810.120816

Orwell
40: [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
rms: 2.589237 symcom: 344.000000 symbad: 890.697699

Miracle
45: [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
rms: 1.637405 symcom: 347.000000 symbad: 568.179603

Hemiwuerschmidt
49: [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]]
rms: .875363 symcom: 611.000000 symbad: 534.846775

{2401/2400, 65625/65536} 11/171
56: [22, -5, 3, -59, -57, 21] [[1, 3, 2, 3], [0, -22, 5, -3]]
rms: .253343 symcom: 1672.000000 symbad: 423.589817

{2401/2400, 48828125/48771072} 83/171
62: [40, 22, 21, -58, -79, -13] [[1, 21, 13, 13], [0, -40, -22, -21]]
rms: .222189 symcom: 3211.000000 symbad: 713.449285

{2401/2400, 32805/32768} 25/171
68: [4, -32, -15, -60, -35, 55] [[1, 1, 7, 5], [0, 4, -32, -15]]
rms: .183810 symcom: 3211.000000 symbad: 590.213786

Ennealimmal
76: [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
rms: .130449 symcom: 1539.000000 symbad: 200.760896

{4375/4374, 52734375/52706752} 62/171
80: [37, 46, 75, -13, 15, 45] [[1, 15, 19, 30], [0, -37, -46, -75]]
rms: .070153 symcom: 11476.000000 symbad: 805.075750

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >squared symmetric
> >complexity, which is an integer. This complexity measure, or else
> >whatever we would get as the dual to Hahn taxicab distance,
>
> What is the difference between symmetric complexity and Hahn
> taxicab distance?

They aren't measuring the same thing. You need to compare symmetric
distance and Hahn distance, or symmetric complexity and Hahn-dual
complexity. Since the former are similar, the latter will be also.

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

> This complexity measure, or else
> whatever we would get as the dual to Hahn taxicab distance,

Wouldn't the complexity measure here be the Hahn taxicab distance
itself? Or at least a Euclidean version of it? Where does duality
come into play?

> seem to be
> the logical ones to use when we are using symmetric, octave
equivalent
> rms error.

Which ones -- taxicab and Euclidean?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >squared symmetric
> > >complexity, which is an integer. This complexity measure, or else
> > >whatever we would get as the dual to Hahn taxicab distance,
> >
> > What is the difference between symmetric complexity and Hahn
> > taxicab distance?
>
> They aren't measuring the same thing. You need to compare symmetric
> distance and Hahn distance, or symmetric complexity and Hahn-dual
> complexity.

You mean the complexity applies to the wedgie? Then how is the dual
to the distance, which applies to monzos? Wouldn't the dual to the
distance be something that operates on vals?

This ordering seems to be good at keeping similar/related temperaments together. It's missing pelogic, injera, and dicot, though. I can understand why pelogic and dicot might be missing, but injera [2, 8, 8, 8, 7, -4] is a good enough temperament that it should have made the list.

Gene Ward Smith wrote:

> 11: [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]]
> rms: 43.659491 symcom: 35.000000 symbad: 1528.082200

Number 13 Father
TOP tuning [1185.869125, 1924.351908, 2819.124589, 3401.317477]

> 12: [2, -2, -2, -8, -9, 1] [[2, 3, 5, 6], [0, 1, -1, -1]]
> rms: 43.142169 symcom: 44.000000 symbad: 1898.255432

Number 62
TOP tuning [1185.468457, 1924.986952, 2816.886876, 3409.621105]
Audibly very similar to Number 13, and has a simpler mapping.

> 13: [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]]
> rms: 41.524693 symcom: 35.000000 symbad: 1453.364254

Number 57
TOP tuning [1185.869125, 1924.351909, 2819.124589, 3333.914203]
Another member of the father temperament family, but the 7:1 approximation is worse than Number 13, and the 7:4 is unrecognizable.

> 14: [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
> rms: 34.566097 symcom: 20.000000 symbad: 691.321943

Number 4 Beep
TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]

> 15: [4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]]
> rms: 23.945252 symcom: 32.000000 symbad: 766.248055

Number 32 Decimal
TOP tuning [1207.657798, 1914.092323, 2768.532858, 3372.361757]

> 16: [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
> rms: 20.163282 symcom: 75.000000 symbad: 1512.246136

Number 7 Dominant Seventh
TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]

> 17: [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]]
> rms: 19.136993 symcom: 48.000000 symbad: 918.575644

Number 17 Diminished
TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404]

> 18: [6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]]
> rms: 18.042924 symcom: 108.000000 symbad: 1948.635783

Number 85
TOP tuning [1194.473353, 1901.955001, 2787.104490, 3384.341166]
Would make a good 12-note keyboard mapping. There aren't many temperaments based on 1/6-octave periods; this is the first one I've seen.

> 19: [6, 0, 3, -14, -12, 7] [[3, 4, 7, 8], [0, 2, 0, 1]]
> rms: 16.786584 symcom: 99.000000 symbad: 1661.871769

Number 75
TOP tuning [1199.400031, 1910.341746, 2798.600074, 3353.970936]
No simpler than Augmented, but sounds a bit more warped.

> 20: [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
> rms: 16.598678 symcom: 99.000000 symbad: 1643.269152

Number 5 Augmented
TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]

> 21: [0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]]
> rms: 15.815352 symcom: 75.000000 symbad: 1186.151431

Number 14 Blackwood
TOP tuning [1195.893464, 1913.429542, 2786.313713, 3348.501698]

> 22: [2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]]
> rms: 12.690078 symcom: 155.000000 symbad: 1966.962143

Number 24 Hemifourths
TOP tuning [1203.668842, 1902.376967, 2794.832500, 3358.526166]

> 23: [6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]]
> rms: 12.273810 symcom: 84.000000 symbad: 1031.000003

Number 27 Kleismic
TOP tuning [1203.187308, 1907.006766, 2792.359613, 3359.878000]

> 24: [4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]]
> rms: 12.188571 symcom: 107.000000 symbad: 1304.177049

Number 28 Negri
TOP tuning [1203.187308, 1907.006766, 2780.900506, 3359.878000]

> 25: [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
> rms: 10.903177 symcom: 108.000000 symbad: 1177.543168

Number 6 Pajara
TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]

> 26: [8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]]
> rms: 10.132266 symcom: 144.000000 symbad: 1459.046340

Number 92
TOP tuning [1198.553882, 1907.135354, 2778.724633, 3378.001574]
Seems to be an alternate 22-ET-type temperament, not as good as Pajara.

> 27: [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
> rms: 8.100679 symcom: 171.000000 symbad: 1385.216092

Number 31 Tripletone
TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420]

> 28: [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]]
> rms: 6.808962 symcom: 276.000000 symbad: 1879.273474

Number 42 Porcupine
TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888]

> 29: [1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]]
> rms: 6.410458 symcom: 280.000000 symbad: 1794.928214

Number 34 Superpythagorean
TOP tuning [1197.596121, 1905.765059, 2780.732078, 3374.046608]

> 30: [2, -9, -4, -19, -12, 16] [[1, 1, 5, 4], [0, 2, -9, -4]]
> rms: 6.245316 symcom: 283.000000 symbad: 1767.424344

Number 79 Beatles
TOP tuning [1197.104145, 1906.544822, 2793.037680, 3369.535226]

> 31: [7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]]
> rms: 5.052932 symcom: 355.000000 symbad: 1793.790776

Number 15 Semisixths
TOP tuning [1198.389531, 1903.732520, 2790.053107, 3364.304748]

> 32: [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
> rms: 4.139051 symcom: 356.000000 symbad: 1473.502082

Number 3 Magic
TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]

> 33: [14, 11, 9, -15, -25, -10] [[1, 5, 5, 5], [0, -14, -11, -9]]
> rms: 4.006991 symcom: 436.000000 symbad: 1747.048215

Not in the 114 list. Seems overly complex to be of much use.

> 34: [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
> rms: 3.665035 symcom: 243.000000 symbad: 890.603432

Almost goes without saying, but....
Number 2 Meantone
TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]

> 35: [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]]
> rms: 3.579262 symcom: 420.000000 symbad: 1503.290125

Number 35 Supermajor seconds
TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099]

> 36: [4, 16, 9, 16, 3, -24] [[1, 3, 8, 6], [0, -4, -16, -9]]
> rms: 3.443812 symcom: 571.000000 symbad: 1966.416662

Number 84 Squares
TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.067656]
Sounds practically identical to Number 35, but with a more complex mapping.

> 37: [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]]
> rms: 3.320167 symcom: 244.000000 symbad: 810.120816

Number 29 Nonkleismic
TOP tuning [1198.828458, 1900.098151, 2789.033948, 3368.077085]

> 38: [9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]]
> rms: 3.065962 symcom: 339.000000 symbad: 1039.361092

Number 30 Quartaminorthirds
TOP tuning [1199.792743, 1900.291122, 2788.751252, 3365.878770]

> 39: [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
> rms: 2.859338 symcom: 603.000000 symbad: 1724.180520

Number 8 Schismic
TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]

> 40: [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
> rms: 2.589237 symcom: 344.000000 symbad: 890.697699

Number 10 Orwell
TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]

> 41: [17, 6, 15, -30, -24, 18] [[1, -5, 0, -3], [0, 17, 6, 15]]
> rms: 2.469727 symcom: 756.000000 symbad: 1867.113518

Now we're starting to get into temperaments that are mostly too complex to be of much interest. This is Number 66 from the big list, and doesn't seem to be enough better than Orwell to justify its complexity. I'll skip most of the rest.

> 45: [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> rms: 1.637405 symcom: 347.000000 symbad: 568.179603

Number 9 Miracle
TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
TOP generators [1200.631014, 116.7206423]
bad: 29.119472 comp: 6.793166 err: .631014

> 49: [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]]
> rms: .875363 symcom: 611.000000 symbad: 534.846775

Number 11 Hemiwuerschmidt
TOP tuning [1199.692003, 1901.466838, 2787.028860, 3368.496143]

> 76: [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
> rms: .130449 symcom: 1539.000000 symbad: 200.760896

Number 1 Ennealimmal
TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]

Injera involves two long chains of fifths, and fifths are just as
long as any other consonance in the symmetric lattice Gene used here.
In a 5-limit version of this list, 2187;2048 would surely score quite
poorly because of the long chain of fifths it involves.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> This ordering seems to be good at keeping similar/related
temperaments
> together. It's missing pelogic, injera, and dicot, though. I can
> understand why pelogic and dicot might be missing, but injera [2,
8, 8,
> 8, 7, -4] is a good enough temperament that it should have made the
list.
>
> Gene Ward Smith wrote:
>
> > 11: [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]]
> > rms: 43.659491 symcom: 35.000000 symbad: 1528.082200
>
> Number 13 Father
> TOP tuning [1185.869125, 1924.351908, 2819.124589, 3401.317477]
>
> > 12: [2, -2, -2, -8, -9, 1] [[2, 3, 5, 6], [0, 1, -1, -1]]
> > rms: 43.142169 symcom: 44.000000 symbad: 1898.255432
>
> Number 62
> TOP tuning [1185.468457, 1924.986952, 2816.886876, 3409.621105]
> Audibly very similar to Number 13, and has a simpler mapping.
>
> > 13: [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]]
> > rms: 41.524693 symcom: 35.000000 symbad: 1453.364254
>
> Number 57
> TOP tuning [1185.869125, 1924.351909, 2819.124589, 3333.914203]
> Another member of the father temperament family, but the 7:1
> approximation is worse than Number 13, and the 7:4 is
unrecognizable.
>
> > 14: [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
> > rms: 34.566097 symcom: 20.000000 symbad: 691.321943
>
> Number 4 Beep
> TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
>
> > 15: [4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]]
> > rms: 23.945252 symcom: 32.000000 symbad: 766.248055
>
> Number 32 Decimal
> TOP tuning [1207.657798, 1914.092323, 2768.532858, 3372.361757]
>
> > 16: [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
> > rms: 20.163282 symcom: 75.000000 symbad: 1512.246136
>
> Number 7 Dominant Seventh
> TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
>
> > 17: [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]]
> > rms: 19.136993 symcom: 48.000000 symbad: 918.575644
>
> Number 17 Diminished
> TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404]
>
> > 18: [6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]]
> > rms: 18.042924 symcom: 108.000000 symbad: 1948.635783
>
> Number 85
> TOP tuning [1194.473353, 1901.955001, 2787.104490, 3384.341166]
> Would make a good 12-note keyboard mapping. There aren't many
> temperaments based on 1/6-octave periods; this is the first one
I've seen.
>
> > 19: [6, 0, 3, -14, -12, 7] [[3, 4, 7, 8], [0, 2, 0, 1]]
> > rms: 16.786584 symcom: 99.000000 symbad: 1661.871769
>
> Number 75
> TOP tuning [1199.400031, 1910.341746, 2798.600074, 3353.970936]
> No simpler than Augmented, but sounds a bit more warped.
>
> > 20: [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
> > rms: 16.598678 symcom: 99.000000 symbad: 1643.269152
>
> Number 5 Augmented
> TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
>
> > 21: [0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]]
> > rms: 15.815352 symcom: 75.000000 symbad: 1186.151431
>
> Number 14 Blackwood
> TOP tuning [1195.893464, 1913.429542, 2786.313713, 3348.501698]
>
> > 22: [2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]]
> > rms: 12.690078 symcom: 155.000000 symbad: 1966.962143
>
> Number 24 Hemifourths
> TOP tuning [1203.668842, 1902.376967, 2794.832500, 3358.526166]
>
> > 23: [6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]]
> > rms: 12.273810 symcom: 84.000000 symbad: 1031.000003
>
> Number 27 Kleismic
> TOP tuning [1203.187308, 1907.006766, 2792.359613, 3359.878000]
>
> > 24: [4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]]
> > rms: 12.188571 symcom: 107.000000 symbad: 1304.177049
>
> Number 28 Negri
> TOP tuning [1203.187308, 1907.006766, 2780.900506, 3359.878000]
>
> > 25: [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
> > rms: 10.903177 symcom: 108.000000 symbad: 1177.543168
>
> Number 6 Pajara
> TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
>
> > 26: [8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]]
> > rms: 10.132266 symcom: 144.000000 symbad: 1459.046340
>
> Number 92
> TOP tuning [1198.553882, 1907.135354, 2778.724633, 3378.001574]
> Seems to be an alternate 22-ET-type temperament, not as good as
Pajara.
>
> > 27: [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
> > rms: 8.100679 symcom: 171.000000 symbad: 1385.216092
>
> Number 31 Tripletone
> TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420]
>
> > 28: [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]]
> > rms: 6.808962 symcom: 276.000000 symbad: 1879.273474
>
> Number 42 Porcupine
> TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888]
>
> > 29: [1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]]
> > rms: 6.410458 symcom: 280.000000 symbad: 1794.928214
>
> Number 34 Superpythagorean
> TOP tuning [1197.596121, 1905.765059, 2780.732078, 3374.046608]
>
> > 30: [2, -9, -4, -19, -12, 16] [[1, 1, 5, 4], [0, 2, -9, -4]]
> > rms: 6.245316 symcom: 283.000000 symbad: 1767.424344
>
> Number 79 Beatles
> TOP tuning [1197.104145, 1906.544822, 2793.037680, 3369.535226]
>
> > 31: [7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]]
> > rms: 5.052932 symcom: 355.000000 symbad: 1793.790776
>
> Number 15 Semisixths
> TOP tuning [1198.389531, 1903.732520, 2790.053107, 3364.304748]
>
> > 32: [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
> > rms: 4.139051 symcom: 356.000000 symbad: 1473.502082
>
> Number 3 Magic
> TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
>
> > 33: [14, 11, 9, -15, -25, -10] [[1, 5, 5, 5], [0, -14, -11, -9]]
> > rms: 4.006991 symcom: 436.000000 symbad: 1747.048215
>
> Not in the 114 list. Seems overly complex to be of much use.
>
> > 34: [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
> > rms: 3.665035 symcom: 243.000000 symbad: 890.603432
>
> Almost goes without saying, but....
> Number 2 Meantone
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
>
> > 35: [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]]
> > rms: 3.579262 symcom: 420.000000 symbad: 1503.290125
>
> Number 35 Supermajor seconds
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099]
>
> > 36: [4, 16, 9, 16, 3, -24] [[1, 3, 8, 6], [0, -4, -16, -9]]
> > rms: 3.443812 symcom: 571.000000 symbad: 1966.416662
>
> Number 84 Squares
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.067656]
> Sounds practically identical to Number 35, but with a more complex
mapping.
>
> > 37: [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]]
> > rms: 3.320167 symcom: 244.000000 symbad: 810.120816
>
> Number 29 Nonkleismic
> TOP tuning [1198.828458, 1900.098151, 2789.033948, 3368.077085]
>
> > 38: [9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]]
> > rms: 3.065962 symcom: 339.000000 symbad: 1039.361092
>
> Number 30 Quartaminorthirds
> TOP tuning [1199.792743, 1900.291122, 2788.751252, 3365.878770]
>
> > 39: [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
> > rms: 2.859338 symcom: 603.000000 symbad: 1724.180520
>
> Number 8 Schismic
> TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
>
> > 40: [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
> > rms: 2.589237 symcom: 344.000000 symbad: 890.697699
>
> Number 10 Orwell
> TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
>
> > 41: [17, 6, 15, -30, -24, 18] [[1, -5, 0, -3], [0, 17, 6, 15]]
> > rms: 2.469727 symcom: 756.000000 symbad: 1867.113518
>
> Now we're starting to get into temperaments that are mostly too
complex
> to be of much interest. This is Number 66 from the big list, and
doesn't
> seem to be enough better than Orwell to justify its complexity.
I'll
> skip most of the rest.
>
> > 45: [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> > rms: 1.637405 symcom: 347.000000 symbad: 568.179603
>
> Number 9 Miracle
> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> TOP generators [1200.631014, 116.7206423]
> bad: 29.119472 comp: 6.793166 err: .631014
>
> > 49: [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]]
> > rms: .875363 symcom: 611.000000 symbad: 534.846775
>
> Number 11 Hemiwuerschmidt
> TOP tuning [1199.692003, 1901.466838, 2787.028860, 3368.496143]
>
> > 76: [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
> > rms: .130449 symcom: 1539.000000 symbad: 200.760896
>
> Number 1 Ennealimmal
> TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
>
> > This complexity measure, or else
> > whatever we would get as the dual to Hahn taxicab distance,
>
> Wouldn't the complexity measure here be the Hahn taxicab distance
> itself?

That's a comma measure; I need a wedgie measure.

Or at least a Euclidean version of it? Where does duality
> come into play?

It's octave-equivalent, so the wedgie measure ends up being for the
first half of the wedgie, which is a mapping, or octave-equivalent
val dual to octave-equivalent monzos. Tossing out 2 makes the 7-limit
in some ways like the 5-limit.

> > seem to be
> > the logical ones to use when we are using symmetric, octave
> equivalent
> > rms error.
>
> Which ones -- taxicab and Euclidean?

Both.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> This ordering seems to be good at keeping similar/related temperaments
> together. It's missing pelogic, injera, and dicot, though. I can
> understand why pelogic and dicot might be missing, but injera [2, 8, 8,
> 8, 7, -4] is a good enough temperament that it should have made the
list.

If I raised the badness cutoff to 2300, it would be on the list; other
things no doubt would be also.

> > 11: [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]]
> > rms: 43.659491 symcom: 35.000000 symbad: 1528.082200
>
> Number 13 Father
> TOP tuning [1185.869125, 1924.351908, 2819.124589, 3401.317477]

TM commas: {16/15, 28/27}

> > 12: [2, -2, -2, -8, -9, 1] [[2, 3, 5, 6], [0, 1, -1, -1]]
> > rms: 43.142169 symcom: 44.000000 symbad: 1898.255432
>
> Number 62
> TOP tuning [1185.468457, 1924.986952, 2816.886876, 3409.621105]
> Audibly very similar to Number 13, and has a simpler mapping.

It has a period of 1/2 octave, which cuts down on how simple the
mapping is. Maybe it should be Bigamist. The numbers game has it as a
little more complex.

TM commas: {16/15, 50/49}

> > 13: [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]]
> > rms: 41.524693 symcom: 35.000000 symbad: 1453.364254
>
> Number 57
> TOP tuning [1185.869125, 1924.351909, 2819.124589, 3333.914203]
> Another member of the father temperament family, but the 7:1
> approximation is worse than Number 13, and the 7:4 is unrecognizable.

Logically we ought to be looking at the rms tuning now, not the TOP
tuning. That gives a decent 7/4 of 981 cents, and of course no longer
has super-flat octaves.

TM commas: {16/15, 49/45} (= {16/15, 49/48})

> > 18: [6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]]
> > rms: 18.042924 symcom: 108.000000 symbad: 1948.635783
>
> Number 85
> TOP tuning [1194.473353, 1901.955001, 2787.104490, 3384.341166]
> Would make a good 12-note keyboard mapping. There aren't many
> temperaments based on 1/6-octave periods; this is the first one I've
seen.

It's come up before, and I think I even mentioned it as a 12-et
tuning; it's two 6-equals 83 cents apart in the rms tuning.

TM commas: {50/49, 128/125}

> Number 6 Pajara
> TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
>
> > 26: [8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]]
> > rms: 10.132266 symcom: 144.000000 symbad: 1459.046340

> Number 92
> TOP tuning [1198.553882, 1907.135354, 2778.724633, 3378.001574]
> Seems to be an alternate 22-ET-type temperament, not as good as Pajara.

It's the 22 and 26 system; pajara is 22 and 12.

TM commas: {50/49, 875/864}

> > 33: [14, 11, 9, -15, -25, -10] [[1, 5, 5, 5], [0, -14, -11, -9]]
> > rms: 4.006991 symcom: 436.000000 symbad: 1747.048215
>
> Not in the 114 list. Seems overly complex to be of much use.

It's not that bad; two generators give a 7/5 and three a 5/3. It might
make more sense as a 13-limit temperament, with a generator of ~13/11
and TM basis {100/99, 196/195, 275/273, 385/384}.

TM commas: {875/864, 2401/2400}

> > 34: [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
> > rms: 3.665035 symcom: 243.000000 symbad: 890.603432
>
> Almost goes without saying, but....
> Number 2 Meantone
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
>
> > 35: [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]]
> > rms: 3.579262 symcom: 420.000000 symbad: 1503.290125
>
> Number 35 Supermajor seconds
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099]
>
> > 36: [4, 16, 9, 16, 3, -24] [[1, 3, 8, 6], [0, -4, -16, -9]]
> > rms: 3.443812 symcom: 571.000000 symbad: 1966.416662
>
> Number 84 Squares
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.067656]
> Sounds practically identical to Number 35, but with a more complex
mapping.

They both are 81/80 temperaments, and have the same 5-limit TOP tuning
as meantone.

> > 41: [17, 6, 15, -30, -24, 18] [[1, -5, 0, -3], [0, 17, 6, 15]]
> > rms: 2.469727 symcom: 756.000000 symbad: 1867.113518
>
> Now we're starting to get into temperaments that are mostly too complex
> to be of much interest.

I'm suspicious of such claims at this level of complexity. I just
finished an ennealimmal piece, and ennealimmal doesn't seem to be too
complex, though it does seem to have reached the area of diminishing
returns so far as tuning goes--as Dave would no doubt point out if he
were officially here, it isn't as much better, earwise, than miracle
or hemiwuerschmidt as the numbers say it should be, because our ears
just aren't that good.

This one has some useful commas and two generators get us to 12/7;
again we could push the prime limit on this one, as the generator is
quite close to 17/13, and the fact that 80 and 111 cover it strongly
suggest putting the limit higher anyway.

Here is the mapping for the 19-limit version of 80&111:

[<1 12 6 12 20 -11 -10 -8|, <0 17 6 15 27 -24 -23 -20|]

TM commas: {1728/1715, 3136/3125}

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

>>Number 62
>>TOP tuning [1185.468457, 1924.986952, 2816.886876, 3409.621105]
>>Audibly very similar to Number 13, and has a simpler mapping.
> > > It has a period of 1/2 octave, which cuts down on how simple the
> mapping is. Maybe it should be Bigamist. The numbers game has it as a
> little more complex. > > TM commas: {16/15, 50/49}

Oops, I should have noticed that! Well, it still has a 16/15 comma, so that explains why it sounds like the other father temperaments.

I have mixed feelings about partial octave temperaments in general. On the one hand, they have higher complexity because they need more notes per octave. But they're also more symmetrical and allow for more transposition. An extreme example would be something like Ennealimmal, where even with the minimum complexity, each chord has 9 possible transpositions. So even though this is slighly more complex than Father [1, -1, 3, -4, 2, 10], it could still be of interest (if any of the father temperaments are of interest).