Gene's private reserve -- 7-limit

I decided to make some top 100 lists based on what I find useful in my
own compositional practice; while people may complain no one should be
interested in microtemperaments, the fact is I am. Moreover, I like
complete o/u-tonalities, which means Graham complexity is the
complexity measure of choice.

Below I give the top 100 with a log-flat badness figure derived from
7-limit Graham complexity and TOP error, with cutoffs of 20 cents for
TOP error and 100 for complexity. The next posting will be the same,
for 9-limit; for me the other interesting limits are 11 and 13, so I
plan to get to them eventually.

1 [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
bad 26.518833 top err .036377 graham 27

2 [22, -5, 3, -59, -57, 21] [[1, 3, 2, 3], [0, -22, 5, -3]]
bad 53.601055 top err .073527 graham 27

3 [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]]
bad 78.847244 top err .307998 graham 16

4 [40, 22, 21, -58, -79, -13] [[1, 21, 13, 13], [0, -40, -22, -21]]
bad 85.360253 top err .053350 graham 40

5 [4, -32, -15, -60, -35, 55] [[1, 1, 7, 5], [0, 4, -32, -15]]
bad 85.691520 top err .066120 graham 36

6 [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]]
bad 93.944647 top err 5.871540 graham 4

7 [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
bad 106.641366 top err .631014 graham 13

8 [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
bad 111.836820 top err 3.106578 graham 6

9 [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
bad 114.473365 top err .946061 graham 11

10 [6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]]
bad 114.743119 top err 3.187309 graham 6

11 [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]]
bad 117.154200 top err 1.171542 graham 10

12 [37, 46, 75, -13, 15, 45] [[1, 15, 19, 30], [0, -37, -46, -75]]
bad 121.726682 top err .021640 graham 75

13 [4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]]
bad 122.524768 top err 7.657798 graham 4

14 [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
bad 127.584947 top err 14.176105 graham 3

15 [38, -3, 8, -93, -94, 27] [[1, -7, 3, 1], [0, 38, -3, 8]]
bad 128.468744 top err .076424 graham 41

16 [14, 59, 33, 61, 13, -89] [[1, -3, -17, -8], [0, 14, 59, 33]]
bad 130.053641 top err .037361 graham 59

17 [33, 78, 90, 47, 50, -10] [[3, 9, 17, 20], [0, -11, -26, -30]]
bad 135.548384 top err .016734 graham 90

18 [19, 19, 57, -14, 37, 79] [[19, 30, 44, 53], [0, 1, 1, 3]]
bad 149.622591 top err .046052 graham 57

19 [9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]]
bad 151.169908 top err 1.049791 graham 12

20 [2, 1, 3, -3, -1, 4] [[1, 1, 2, 2], [0, 2, 1, 3]]
bad 151.541080 top err 16.837898 graham 3

21 [2, 1, -1, -3, -7, -5] [[1, 1, 2, 3], [0, 2, 1, -1]]
bad 151.866975 top err 16.874108 graham 3

22 [55, 73, 93, -12, -7, 11] [[1, -19, -25, -32], [0, 55, 73, 93]]
bad 153.712613 top err .017772 graham 93

23 [4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]]
bad 156.178134 top err 3.187309 graham 7

24 [34, 29, 23, -33, -59, -28] [[1, -7, -5, -3], [0, 34, 29, 23]]
bad 161.265468 top err .139503 graham 34

25 [58, 49, 39, -57, -101, -47] [[1, -13, -10, -7], [0, 58, 49, 39]]
bad 161.751212 top err .048083 graham 58

26 [96, 99, 6, -66, -260, -264] [[3, -2, 0, 8], [0, 32, 33, 2]]
bad 163.318824 top err .016663 graham 99

27 [1, -8, 39, -15, 59, 113] [[1, 2, -1, 19], [0, -1, 8, -39]]
bad 164.610262 top err .074518 graham 47

28 [24, 20, 16, -24, -42, -19] [[4, 6, 9, 11], [0, 6, 5, 4]]
bad 167.638136 top err .291038 graham 24

29 [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
bad 169.852100 top err 1.698521 graham 10

30 [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
bad 171.757764 top err 4.771049 graham 6

31 [17, 6, 15, -30, -24, 18] [[1, -5, 0, -3], [0, 17, 6, 15]]
bad 179.406875 top err .620785 graham 17

32 [0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]]
bad 180.990733 top err 7.239629 graham 5

33 [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
bad 183.851136 top err 1.276744 graham 12

34 [78, 72, -12, -67, -238, -230] [[6, 15, 19, 16], [0, -13, -12, 2]]
bad 185.389623 top err .022888 graham 90

35 [60, -8, 11, -152, -151, 48] [[1, 19, 0, 6], [0, -60, 8, -11]]
bad 185.994225 top err .040224 graham 68

36 [2, 25, 13, 35, 15, -40] [[1, 1, -5, -1], [0, 2, 25, 13]]
bad 187.279432 top err .299647 graham 25

37 [23, -1, 13, -55, -44, 33] [[1, 9, 2, 7], [0, -23, 1, -13]]
bad 189.152064 top err .328389 graham 24

38 [15, 51, 72, 46, 72, 24] [[3, 3, 1, 0], [0, 5, 17, 24]]
bad 190.153222 top err .036681 graham 72

39 [6, 0, 3, -14, -12, 7] [[3, 4, 7, 8], [0, 2, 0, 1]]
bad 190.492116 top err 5.291448 graham 6

40 [15, -2, -5, -38, -50, -6] [[1, 4, 2, 2], [0, -15, 2, 5]]
bad 191.882395 top err .479706 graham 20

41 [56, 24, 26, -92, -116, -7] [[2, 4, 5, 6], [0, -28, -12, -13]]
bad 193.718370 top err .061772 graham 56

42 [4, 4, 0, -3, -11, -11] [[4, 6, 9, 11], [0, 1, 1, 0]]
bad 198.154432 top err 12.384652 graham 4

43 [6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]]
bad 198.959296 top err 5.526647 graham 6

44 [20, -30, -10, -94, -72, 61] [[10, 16, 23, 28], [0, -2, 3, 1]]
bad 200.488970 top err .080196 graham 50

45 [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
bad 205.403314 top err .912903 graham 15

46 [8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]]
bad 209.180089 top err 3.268439 graham 8

47 [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
bad 211.351646 top err 5.870879 graham 6

48 [3, -24, -54, -45, -94, -58] [[3, 5, 5, 4], [0, -1, 8, 18]]
bad 211.576339 top err .065120 graham 57

49 [41, 14, 60, -73, -20, 100] [[1, -14, -3, -20], [0, 41, 14, 60]]
bad 217.304404 top err .060362 graham 60

50 [39, 42, 6, -24, -100, -104] [[3, 2, 4, 8], [0, 13, 14, 2]]
bad 221.669129 top err .125663 graham 42

51 [23, -13, 42, -74, 2, 134] [[1, 11, -3, 20], [0, -23, 13, -42]]
bad 224.748425 top err .074297 graham 55

52 [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]]
bad 226.094010 top err 14.130876 graham 4

53 [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]]
bad 226.094010 top err 14.130876 graham 4

54 [1, 4, 3, 4, 2, -4] [[1, 2, 4, 4], [0, -1, -4, -3]]
bad 228.058272 top err 14.253642 graham 4

55 [17, 35, -21, 16, -81, -147] [[1, -1, -3, 6], [0, 17, 35, -21]]
bad 229.263286 top err .073107 graham 56

56 [2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]]
bad 229.293222 top err 3.582707 graham 8

57 [9, 10, -3, -5, -30, -35] [[1, 4, 5, 2], [0, -9, -10, 3]]
bad 231.854142 top err 1.371918 graham 13

58 [2, -2, -2, -8, -9, 1] [[2, 3, 5, 6], [0, 1, -1, -1]]
bad 232.504691 top err 14.531543 graham 4

59 [57, 57, 0, -42, -160, -160] [[57, 90, 132, 160], [0, 1, 1, 0]]
bad 234.286461 top err .072110 graham 57

60 [2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]]
bad 234.805888 top err 3.668842 graham 8

61 [23, 40, 1, 10, -63, -110] [[1, 6, 10, 3], [0, -23, -40, -1]]
bad 237.044800 top err .148153 graham 40

62 [78, 19, 29, -151, -173, 14] [[1, 29, 9, 13], [0, -78, -19, -29]]
bad 237.317411 top err .039007 graham 78

63 [39, 30, -18, -43, -138, -126] [[3, 10, 11, 6], [0, -13, -10, 6]]
bad 237.524367 top err .073107 graham 57

64 [9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]]
bad 238.136875 top err 2.939961 graham 9

65 [9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]]
bad 238.136875 top err 2.939961 graham 9

66 [6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]]
bad 239.419651 top err 3.740932 graham 8

67 [21, 3, -36, -44, -116, -92] [[3, 5, 7, 8], [0, -7, -1, 12]]
bad 243.155160 top err .074840 graham 57

68 [62, 17, 24, -117, -136, 8] [[1, 15, 6, 8], [0, -62, -17, -24]]
bad 243.206036 top err .063269 graham 62

69 [1, -3, -4, -7, -9, -1] [[1, 2, 1, 1], [0, -1, 3, 4]]
bad 243.351404 top err 9.734056 graham 5

70 [42, -35, -7, -153, -129, 82] [[7, 9, 18, 20], [0, 6, -5, -1]]
bad 245.608557 top err .041425 graham 77

71 [40, 75, -20, 26, -144, -257] [[5, 6, 8, 15], [0, 8, 15, -4]]
bad 246.431399 top err .027305 graham 95

72 [21, 15, -12, -25, -78, -70] [[3, 2, 5, 10], [0, 7, 5, -4]]
bad 247.770369 top err .227521 graham 33

73 [5, 13, -17, 9, -41, -76] [[1, 3, 6, -2], [0, -5, -13, 17]]
bad 248.495400 top err .276106 graham 30

74 [18, 15, -6, -18, -60, -56] [[3, 6, 8, 8], [0, -6, -5, 2]]
bad 258.439259 top err .448679 graham 24

75 [6, 5, 22, -6, 18, 37] [[1, 0, 1, -3], [0, 6, 5, 22]]
bad 259.596304 top err .536356 graham 22

76 [12, 10, -9, -12, -48, -49] [[1, 0, 1, 4], [0, 12, 10, -9]]
bad 259.647200 top err .588769 graham 21

77 [32, 86, 51, 62, -9, -123] [[1, 13, 33, 21], [0, -32, -86, -51]]
bad 260.834732 top err .035267 graham 86

78 [76, 76, 57, -56, -123, -81] [[19, 31, 45, 54], [0, -4, -4, -3]]
bad 261.710560 top err .045310 graham 76

79 [14, 11, 9, -15, -25, -10] [[1, 5, 5, 5], [0, -14, -11, -9]]
bad 262.233496 top err 1.337926 graham 14

80 [13, 9, 1, -16, -35, -23] [[1, 4, 4, 3], [0, -13, -9, -1]]
bad 267.037238 top err 1.580102 graham 13

81 [7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]]
bad 272.169318 top err 1.610469 graham 13

82 [74, 51, 44, -91, -138, -41] [[1, -25, -16, -13], [0, 74, 51, 44]]
bad 273.596225 top err .049963 graham 74

83 [63, 9, 63, -132, -77, 121] [[9, 15, 21, 26], [0, -7, -1, -7]]
bad 273.634592 top err .068943 graham 63

84 [59, 41, 78, -72, -42, 66] [[1, 4, 4, 6], [0, -59, -41, -78]]
bad 274.339233 top err .045092 graham 78

85 [11, -6, 10, -35, -15, 40] [[1, 4, 1, 5], [0, -11, 6, -10]]
bad 274.666756 top err .950404 graham 17

86 [52, 56, 41, -32, -81, -62] [[1, -21, -22, -15], [0, 52, 56, 41]]
bad 274.867264 top err .087649 graham 56

87 [60, 45, -30, -68, -216, -196] [[15, 24, 35, 42], [0, -4, -3, 2]]
bad 277.368300 top err .034243 graham 90

88 [26, -37, -12, -119, -92, 76] [[1, -1, 6, 4], [0, 26, -37, -12]]
bad 278.973072 top err .070288 graham 63

89 [3, 2, -1, -4, -10, -8] [[1, 1, 2, 3], [0, 3, 2, -1]]
bad 281.038694 top err 17.564918 graham 4

90 [12, 22, -4, 7, -40, -71] [[2, 5, 8, 5], [0, -6, -11, 2]]
bad 284.925888 top err .421488 graham 26

91 [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]]
bad 287.050049 top err 1.698521 graham 13

92 [3, 17, -1, 20, -10, -50] [[1, 1, -1, 3], [0, 3, 17, -1]]
bad 289.868296 top err .894655 graham 18

93 [1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]]
bad 290.869402 top err 2.403879 graham 11

94 [2, -57, -28, -95, -50, 95] [[1, 1, 19, 11], [0, 2, -57, -28]]
bad 292.014128 top err .083888 graham 59

95 [45, -18, 45, -133, -55, 155] [[9, 14, 21, 25], [0, 5, -2, 5]]
bad 294.245784 top err .074136 graham 63

96 [5, -11, -12, -29, -33, 3] [[1, 4, -3, -3], [0, -5, 11, 12]]
bad 295.707979 top err 1.023211 graham 17

97 [11, 18, 5, 3, -23, -39] [[1, 2, 3, 3], [0, -11, -18, -5]]
bad 296.963913 top err .916555 graham 18

98 [1, -3, -2, -7, -6, 4] [[1, 2, 1, 2], [0, -1, 3, 2]]
bad 298.143022 top err 18.633939 graham 4

99 [5, -40, 24, -75, 24, 168] [[1, 1, 7, 0], [0, 5, -40, 24]]
bad 299.445924 top err .073107 graham 64

100 [13, -10, 6, -46, -27, 42] [[1, 2, 2, 3], [0, -13, 10, -6]]
bad 300.099731 top err .567296 graham 23

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

Rats--two temperaments with identical TOP errors *and* Graham
complexities. I was aware this could happen, but guessed, apparently
incorrectly, that it was quite unlikely and then forgot to check. The
missing one gets 64th place since it is better in the 9-limit.

64 [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
bad 238.136875 top err 2.939961 graham 9

> 64 [9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]]
> bad 238.136875 top err 2.939961 graham 9
>
> 65 [9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]]
> bad 238.136875 top err 2.939961 graham 9

Gene Ward Smith wrote:

> Rats--two temperaments with identical TOP errors *and* Graham
> complexities. I was aware this could happen, but guessed, apparently
> incorrectly, that it was quite unlikely and then forgot to check. The
> missing one gets 64th place since it is better in the 9-limit.

I've verified that, contorsion aside, each temperament I look at does have a unique badness. I need this to test my temperament finder (now in three languages!). I can't prove it but it's always worked so far.

> 64 [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
> bad 238.136875 top err 2.939961 graham 9
> > >>64 [9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]]
>>bad 238.136875 top err 2.939961 graham 9

Oh yes, that'll be because it's a kind of minimax. The 7 mapping presumable doesn't contribute to the TOP error, and the complexities happen to be the same. They should be distinct if you switch to some kind of RMS.

Graham