If a 7-limit octave equivalent scale has four distinct step sizes, with ratios W, X, Y and Z, the octave can be represented as a linear combination of these four step sizes:

2/1 = W^a * X^b * Y^c * Z^d

Each of these ratios W, X, Y and Z has a monzo* which is a four-dimensional vector; the above equation in monzo notation looks like:

[1 0 0 0] = a*monzo(W) + b*monzo(X) + c*monzo(Y) + d*monzo(Z)

Here the monzo of a 7 limit interval with ratio 2^p 3^q 5^r 7^s is defined as the vector [p q r s]. So in vector notation,

[Monzo (W) ]
[Monzo (X) ]
[1 0 0 0 ] =[a b c d ]*[Monzo (Y) ]
[Monzo (Z) ]

If the original step sizes are linearly independent, i.e. they span the whole 7 limit lattice, the matrix of monzos is invertible, and the entries of the top row of its inverse are the values of a, b, c and d. Adding them gives the number of notes in the scale.

By choosing any 7-limit interval other than 2/1 and using its representation as a product of primes instead of [1 0 0 0], one can likewise solve for the number of steps in the interval. This means the scale is a constant structure because every occurrence of a given interval has the same number of steps.

If one or more of the values are negative, then a scale can’t be built from those step sizes; and zero values correspond to unneeded steps.

THE 182 CONSTANT STRUCTURES
with 4 distinct, linearly independent, superparticular 7 limit steps with sizes between 5/4 and 126/125

1: 4 notes, 1 step of size 5/4, 1 step of size 6/5, 1 step of size 7/6, and 1 step of size 8/7. There are 6 such scales.

2: 5 notes, 1 step of size 5/4, 1 step of size 6/5, 2 steps of size 8/7, and 1 step of size 49/48. There are 12 such scales.

3: 5 notes, 1 step of size 5/4, 2 steps of size 6/5, 1 step of size 15/14, and 1 step of size 28/27. There are 12 such scales.

4: 5 notes, 2 steps of size 5/4, 1 step of size 6/5, 1 step of size 21/20, and 1 step of size 64/63. There are 12 such scales.

5: 5 notes, 2 steps of size 5/4, 1 step of size 6/5, 1 step of size 28/27, and 1 step of size 36/35. There are 12 such scales.

6: 5 notes, 1 step of size 5/4, 1 step of size 7/6, 2 steps of size 8/7, and 1 step of size 21/20. There are 12 such scales.

7: 5 notes, 1 step of size 5/4, 2 steps of size 7/6, 1 step of size 8/7, and 1 step of size 36/35. There are 12 such scales.

8: 5 notes, 2 steps of size 5/4, 1 step of size 7/6, 1 step of size 16/15, and 1 step of size 36/35. There are 12 such scales.

9: 5 notes, 2 steps of size 5/4, 1 step of size 8/7, 1 step of size 16/15, and 1 step of size 21/20. There are 12 such scales.

10: 5 notes, 1 step of size 6/5, 2 steps of size 7/6, 1 step of size 8/7, and 1 step of size 15/14. There are 12 such scales.

11: 5 notes, 2 steps of size 6/5, 1 step of size 7/6, 1 step of size 8/7, and 1 step of size 25/24. There are 12 such scales.

12: 5 notes, 2 steps of size 6/5, 1 step of size 7/6, 1 step of size 10/9, and 1 step of size 15/14. There are 12 such scales.

13: 6 notes, 1 step of size 5/4, 2 steps of size 7/6, 1 step of size 10/9, and 2 steps of size 36/35. There are 30 such scales.

14: 6 notes, 2 steps of size 5/4, 1 step of size 7/6, 1 step of size 28/27, and 2 steps of size 36/35. There are 30 such scales.

15: 6 notes, 1 step of size 5/4, 2 steps of size 8/7, 1 step of size 10/9, and 2 steps of size 21/20. There are 30 such scales.

16: 6 notes, 1 step of size 5/4, 3 steps of size 8/7, 1 step of size 21/20, and 1 step of size 49/48. There are 20 such scales.

17: 6 notes, 2 steps of size 5/4, 1 step of size 8/7, 2 steps of size 21/20, and 1 step of size 64/63. There are 30 such scales.

18: 6 notes, 2 steps of size 5/4, 1 step of size 15/14, 2 steps of size 16/15, and 1 step of size 21/20. There are 30 such scales.

19: 6 notes, 1 step of size 6/5, 2 steps of size 7/6, 2 steps of size 15/14, and 1 step of size 16/15. There are 30 such scales.

20: 6 notes, 2 steps of size 6/5, 1 step of size 7/6, 2 steps of size 15/14, and 1 step of size 28/27. There are 30 such scales.

21: 6 notes, 2 steps of size 6/5, 2 steps of size 8/7, 1 step of size 25/24, and 1 step of size 49/48. There are 30 such scales.

22: 6 notes, 2 steps of size 6/5, 2 steps of size 10/9, 1 step of size 15/14, and 1 step of size 21/20. There are 30 such scales.

23: 6 notes, 3 steps of size 6/5, 1 step of size 15/14, 1 step of size 25/24, and 1 step of size 28/27. There are 20 such scales.

24: 6 notes, 2 steps of size 7/6, 2 steps of size 8/7, 1 step of size 10/9, and 1 step of size 81/80. There are 30 such scales.

25: 6 notes, 2 steps of size 7/6, 2 steps of size 8/7, 1 step of size 15/14, and 1 step of size 21/20. There are 30 such scales.

26: 6 notes, 3 steps of size 7/6, 1 step of size 8/7, 1 step of size 15/14, and 1 step of size 36/35. There are 20 such scales.

27: 7 notes, 1 step of size 5/4, 3 steps of size 8/7, 1 step of size 36/35, and 2 steps of size 49/48. There are 60 such scales.

28: 7 notes, 2 steps of size 5/4, 1 step of size 9/8, 2 steps of size 21/20, and 2 steps of size 64/63. There are 90 such scales.

29: 7 notes, 2 steps of size 5/4, 1 step of size 9/8, 2 steps of size 28/27, and 2 steps of size 36/35. There are 90 such scales.

30: 7 notes, 2 steps of size 5/4, 2 steps of size 16/15, 2 steps of size 21/20, and 1 step of size 50/49. There are 90 such scales.

31: 7 notes, 1 step of size 6/5, 3 steps of size 8/7, 1 step of size 15/14, and 2 steps of size 49/48. There are 60 such scales.

32: 7 notes, 2 steps of size 6/5, 2 steps of size 8/7, 2 steps of size 49/48, and 1 step of size 50/49. There are 90 such scales.

33: 7 notes, 2 steps of size 6/5, 1 step of size 9/8, 2 steps of size 15/14, and 2 steps of size 28/27. There are 90 such scales.

34: 7 notes, 2 steps of size 6/5, 2 steps of size 10/9, 2 steps of size 21/20, and 1 step of size 50/49. There are 90 such scales.

35: 7 notes, 3 steps of size 6/5, 1 step of size 21/20, 2 steps of size 25/24, and 1 step of size 64/63. There are 60 such scales.

36: 7 notes, 3 steps of size 6/5, 2 steps of size 25/24, 1 step of size 28/27, and 1 step of size 36/35. There are 60 such scales.

37: 7 notes, 1 step of size 7/6, 3 steps of size 8/7, 2 steps of size 21/20, and 1 step of size 25/24. There are 60 such scales.

38: 7 notes, 2 steps of size 7/6, 2 steps of size 8/7, 2 steps of size 21/20, and 1 step of size 50/49. There are 90 such scales.

39: 7 notes, 3 steps of size 7/6, 1 step of size 8/7, 1 step of size 25/24, and 2 steps of size 36/35. There are 60 such scales.

40: 7 notes, 2 steps of size 7/6, 1 step of size 9/8, 2 steps of size 10/9, and 2 steps of size 36/35. There are 90 such scales.

41: 7 notes, 2 steps of size 7/6, 1 step of size 9/8, 2 steps of size 15/14, and 2 steps of size 16/15. There are 90 such scales.

42: 7 notes, 3 steps of size 7/6, 1 step of size 10/9, 1 step of size 15/14, and 2 steps of size 36/35. There are 60 such scales.

43: 7 notes, 3 steps of size 7/6, 2 steps of size 15/14, 1 step of size 16/15, and 1 step of size 36/35. There are 60 such scales.

44: 7 notes, 2 steps of size 8/7, 1 step of size 9/8, 2 steps of size 10/9, and 2 steps of size 21/20. There are 90 such scales.

45: 8 notes, 1 step of size 5/4, 3 steps of size 10/9, 2 steps of size 15/14, and 2 steps of size 126/125. There are 210 such scales.

46: 8 notes, 1 step of size 5/4, 3 steps of size 10/9, 2 steps of size 21/20, and 2 steps of size 36/35. There are 210 such scales.

47: 8 notes, 2 steps of size 5/4, 1 step of size 15/14, 3 steps of size 21/20, and 2 steps of size 64/63. There are 210 such scales.

48: 8 notes, 2 steps of size 6/5, 3 steps of size 15/14, 1 step of size 21/20, and 2 steps of size 28/27. There are 210 such scales.

49: 8 notes, 3 steps of size 6/5, 3 steps of size 25/24, 1 step of size 64/63, and 1 step of size 126/125. There are 140 such scales.

50: 8 notes, 3 steps of size 7/6, 1 step of size 10/9, 1 step of size 25/24, and 3 steps of size 36/35. There are 140 such scales.

51: 8 notes, 2 steps of size 7/6, 3 steps of size 10/9, 2 steps of size 36/35, and 1 step of size 81/80. There are 210 such scales.

52: 8 notes, 2 steps of size 7/6, 3 steps of size 15/14, 2 steps of size 16/15, and 1 step of size 21/20. There are 210 such scales.

53: 8 notes, 3 steps of size 7/6, 2 steps of size 15/14, 1 step of size 28/27, and 2 steps of size 36/35. There are 210 such scales.

54: 8 notes, 2 steps of size 8/7, 2 steps of size 10/9, 1 step of size 15/14, and 3 steps of size 21/20. There are 210 such scales.

55: 8 notes, 3 steps of size 8/7, 1 step of size 10/9, 3 steps of size 21/20, and 1 step of size 25/24. There are 140 such scales.

56: 8 notes, 2 steps of size 8/7, 3 steps of size 10/9, 2 steps of size 21/20, and 1 step of size 81/80. There are 210 such scales.

57: 8 notes, 4 steps of size 8/7, 1 step of size 10/9, 2 steps of size 49/48, and 1 step of size 81/80. There are 105 such scales.

58: 8 notes, 4 steps of size 8/7, 1 step of size 15/14, 1 step of size 21/20, and 2 steps of size 49/48. There are 105 such scales.

59: 8 notes, 4 steps of size 8/7, 2 steps of size 21/20, 1 step of size 25/24, and 1 step of size 49/48. There are 105 such scales.

60: 9 notes, 2 steps of size 5/4, 4 steps of size 21/20, 1 step of size 50/49, and 2 steps of size 64/63. There are 420 such scales.

61: 9 notes, 2 steps of size 6/5, 3 steps of size 15/14, 3 steps of size 28/27, and 1 step of size 81/80. There are 560 such scales.

62: 9 notes, 1 step of size 7/6, 3 steps of size 8/7, 3 steps of size 25/24, and 2 steps of size 126/125. There are 560 such scales.

63: 9 notes, 1 step of size 7/6, 3 steps of size 10/9, 3 steps of size 15/14, and 2 steps of size 126/125. There are 560 such scales.

64: 9 notes, 3 steps of size 7/6, 1 step of size 16/15, 2 steps of size 25/24, and 3 steps of size 36/35. There are 560 such scales.

65: 9 notes, 2 steps of size 8/7, 2 steps of size 10/9, 4 steps of size 21/20, and 1 step of size 50/49. There are 420 such scales.

66: 9 notes, 4 steps of size 8/7, 1 step of size 15/14, 1 step of size 36/35, and 3 steps of size 49/48. There are 280 such scales.

67: 9 notes, 3 steps of size 8/7, 1 step of size 16/15, 3 steps of size 21/20, and 2 steps of size 25/24. There are 560 such scales.

68: 9 notes, 4 steps of size 8/7, 2 steps of size 21/20, 2 steps of size 49/48, and 1 step of size 50/49. There are 420 such scales.

69: 9 notes, 1 step of size 9/8, 4 steps of size 10/9, 2 steps of size 15/14, and 2 steps of size 126/125. There are 420 such scales.

70: 9 notes, 1 step of size 9/8, 4 steps of size 10/9, 2 steps of size 21/20, and 2 steps of size 36/35. There are 420 such scales.

71: 9 notes, 3 steps of size 9/8, 2 steps of size 10/9, 2 steps of size 21/20, and 2 steps of size 64/63. There are 840 such scales.

72: 9 notes, 3 steps of size 9/8, 2 steps of size 10/9, 2 steps of size 28/27, and 2 steps of size 36/35. There are 840 such scales.

73: 9 notes, 3 steps of size 9/8, 2 steps of size 15/14, 2 steps of size 16/15, and 2 steps of size 28/27. There are 840 such scales.

74: 10 notes, 1 step of size 5/4, 3 steps of size 10/9, 4 steps of size 36/35, and 2 steps of size 49/48. There are 1260 such scales.

75: 10 notes, 1 step of size 6/5, 4 steps of size 15/14, 3 steps of size 16/15, and 2 steps of size 49/48. There are 1260 such scales.

76: 10 notes, 2 steps of size 7/6, 3 steps of size 10/9, 2 steps of size 64/63, and 3 steps of size 81/80. There are 2520 such scales.

77: 10 notes, 2 steps of size 7/6, 3 steps of size 15/14, 3 steps of size 21/20, and 2 steps of size 64/63. There are 2520 such scales.

78: 10 notes, 3 steps of size 7/6, 2 steps of size 25/24, 1 step of size 28/27, and 4 steps of size 36/35. There are 1260 such scales.

79: 10 notes, 2 steps of size 8/7, 3 steps of size 10/9, 2 steps of size 28/27, and 3 steps of size 81/80. There are 2520 such scales.

80: 10 notes, 2 steps of size 8/7, 3 steps of size 15/14, 3 steps of size 21/20, and 2 steps of size 28/27. There are 2520 such scales.

81: 10 notes, 3 steps of size 8/7, 4 steps of size 21/20, 2 steps of size 25/24, and 1 step of size 64/63. There are 1260 such scales.

82: 10 notes, 4 steps of size 8/7, 1 step of size 25/24, 2 steps of size 36/35, and 3 steps of size 49/48. There are 1260 such scales.

83: 10 notes, 4 steps of size 8/7, 3 steps of size 25/24, 1 step of size 49/48, and 2 steps of size 126/125. There are 1260 such scales.

84: 10 notes, 2 steps of size 10/9, 3 steps of size 15/14, 2 steps of size 16/15, and 3 steps of size 21/20. There are 2520 such scales.

85: 10 notes, 4 steps of size 10/9, 1 step of size 15/14, 3 steps of size 21/20, and 2 steps of size 36/35. There are 1260 such scales.

86: 10 notes, 4 steps of size 10/9, 3 steps of size 15/14, 1 step of size 21/20, and 2 steps of size 126/125. There are 1260 such scales.

87: 10 notes, 5 steps of size 10/9, 2 steps of size 15/14, 1 step of size 81/80, and 2 steps of size 126/125. There are 756 such scales.

88: 10 notes, 5 steps of size 10/9, 2 steps of size 21/20, 2 steps of size 36/35, and 1 step of size 81/80. There are 756 such scales

89: 11 notes, 1 step of size 5/4, 5 steps of size 15/14, 3 steps of size 28/27, and 2 steps of size 126/125. There are 2520 such scales.

90: 11 notes, 2 steps of size 6/5, 4 steps of size 21/20, 2 steps of size 28/27, and 3 steps of size 50/49. There are 6300 such scales.

91: 11 notes, 2 steps of size 7/6, 2 steps of size 16/15, 4 steps of size 21/20, and 3 steps of size 50/49. There are 6300 such scales.

92: 11 notes, 3 steps of size 8/7, 1 step of size 10/9, 4 steps of size 25/24, and 3 steps of size 126/125. There are 4200 such scales.

93: 11 notes, 4 steps of size 8/7, 2 steps of size 36/35, 4 steps of size 49/48, and 1 step of size 50/49. There are 3150 such scales.

94: 11 notes, 1 step of size 9/8, 4 steps of size 10/9, 4 steps of size 36/35, and 2 steps of size 49/48. There are 3150 such scales.

95: 11 notes, 1 step of size 9/8, 4 steps of size 15/14, 4 steps of size 16/15, and 2 steps of size 49/48. There are 3150 such scales.

96: 11 notes, 3 steps of size 9/8, 2 steps of size 15/14, 4 steps of size 28/27, and 2 steps of size 36/35. There are 6300 such scales.

97: 11 notes, 4 steps of size 10/9, 3 steps of size 15/14, 1 step of size 25/24, and 3 steps of size 126/125. There are 4200 such scales.

98: 11 notes, 4 steps of size 10/9, 3 steps of size 21/20, 1 step of size 25/24, and 3 steps of size 36/35. There are 4200 such scales.

99: 11 notes, 4 steps of size 10/9, 4 steps of size 21/20, 2 steps of size 36/35, and 1 step of size 50/49. There are 3150 such scales.

100: 12 notes, 1 step of size 7/6, 6 steps of size 15/14, 3 steps of size 28/27, and 2 steps of size 126/125. There are 4620 such scales.

101: 12 notes, 3 steps of size 8/7, 1 step of size 16/15, 5 steps of size 25/24, and 3 steps of size 126/125. There are 9240 such scales.

102: 12 notes, 2 steps of size 10/9, 3 steps of size 15/14, 5 steps of size 21/20, and 2 steps of size 64/63. There are 13860 such scales.

103: 12 notes, 5 steps of size 10/9, 2 steps of size 21/20, 2 steps of size 64/63, and 3 steps of size 81/80. There are 13860 such scales.

104: 12 notes, 5 steps of size 10/9, 2 steps of size 28/27, 2 steps of size 36/35, and 3 steps of size 81/80. There are 13860 such scales.

105: 12 notes, 5 steps of size 10/9, 4 steps of size 36/35, 2 steps of size 49/48, and 1 step of size 81/80. There are 6930 such scales.

106: 12 notes, 3 steps of size 15/14, 4 steps of size 16/15, 3 steps of size 21/20, and 2 steps of size 25/24. There are 23100 such scales.

107: 12 notes, 5 steps of size 15/14, 2 steps of size 16/15, 3 steps of size 21/20, and 2 steps of size 28/27. There are 13860 such scales.

108: 12 notes, 5 steps of size 15/14, 4 steps of size 16/15, 1 step of size 21/20, and 2 steps of size 49/48. There are 6930 such scales.

109: 13 notes, 2 steps of size 7/6, 6 steps of size 21/20, 3 steps of size 50/49, and 2 steps of size 64/63. There are 27720 such scales.

110: 13 notes, 2 steps of size 8/7, 3 steps of size 15/14, 5 steps of size 28/27, and 3 steps of size 81/80. There are 55440 such scales.

111: 13 notes, 2 steps of size 8/7, 6 steps of size 21/20, 2 steps of size 28/27, and 3 steps of size 50/49. There are 27720 such scales.

112: 13 notes, 4 steps of size 8/7, 4 steps of size 49/48, 3 steps of size 50/49, and 2 steps of size 126/125. There are 69300 such scales.

113: 13 notes, 1 step of size 9/8, 6 steps of size 15/14, 4 steps of size 28/27, and 2 steps of size 126/125. There are 13860 such scales.

114: 13 notes, 3 steps of size 9/8, 4 steps of size 21/20, 2 steps of size 25/24, and 4 steps of size 64/63. There are 69300 such scales.

115: 13 notes, 3 steps of size 9/8, 2 steps of size 25/24, 4 steps of size 28/27, and 4 steps of size 36/35. There are 69300 such scales.

116: 13 notes, 4 steps of size 10/9, 1 step of size 15/14, 5 steps of size 36/35, and 3 steps of size 49/48. There are 27720 such scales.

117: 13 notes, 2 steps of size 10/9, 2 steps of size 16/15, 6 steps of size 21/20, and 3 steps of size 50/49. There are 27720 such scales.

118: 13 notes, 4 steps of size 10/9, 4 steps of size 21/20, 3 steps of size 50/49, and 2 steps of size 126/125. There are 69300 such scales.

119: 13 notes, 5 steps of size 15/14, 4 steps of size 16/15, 1 step of size 36/35, and 3 steps of size 49/48. There are 27720 such scales.

120: 14 notes, 3 steps of size 8/7, 6 steps of size 25/24, 1 step of size 64/63, and 4 steps of size 126/125. There are 60060 such scales.

121: 14 notes, 4 steps of size 10/9, 1 step of size 25/24, 6 steps of size 36/35, and 3 steps of size 49/48. There are 60060 such scales.

122: 14 notes, 4 steps of size 10/9, 4 steps of size 25/24, 3 steps of size 36/35, and 3 steps of size 126/125. There are 300300 such scales.

123: 14 notes, 5 steps of size 10/9, 2 steps of size 28/27, 2 steps of size 64/63, and 5 steps of size 81/80. There are 108108 such scales.

124: 14 notes, 5 steps of size 15/14, 3 steps of size 21/20, 4 steps of size 28/27, and 2 steps of size 36/35. There are 180180 such scales.

125: 14 notes, 5 steps of size 15/14, 5 steps of size 21/20, 2 steps of size 28/27, and 2 steps of size 64/63. There are 108108 such scales.

126: 14 notes, 7 steps of size 15/14, 1 step of size 21/20, 4 steps of size 28/27, and 2 steps of size 126/125. There are 25740 such scales.

127: 15 notes, 2 steps of size 6/5, 6 steps of size 28/27, 3 steps of size 50/49, and 4 steps of size 81/80. There are 420420 such scales.

128: 15 notes, 2 steps of size 10/9, 8 steps of size 21/20, 3 steps of size 50/49, and 2 steps of size 64/63. There are 90090 such scales.

129: 15 notes, 4 steps of size 10/9, 6 steps of size 36/35, 4 steps of size 49/48, and 1 step of size 50/49. There are 210210 such scales.

130: 15 notes, 3 steps of size 15/14, 4 steps of size 16/15, 5 steps of size 25/24, and 3 steps of size 126/125. There are 840840 such scales.

131: 15 notes, 5 steps of size 15/14, 2 steps of size 16/15, 5 steps of size 28/27, and 3 steps of size 81/80. There are 504504 such scales.

132: 15 notes, 7 steps of size 15/14, 1 step of size 25/24, 4 steps of size 28/27, and 3 steps of size 126/125. There are 120120 such scales.

133: 15 notes, 7 steps of size 15/14, 5 steps of size 28/27, 1 step of size 81/80, and 2 steps of size 126/125. There are 72072 such scales.

134: 15 notes, 4 steps of size 16/15, 3 steps of size 21/20, 5 steps of size 25/24, and 3 steps of size 36/35. There are 840840 such scales.

135: 15 notes, 4 steps of size 16/15, 6 steps of size 21/20, 2 steps of size 25/24, and 3 steps of size 50/49. There are 420420 such scales.

136: 16 notes, 5 steps of size 10/9, 2 steps of size 49/48, 4 steps of size 64/63, and 5 steps of size 81/80. There are 1891890 such scales.

137: 16 notes, 3 steps of size 15/14, 7 steps of size 21/20, 2 steps of size 25/24, and 4 steps of size 64/63. There are 900900 such scales.

138: 16 notes, 5 steps of size 15/14, 5 steps of size 21/20, 2 steps of size 49/48, and 4 steps of size 64/63. There are 1891890 such scales.

139: 17 notes, 3 steps of size 9/8, 6 steps of size 25/24, 4 steps of size 64/63, and 4 steps of size 126/125. There are 8408400 such scales.

140: 17 notes, 4 steps of size 10/9, 4 steps of size 25/24, 3 steps of size 50/49, and 6 steps of size 126/125. There are 8408400 such scales.

141: 17 notes, 5 steps of size 15/14, 4 steps of size 28/27, 5 steps of size 36/35, and 3 steps of size 49/48. There are 10090080 such scales.

142: 17 notes, 5 steps of size 15/14, 7 steps of size 28/27, 2 steps of size 36/35, and 3 steps of size 81/80. There are 2882880 such scales.

143: 17 notes, 2 steps of size 16/15, 8 steps of size 21/20, 2 steps of size 28/27, and 5 steps of size 50/49. There are 1081080 such scales.

144: 17 notes, 4 steps of size 16/15, 6 steps of size 21/20, 2 steps of size 49/48, and 5 steps of size 50/49. There are 5045040 such scales.

145: 18 notes, 4 steps of size 16/15, 5 steps of size 25/24, 6 steps of size 36/35, and 3 steps of size 49/48. There are 28588560 such scales.

146: 18 notes, 4 steps of size 16/15, 8 steps of size 25/24, 3 steps of size 36/35, and 3 steps of size 126/125. There are 10210200 such scales.

147: 19 notes, 2 steps of size 8/7, 8 steps of size 28/27, 3 steps of size 50/49, and 6 steps of size 81/80. There are 18378360 such scales.

148: 19 notes, 5 steps of size 15/14, 7 steps of size 28/27, 2 steps of size 64/63, and 5 steps of size 81/80. There are 44108064 such scales.

149: 19 notes, 3 steps of size 21/20, 5 steps of size 25/24, 4 steps of size 28/27, and 7 steps of size 36/35. There are 73513440 such scales.

150: 19 notes, 7 steps of size 21/20, 5 steps of size 25/24, 3 steps of size 36/35, and 4 steps of size 64/63. There are 73513440 such scales.

151: 19 notes, 10 steps of size 21/20, 2 steps of size 25/24, 3 steps of size 50/49, and 4 steps of size 64/63. There are 6126120 such scales.

152: 19 notes, 8 steps of size 21/20, 4 steps of size 28/27, 2 steps of size 36/35, and 5 steps of size 50/49. There are 27567540 such scales.

153: 19 notes, 10 steps of size 21/20, 2 steps of size 28/27, 5 steps of size 50/49, and 2 steps of size 64/63. There are 3675672 such scales.

154: 21 notes, 4 steps of size 10/9, 4 steps of size 49/48, 7 steps of size 50/49, and 6 steps of size 126/125. There are 1163962800 such scales.

155: 21 notes, 5 steps of size 15/14, 5 steps of size 36/35, 7 steps of size 49/48, and 4 steps of size 64/63. There are 1396755360 such scales.

156: 21 notes, 4 steps of size 16/15, 8 steps of size 25/24, 3 steps of size 50/49, and 6 steps of size 126/125. There are 581981400 such scales.

157: 21 notes, 8 steps of size 21/20, 4 steps of size 28/27, 7 steps of size 50/49, and 2 steps of size 126/125. There are 249420600 such scales.

158: 21 notes, 10 steps of size 21/20, 2 steps of size 49/48, 5 steps of size 50/49, and 4 steps of size 64/63. There are 116396280 such scales.

159: 22 notes, 7 steps of size 21/20, 5 steps of size 25/24, 7 steps of size 64/63, and 3 steps of size 81/80. There are 2793510720 such scales.

160: 22 notes, 5 steps of size 25/24, 4 steps of size 28/27, 10 steps of size 36/35, and 3 steps of size 49/48. There are 814773960 such scales.

161: 22 notes, 5 steps of size 25/24, 7 steps of size 28/27, 7 steps of size 36/35, and 3 steps of size 81/80. There are 2793510720 such scales.

162: 22 notes, 8 steps of size 25/24, 4 steps of size 28/27, 7 steps of size 36/35, and 3 steps of size 126/125. There are 1745944200 such scales.

163: 23 notes, 3 steps of size 15/14, 9 steps of size 25/24, 4 steps of size 64/63, and 7 steps of size 126/125. There are 4267863600 such scales.

164: 23 notes, 4 steps of size 16/15, 6 steps of size 36/35, 8 steps of size 49/48, and 5 steps of size 50/49. There are 13443770340 such scales.

165: 25 notes, 2 steps of size 16/15, 10 steps of size 28/27, 5 steps of size 50/49, and 8 steps of size 81/80. There are 17668955304 such scales.

166: 26 notes, 5 steps of size 15/14, 7 steps of size 49/48, 9 steps of size 64/63, and 5 steps of size 81/80. There are 588965176800 such scales.

167: 26 notes, 5 steps of size 25/24, 10 steps of size 36/35, 7 steps of size 49/48, and 4 steps of size 64/63. There are 294482588400 such scales.

168: 26 notes, 12 steps of size 25/24, 3 steps of size 36/35, 4 steps of size 64/63, and 7 steps of size 126/125. There are 44618574000 such scales.

169: 27 notes, 4 steps of size 28/27, 10 steps of size 36/35, 8 steps of size 49/48, and 5 steps of size 50/49. There are 957068412300 such scales.

170: 27 notes, 12 steps of size 28/27, 2 steps of size 36/35, 5 steps of size 50/49, and 8 steps of size 81/80. There are 87006219300 such scales.

171: 29 notes, 4 steps of size 16/15, 8 steps of size 49/48, 11 steps of size 50/49, and 6 steps of size 126/125. There are 10962783631800 such scales.

172: 29 notes, 8 steps of size 25/24, 4 steps of size 28/27, 7 steps of size 50/49, and 10 steps of size 126/125. There are 17227231421400 such scales.

173: 29 notes, 5 steps of size 25/24, 7 steps of size 28/27, 7 steps of size 64/63, and 10 steps of size 81/80. There are 27563570274240 such scales.

174: 29 notes, 12 steps of size 25/24, 3 steps of size 50/49, 4 steps of size 64/63, and 10 steps of size 126/125. There are 1218087070200 such scales.

175: 29 notes, 12 steps of size 25/24, 7 steps of size 64/63, 3 steps of size 81/80, and 7 steps of size 126/125. There are 4176298526400 such scales.

176: 29 notes, 12 steps of size 28/27, 5 steps of size 50/49, 2 steps of size 64/63, and 10 steps of size 81/80. There are 730852242120 such scales.

177: 29 notes, 12 steps of size 28/27, 7 steps of size 50/49, 8 steps of size 81/80, and 2 steps of size 126/125. There are 1566111947400 such scales.

178: 31 notes, 10 steps of size 36/35, 12 steps of size 49/48, 5 steps of size 50/49, and 4 steps of size 64/63. There are 52986787553700 such scales.

179: 36 notes, 5 steps of size 25/24, 7 steps of size 49/48, 14 steps of size 64/63, and 10 steps of size 81/80. There are 54006928782206400 such scales.

180: 37 notes, 4 steps of size 28/27, 8 steps of size 49/48, 15 steps of size 50/49, and 10 steps of size 126/125. There are 81010393173309600 such scales.

181: 41 notes, 12 steps of size 49/48, 5 steps of size 50/49, 14 steps of size 64/63, and 10 steps of size 81/80. There are 44869938376416753600 such scales.

182: 41 notes, 12 steps of size 49/48, 15 steps of size 50/49, 4 steps of size 64/63, and 10 steps of size 126/125. There are 14956646125472251200 such scales.

In total, there are 59961720275076322577 constant structures in the 7 limit, up to cyclic permutation, with 4 distinct superparticular steps with sizes between 5/4 and 126/125 that span the whole 7 limit lattice.

  • a monzo is a factorial expression of a JI ratio in terms of the simple ratios that make it up. In terms of this article, harmonics 1, 3, 5, and 7-ed.

May 16,2016

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