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