RE: Harry Partch's 43-Tone Scale

Speaking of Harry Partch, here is an analysis of his scale:

Harry Partch's 43 Tone Scale

LCM: 2^11*3^8*5^2*7^2*11^2

43 1971200 160/ 81
42 1935360 64/ 33
41 1900800 40/ 21
40 1871100 15/ 8 Major Seventh
39 1829520 11/ 6
38 1814400 20/ 11
37 1796256 9/ 5 Minor Seventh
36 1774080 16/ 9 Grave Minor Seventh
35 1746360 7/ 4 Harmonic Minor Seventh
34 1710720 12/ 7
33 1683990 27/ 16 Pyth Major Sixth
32 1663200 5/ 3 Major Sixth
31 1632960 18/ 11
30 1596672 8/ 5 Minor Sixth
29 1568160 11/ 7
28 1552320 14/ 9
27 1520640 32/ 21
26 1496880 3/ 2 Perfect Fifth
25 1478400 40/ 27
24 1451520 16/ 11
23 1425600 10/ 7
22 1397088 7/ 5
21 1372140 11/ 8
20 1347192 27/ 20
19 1330560 4/ 3 Perfect Fourth
18 1309770 21/ 16
17 1283040 9/ 7
16 1270080 14/ 11
15 1247400 5/ 4 Major Third
14 1219680 11/ 9
13 1197504 6/ 5 Minor Third
12 1182720 32/ 27 Pyth Minor Third
11 1164240 7/ 6
10 1140480 8/ 7
9 1122660 9/ 8 Major Tone
8 1108800 10/ 9 Minor Tone
7 1097712 11/ 10
6 1088640 12/ 11
5 1064448 16/ 15 Semitone
4 1047816 21/ 20
3 1029105 33/ 32
2 1010394 81/ 80
1 997920 1/ 1 Unison

Minimum Chords:

1 60 3: 4: 5 1- 19- 32
2 60 3: 4: 5 2- 20- 33
3 60 3: 4: 5 4- 22- 35
4 60 3: 4: 5 6- 24- 38
5 60 3: 4: 5 9- 26- 40
6 60 3: 4: 5 10- 27- 41
7 60 4: 5: 6 1- 15- 26
8 60 4: 5: 6 5- 19- 30
9 60 4: 5: 6 10- 23- 34
10 60 4: 5: 6 12- 25- 36
11 60 4: 5: 6 13- 26- 37
12 60 10: 12: 15 8- 19- 32
13 60 10: 12: 15 11- 22- 35
14 60 12: 15: 20 4- 18- 35
15 60 12: 15: 20 7- 21- 39
16 60 12: 15: 20 12- 25- 43
17 72 6: 8: 9 6- 24- 31
18 72 6: 8: 9 11- 28- 35
19 72 8: 9: 12 10- 17- 34
20 72 8: 9: 12 14- 21- 39
21 126 6: 7: 9 6- 16- 31
22 126 14: 18: 21 14- 29- 39
23 144 9: 12: 16 3- 21- 39
24 144 9: 12: 16 6- 24- 42

Chord List: LCM's to 280

1 60 3: 4: 5 1- 19- 32
2 60 3: 4: 5 2- 20- 33
3 60 3: 4: 5 4- 22- 35
4 60 3: 4: 5 6- 24- 38
5 60 3: 4: 5 9- 26- 40
6 60 3: 4: 5 10- 27- 41
7 60 4: 5: 6 1- 15- 26
8 60 4: 5: 6 5- 19- 30
9 60 4: 5: 6 10- 23- 34
10 60 4: 5: 6 12- 25- 36
11 60 4: 5: 6 13- 26- 37
12 60 10: 12: 15 1- 13- 26
13 60 10: 12: 15 8- 19- 32
14 60 10: 12: 15 9- 20- 33
15 60 10: 12: 15 11- 22- 35
16 60 10: 12: 15 15- 26- 40
17 60 12: 15: 20 1- 15- 32
18 60 12: 15: 20 4- 18- 35
19 60 12: 15: 20 5- 19- 36
20 60 12: 15: 20 7- 21- 39
21 60 12: 15: 20 10- 23- 41
22 60 12: 15: 20 12- 25- 43
23 72 6: 8: 9 1- 19- 26
24 72 6: 8: 9 6- 24- 31
25 72 6: 8: 9 8- 25- 32
26 72 6: 8: 9 9- 26- 33
27 72 6: 8: 9 10- 27- 34
28 72 6: 8: 9 11- 28- 35
29 72 6: 8: 9 13- 30- 37
30 72 6: 8: 9 15- 32- 40
31 72 8: 9: 12 1- 9- 26
32 72 8: 9: 12 5- 13- 30
33 72 8: 9: 12 8- 15- 32
34 72 8: 9: 12 10- 17- 34
35 72 8: 9: 12 11- 18- 35
36 72 8: 9: 12 12- 19- 36
37 72 8: 9: 12 13- 20- 37
38 72 8: 9: 12 14- 21- 39
39 84 4: 6: 7 1- 26- 35
40 84 12: 14: 21 1- 11- 35
41 90 5: 6: 9 1- 13- 37
42 90 6: 9: 10 1- 26- 32
43 90 6: 9: 10 5- 30- 36
44 90 6: 9: 10 6- 31- 38
45 90 6: 9: 10 9- 33- 40
46 90 6: 9: 10 10- 34- 41
47 90 6: 9: 10 12- 36- 43
48 90 9: 10: 15 1- 8- 32
49 90 9: 10: 15 2- 9- 33
50 90 9: 10: 15 4- 11- 35
51 90 9: 10: 15 5- 12- 36
52 90 9: 10: 15 7- 14- 39
53 90 9: 10: 15 9- 15- 40
54 90 10: 15: 18 1- 26- 37
55 120 5: 6: 8 1- 13- 30
56 120 5: 6: 8 8- 19- 36
57 120 5: 6: 8 9- 20- 37
58 120 8: 12: 15 1- 26- 40
59 120 8: 10: 15 1- 15- 40
60 120 15: 20: 24 1- 19- 30
61 120 15: 20: 24 8- 25- 36
62 120 15: 20: 24 9- 26- 37
63 126 6: 7: 9 1- 11- 26
64 126 6: 7: 9 6- 16- 31
65 126 6: 7: 9 9- 18- 33
66 126 6: 7: 9 10- 19- 34
67 126 6: 7: 9 13- 22- 37
68 126 14: 18: 21 1- 17- 26
69 126 14: 18: 21 8- 23- 32
70 126 14: 18: 21 11- 26- 35
71 126 14: 18: 21 12- 27- 36
72 126 14: 18: 21 14- 29- 39
73 140 4: 5: 7 1- 15- 35
74 140 20: 28: 35 1- 22- 35
75 144 9: 12: 16 1- 19- 36
76 144 9: 12: 16 2- 20- 37
77 144 9: 12: 16 3- 21- 39
78 144 9: 12: 16 6- 24- 42
79 144 9: 12: 16 8- 25- 43
80 168 6: 7: 8 1- 11- 19
81 168 6: 7: 8 6- 16- 24
82 168 6: 7: 8 9- 18- 26
83 168 6: 7: 8 10- 19- 27
84 168 6: 7: 8 13- 22- 30
85 168 6: 7: 8 17- 26- 34
86 168 6: 7: 8 19- 28- 36
87 168 6: 7: 8 23- 32- 41
88 168 7: 8: 12 1- 10- 34
89 168 7: 8: 12 4- 13- 37
90 168 14: 21: 24 1- 26- 34
91 168 14: 21: 24 8- 32- 41
92 168 21: 24: 28 1- 10- 19
93 168 21: 24: 28 4- 13- 22
94 168 21: 24: 28 9- 17- 26
95 168 21: 24: 28 11- 19- 28
96 168 21: 24: 28 15- 23- 32
97 168 21: 24: 28 18- 26- 35
98 168 21: 24: 28 19- 27- 36
99 168 21: 24: 28 21- 29- 39
100 180 9: 10: 12 1- 8- 19
101 180 9: 10: 12 2- 9- 20
102 180 9: 10: 12 4- 11- 22
103 180 9: 10: 12 9- 15- 26
104 180 9: 10: 12 13- 19- 30
105 180 9: 10: 12 17- 23- 34
106 180 9: 10: 12 19- 25- 36
107 180 9: 10: 12 20- 26- 37
108 180 15: 18: 20 1- 13- 19
109 180 15: 18: 20 8- 19- 25
110 180 15: 18: 20 9- 20- 26
111 180 15: 18: 20 11- 22- 28
112 180 15: 18: 20 15- 26- 32
113 180 15: 18: 20 19- 30- 36
114 180 15: 18: 20 23- 34- 41
115 180 15: 18: 20 25- 36- 43
116 198 6: 9: 11 1- 26- 39
117 198 18: 22: 33 1- 14- 39
118 210 5: 6: 7 1- 13- 22
119 210 5: 6: 7 8- 19- 28
120 210 5: 6: 7 15- 26- 35
121 210 6: 7: 10 1- 11- 32
122 210 6: 7: 10 6- 16- 38
123 210 6: 7: 10 9- 18- 40
124 210 6: 7: 10 10- 19- 41
125 210 10: 14: 15 1- 22- 26
126 210 10: 14: 15 8- 28- 32
127 210 10: 14: 15 10- 30- 34
128 210 10: 14: 15 15- 35- 40
129 210 14: 15: 21 5- 10- 30
130 210 14: 15: 21 11- 15- 35
131 210 14: 15: 21 13- 17- 37
132 210 21: 30: 35 1- 23- 32
133 210 21: 30: 35 4- 26- 35
134 210 21: 30: 35 5- 27- 36
135 210 21: 30: 35 7- 29- 39
136 210 30: 35: 42 1- 11- 22
137 210 30: 35: 42 10- 19- 30
138 210 30: 35: 42 17- 26- 37
139 240 10: 15: 16 1- 26- 30
140 240 10: 15: 16 8- 32- 36
141 240 10: 15: 16 9- 33- 37
142 240 12: 15: 16 1- 15- 19
143 240 12: 15: 16 4- 18- 22
144 240 12: 15: 16 10- 23- 27
145 240 12: 15: 16 13- 26- 30
146 240 12: 15: 16 19- 32- 36
147 240 12: 15: 16 20- 33- 37
148 240 12: 15: 16 24- 38- 42
149 240 15: 16: 20 1- 5- 19
150 240 15: 16: 24 1- 5- 30
151 240 15: 16: 20 3- 7- 21
152 240 15: 16: 24 8- 12- 36
153 240 15: 16: 20 8- 12- 25
154 240 15: 16: 24 9- 13- 37
155 240 15: 16: 20 9- 13- 26
156 240 15: 16: 20 15- 19- 32
157 240 15: 16: 20 18- 22- 35
158 240 15: 16: 20 23- 27- 41
159 252 7: 9: 12 1- 17- 34
160 252 7: 9: 12 4- 20- 37
161 252 7: 9: 12 8- 23- 41
162 252 9: 12: 14 1- 19- 28
163 252 9: 12: 14 9- 26- 35
164 252 9: 12: 14 10- 27- 36
165 252 18: 21: 28 1- 11- 28
166 252 18: 21: 28 9- 18- 35
167 252 18: 21: 28 10- 19- 36
168 252 21: 28: 36 1- 19- 34
169 252 21: 28: 36 4- 22- 37
170 252 21: 28: 36 8- 25- 41
171 264 6: 8: 11 1- 19- 39
172 264 8: 11: 12 1- 21- 26
173 264 8: 11: 12 6- 26- 31
174 264 8: 11: 12 10- 29- 34
175 264 22: 24: 33 1- 6- 26
176 264 22: 24: 33 11- 16- 35
177 264 22: 24: 33 14- 19- 39
178 264 24: 33: 44 1- 21- 39
179 280 5: 7: 8 1- 22- 30
180 280 5: 7: 8 8- 28- 36
181 280 7: 8: 10 1- 10- 23
182 280 7: 8: 10 4- 13- 26
183 280 7: 8: 10 11- 19- 32
184 280 7: 8: 10 16- 24- 38
185 280 7: 8: 10 18- 26- 40
186 280 7: 8: 10 19- 27- 41
187 280 28: 35: 40 1- 15- 23
188 280 28: 35: 40 4- 18- 26
189 280 28: 35: 40 5- 19- 27
190 280 28: 35: 40 7- 21- 29
191 280 28: 35: 40 13- 26- 34
192 280 28: 35: 40 19- 32- 41
193 280 35: 40: 56 1- 10- 30
194 280 35: 40: 56 9- 17- 37


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I'm not sure what everybody says about this, but according to Harry Partch the
12-tone scale is derived from both the partials themselves and their relations
to each other. For instance, the perfect fourth is derived from the fourth
partial's relation to the third partial, transposed to be in relation to the
fundamental. The seconds and sevenths must be derived by partials of partials,
for instance the minor sixth is derived from the fifth partial's relationship
to the eighth partial (which is found as the fourth partial of the second
partial's overtone series, thus staying within the five-partial limit).
If your fundamental is C, then, you get an "A flat" in your scale because it is
a minor sixth up from the fundamental, even though that tone does not occur in
the harmonic series.
I assume that Partch's thinking is in line with other theorists, but
with Partch you never know. The information above I got from _Interval_,
when they published an early draft of an intro to _Genesis_ called
"Exposition of Monophony." I hope this is some of what you were looking to
find out, and I realize this may be stuff you already knew.
Morgan Smith

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