Re: [tuning-math] Digest Number 185

Gene et al. Here's the whole article on superparticulars as an asci file. Ignore
the first column of numbers, they were an internal check.

The Number of 23-Prime-Limit Superparticular
Ratios Less than 10,000,000

It has been conjectured that there are only 10 superparticular (epimore) ratios
whose terms are factorable by 2, 3 and 5 1. These ratios are 2/1, 3/2, 4/3, 5/4,
6/5, 9/8, 10/9, 16/15, 25/24, and 81/80 2 and are well-known in music theory.
Computer searches have verified this conjecture with numerators up to 1 x 1012,
but a general proof is not known, though it has been claimed that one was known
to the ancient Pythagoreans 3.

I find this conjecture astonishing and have repeated the search on a Mac SE/30
up to 107 with a Microsoft QuickBASIC program I recently wrote4 . Out of
curiosity, I decided to extend the search to include each prime limit up to 23
inclusive with numerators less than or equal to 107, which seems to be the
practical limit for my program and system. I have found 240 ratios, the list of
which I have appended below as Table 1. Needless to say, the largest intervals
at each prime limit, including the recently discovered "ragisma," 4375/4374 5
have been exploited by musicians and theorists.

John H. Chalmers
Rancho Santa Fe, California
January, 26, 1997

Table 1.

Numbers of Superparticular Ratios Less
Than 107 at the 23 Prime-Limit

The number of New Ratios at each new prime limit is indicated as well as the
Cumulative Total. The numbers in the first column are the order of generation
numbers of the ratios as they are found by my program and are perhaps of less
general interest.

Ratios of 2
1 2 / 1
New Ratios = 1
Cumulative Total = 1

Ratios of 3
2 3 / 2
3 4 / 3
8 9 / 8
New Ratios = 3
Cumulative Total = 4

Ratios of 5
4 5 / 4
5 6 / 5
9 10 / 9
15 16 / 15
24 25 / 24
51 81 / 80
New Ratios = 6
Cumulative Total = 10

Ratios of 7
6 7 / 6
7 8 / 7
14 15 / 14
20 21 / 20
27 28 / 27
31 36 / 35
36 49 / 48
37 50 / 49
43 64 / 63
62 126 / 125
80 225 / 224
153 2401 / 2400
171 4375 / 4374
New Ratios = 13
Cumulative Total = 23

Ratios of 11
10 11 / 10
11 12 / 11
21 22 / 21
28 33 / 32
34 45 / 44
40 55 / 54
41 56 / 55
56 99 / 98
57 100 / 99
61 121 / 120
73 176 / 175
82 243 / 242
99 385 / 384
103 441 / 440
113 540 / 539
161 3025 / 3024
186 9801 / 9800
New Ratios = 17
Cumulative Total = 40

Ratios of 13
12 13 / 12
13 14 / 13
25 26 / 25
26 27 / 26
33 40 / 39
44 65 / 64
45 66 / 65
50 78 / 77
53 91 / 90
58 105 / 104
65 144 / 143
70 169 / 168
75 196 / 195
92 325 / 324
94 351 / 350
95 352 / 351
97 364 / 363
117 625 / 624
118 676 / 675
120 729 / 728
128 1001 / 1000
143 1716 / 1715
149 2080 / 2079
168 4096 / 4095
170 4225 / 4224
181 6656 / 6655
189 10648 / 10647
223 123201 / 123200
New Ratios = 28
Cumulative Total = 68

Ratios of 17
16 17 / 16
17 18 / 17
29 34 / 33
30 35 / 34
38 51 / 50
39 52 / 51
52 85 / 84
60 120 / 119
64 136 / 135
67 154 / 153
71 170 / 169
79 221 / 220
84 256 / 255
85 273 / 272
88 289 / 288
98 375 / 374
104 442 / 441
114 561 / 560
116 595 / 594
119 715 / 714
123 833 / 832
126 936 / 935
129 1089 / 1088
131 1156 / 1155
134 1225 / 1224
135 1275 / 1274
142 1701 / 1700
148 2058 / 2057
154 2431 / 2430
156 2500 / 2499
157 2601 / 2600
174 4914 / 4913
177 5832 / 5831
194 12376 / 12375
199 14400 / 14399
209 28561 / 28560
211 31213 / 31212
212 37180 / 37179
227 194481 / 194480
233 336141 / 336140
New Ratios = 40
Cumulative Total = 108

Ratios of 19
18 19 / 18
19 20 / 19
32 39 / 38
42 57 / 56
48 76 / 75
49 77 / 76
55 96 / 95
63 133 / 132
66 153 / 152
72 171 / 170
74 190 / 189
77 209 / 208
78 210 / 209
87 286 / 285
91 324 / 323
93 343 / 342
96 361 / 360
102 400 / 399
105 456 / 455
107 476 / 475
109 495 / 494
111 513 / 512
127 969 / 968
133 1216 / 1215
137 1331 / 1330
138 1445 / 1444
140 1521 / 1520
141 1540 / 1539
144 1729 / 1728
152 2376 / 2375
155 2432 / 2431
160 2926 / 2925
163 3136 / 3135
164 3250 / 3249
169 4200 / 4199
176 5776 / 5775
178 5929 / 5928
179 5985 / 5984
180 6175 / 6174
182 6860 / 6859
187 10241 / 10240
190 10830 / 10829
195 12636 / 12635
196 13377 / 13376
197 14080 / 14079
198 14365 / 14364
205 23409 / 23408
208 27456 / 27455
210 28900 / 28899
214 43681 / 43680
219 89376 / 89375
221 104976 / 104975
226 165376 / 165375
229 228096 / 228095
234 601426 / 601425
235 633556 / 633555
236 709632 / 709631
240 5909761 / 5909760
New Ratios = 58
Cumulative Total = 166

Ratios of 23
22 23 / 22
23 24 / 23
35 46 / 45
46 69 / 68
47 70 / 69
54 92 / 91
59 115 / 114
68 161 / 160
69 162 / 161
76 208 / 207
81 231 / 230
83 253 / 252
86 276 / 275
89 300 / 299
90 323 / 322
100 391 / 390
101 392 / 391
106 460 / 459
108 484 / 483
110 507 / 506
112 529 / 528
115 576 / 575
121 736 / 735
122 760 / 759
124 875 / 874
125 897 / 896
130 1105 / 1104
132 1197 / 1196
136 1288 / 1287
139 1496 / 1495
145 1863 / 1862
146 2024 / 2023
147 2025 / 2024
150 2185 / 2184
151 2300 / 2299
158 2646 / 2645
159 2737 / 2736
162 3060 / 3059
165 3381 / 3380
166 3520 / 3519
167 3888 / 3887
172 4693 / 4692
173 4761 / 4760
175 5083 / 5082
183 7866 / 7865
184 8281 / 8280
185 8625 / 8624
188 10626 / 10625
191 11271 / 11270
192 11662 / 11661
193 12168 / 12167
200 16929 / 16928
201 19551 / 19550
202 21505 / 21504
203 21736 / 21735
204 23276 / 23275
206 25025 / 25024
207 25921 / 25920
213 43264 / 43263
215 52326 / 52325
216 71875 / 71874
217 75141 / 75140
218 76545 / 76544
220 104329 / 104328
222 122452 / 122451
224 126225 / 126224
225 152881 / 152880
228 202125 / 202124
230 264385 / 264384
231 282625 / 282624
232 328510 / 328509
237 2023425 / 2023424
238 4096576 / 4096575
239 5142501 / 5142500
New Ratios = 74
Cumulative Total = 240
1 Streitberg, Bernd and Klaus Balzer. 1988. The Sound of Mathematics.
Proceedings of the 1988 International Computer Music Conference: 158-165.
2 ibid.
3 ibid.
4 This program is an update of one I wrote in 1993 and mentioned in a letter to
Ervin Wilson who persuaded me to write up my results for Xenharmonikon 17.
5 Personal communication, Erv Wilson.