Scala analysis: xen18-ayers-table-37
7 Iterated Harmonic Square Means between 1/1 and 2/1
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 16384/14321 232.986461 2: 4096/3281 384.096146 3: 32768/24305 517.241384 4: 128/89 629.119883 5: 4096/2705 718.304546 6: 64/41 770.937594 7: 8/5 813.686286 minor sixth 8: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 5/4 386.3137 major third 1: 16384/14321 232.9865 2: 14321/13124 151.1097 3: 26248/24305 133.1452 4: 24305/22784 111.8785 5: 2848/2705 89.1847 6: 2705/2624 52.6330 7: 41/40 42.7487 8: 5/4 386.3137 major third
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 41/40 42.749 cents 1: 1 2705/2624 52.633 cents 1: 1 2848/2705 89.185 cents 1: 1 24305/22784 111.878 cents 1: 1 26248/24305 133.145 cents 1: 1 14321/13124 151.110 cents 1: 1 16384/14321 232.986 cents 1: 1 5/4 386.314 cents major third 2: 1 541/512 95.382 cents 2: 1 89/82 141.818 cents 2: 1 4861/4328 201.063 cents 2: 1 3281/2848 245.024 cents 2: 1 28642/24305 284.255 cents 2: 1 4096/3281 384.096 cents 2: 1 41/32 429.062 cents 2: 1 20480/14321 619.300 cents 3: 1 89/80 184.566 cents 3: 1 24305/20992 253.696 cents 3: 1 3281/2705 334.208 cents 3: 1 14321/11392 396.133 cents 3: 1 2705/2048 481.695 cents 3: 1 32768/24305 517.241 cents 3: 1 20992/14321 662.049 cents 3: 1 5120/3281 770.410 cents 4: 1 4861/4096 296.445 cents 4: 1 3281/2624 386.841 cents 4: 1 14321/10820 485.318 cents 4: 1 89/64 570.880 cents 4: 1 128/89 629.120 cents 4: 1 21640/14321 714.682 cents 4: 1 5248/3281 813.159 cents 4: 1 8192/4861 903.555 cents 5: 1 3281/2560 429.590 cents 5: 1 14321/10496 537.951 cents 5: 1 24305/16384 682.759 cents 5: 1 4096/2705 718.305 cents 5: 1 22784/14321 803.867 cents 5: 1 5410/3281 865.792 cents 5: 1 41984/24305 946.304 cents 5: 1 160/89 1015.434 cents 6: 1 14321/10240 580.700 cents 6: 1 64/41 770.938 cents 6: 1 3281/2048 815.904 cents 6: 1 24305/14321 915.745 cents 6: 1 5696/3281 954.976 cents 6: 1 8656/4861 998.937 cents 6: 1 164/89 1058.182 cents 6: 1 1024/541 1104.618 cents 7: 1 8/5 813.686 cents minor sixth 7: 1 14321/8192 967.014 cents 7: 1 26248/14321 1048.890 cents 7: 1 24305/13124 1066.855 cents 7: 1 45568/24305 1088.122 cents 7: 1 2705/1424 1110.815 cents 7: 1 5248/2705 1147.367 cents 7: 1 80/41 1157.251 cents Highest number of different intervals for one interval class: 8 Average number of different intervals per interval class: 8.00000 = 8
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 0.0 : 233.0 384.1 517.2 629.1 718.3 770.9 813.7 1200.0 233.0 : 151.1 284.3 396.1 485.3 538.0 580.7 967.0 1200.0 384.1 : 133.1 245.0 334.2 386.8 429.6 815.9 1048.9 1200.0 517.2 : 111.9 201.1 253.7 296.4 682.8 915.7 1066.9 1200.0 629.1 : 89.2 141.8 184.6 570.9 803.9 955.0 1088.1 1200.0 718.3 : 52.6 95.4 481.7 714.7 865.8 998.9 1110.8 1200.0 770.9 : 42.7 429.1 662.0 813.2 946.3 1058.2 1147.4 1200.0 813.7 : 386.3 619.3 770.4 903.6 1015.4 1104.6 1157.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 42.7487 .. 386.3137 cents 50/41, 343.5650 cents 41/40, 42.7487 cents 2: 95.3817 .. 619.3002 cents 10485760/7747661, 523.9184 cents 541/640,-290.9320 cents 3: 184.5664 .. 770.4099 cents 409600/292009, 585.8435 cents 1274569/1638400,-434.7338 cents 4: 296.4449 .. 903.5551 cents 33554432/23629321, 607.1102 cents 15948941/20971520,-473.9650 cents 5: 429.5901 .. 1015.4336 cents 409600/292009, 585.8435 cents 15948941/20971520,-473.9650 cents 6: 580.6998 .. 1104.6183 cents 10485760/7747661, 523.9184 cents 1274569/1638400,-434.7338 cents 7: 813.6863 .. 1157.2513 cents 50/41, 343.5650 cents 541/640,-290.9320 cents
SHOW DATA
Number of notes : 8 -- Interval properties -- Smallest interval : 41/40, 42.7487 cents, class 1 Average step (divided formal octave): 150.0000 cents Largest one step interval : 5/4, 386.3137 cents Average / Smallest step : 3.508879 Largest / Average step : 2.575425 Largest / Smallest step : 9.036855 Median interval of one step : 122.51187 cents, amount: 0 Least squares average step : 140.10948 cents, oct.: 1120.87587 cents Scale is not proper Step pattern alph. order: ABCDEFGH Step pattern size order : GFEDCBAH Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 0.52773 cents Scale diversity : 1.745665 Lumma stability : 0.071248 Lumma impropriety factor : 0.841030 Rothenberg efficiency : 0.250000 redundancy: 0.750000 Efficiency x scale size : 2.000000 Number of different interval sizes : 56 = 8.00000 / class Number of one step interval sizes : 8 Highest interval variety : 8 Mean interval variety : 8.00000 = 8 Median interval variety : 8 Lowest interval variety : 8 Smallest interval difference : 3281/3280, 0.5277 cents Number of recognisable fifths : 2, average 716.4932 cents Number of appreciable fifths : 0 Formal octave complements present : 1 = 12.5000% -- Rational properties -- Prime limit : 14321 Odd number limit : 24305 (O: 24305 U: 24305) Highest odd numerator or denominator: 24305 Harmonicity values : limit above prime count Fundamental : too low to compute Guide tone : 32768, 15.0000 octaves, 8572946.524 Hz. Sum of Mann's harmonic distance : 51147.0, average 6393.37500 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 19.75099, average 2.46887 Sum of Tenney's harmonic distance : 40.82309, average 5.10289 Vogel's harmonic complexity : too high to compute Wille's k value : too high to compute Lattice compactness : 250.18079, average 6.94947 Lattice compactness (without 2's) : 24.24264, average 0.67341 Number of different primes : 9 Prime exponents' range, average, count, tones@limit: 2: 1 .. 15 8.75000 70 1 5: -1 .. 0 -0.37500 3 1 17: -1 .. 0 -0.12500 1 41: -1 .. 0 -0.12500 1 1 89: -1 .. 0 -0.12500 1 1 193: -1 .. 0 -0.12500 1 1 541: -1 .. 0 -0.12500 1 1 4861: -1 .. 0 -0.12500 1 1 14321: -1 .. 0 -0.12500 1 1 Average exponent except 2's :-10 / 8 =-1.25000 Average absolute exponent except 2's: 10 / 8 = 1.25000
FIT/MODE
8: 2 1 0 1 1 0 0 3 N SD: 49.6266 c. M: 67.2414 c. 9: 2 1 1 1 0 1 0 3 N SD: 29.0569 c. M: 51.6379 c. 10: 2 1 1 1 1 0 1 3 N SD: 27.7368 c. M: 50.9376 c. 12: 2 2 1 1 1 1 0 4 N SD: 21.9467 c. M: 32.9865 c. 15: 3 2 1 2 1 1 0 5 N SD: 18.8488 c. M: 37.2414 c. 16: 3 2 2 1 2 0 1 5 N SD: 18.1165 c. M:-31.6955 c. 17: 3 2 2 2 1 1 0 6 N SD: 21.1061 c. M: 37.2157 c. 19: 4 2 2 2 1 1 1 6 N SD: 12.9508 c. M: 23.5677 c. 22: 4 3 2 2 2 1 1 7 N SD: 15.5086 c. M: 29.1199 c. 23: 4 3 3 2 2 1 1 7 N SD: 14.5958 c. M: 24.2908 c. 24: 5 3 2 3 1 1 1 8 N SD: 16.7197 c. M: 20.9376 c. 25: 5 3 3 2 2 1 1 8 N SD: 5.0992 c. M:-10.7586 c. 40: 8 5 4 4 3 2 1 13 N SD: 5.4296 c. M:-9.0624 c. 47: 9 6 5 5 3 2 2 15 N SD: 4.8332 c. M:-9.1780 c. 53: 10 7 6 5 4 2 2 17 N SD: 3.8990 c. M: 6.5714 c. 62: 12 8 7 6 4 3 2 20 N SD: 4.2713 c. M:-9.5898 c. 65: 13 8 7 6 5 3 2 21 N SD: 3.3319 c. M:-7.0135 c. 68: 13 9 7 7 5 3 2 22 N SD: 4.4670 c. M:-6.1742 c. 69: 13 9 8 6 5 3 3 22 N SD: 4.3785 c. M: 6.8995 c. 72: 14 9 8 7 5 3 3 23 N SD: 2.4645 c. M: 4.2709 c. 84: 16 11 9 8 6 4 3 27 N SD: 2.4469 c. M: 4.4150 c. 93: 18 12 10 9 7 4 3 30 N SD: 2.5030 c. M:-4.2761 c. 97: 19 12 11 9 7 4 4 31 N SD: 2.1594 c. M: 3.9273 c. 109: 21 14 12 10 8 5 4 35 N SD: 1.4014 c. M: 2.7082 c. 134: 26 17 15 12 10 6 5 43 N SD: 1.4340 c. M: 2.2542 c. 162: 31 21 18 15 12 7 6 52 N SD: 1.4136 c. M: 3.3568 c. 174: 34 22 19 16 13 8 6 56 N SD: 1.2433 c. M:-2.1108 c. 190: 37 24 21 18 14 8 7 61 N SD: 1.2485 c. M:-2.4591 c. 202: 39 26 22 19 15 9 7 65 N SD: 1.0284 c. M:-2.0425 c. 206: 40 26 23 19 15 9 8 66 N SD: 1.2382 c. M: 2.0056 c. 212: 41 27 23 20 16 9 8 68 N SD: 1.1365 c. M: 2.1470 c. 215: 42 27 24 20 16 9 8 69 N SD: 1.3100 c. M:-1.8284 c. 221: 43 28 24 21 16 10 8 71 N SD: 0.9927 c. M: 1.5625 c. 227: 44 29 25 21 17 10 8 73 N SD: 0.8231 c. M:-1.8069 c. 237: 46 30 26 22 18 10 9 76 N SD: 0.9495 c. M:-1.5036 c. 246: 48 31 27 23 18 11 9 79 N SD: 0.8265 c. M:-1.2697 c. 257: 50 32 29 24 19 11 9 83 N SD: 0.9139 c. M: 1.2349 c. 269: 52 34 30 25 20 12 9 87 N SD: 0.8043 c. M: 1.7904 c. 279: 54 35 31 26 21 12 10 90 N SD: 0.9026 c. M: 1.3004 c. 288: 56 36 32 27 21 13 10 93 N SD: 0.8012 c. M: 1.6379 c. 294: 57 37 33 27 22 13 10 95 N SD: 0.7229 c. M: 1.4414 c. 299: 58 38 33 28 22 13 11 96 N SD: 0.6943 c. M:-1.1881 c. 341: 66 43 38 32 25 15 12 110 N SD: 0.5346 c. M:-0.7921 c. 366: 71 46 41 34 27 16 13 118 N SD: 0.4550 c. M:-0.7914 c. 406: 79 51 45 38 30 18 14 131 N SD: 0.4316 c. M: 0.8784 c. 469: 91 59 52 44 35 20 17 151 N SD: 0.4240 c. M: 0.7883 c. 490: 95 62 54 46 36 22 17 158 N SD: 0.4736 c. M: 0.7535 c. 494: 96 62 55 46 37 21 18 159 N SD: 0.4320 c. M: 0.8971 c. 506: 98 64 56 47 38 22 18 163 N SD: 0.3549 c. M: 0.6614 c. 515: 100 65 57 48 38 23 18 166 N SD: 0.3316 c. M: 0.6346 c. 540: 105 68 60 50 40 24 19 174 N SD: 0.3562 c. M:-0.5364 c. 568: 110 72 63 53 42 25 20 183 N SD: 0.3519 c. M: 0.5921 c. 578: 112 73 64 54 43 25 21 186 N SD: 0.3174 c. M: 0.6954 c. 618: 120 78 68 58 46 27 22 199 N SD: 0.2985 c. M: 0.7365 c. 643: 125 81 71 60 48 28 23 207 N SD: 0.2246 c. M:-0.3518 c. 912: 177 115 101 85 68 40 32 294 N SD: 0.2167 c. M: 0.5284 c. 937: 182 118 104 87 70 41 33 302 N SD: 0.2148 c. M: 0.4526 c. 979: 190 123 109 91 73 43 35 315 N SD: 0.2084 c. M: 0.4394 c. 984: 191 124 109 92 73 43 35 317 N SD: 0.1469 c. M: 0.2717 c. 1009: 196 127 112 94 75 44 36 325 N SD: 0.1347 c. M: 0.2736 c. 1206: 234 152 134 112 90 53 43 388 N SD: 0.1714 c. M: 0.2642 c. 1211: 235 153 134 113 90 53 43 390 N SD: 0.1598 c. M:-0.3795 c. 1278: 248 161 142 119 95 56 46 411 N SD: 0.1566 c. M:-0.3982 c. 1318: 256 166 146 123 98 58 47 424 N SD: 0.1434 c. M:-0.2743 c. 1385: 269 174 154 129 103 61 49 446 N SD: 0.1301 c. M: 0.2694 c. 1438: 279 181 160 134 107 63 51 463 N SD: 0.1441 c. M: 0.2297 c. 1457: 283 183 162 136 108 64 52 469 N SD: 0.1256 c. M: 0.2938 c. 1575: 306 198 175 147 117 69 56 507 N SD: 0.1275 c. M:-0.2135 c. 1587: 308 200 176 148 118 70 56 511 N SD: 0.1255 c. M:-0.3289 c. 1612: 313 203 179 150 120 71 57 519 N SD: 0.1158 c. M:-0.2783 c. 1659: 322 209 184 155 123 73 59 534 N SD: 0.0879 c. M:-0.1749 c. 1684: 327 212 187 157 125 74 60 542 N SD: 0.0665 c. M:-0.0983 c. 2190: 425 276 243 204 163 96 78 705 N SD: 0.0525 c. M: 0.1097 c. 2327: 452 293 258 217 173 102 83 749 N SD: 0.0565 c. M:-0.1034 c.
FIT/HARMONIC
1 x x x x x x x 2 S SD: 0.0000 cents 2 x x x x 3 x x 4 S SD: 8.1748 cents 3 x x 4 x x x 5:6 S SD: 24.4110 cents 4 x 5 x x 6 x x 8 S SD: 5.4997 cents 5 x 6 x 7 x x 8:10 S SD: 20.7038 cents 6:7 x 8 x 9 x 10:12 S SD: 16.4664 cents 7:8:9 x 10 x 11 x 14 SD: 10.7180 cents 8:9:10:11 x 12 x 13:16 SD: 9.1277 cents 9:10:11:12:13 x 14 x 18 SD: 11.0121 cents 10:11:12:13:14:15 x 16:20 S SD: 17.9112 cents 11:13:14:15:16 x 17:18:22 SD: 11.8335 cents 12:14:15:16:17:18 x 19:24 SD: 7.5589 cents 13:15:16:18:19 x 20:21:26 SD: 9.7275 cents 14:16:17:19:20:21:22 x 28 SD: 7.7870 cents 15:17:19:20:22:23 x 24:30 SD: 7.6782 cents 16:18:20:22:23:24:25:26:32 SD: 6.8498 cents 17:19:21:23:24:26 x 27:34 SD: 8.4496 cents 18:21:22:24:26:27:28:29:36 SD: 7.2522 cents 19:22:24:26:27:29:30 x 38 SD: 7.1785 cents 20:23:25:27:29:30:31:32:40 SD: 3.3245 cents 29:33:36:39:42:44:45:46:58 SD: 3.2807 cents 35:40:44:47:50:53:55:56:70 SD: 2.6965 cents 37:42:46:50:53:56:58:59:74 SD: 2.4620 cents 41:47:51:55:59:62:64:66:82 SD: 1.9501 cents 43:49:54:58:62:65:67:69:86 SD: 1.8376 cents 48:55:60:65:69:73:75:77:96 SD: 1.5368 cents 55:63:69:74:79:83:86:88:110 SD: 1.4657 cents 57:65:71:77:82:86:89:91:114 SD: 1.3245 cents 68:78:85:92:98:103:106:109:136 SD: 1.1949 cents 77:88:96:104:111:117:120:123:154 SD: 1.1616 cents 84:96:105:113:121:127:131:134:168 SD: 1.0193 cents 89:102:111:120:128:135:139:142:178 SD: 0.8442 cents 105:120:131:142:151:159:164:168:210 SD: 0.7318 cents 112:128:140:151:161:170:175:179:224 SD: 0.7141 cents 130:149:162:175:187:197:203:208:260 SD: 0.6723 cents 132:151:165:178:190:200:206:211:264 SD: 0.4164 cents 173:198:216:233:249:262:270:277:346 SD: 0.3342 cents 210:240:262:283:302:318:328:336:420 SD: 0.3086 cents 230:263:287:310:331:348:359:368:460 SD: 0.2720 cents 278:318:347:375:400:421:434:445:556 SD: 0.1887 cents 410:469:512:553:590:621:640:656:820 SD: 0.1801 cents 422:483:527:569:607:639:659:675:844 SD: 0.1648 cents 500:572:624:674:719:757:780:800:1000 SD: 0.1621 cents 520:595:649:701:748:787:812:832:1040 SD: 0.1562 cents 543:621:678:732:781:822:848:869:1086 SD: 0.1553 cents 548:627:684:739:788:830:855:877:1096 SD: 0.1498 cents 568:650:709:766:817:860:887:909:1136 SD: 0.1400 cents