Scala analysis: rat12
72-et Hahn reduced 12-fairly-equal well-temperament
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 200/189 97.936519 2: 9/8 203.910002 major whole tone 3: 25/21 301.846520 BP second, quasi-equal minor third 4: 63/50 400.108480 quasi-equal major third 5: 4/3 498.044999 perfect fourth 6: 625/441 603.693041 two (BP second, quasi-equal minor third) 7: 3/2 701.955001 perfect fifth 8: 100/63 799.891520 quasi-equal minor sixth 9: 42/25 898.153480 quasi-equal major sixth 10: 16/9 996.089998 Pythagorean minor seventh 11: 189/100 1102.063481 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 200/189 97.9365 1: 200/189 97.9365 2: 1701/1600 105.9735 3: 200/189 97.9365 4: 1323/1250 98.2620 5: 200/189 97.9365 6: 625/588 105.6480 7: 1323/1250 98.2620 8: 200/189 97.9365 9: 1323/1250 98.2620 10: 200/189 97.9365 11: 1701/1600 105.9735 12: 200/189 97.9365
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 6 200/189 97.937 cents 1: 3 1323/1250 98.262 cents 1: 1 625/588 105.648 cents 1: 2 1701/1600 105.973 cents 2: 1 40000/35721 195.873 cents 2: 5 28/25 196.198 cents middle second 2: 1 31250/27783 203.585 cents 2: 5 9/8 203.910 cents major whole tone 3: 2 32/27 294.135 cents Pythagorean minor third 3: 1 18522/15625 294.460 cents 3: 7 25/21 301.847 cents BP second, quasi-equal minor third 3: 2 11907/10000 302.172 cents 4: 1 784/625 392.397 cents two (middle second) 4: 4 5000/3969 399.783 cents 4: 6 63/50 400.108 cents quasi-equal major third 4: 1 81/64 407.820 cents Pythagorean major third 5: 8 4/3 498.045 cents perfect fourth 5: 1 83349/62500 498.370 cents 5: 3 75/56 505.757 cents marvelous fourth 6: 4 800/567 595.982 cents 6: 2 882/625 596.307 cents 6: 2 625/441 603.693 cents two (BP second, quasi-equal minor third) 6: 4 567/400 604.018 cents 7: 3 112/75 694.243 cents marvelous fifth 7: 1 125000/83349 701.630 cents 7: 8 3/2 701.955 cents perfect fifth 8: 1 128/81 792.180 cents Pythagorean minor sixth 8: 6 100/63 799.892 cents quasi-equal minor sixth 8: 4 3969/2500 800.217 cents two (quasi-equal major third) 8: 1 625/392 807.603 cents 9: 2 20000/11907 897.828 cents 9: 7 42/25 898.153 cents quasi-equal major sixth 9: 1 15625/9261 905.540 cents 9: 2 27/16 905.865 cents Pythagorean major sixth 10: 5 16/9 996.090 cents Pythagorean minor seventh 10: 1 27783/15625 996.415 cents 10: 5 25/14 1003.802 cents middle minor seventh 10: 1 35721/20000 1004.127 cents 11: 2 3200/1701 1094.027 cents 11: 1 1176/625 1094.352 cents 11: 3 2500/1323 1101.738 cents 11: 6 189/100 1102.063 cents Highest number of different intervals for one interval class: 4 Average number of different intervals per interval class: 3.81818 = 42/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 97.9 203.9 301.8 400.1 498.0 603.7 702.0 799.9 898.2 996.1 1102.1 1200.0 97.9 : 106.0 203.9 302.2 400.1 505.8 604.0 702.0 800.2 898.2 1004.1 1102.1 1200.0 203.9 : 97.9 196.2 294.1 399.8 498.0 596.0 694.2 792.2 898.2 996.1 1094.0 1200.0 301.8 : 98.3 196.2 301.8 400.1 498.0 596.3 694.2 800.2 898.2 996.1 1102.1 1200.0 400.1 : 97.9 203.6 301.8 399.8 498.0 596.0 702.0 799.9 897.8 1003.8 1101.7 1200.0 498.0 : 105.6 203.9 301.8 400.1 498.0 604.0 702.0 799.9 905.9 1003.8 1102.1 1200.0 603.7 : 98.3 196.2 294.5 392.4 498.4 596.3 694.2 800.2 898.2 996.4 1094.4 1200.0 702.0 : 97.9 196.2 294.1 400.1 498.0 596.0 702.0 799.9 898.2 996.1 1101.7 1200.0 799.9 : 98.3 196.2 302.2 400.1 498.0 604.0 702.0 800.2 898.2 1003.8 1102.1 1200.0 898.2 : 97.9 203.9 301.8 399.8 505.8 603.7 702.0 799.9 905.5 1003.8 1101.7 1200.0 996.1 : 106.0 203.9 301.8 407.8 505.8 604.0 702.0 807.6 905.9 1003.8 1102.1 1200.0 1102.1: 97.9 195.9 301.8 399.8 498.0 596.0 701.6 799.9 897.8 996.1 1094.0 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 97.9365 .. 105.9735 cents 321489/320000, 8.0370 cents 200/189, 97.9365 cents 2: 195.8730 .. 203.9100 cents 321489/320000, 8.0370 cents 64000000/60761421, 89.8996 cents 3: 294.1350 .. 302.1720 cents 321489/320000, 8.0370 cents 256/243, 90.2250 cents 4: 392.3970 .. 407.8200 cents 50625/50176, 15.4230 cents 256/243, 90.2250 cents 5: 498.0450 .. 505.7565 cents 225/224, 7.7115 cents 256/243, 90.2250 cents 6: 595.9815 .. 604.0185 cents 321489/320000, 8.0370 cents 256/243, 90.2250 cents 7: 694.2435 .. 701.9550 cents 225/224, 7.7115 cents 256/243, 90.2250 cents 8: 792.1800 .. 807.6030 cents 50625/50176, 15.4230 cents 256/243, 90.2250 cents 9: 897.8280 .. 905.8650 cents 321489/320000, 8.0370 cents 256/243, 90.2250 cents 10: 996.0900 .. 1004.1270 cents 321489/320000, 8.0370 cents 256/243, 90.2250 cents 11: 1094.0265 .. 1102.0635 cents 321489/320000, 8.0370 cents 64000000/60761421, 89.8996 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 200/189, 97.9365 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 1701/1600, 105.9735 cents Average / Smallest step : 1.021070 Largest / Average step : 1.059735 Largest / Smallest step : 1.082063 Median interval of one step : 98.09924 cents, amount: 0 Most common interval of one step : 200/189, 97.9365 cents, amount: 6 Least squares average step : 100.00602 cents, oct.: 1200.07218 cents Scale is strictly proper Step pattern alph. order: ABACADCACABA Step pattern size order : ADABACBABADA Scale is a Constant Structure, by a margin of 89.89955 cents Scale diversity : 1.068602 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.914560 Rothenberg efficiency : 0.358063 redundancy: 0.641937 Efficiency x scale size : 4.296753 Number of different interval sizes : 42 = 3.81818 / class Number of one step interval sizes : 4 Highest interval variety : 4 Mean interval variety : 3.81818 = 42/11 Median interval variety : 3 Lowest interval variety : 3 Smallest interval difference : 250047/250000, 0.3254 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 8 Most common triad is 0.0 498.045 701.955 cents, amount: 5 Number of recognisable fifths : 12, average 700.0000 cents Scale is a well-temperament. It has 3 different sizes of fifth Scale has monotonic third-sizes over circle of fifths Best fifths form a closed circle Scale contains a complete diamond : 21 25 63 189 Formal octave complements present : 11 = 91.6667% Limited transpositions with margin 8.0370 cents: 1 2 3 4 5 6 7 8 9 10 11 Limited inverse transpositions with margin 8.0370 cents: 1 2 3 4 5 6 7 8 9 10 11 -- Rational properties -- Prime limit : 7 Odd number limit : 83349 (O: 83349 U: 83349) Highest odd numerator or denominator: 625 Scale harmonicity : 0.010839 Average absolute harmonicity : 0.161350 Specific harmonicity : 0.069223 Fundamental : 1/264600, -18.0135 octaves, 0.0010 Hz. Guide tone : 1890000, 20.8500 octaves, 494472318.419 Hz. Exponens Consonantiae : 5.000940E+11, 38.86341 octaves Euler's gradus suavitatis : 62 Sum of Mann's harmonic distance : 1095.0, average 91.25000 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 16.60609, average 1.38384 Sum of Tenney's harmonic distance : 33.50733, average 2.79228 Vogel's harmonic complexity : 36.50000 Wille's k value : 148837 Wilson's harmonic complexity : 69 Rectangular lattice diameter : 15 Triangular lattice diameter : 8 Lattice compactness : 340.18623, average 4.36136 Lattice compactness (without 2's) : 266.12442, average 3.41185 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -3 .. 4 0.50000 20 1 3: -3 .. 3 -0.16667 20 4 5: -2 .. 4 0.33333 16 7: -2 .. 1 -0.16667 8 7 Average exponent except 2's : 0 / 12 = 0.00000 Average absolute exponent except 2's: 44 / 12 = 3.66667 Scale is weakly epimorphic with val: <12 17 25 36| Scale is weakly epimorphic with val: <12 17 29 32| Scale is weakly epimorphic with val: <12 17 30 34| Scale is weakly epimorphic with val: <12 17 31 36| Scale is JI-epimorphic with val: <12 19 27 32| Scale is JI-epimorphic with val: <12 19 28 34| = patent Scale is JI-epimorphic with val: <12 19 29 36|
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 2.3667 c. M:-3.9100 c. Twelve-tone Chromatic (1/11-comma) 87: 7 8 7 7 7 8 7 7 7 7 8 7 SP D3 SD: 1.8608 c. M:-3.2035 c. 99: 8 9 8 8 8 9 8 8 8 8 9 8 SP D3 SD: 1.3498 c. M:-2.3676 c. 111: 9 10 9 9 9 10 9 9 9 9 10 9 SP D3 SD: 0.9500 c. M:-1.7124 c. 123: 10 11 10 10 10 11 10 10 10 10 11 10 SP D3 SD: 0.6295 c. M:-1.1850 c. 135: 11 12 11 11 11 12 11 11 11 11 12 11 SP D3 SD: 0.3691 c. M:-0.7514 c. 147: 12 13 12 12 12 13 12 12 12 12 13 12 SP D3 SD: 0.1645 c. M:-0.3886 c. 159: 13 14 13 13 13 14 13 13 13 13 14 13 SP D3 SD: 0.1089 c. M:-0.1767 c. 306: 25 27 25 25 25 27 25 25 25 25 27 25 SP D3 SD: 0.1014 c. M:-0.2285 c. 465: 38 41 38 38 38 41 38 38 38 38 41 38 SP D3 SD: 0.0947 c. M:-0.1779 c. 624: 51 55 51 51 51 55 51 51 51 51 55 51 SP D3 SD: 0.0950 c. M:-0.1531 c. 783: 64 69 64 64 64 69 64 64 64 64 69 64 SP D3 SD: 0.0963 c. M:-0.1478 c. 1089: 89 96 89 89 89 96 89 89 89 89 96 89 SP D3 SD: 0.0946 c. M:-0.1637 c. 1407: 115 124 115 115 115 124 115 115 115 115 124 115 SP D3 SD: 0.0956 c. M:-0.1449 c. 1554: 127 137 127 127 127 137 127 127 127 127 137 127 SP D3 SD: 0.0945 c. M:-0.1680 c. 1865: 152 165 152 153 152 164 153 152 153 152 165 152 SP SD: 0.0926 c. M: 0.1542 c. 1877: 153 166 153 154 153 165 154 153 154 153 166 153 SP SD: 0.0916 c. M: 0.1768 c. 2000: 163 177 163 164 163 176 164 163 164 163 177 163 SP SD: 0.0853 c. M:-0.1365 c. 2012: 164 178 164 165 164 177 165 164 165 164 178 164 SP SD: 0.0803 c. M:-0.1234 c. 2024: 165 179 165 166 165 178 166 165 166 165 179 165 SP SD: 0.0777 c. M: 0.1357 c. 2036: 166 180 166 167 166 179 167 166 167 166 180 166 SP SD: 0.0776 c. M: 0.1567 c. 2147: 175 190 175 176 175 189 176 175 176 175 190 175 SP SD: 0.0774 c. M:-0.1256 c. 2159: 176 191 176 177 176 190 177 176 177 176 191 176 SP SD: 0.0711 c. M:-0.1135 c. 2171: 177 192 177 178 177 191 178 177 178 177 192 177 SP SD: 0.0669 c. M:-0.1014 c. 2183: 178 193 178 179 178 192 179 178 179 178 193 178 SP SD: 0.0652 c. M: 0.1200 c. 2306: 188 204 188 189 188 203 189 188 189 188 204 188 SP SD: 0.0646 c. M:-0.1048 c. 2318: 189 205 189 190 189 204 190 189 190 189 205 189 SP SD: 0.0589 c. M:-0.0936 c. 2330: 190 206 190 191 190 205 191 190 191 190 206 190 SP SD: 0.0554 c. M: 0.0879 c. 2342: 191 207 191 192 191 206 192 191 192 191 207 191 SP SD: 0.0546 c. M: 0.1064 c.
FIT/HARMONIC
1 x x x x x x x x x x x 2 S SD: 0.0000 cents 2 x x x x x x 3 x x x x 4 S SD: 0.0000 cents 3 x x x x 4 x x x 5 x x 6 S SD: 4.5983 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 7.6388 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 7.1882 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 10.7896 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 9.6941 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 9.2857 cents 9 x 10:11 x 12:13 x 14:15:16:17:18 S SD: 7.8858 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.2317 cents 11:12 x 13:14:15:16 x 17:18:20:21:22 S SD: 11.3177 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 6.1699 cents 13:14:15 x 16:17:18:20:21:22:23:25:26 SD: 9.6311 cents 14:15:16:17:18:19:20:21:22:24:25:26:28 S SD: 7.0912 cents 15:16:17:18:19:20:21:23:24:25:27:28:30 SD: 5.1569 cents 16:17:18:19:20:21:23:24:25:27:28:30:32 SD: 4.8177 cents 22:23:25:26:28:29:31:33:35:37:39:42:44 SD: 3.7660 cents 26:28:29:31:33:35:37:39:41:44:46:49:52 SD: 3.7429 cents 27:29:30:32:34:36:38:41:43:45:48:51:54 SD: 3.7345 cents 30:32:34:36:38:40:43:45:48:50:53:57:60 SD: 3.3246 cents 31:33:35:37:39:41:44:47:49:52:55:59:62 SD: 2.5488 cents 32:34:36:38:40:43:45:48:51:54:57:60:64 SD: 2.5437 cents 36:38:41:43:45:48:51:54:57:60:64:68:72 SD: 2.5057 cents 38:40:43:45:48:51:54:57:60:64:68:72:76 SD: 2.1860 cents 46:49:52:55:58:61:65:69:73:77:82:87:92 SD: 1.7416 cents 48:51:54:57:60:64:68:72:76:81:85:91:96 SD: 1.6903 cents 53:56:60:63:67:71:75:80:84:89:94:100:106 SD: 1.6409 cents 54:57:61:64:68:72:77:81:86:91:96:102:108 SD: 1.4544 cents 63:67:71:75:79:84:89:95:100:106:112:119:126 SD: 1.3819 cents 69:73:78:82:87:92:98:104:110:116:123:130:138 SD: 1.3779 cents 70:74:79:83:88:93:99:105:111:118:124:132:140 SD: 1.2895 cents 72:76:81:86:91:96:102:108:114:121:128:136:144 SD: 0.8377 cents 104:110:117:124:131:139:147:156:165:175:185:197:208 SD: 0.7025 cents 120:127:135:143:151:160:170:180:190:202:213:227:240 SD: 0.5832 cents 136:144:153:162:171:181:193:204:216:228:242:257:272 SD: 0.5753 cents 138:146:155:164:174:184:196:207:219:232:245:261:276 SD: 0.5328 cents 153:162:172:182:193:204:217:230:243:257:272:289:306 SD: 0.4282 cents 184:195:207:219:232:245:261:276:292:309:327:348:368 SD: 0.3581 cents 216:229:243:257:272:288:306:324:343:363:384:408:432 SD: 0.3211 cents 240:254:270:286:302:320:340:360:381:403:427:454:480 SD: 0.3086 cents 242:256:272:288:305:323:343:363:384:407:430:457:484 SD: 0.3041 cents 257:272:289:306:324:343:364:386:408:432:457:486:514 SD: 0.2980 cents 262:277:295:312:330:349:371:393:416:440:466:495:524 SD: 0.2873 cents 273:289:307:325:344:364:387:410:433:459:485:516:546 SD: 0.2718 cents 288:305:324:343:363:384:408:432:457:484:512:544:576 SD: 0.1828 cents 378:400:425:450:476:504:536:567:600:635:672:714:756 SD: 0.1661 cents 400:423:450:476:504:533:567:600:635:672:711:756:800 SD: 0.1489 cents 446:472:502:531:562:595:632:669:708:749:793:843:892 SD: 0.1270 cents 546:578:614:650:688:728:774:819:867:917:971:1032:1092 SD: 0.1241 cents 582:616:655:693:733:776:825:873:924:978:1035:1100:1164 SD: 0.1181 cents 584:618:657:695:736:779:828:876:927:981:1038:1104:1168 SD: 0.1137 cents 600:635:675:714:756:800:850:900:952:1008:1067:1134:1200 SD: 0.1111 cents