Scala analysis: ratwell
7-limit rational well-temperament
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 256/243 90.224996 limma, Pythagorean minor second 2: 28/25 196.198479 middle second 3: 32/27 294.134997 Pythagorean minor third 4: 175616/140625 384.685434 5: 4/3 498.044999 perfect fourth 6: 1024/729 588.269995 Pythagorean diminished fifth 7: 3/2 701.955001 perfect fifth 8: 128/81 792.179997 Pythagorean minor sixth 9: 3136/1875 890.441957 10: 16/9 996.089998 Pythagorean minor seventh 11: 4096/2187 1086.314994 Pythagorean diminished octave 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 2187/2048 113.6850 apotome 1: 256/243 90.2250 limma, Pythagorean minor second 2: 1701/1600 105.9735 3: 200/189 97.9365 4: 16464/15625 90.5504 5: 46875/43904 113.3596 6: 256/243 90.2250 limma, Pythagorean minor second 7: 2187/2048 113.6850 apotome 8: 256/243 90.2250 limma, Pythagorean minor second 9: 1323/1250 98.2620 10: 625/588 105.6480 11: 256/243 90.2250 limma, Pythagorean minor second 12: 2187/2048 113.6850 apotome
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 4 256/243 90.225 cents limma, Pythagorean minor second 1: 1 16464/15625 90.550 cents 1: 1 200/189 97.937 cents 1: 1 1323/1250 98.262 cents 1: 1 625/588 105.648 cents 1: 1 1701/1600 105.973 cents 1: 1 46875/43904 113.360 cents 1: 2 2187/2048 113.685 cents apotome 2: 2 6272/5625 188.487 cents 2: 1 40000/35721 195.873 cents 2: 1 28/25 196.198 cents middle second 2: 1 31250/27783 203.585 cents 2: 7 9/8 203.910 cents major whole tone 3: 6 32/27 294.135 cents Pythagorean minor third 3: 1 18522/15625 294.460 cents 3: 1 25/21 301.847 cents BP second, quasi-equal minor third 3: 1 11907/10000 302.172 cents 3: 1 1875/1568 309.558 cents 3: 1 15309/12800 309.883 cents 3: 1 421875/351232 317.270 cents 4: 1 8192/6561 384.360 cents Pythagorean diminished fourth 4: 1 175616/140625 384.685 cents 4: 1 6400/5103 392.072 cents 4: 1 784/625 392.397 cents two (middle second) 4: 1 5000/3969 399.783 cents 4: 1 63/50 400.108 cents quasi-equal major third 4: 1 15625/12348 407.495 cents 4: 5 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: 3 1024/729 588.270 cents Pythagorean diminished fifth 6: 1 21952/15625 588.595 cents 6: 1 800/567 595.982 cents 6: 1 882/625 596.307 cents 6: 1 625/441 603.693 cents two (BP second, quasi-equal minor third) 6: 1 567/400 604.018 cents 6: 1 15625/10976 611.405 cents 6: 3 729/512 611.730 cents Pythagorean tritone 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: 5 128/81 792.180 cents Pythagorean minor sixth 8: 1 24696/15625 792.505 cents 8: 1 100/63 799.892 cents quasi-equal minor sixth 8: 1 3969/2500 800.217 cents two (quasi-equal major third) 8: 1 625/392 807.603 cents 8: 1 5103/3200 807.928 cents 8: 1 140625/87808 815.315 cents 8: 1 6561/4096 815.640 cents Pythagorean augmented fifth 9: 1 702464/421875 882.730 cents 9: 1 25600/15309 890.117 cents 9: 1 3136/1875 890.442 cents 9: 1 20000/11907 897.828 cents 9: 1 42/25 898.153 cents quasi-equal major sixth 9: 1 15625/9261 905.540 cents 9: 6 27/16 905.865 cents Pythagorean major sixth 10: 7 16/9 996.090 cents Pythagorean minor seventh 10: 1 27783/15625 996.415 cents 10: 1 25/14 1003.802 cents middle minor seventh 10: 1 35721/20000 1004.127 cents 10: 2 5625/3136 1011.513 cents two (marvelous fourth) 11: 2 4096/2187 1086.315 cents Pythagorean diminished octave 11: 1 87808/46875 1086.640 cents 11: 1 3200/1701 1094.027 cents 11: 1 1176/625 1094.352 cents 11: 1 2500/1323 1101.738 cents 11: 1 189/100 1102.063 cents 11: 1 15625/8232 1109.450 cents 11: 4 243/128 1109.775 cents Pythagorean major seventh Highest number of different intervals for one interval class: 8 Average number of different intervals per interval class: 6.36364 = 70/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 90.2 196.2 294.1 384.7 498.0 588.3 702.0 792.2 890.4 996.1 1086.3 1200.0 90.2 : 106.0 203.9 294.5 407.8 498.0 611.7 702.0 800.2 905.9 996.1 1109.8 1200.0 196.2 : 97.9 188.5 301.8 392.1 505.8 596.0 694.2 799.9 890.1 1003.8 1094.0 1200.0 294.1 : 90.6 203.9 294.1 407.8 498.0 596.3 702.0 792.2 905.9 996.1 1102.1 1200.0 384.7 : 113.4 203.6 317.3 407.5 505.8 611.4 701.6 815.3 905.5 1011.5 1109.4 1200.0 498.0 : 90.2 203.9 294.1 392.4 498.0 588.3 702.0 792.2 898.2 996.1 1086.6 1200.0 588.3 : 113.7 203.9 302.2 407.8 498.0 611.7 702.0 807.9 905.9 996.4 1109.8 1200.0 702.0 : 90.2 188.5 294.1 384.4 498.0 588.3 694.2 792.2 882.7 996.1 1086.3 1200.0 792.2 : 98.3 203.9 294.1 407.8 498.0 604.0 702.0 792.5 905.9 996.1 1109.8 1200.0 890.4 : 105.6 195.9 309.6 399.8 505.8 603.7 694.2 807.6 897.8 1011.5 1101.7 1200.0 996.1 : 90.2 203.9 294.1 400.1 498.0 588.6 702.0 792.2 905.9 996.1 1094.4 1200.0 1086.3: 113.7 203.9 309.9 407.8 498.4 611.7 702.0 815.6 905.9 1004.1 1109.8 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.2250 .. 113.6850 cents 531441/524288, 23.4600 cents 256/243, 90.2250 cents 2: 188.4870 .. 203.9100 cents 50625/50176, 15.4230 cents 12845056/12301875, 74.8019 cents 3: 294.1350 .. 317.2696 cents 11390625/11239424, 23.1346 cents 256/243, 90.2250 cents 4: 384.3600 .. 407.8200 cents 531441/524288, 23.4600 cents 67.09043 cents 5: 498.0450 .. 505.7565 cents 225/224, 7.7115 cents 256/243, 90.2250 cents 6: 588.2700 .. 611.7300 cents 531441/524288, 23.4600 cents 57344/54675, 82.5135 cents 7: 694.2435 .. 701.9550 cents 225/224, 7.7115 cents 57344/54675, 82.5135 cents 8: 792.1800 .. 815.6400 cents 531441/524288, 23.4600 cents 256/243, 90.2250 cents 9: 882.7304 .. 905.8650 cents 11390625/11239424, 23.1346 cents 67.09043 cents 10: 996.0900 .. 1011.5130 cents 50625/50176, 15.4230 cents 256/243, 90.2250 cents 11: 1086.3150 .. 1109.7750 cents 531441/524288, 23.4600 cents 12845056/12301875, 74.8019 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 256/243, 90.2250 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 2187/2048, 113.6850 cents Average / Smallest step : 1.108340 Largest / Average step : 1.136850 Largest / Smallest step : 1.260017 Median interval of one step : 98.09924 cents, amount: 0 Most common interval of one step : 256/243, 90.2250 cents, amount: 4 Least squares average step : 99.22936 cents, oct.: 1190.75227 cents Scale is strictly proper Step pattern alph. order: ABCDEAFAGHAF Step pattern size order : AFCBGAHADEAH Scale is a Constant Structure, by a margin of 67.09043 cents Scale diversity : 1.142917 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.825135 Rothenberg efficiency : 0.280682 redundancy: 0.719318 Efficiency x scale size : 3.368182 Number of different interval sizes : 70 = 6.36364 / class Number of one step interval sizes : 8 Highest interval variety : 8 Mean interval variety : 6.36364 = 70/11 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: 7 Number of recognisable fifths : 12, average 700.0000 cents Best fifths form a closed circle Scale contains a complete diamond : 1 3 Formal octave complements present : 3 = 25.0000% Limited transpositions with margin 23.4600 cents: 2 5 7 9 10 Limited inverse transpositions with margin 23.4600 cents: 1 2 4 6 9 11 Inversional symmetry on intervals : 5-6 11-12 -- Rational properties -- Prime limit : 7 Odd number limit : 421875 (O: 421875 U: 421875) Highest odd numerator or denominator: 140625 Scale harmonicity : 0.009653 Average absolute harmonicity : 0.159640 Specific harmonicity : 0.068398 Fundamental : 1/68343750, -26.0263 octaves, 0.0000 Hz. Guide tone : 4214784, 22.0070 octaves, 1102695246.622 Hz. Exponens Consonantiae : 2.880541E+14, 48.03333 octaves Euler's gradus suavitatis : 72 Sum of Mann's harmonic distance : 165074.0, average 13756.16667 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 23.65552, average 1.97129 Sum of Tenney's harmonic distance : 48.89512, average 4.07459 Vogel's harmonic complexity : 48.41667 Wille's k value : 51257812 Wilson's harmonic complexity : 75 Rectangular lattice diameter : 17 Triangular lattice diameter : 9 Lattice compactness : 516.11457, average 6.61685 Lattice compactness (without 2's) : 324.46328, average 4.15979 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -1 .. 12 5.41667 67 1 3: -7 .. 1 -2.50000 32 8 5: -6 .. 0 -1.00000 12 7: 0 .. 3 0.50000 6 3 Average exponent except 2's :-36 / 12 =-3.00000 Average absolute exponent except 2's: 50 / 12 = 4.16667 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: 8.5598 c. M:-15.3146 c. Twelve-tone Chromatic (1/11-comma) 41: 3 4 3 3 4 3 4 3 3 4 3 4 SP M ME SD: 4.8268 c. M: 12.3932 c. Schismic-12 45: 3 4 4 3 5 3 4 4 3 4 4 4 SP D3 SD: 8.0850 c. M: 11.3521 c. 49: 4 4 4 4 4 4 5 3 4 5 3 5 SP D3 SD: 6.6964 c. M: 8.8093 c. 50: 4 4 4 4 5 4 4 4 4 5 3 5 SP D3 SD: 6.3595 c. M:-11.9100 c. 51: 4 4 4 4 5 4 5 4 4 4 4 5 SP D3 SD: 6.2435 c. M: 11.7821 c. 53: 4 5 4 4 5 4 5 4 4 5 4 5 SP M ME SD: 3.0712 c. M:-7.5751 c. Schismic-12 65: 5 6 5 5 6 5 6 5 5 6 5 6 SP M ME SD: 2.8753 c. M:-6.8784 c. 77: 6 7 6 6 7 6 7 6 6 7 6 7 SP M ME SD: 3.2974 c. M:-6.3989 c. 82: 6 7 7 6 8 6 8 6 7 7 6 8 SP D3 SD: 2.7558 c. M: 5.9546 c. 93: 7 8 8 7 9 7 8 7 8 8 7 9 SP SD: 3.4113 c. M:-5.2784 c. 94: 7 8 8 7 9 7 9 7 8 8 7 9 SP D3 SD: 1.8114 c. M: 4.7091 c. 102: 8 9 8 8 9 8 10 7 9 9 7 10 SP S SD: 3.3306 c. M: 3.9621 c. 104: 8 9 8 8 10 8 10 8 8 9 8 10 SP SD: 2.8279 c. M: 5.6735 c. 106: 8 9 9 8 10 8 10 8 9 9 8 10 SP D3 SD: 1.5792 c. M:-3.8977 c. 133: 10 12 11 10 12 10 13 10 11 11 10 13 SP SD: 2.5158 c. M:-3.6094 c. 135: 10 12 11 10 13 10 13 10 11 12 10 13 SP SD: 1.2570 c. M: 2.4632 c. 147: 11 13 12 11 14 11 14 11 12 13 11 14 SP SD: 0.4617 c. M: 1.0120 c. 159: 12 14 13 12 15 12 15 12 13 14 12 15 SP SD: 0.2451 c. M:-0.4775 c. 306: 23 27 25 23 29 23 29 23 25 27 23 29 SP SD: 0.1347 c. M: 0.3717 c. 465: 35 41 38 35 44 35 44 35 38 41 35 44 SP SD: 0.0919 c. M: 0.1693 c. 1701: 128 150 139 128 161 128 161 128 139 150 128 161 SP SD: 0.0914 c. M: 0.2057 c. 1877: 141 166 153 142 177 141 178 141 154 165 141 178 SP SD: 0.0879 c. M:-0.1840 c. 2019: 152 178 165 152 191 152 191 152 165 178 152 191 SP SD: 0.0957 c. M:-0.1635 c. 2024: 152 179 165 153 191 152 192 152 166 178 152 192 SP SD: 0.0820 c. M: 0.1490 c. 2036: 153 180 166 154 192 153 193 153 167 179 153 193 SP SD: 0.0759 c. M:-0.1869 c. 2171: 163 192 177 164 205 163 206 163 178 191 163 206 SP SD: 0.0887 c. M: 0.1796 c. 2183: 164 193 178 165 206 164 207 164 179 192 164 207 SP SD: 0.0644 c. M:-0.1061 c. 2330: 175 206 190 176 220 175 221 175 191 205 175 221 SP SD: 0.0677 c. M: 0.1347 c. 2342: 176 207 191 177 221 176 222 176 192 206 176 222 SP SD: 0.0524 c. M:-0.1139 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: 2.0277 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 6.8282 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 7.5385 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 7.7204 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 11.7340 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 8.3966 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 7.7895 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.5693 cents 11 x 12:13:14:15 x x 17:18:20:21:22 SD: 11.2336 cents 12:13 x 14:15:16:17:18:19:20:21:22:24 S SD: 6.7160 cents 13:14 x 15:16:17:18:20:21:22:23:24:26 S SD: 9.2981 cents 14:15:16:17 x 19:20:21:22:23:25:26:28 SD: 7.5846 cents 15:16:17:18:19:20:21:23:24:25:27:28:30 SD: 5.5429 cents 16:17:18:19:20:21:22:24:25:27:28:30:32 SD: 5.1144 cents 17:18:19:20:21:23:24:26:27:28:30:32:34 SD: 4.9126 cents 22:23:25:26:27:29:31:33:35:37:39:41:44 SD: 4.1545 cents 24:25:27:28:30:32:34:36:38:40:43:45:48 SD: 3.3732 cents 32:34:36:38:40:43:45:48:51:54:57:60:64 SD: 2.5630 cents 33:35:37:39:41:44:46:50:52:55:59:62:66 SD: 2.5116 cents 36:38:40:43:45:48:51:54:57:60:64:67:72 SD: 2.3131 cents 40:42:45:47:50:53:56:60:63:67:71:75:80 SD: 1.9009 cents 48:51:54:57:60:64:67:72:76:80:85:90:96 SD: 1.8780 cents 52:55:58:62:65:69:73:78:82:87:92:97:104 SD: 1.6890 cents 57:60:64:68:71:76:80:86:90:95:101:107:114 SD: 1.5831 cents 59:62:66:70:74:79:83:89:93:99:105:111:118 SD: 1.5663 cents 60:63:67:71:75:80:84:90:95:100:107:112:120 SD: 1.2181 cents 72:76:81:85:90:96:101:108:114:120:128:135:144 SD: 1.1090 cents 76:80:85:90:95:101:107:114:120:127:135:142:152 SD: 0.7661 cents 108:114:121:128:135:144:152:162:171:181:192:202:216 SD: 0.6176 cents 126:133:141:149:157:168:177:189:199:211:224:236:252 SD: 0.5841 cents 134:141:150:159:167:179:188:201:212:224:238:251:268 SD: 0.5533 cents 150:158:168:178:187:200:211:225:237:251:267:281:300 SD: 0.4201 cents 168:177:188:199:210:224:236:252:265:281:299:315:336 SD: 0.4015 cents 190:200:213:225:237:253:267:285:300:318:338:356:380 SD: 0.3892 cents 210:221:235:249:262:280:295:315:332:351:373:393:420 SD: 0.3234 cents 242:255:271:287:302:323:340:363:382:405:430:453:484 SD: 0.2965 cents 260:274:291:308:325:347:365:390:411:435:462:487:520 SD: 0.2667 cents 282:297:316:334:352:376:396:423:446:472:501:528:564 SD: 0.2436 cents 302:318:338:358:377:403:424:453:477:505:537:566:604 SD: 0.2354 cents 314:331:352:372:392:419:441:471:496:525:558:588:628 SD: 0.2314 cents 324:341:363:384:405:432:455:486:512:542:576:607:648 SD: 0.2108 cents 378:398:423:448:472:504:531:567:597:632:672:708:756 SD: 0.1760 cents 410:432:459:486:512:547:576:615:648:686:729:768:820 SD: 0.1328 cents 450:474:504:533:562:600:632:675:711:753:800:843:900 SD: 0.1252 cents 486:512:544:576:607:648:683:729:768:813:864:910:972 SD: 0.1199 cents 558:588:625:661:697:744:784:837:882:933:992:1045:1116 SD: 0.1107 cents 598:630:670:709:747:797:840:897:945:1000:1063:1120:1196 SD: 0.1073 cents