Scala analysis: LummaVRWT
Carl Lumma's (2,3,17,19) VRWT (transposed 1/2-step downward)
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 18/17 98.954592 Arabic lute index finger 2: 64/57 200.531983 3: 384/323 299.486575 4: 64/51 393.089590 5: 4/3 498.044999 perfect fourth 6: 24/17 596.999591 1st septendecimal tritone 7: 3/2 701.955001 perfect fifth 8: 512/323 797.531574 9: 57/34 894.512607 10: 576/323 1001.441576 11: 32/17 1095.044590 17th subharmonic 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 17/16 104.9554 17th harmonic 1: 18/17 98.9546 Arabic lute index finger 2: 544/513 101.5774 3: 18/17 98.9546 Arabic lute index finger 4: 19/18 93.6030 undevicesimal semitone 5: 17/16 104.9554 17th harmonic 6: 18/17 98.9546 Arabic lute index finger 7: 17/16 104.9554 17th harmonic 8: 1024/969 95.5766 9: 1083/1024 96.9810 10: 384/361 106.9290 11: 19/18 93.6030 undevicesimal semitone 12: 17/16 104.9554 17th harmonic
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 19/18 93.603 cents undevicesimal semitone 1: 1 1024/969 95.577 cents 1: 1 1083/1024 96.981 cents 1: 3 18/17 98.955 cents Arabic lute index finger 1: 1 544/513 101.577 cents 1: 3 17/16 104.955 cents 17th harmonic 1: 1 384/361 106.929 cents 2: 2 19/17 192.558 cents quasi-meantone 2: 2 323/288 198.558 cents 2: 4 64/57 200.532 cents 2: 4 9/8 203.910 cents major whole tone 3: 1 32/27 294.135 cents Pythagorean minor third 3: 5 19/16 297.513 cents 19th harmonic 3: 3 384/323 299.487 cents 3: 2 68/57 305.487 cents 3: 1 153/128 308.865 cents 4: 2 64/51 393.090 cents 4: 2 171/136 396.468 cents 4: 2 34/27 399.090 cents septendecimal major third 4: 2 323/256 402.468 cents 4: 4 24/19 404.442 cents smaller undevicesimal major third 5: 7 4/3 498.045 cents perfect fourth 5: 3 171/128 501.423 cents 5: 1 432/323 503.397 cents 5: 1 4352/3249 506.019 cents 6: 1 38/27 591.648 cents 6: 1 361/256 595.026 cents two (19th harmonic) 6: 4 24/17 597.000 cents 1st septendecimal tritone 6: 4 17/12 603.000 cents 2nd septendecimal tritone 6: 1 512/361 604.974 cents 6: 1 27/19 608.352 cents 7: 1 3249/2176 693.981 cents 7: 1 323/216 696.603 cents 7: 3 256/171 698.577 cents 7: 7 3/2 701.955 cents perfect fifth 8: 4 19/12 795.558 cents undevicesimal minor sixth 8: 2 512/323 797.532 cents 8: 2 27/17 800.910 cents septendecimal minor sixth 8: 2 272/171 803.532 cents 8: 2 51/32 806.910 cents 9: 1 256/153 891.135 cents 9: 2 57/34 894.513 cents 9: 3 323/192 900.513 cents 9: 5 32/19 902.487 cents 19th subharmonic 9: 1 27/16 905.865 cents Pythagorean major sixth 10: 4 16/9 996.090 cents Pythagorean minor seventh 10: 4 57/32 999.468 cents undevicesimal minor seventh 10: 2 576/323 1001.442 cents 10: 2 34/19 1007.442 cents quasi-mean seventh 11: 1 361/192 1093.071 cents 11: 3 32/17 1095.045 cents 17th subharmonic 11: 1 513/272 1098.423 cents 11: 3 17/9 1101.045 cents septendecimal major seventh 11: 1 2048/1083 1103.019 cents 11: 1 969/512 1104.423 cents 11: 2 36/19 1106.397 cents smaller undevicesimal major seventh Highest number of different intervals for one interval class: 7 Average number of different intervals per interval class: 5.09091 = 56/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 99.0 200.5 299.5 393.1 498.0 597.0 702.0 797.5 894.5 1001.4 1095.0 1200.0 99.0 : 101.6 200.5 294.1 399.1 498.0 603.0 698.6 795.6 902.5 996.1 1101.0 1200.0 200.5 : 99.0 192.6 297.5 396.5 501.4 597.0 694.0 800.9 894.5 999.5 1098.4 1200.0 299.5 : 93.6 198.6 297.5 402.5 498.0 595.0 702.0 795.6 900.5 999.5 1101.0 1200.0 393.1 : 105.0 203.9 308.9 404.4 501.4 608.4 702.0 806.9 905.9 1007.4 1106.4 1200.0 498.0 : 99.0 203.9 299.5 396.5 503.4 597.0 702.0 800.9 902.5 1001.4 1095.0 1200.0 597.0 : 105.0 200.5 297.5 404.4 498.0 603.0 702.0 803.5 902.5 996.1 1101.0 1200.0 702.0 : 95.6 192.6 299.5 393.1 498.0 597.0 698.6 797.5 891.1 996.1 1095.0 1200.0 797.5 : 97.0 203.9 297.5 402.5 501.4 603.0 702.0 795.6 900.5 999.5 1104.4 1200.0 894.5 : 106.9 200.5 305.5 404.4 506.0 605.0 698.6 803.5 902.5 1007.4 1103.0 1200.0 1001.4: 93.6 198.6 297.5 399.1 498.0 591.6 696.6 795.6 900.5 996.1 1093.1 1200.0 1095.0: 105.0 203.9 305.5 404.4 498.0 603.0 702.0 806.9 902.5 999.5 1106.4 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 93.6030 .. 106.9290 cents 6912/6859, 13.3260 cents 19/18, 93.6030 cents 2: 192.5576 .. 203.9100 cents 153/152, 11.3524 cents 6859/6528, 85.6286 cents 3: 294.1350 .. 308.8654 cents 4131/4096, 14.7304 cents 256/243, 90.2250 cents 4: 393.0896 .. 404.4420 cents 153/152, 11.3524 cents 8192/7803, 84.2242 cents 5: 498.0450 .. 506.0194 cents 1088/1083, 7.9744 cents 19/18, 93.6030 cents 6: 591.6480 .. 608.3520 cents 729/722, 16.7040 cents 6859/6528, 85.6286 cents 7: 693.9806 .. 701.9550 cents 1088/1083, 7.9744 cents 6859/6528, 85.6286 cents 8: 795.5580 .. 806.9104 cents 153/152, 11.3524 cents 19/18, 93.6030 cents 9: 891.1346 .. 905.8650 cents 4131/4096, 14.7304 cents 8192/7803, 84.2242 cents 10: 996.0900 .. 1007.4424 cents 153/152, 11.3524 cents 256/243, 90.2250 cents 11: 1093.0710 .. 1106.3970 cents 6912/6859, 13.3260 cents 6859/6528, 85.6286 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 19/18, 93.6030 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 384/361, 106.9290 cents Average / Smallest step : 1.068342 Largest / Average step : 1.069290 Largest / Smallest step : 1.142367 Median interval of one step : 18/17, 98.9546 cents, amount: 3 Least squares average step : 99.76541 cents, oct.: 1197.18492 cents Scale is strictly proper Step pattern alph. order: ABACDADEFGCD Step pattern size order : DEDAFDFBCGAF Scale is a Constant Structure, by a margin of 84.22418 cents Scale diversity : 1.092686 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.888187 Rothenberg efficiency : 0.294907 redundancy: 0.705093 Efficiency x scale size : 3.538889 Number of different interval sizes : 56 = 5.09091 / class Number of one step interval sizes : 7 Highest interval variety : 7 Mean interval variety : 5.09091 = 56/11 Median interval variety : 4 Lowest interval variety : 4 Smallest interval difference : 1049427/1048576, 1.4045 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 7 Most common triad is 0.0 498.045 701.955 cents, amount: 4 Number of recognisable fifths : 12, average 700.0000 cents Scale is a well-temperament. It has 4 different sizes of fifth Scale has monotonic third-sizes over circle of fifths Best fifths form a closed circle Scale contains a complete diamond : 1 3 Length of longest harmonic part : 2 at degree 2 Length of longest subharmonic part : 2 at degree 5 Formal octave complements present : 3 = 25.0000% Limited transpositions with margin 11.3524 cents: 1 2 3 4 5 6 7 8 9 10 11 Limited inverse transpositions with margin 11.3524 cents: 1 2 3 4 5 6 7 8 9 10 11 -- Rational properties -- Prime limit : 19 Odd number limit : 3249 (O: 3249 U: 3249) Highest odd numerator or denominator: 323 Scale harmonicity : 0.008596 Average absolute harmonicity : 0.139598 Specific harmonicity : 0.049758 Fundamental : 1/1938, -10.9204 octaves, 0.1350 Hz. Guide tone : 87552, 16.4179 octaves, 22905841.493 Hz. Exponens Consonantiae : 1.696758E+08, 27.33821 octaves Euler's gradus suavitatis : 69 Sum of Mann's harmonic distance : 1454.0, average 121.16667 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 17.39252, average 1.44938 Sum of Tenney's harmonic distance : 35.93513, average 2.99459 Vogel's harmonic complexity : 28.83333 Wille's k value : 1624 Wilson's harmonic complexity : 64 Rectangular lattice diameter : 6 Triangular lattice diameter : 4 Lattice compactness : 349.12899, average 4.47601 Lattice compactness (without 2's) : 134.16945, average 1.72012 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -1 .. 9 3.66667 48 1 3: -1 .. 2 0.41667 11 2 17: -1 .. 0 -0.66667 8 4 19: -1 .. 1 -0.25000 5 5 Average exponent except 2's :-6 / 12 =-0.50000 Average absolute exponent except 2's: 24 / 12 = 2.00000 Scale is JI-epimorphic with val: <12 19 49 51| = patent
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 3.2772 c. M:-6.9104 c. Twelve-tone Chromatic (1/11-comma) 77: 6 7 6 6 7 6 7 6 6 7 6 7 SP M ME SD: 3.6585 c. M: 6.2009 c. 80: 7 6 7 6 7 7 7 6 7 7 6 7 SP D3 SD: 3.5788 c. M:-6.0454 c. 89: 7 8 7 7 8 7 8 7 7 8 7 8 SP M ME SD: 2.8764 c. M: 4.6250 c. 101: 8 9 8 8 9 8 9 8 8 9 8 9 SP M ME SD: 2.3178 c. M: 3.9051 c. 113: 9 10 9 9 10 9 10 9 9 10 9 10 SP M ME SD: 1.9219 c. M: 3.3794 c. 125: 10 11 10 10 11 10 11 10 10 11 10 11 SP M ME SD: 1.6516 c. M: 3.0416 c. 137: 11 12 11 11 12 11 12 11 11 12 11 12 SP M ME SD: 1.4810 c. M: 2.9014 c. 149: 12 13 12 12 13 12 13 12 12 13 12 13 SP M ME SD: 1.3886 c. M: 2.7839 c. 157: 13 13 13 12 14 13 14 12 13 14 12 14 SP SD: 1.6083 c. M: 3.2807 c. 161: 13 14 13 13 14 13 14 13 13 14 13 14 SP M ME SD: 1.3543 c. M: 2.6838 c. 169: 14 14 14 13 15 14 15 13 14 15 13 15 SP SD: 1.3302 c. M: 2.5570 c. 181: 15 15 15 14 16 15 16 14 15 16 14 16 SP SD: 1.1106 c. M: 1.9515 c. 193: 16 16 16 15 17 16 17 15 16 17 15 17 SP SD: 0.9466 c. M: 1.6767 c. 205: 17 17 17 16 18 17 18 16 17 18 16 18 SP SD: 0.8370 c. M: 1.5076 c. 217: 18 18 18 17 19 18 19 17 18 19 17 19 SP SD: 0.7792 c. M: 1.4536 c. 229: 19 19 19 18 20 19 20 18 19 20 18 20 SP SD: 0.7668 c. M:-1.5573 c. 306: 25 26 25 24 27 25 27 24 25 27 24 27 SP SD: 0.9192 c. M: 1.4531 c. 318: 26 27 26 25 28 26 28 25 26 28 25 28 SP SD: 0.8265 c. M: 1.4416 c. 330: 27 28 27 26 29 27 29 26 27 29 26 29 SP SD: 0.7554 c. M: 1.4416 c. 339: 28 29 28 26 30 28 29 27 28 30 26 30 SP SD: 0.9690 c. M:-1.3984 c. 340: 28 29 28 26 30 28 30 27 27 31 26 30 SP SD: 0.7757 c. M: 1.5714 c. 342: 28 29 28 27 30 28 30 27 28 30 27 30 SP SD: 0.7060 c. M: 1.4416 c. 343: 28 29 29 26 30 29 30 27 28 30 27 30 SP SD: 1.0177 c. M:-1.3881 c. 344: 28 29 29 27 30 28 30 28 27 31 27 30 SP SD: 0.9816 c. M: 1.6948 c. 345: 28 30 28 27 30 29 30 27 28 31 27 30 SP SD: 0.8323 c. M: 1.5633 c. 351: 29 30 29 27 31 29 30 28 29 31 27 31 SP SD: 0.9527 c. M:-1.3681 c. 352: 29 30 29 27 31 29 31 28 28 32 27 31 SP SD: 0.6535 c. M: 1.3308 c. 364: 30 31 30 28 32 30 32 29 29 33 28 32 SP SD: 0.5450 c. M: 1.1060 c. 376: 31 32 31 29 33 31 33 30 30 34 29 33 SP SD: 0.4520 c. M: 0.8956 c. 388: 32 33 32 30 34 32 34 31 31 35 30 34 SP SD: 0.3780 c. M: 0.6982 c. 400: 33 34 33 31 35 33 35 32 32 36 31 35 SP SD: 0.3282 c. M:-0.5584 c. 412: 34 35 34 32 36 34 36 33 33 37 32 36 SP SD: 0.3075 c. M:-0.5267 c. 424: 35 36 35 33 37 35 37 34 34 38 33 37 SP SD: 0.3161 c. M:-0.5816 c. 436: 36 37 36 34 38 36 38 35 35 39 34 38 SP SD: 0.3475 c. M:-0.6336 c. 581: 48 49 48 45 51 48 51 46 47 52 45 51 SP SD: 0.2914 c. M: 0.6627 c. 593: 49 50 49 46 52 49 52 47 48 53 46 52 SP SD: 0.2318 c. M: 0.5095 c. 605: 50 51 50 47 53 50 53 48 49 54 47 53 SP SD: 0.1828 c. M: 0.3623 c. 617: 51 52 51 48 54 51 54 49 50 55 48 54 SP SD: 0.1508 c. M:-0.2350 c. 629: 52 53 52 49 55 52 55 50 51 56 49 55 SP SD: 0.1437 c. M:-0.2505 c. 1029: 85 87 85 80 90 85 90 82 83 92 80 90 SP SD: 0.1373 c. M:-0.3077 c. 1041: 86 88 86 81 91 86 91 83 84 93 81 91 SP SD: 0.1353 c. M:-0.2875 c. 1053: 87 89 87 82 92 87 92 84 85 94 82 92 SP SD: 0.1435 c. M:-0.2678 c. 1142: 94 97 94 89 100 94 100 91 92 102 89 100 SP SD: 0.1282 c. M: 0.2919 c. 1154: 95 98 95 90 101 95 101 92 93 103 90 101 SP SD: 0.1069 c. M: 0.2318 c. 1166: 96 99 96 91 102 96 102 93 94 104 91 102 SP SD: 0.0943 c. M: 0.1730 c. 1178: 97 100 97 92 103 97 103 94 95 105 92 103 SP SD: 0.0933 c. M:-0.1471 c. 1359: 112 115 112 106 119 112 119 108 110 121 106 119 SP SD: 0.1039 c. M: 0.1806 c. 1371: 113 116 113 107 120 113 120 109 111 122 107 120 SP SD: 0.0876 c. M: 0.1574 c. 1383: 114 117 114 108 121 114 121 110 112 123 108 121 SP SD: 0.0792 c. M: 0.1401 c. 1759: 145 149 145 137 154 145 154 140 142 157 137 154 SP SD: 0.0720 c. M: 0.1408 c. 1771: 146 150 146 138 155 146 155 141 143 158 138 155 SP SD: 0.0499 c. M: 0.1027 c. 1783: 147 151 147 139 156 147 156 142 144 159 139 156 SP SD: 0.0289 c. M: 0.0650 c. 1795: 148 152 148 140 157 148 157 143 145 160 140 157 SP SD: 0.0135 c. M: 0.0279 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: 3.3846 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 8.3280 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 6.3018 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 9.5069 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 10.4828 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 8.9290 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 8.0251 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.0848 cents 11 x 12:13:14:15:16 x 17:18:20:21:22 S SD: 11.3059 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 6.3403 cents 13:14:15 x 16:17:18:20:21:22:23:24:26 S SD: 9.5583 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 7.3738 cents 15:16:17:18:19:20:21:23:24:25:27:28:30 SD: 4.8868 cents 16:17:18:19:20:21:23:24:25:27:29:30:32 SD: 4.8629 cents 22:23:25:26:28:29:31:33:35:37:39:41:44 SD: 4.1622 cents 24:25:27:29:30:32:34:36:38:40:43:45:48 SD: 3.6359 cents 27:29:30:32:34:36:38:41:43:45:48:51:54 SD: 3.4559 cents 31:33:35:37:39:41:44:47:49:52:55:58:62 SD: 2.7418 cents 32:34:36:38:40:43:45:48:51:54:57:60:64 SD: 2.0264 cents 46:49:52:55:58:61:65:69:73:77:82:87:92 SD: 2.0090 cents 48:51:54:57:60:64:68:72:76:80:86:90:96 SD: 1.5637 cents 58:61:65:69:73:77:82:87:92:97:103:109:116 SD: 1.4789 cents 63:67:71:75:79:84:89:95:100:106:112:119:126 SD: 1.4451 cents 66:70:74:78:83:88:93:99:105:111:118:124:132 SD: 1.3046 cents 68:72:76:81:85:91:96:102:108:114:121:128:136 SD: 1.1391 cents 80:85:90:95:100:107:113:120:127:134:143:151:160 SD: 1.1032 cents 84:89:94:100:105:112:119:126:133:141:150:158:168 SD: 0.9802 cents 90:95:101:107:113:120:127:135:143:151:160:169:180 SD: 0.8159 cents 102:108:115:121:128:136:144:153:162:171:182:192:204 SD: 0.7342 cents 106:112:119:126:133:141:150:159:168:178:189:200:212 SD: 0.7056 cents 121:128:136:144:152:161:171:182:192:203:216:228:242 SD: 0.6560 cents 137:145:154:163:172:183:193:206:217:230:244:258:274 SD: 0.6519 cents 138:146:155:164:173:184:195:207:219:231:246:260:276 SD: 0.3917 cents 204:216:229:243:256:272:288:306:323:342:364:384:408 SD: 0.3382 cents 238:252:267:283:299:317:336:357:377:399:424:448:476 SD: 0.3089 cents 254:269:285:302:319:339:359:381:403:426:453:478:508 SD: 0.3077 cents 260:275:292:309:326:347:367:390:412:436:464:489:520 SD: 0.3032 cents 270:286:303:321:339:360:381:405:428:453:481:508:540 SD: 0.2401 cents 306:324:344:364:384:408:432:459:485:513:546:576:612 SD: 0.2131 cents 318:337:357:378:399:424:449:477:504:533:567:599:636 SD: 0.1713 cents 408:432:458:485:512:544:576:612:647:684:728:768:816 SD: 0.1084 cents