Scala analysis: 12_lumma_vrwt
[2 3 17 19] well temperament.
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 19/18 93.603014 undevicesimal semitone 2: 323/288 198.558424 3: 19/16 297.513016 19th harmonic 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: 19/12 795.558015 undevicesimal minor sixth 9: 57/34 894.512607 10: 16/9 996.089998 Pythagorean minor seventh 11: 32/17 1095.044590 17th subharmonic 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 17/16 104.9554 17th harmonic 1: 19/18 93.6030 undevicesimal semitone 2: 17/16 104.9554 17th harmonic 3: 18/17 98.9546 Arabic lute index finger 4: 1024/969 95.5766 5: 17/16 104.9554 17th harmonic 6: 18/17 98.9546 Arabic lute index finger 7: 17/16 104.9554 17th harmonic 8: 19/18 93.6030 undevicesimal semitone 9: 18/17 98.9546 Arabic lute index finger 10: 544/513 101.5774 11: 18/17 98.9546 Arabic lute index finger 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: 4 18/17 98.955 cents Arabic lute index finger 1: 1 544/513 101.577 cents 1: 4 17/16 104.955 cents 17th harmonic 2: 1 19/17 192.558 cents quasi-meantone 2: 1 6144/5491 194.531 cents 2: 3 323/288 198.558 cents 2: 3 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: 4 19/16 297.513 cents 19th harmonic 3: 4 384/323 299.487 cents 3: 1 5491/4608 303.514 cents 3: 1 68/57 305.487 cents 3: 1 153/128 308.865 cents 4: 2 64/51 393.090 cents 4: 1 171/136 396.468 cents 4: 1 6912/5491 398.441 cents 4: 2 34/27 399.090 cents septendecimal major third 4: 3 323/256 402.468 cents 4: 3 24/19 404.442 cents smaller undevicesimal major third 5: 7 4/3 498.045 cents perfect fourth 5: 2 171/128 501.423 cents 5: 2 432/323 503.397 cents 5: 1 289/216 504.046 cents 6: 1 38/27 591.648 cents 6: 5 24/17 597.000 cents 1st septendecimal tritone 6: 5 17/12 603.000 cents 2nd septendecimal tritone 6: 1 27/19 608.352 cents 7: 1 432/289 695.954 cents 7: 2 323/216 696.603 cents 7: 2 256/171 698.577 cents 7: 7 3/2 701.955 cents perfect fifth 8: 3 19/12 795.558 cents undevicesimal minor sixth 8: 3 512/323 797.532 cents 8: 2 27/17 800.910 cents septendecimal minor sixth 8: 1 5491/3456 801.559 cents 8: 1 272/171 803.532 cents 8: 2 51/32 806.910 cents 9: 1 256/153 891.135 cents 9: 1 57/34 894.513 cents 9: 1 9216/5491 896.486 cents 9: 4 323/192 900.513 cents 9: 4 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: 3 57/32 999.468 cents undevicesimal minor seventh 10: 3 576/323 1001.442 cents 10: 1 5491/3072 1005.469 cents 10: 1 34/19 1007.442 cents quasi-mean seventh 11: 4 32/17 1095.045 cents 17th subharmonic 11: 1 513/272 1098.423 cents 11: 4 17/9 1101.045 cents septendecimal major seventh 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: 6 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 : 93.6 198.6 297.5 393.1 498.0 597.0 702.0 795.6 894.5 996.1 1095.0 1200.0 93.6 : 105.0 203.9 299.5 404.4 503.4 608.4 702.0 800.9 902.5 1001.4 1106.4 1200.0 198.6 : 99.0 194.5 299.5 398.4 503.4 597.0 696.0 797.5 896.5 1001.4 1095.0 1200.0 297.5 : 95.6 200.5 299.5 404.4 498.0 597.0 698.6 797.5 902.5 996.1 1101.0 1200.0 393.1 : 105.0 203.9 308.9 402.5 501.4 603.0 702.0 806.9 900.5 1005.5 1104.4 1200.0 498.0 : 99.0 203.9 297.5 396.5 498.0 597.0 702.0 795.6 900.5 999.5 1095.0 1200.0 597.0 : 105.0 198.6 297.5 399.1 498.0 603.0 696.6 801.6 900.5 996.1 1101.0 1200.0 702.0 : 93.6 192.6 294.1 393.1 498.0 591.6 696.6 795.6 891.1 996.1 1095.0 1200.0 795.6 : 99.0 200.5 299.5 404.4 498.0 603.0 702.0 797.5 902.5 1001.4 1106.4 1200.0 894.5 : 101.6 200.5 305.5 399.1 504.0 603.0 698.6 803.5 902.5 1007.4 1101.0 1200.0 996.1 : 99.0 203.9 297.5 402.5 501.4 597.0 702.0 800.9 905.9 999.5 1098.4 1200.0 1095.0: 105.0 198.6 303.5 402.5 498.0 603.0 702.0 806.9 900.5 999.5 1101.0 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 93.6030 .. 104.9554 cents 153/152, 11.3524 cents 19/18, 93.6030 cents 2: 192.5576 .. 203.9100 cents 153/152, 11.3524 cents 304/289, 87.6022 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 .. 504.0458 cents 289/288, 6.0008 cents 19/18, 93.6030 cents 6: 591.6480 .. 608.3520 cents 729/722, 16.7040 cents 304/289, 87.6022 cents 7: 695.9542 .. 701.9550 cents 289/288, 6.0008 cents 304/289, 87.6022 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: 1095.0446 .. 1106.3970 cents 153/152, 11.3524 cents 304/289, 87.6022 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 : 17/16, 104.9554 cents Average / Smallest step : 1.068342 Largest / Average step : 1.049554 Largest / Smallest step : 1.121282 Median interval of one step : 18/17, 98.9546 cents, amount: 4 Least squares average step : 99.63537 cents, oct.: 1195.62448 cents Scale is strictly proper Step pattern alph. order: ABCDBCBACECB Step pattern size order : AECBECEACDCE Scale is a Constant Structure, by a margin of 84.22418 cents Scale diversity : 1.087168 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.894766 Rothenberg efficiency : 0.293476 redundancy: 0.706524 Efficiency x scale size : 3.521717 Number of different interval sizes : 56 = 5.09091 / class Number of one step interval sizes : 5 Highest interval variety : 6 Mean interval variety : 5.09091 = 56/11 Median interval variety : 4 Lowest interval variety : 4 Smallest interval difference : 93347/93312, 0.6492 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 1 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 : 5491 (O: 5491 U: 5491) Highest odd numerator or denominator: 323 Scale harmonicity : 0.008823 Average absolute harmonicity : 0.148438 Specific harmonicity : 0.049825 Fundamental : 1/4896, -12.2574 octaves, 0.0534 Hz. Guide tone : 62016, 15.9204 octaves, 16224971.058 Hz. Exponens Consonantiae : 3.036303E+08, 28.17774 octaves Euler's gradus suavitatis : 68 Sum of Mann's harmonic distance : 525.0, average 43.75000 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 14.17829, average 1.18152 Sum of Tenney's harmonic distance : 28.80752, average 2.40063 Vogel's harmonic complexity : 30.75000 Wille's k value : 2745 Wilson's harmonic complexity : 62 Rectangular lattice diameter : 6 Triangular lattice diameter : 4 Lattice compactness : 355.33316, average 4.55555 Lattice compactness (without 2's) : 139.25016, average 1.78526 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -5 .. 6 0.58333 35 1 3: -2 .. 1 -0.50000 12 3 17: -1 .. 1 -0.25000 5 3 19: 0 .. 1 0.41667 5 5 Average exponent except 2's :-4 / 12 =-0.33333 Average absolute exponent except 2's: 22 / 12 = 1.83333 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: 4.1157 c. M:-6.9104 c. Twelve-tone Chromatic (1/11-comma) 65: 5 6 5 5 6 5 6 5 5 6 5 6 SP M ME SD: 4.1050 c. M: 8.3588 c. 73: 6 6 6 6 6 6 7 5 6 7 6 6 SP D3 SD: 4.8329 c. M: 6.8414 c. 75: 6 6 7 6 6 6 7 6 6 6 6 7 SP D3 SD: 4.6570 c. M: 7.0446 c. 77: 6 7 6 6 7 6 7 6 6 7 6 7 SP M ME SD: 3.0561 c. M: 6.2009 c. 80: 6 7 7 6 7 7 7 6 7 6 7 7 SP D3 SD: 3.3719 c. M: 6.0900 c. 89: 7 8 7 7 8 7 8 7 7 8 7 8 SP M ME SD: 2.3685 c. M: 4.6250 c. 101: 8 9 8 8 9 8 9 8 8 9 8 9 SP M ME SD: 1.9395 c. M: 3.4235 c. 113: 9 10 9 9 10 9 10 9 9 10 9 10 SP M ME SD: 1.7066 c. M:-3.2115 c. 125: 10 11 10 10 11 10 11 10 10 11 10 11 SP M ME SD: 1.6176 c. M:-3.0416 c. 137: 11 12 11 11 12 11 12 11 11 12 11 12 SP M ME SD: 1.6233 c. M:-2.9014 c. 149: 12 13 12 12 13 12 13 12 12 13 12 13 SP M ME SD: 1.6833 c. M:-3.0413 c. 157: 12 14 13 12 14 13 14 12 13 13 13 14 SP SD: 1.5575 c. M: 3.2807 c. 169: 13 15 14 13 15 14 15 13 14 14 14 15 SP SD: 1.2131 c. M: 2.5570 c. 181: 14 16 15 14 16 15 16 14 15 15 15 16 SP SD: 0.9432 c. M: 1.9294 c. 193: 15 17 16 15 17 16 17 15 16 16 16 17 SP SD: 0.7533 c. M: 1.3797 c. 205: 16 18 17 16 18 17 18 16 17 17 17 18 SP SD: 0.6546 c. M:-1.0971 c. 217: 17 19 18 17 19 18 19 17 18 18 18 19 SP SD: 0.6478 c. M:-1.3399 c. 229: 18 20 19 18 20 19 20 18 19 19 19 20 SP SD: 0.7093 c. M:-1.5573 c. 306: 24 27 25 24 27 25 27 24 25 26 25 27 SP SD: 0.6846 c. M:-1.4416 c. 318: 25 28 26 25 28 26 28 25 26 27 26 28 SP SD: 0.6446 c. M:-1.4416 c. 330: 26 29 27 26 29 27 29 26 27 28 27 29 SP SD: 0.6430 c. M:-1.4416 c. 364: 28 32 30 29 32 30 32 28 30 31 30 32 SP SD: 0.7405 c. M: 1.2953 c. 376: 29 33 31 30 33 31 33 29 31 32 31 33 SP SD: 0.5969 c. M: 1.0498 c. 388: 30 34 32 31 34 32 34 30 32 33 32 34 SP SD: 0.4681 c. M: 0.8195 c. 400: 31 35 33 32 35 33 35 31 33 34 33 35 SP SD: 0.3576 c. M: 0.6030 c. 412: 32 36 34 33 36 34 36 32 34 35 34 36 SP SD: 0.2749 c. M: 0.5002 c. 424: 33 37 35 34 37 35 37 33 35 36 35 37 SP SD: 0.2367 c. M: 0.4452 c. 605: 47 53 50 48 53 50 53 47 50 51 50 53 SP SD: 0.2255 c. M: 0.3875 c. 617: 48 54 51 49 54 51 54 48 51 52 51 54 SP SD: 0.1644 c. M: 0.3039 c. 629: 49 55 52 50 55 52 55 49 52 53 52 55 SP SD: 0.1315 c. M:-0.2410 c. 641: 50 56 53 51 56 53 56 50 53 54 53 56 SP SD: 0.1419 c. M:-0.3392 c. 1029: 80 90 85 82 90 85 90 80 85 87 85 90 SP SD: 0.1633 c. M: 0.3086 c. 1041: 81 91 86 83 91 86 91 81 86 88 86 91 SP SD: 0.1344 c. M: 0.2875 c. 1053: 82 92 87 84 92 87 92 82 87 89 87 92 SP SD: 0.1194 c. M: 0.2678 c. 1077: 84 94 89 86 94 89 94 84 89 91 89 94 SP SD: 0.1405 c. M: 0.2297 c. 1130: 88 99 93 90 99 93 99 88 93 96 93 99 SP SD: 0.1573 c. M: 0.3533 c. 1142: 89 100 94 91 100 94 100 89 94 97 94 100 SP SD: 0.1218 c. M: 0.2919 c. 1154: 90 101 95 92 101 95 101 90 95 98 95 101 SP SD: 0.0947 c. M: 0.2318 c. 1166: 91 102 96 93 102 96 102 91 96 99 96 102 SP SD: 0.0833 c. M: 0.1730 c. 1359: 106 119 112 108 119 112 119 106 112 115 112 119 SP SD: 0.0772 c. M: 0.1536 c. 1371: 107 120 113 109 120 113 120 107 113 116 113 120 SP SD: 0.0644 c. M:-0.1287 c. 1383: 108 121 114 110 121 114 121 108 114 117 114 121 SP SD: 0.0704 c. M:-0.1401 c. 1771: 138 155 146 141 155 146 155 138 146 150 146 155 SP SD: 0.0629 c. M: 0.1027 c. 1783: 139 156 147 142 156 147 156 139 147 151 147 156 SP SD: 0.0355 c. M: 0.0650 c. 1795: 140 157 148 143 157 148 157 140 148 152 148 157 SP SD: 0.0122 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: 7.0234 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 7.2929 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 9.3227 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 10.5501 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 8.8148 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 7.8532 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.2249 cents 11 x 12:13:14:15:16 x 17:18:20:21:22 S SD: 11.3009 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 6.3426 cents 13:14:15 x 16:17:18:20:21:22:23:24:26 S SD: 9.7923 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 7.6146 cents 15:16:17:18:19:20:21:23:24:25:27:28:30 SD: 5.2762 cents 16:17:18:19:20:21:23:24:25:27:28:30:32 SD: 4.8368 cents 22:23:25:26:28:29:31:33:35:37:39:41:44 SD: 3.9830 cents 24:25:27:29:30:32:34:36:38:40:43:45:48 SD: 3.6070 cents 31:33:35:37:39:41:44:47:49:52:55:58:62 SD: 2.8885 cents 32:34:36:38:40:43:45:48:51:54:57:60:64 SD: 2.2760 cents 36:38:40:43:45:48:51:54:57:60:64:68:72 SD: 2.0137 cents 48:51:54:57:60:64:68:72:76:80:85:90:96 S SD: 1.7368 cents 58:61:65:69:73:77:82:87:92:97:103:109:116 SD: 1.1127 cents 74:78:83:88:93:99:104:111:117:124:132:139:148 SD: 1.0804 cents 75:79:84:89:94:100:106:113:119:126:133:141:150 SD: 0.9778 cents 90:95:101:107:113:120:127:135:143:151:160:169:180 SD: 0.6616 cents 122:129:137:145:153:163:172:183:193:205:217:230:244 SD: 0.6523 cents 126:133:141:150:158:168:178:189:200:211:224:237:252 SD: 0.6487 cents 138:146:155:164:173:184:195:207:219:231:245:260:276 SD: 0.6464 cents 144:152:162:171:181:192:203:216:228:241:256:271:288 SD: 0.5993 cents 146:154:164:173:183:195:206:219:231:245:260:275:292 SD: 0.5964 cents 148:156:166:176:186:197:209:222:234:248:263:279:296 SD: 0.5356 cents 165:174:185:196:207:220:233:248:261:277:293:311:330 SD: 0.4826 cents 180:190:202:214:226:240:254:270:285:302:320:339:360 SD: 0.2546 cents 288:304:323:342:361:384:407:432:456:483:512:542:576 SD: 0.2283 cents 306:323:343:363:384:408:432:459:485:513:544:576:612 SD: 0.2248 cents 362:382:406:430:454:483:511:543:573:607:644:681:724 SD: 0.2035 cents 384:405:431:456:482:512:542:576:608:644:683:723:768 SD: 0.1943 cents 396:418:444:470:497:528:559:594:627:664:704:745:792 SD: 0.1227 cents 486:513:545:577:610:648:686:729:770:815:864:915:972 SD: 0.1180 cents 576:608:646:684:723:768:813:864:912:966:1024:1084:1152 SD: 0.0773 cents