Scala analysis: val-werck
Vallotti-Young and Werckmeister III, 10 cents least squares optimized
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 87.164 cents 87.164120 2: 192.018 cents 192.017930 3: 296.872 cents 296.871740 4: 384.036 cents 384.035860 5: 502.601 cents 502.600700 6: 582.310 cents 582.310310 7: 696.704 cents 696.704130 8: 792.018 cents 792.017930 9: 887.332 cents 887.331730 10: 1001.726 cents 1001.725550 11: 1081.435 cents 1081.435170 12: 1200.000 cents 1200.000000
SHOW/INTERVAL
0: 118.5648 cents 118.5648 1: 87.1641 cents 87.1641 2: 104.8538 cents 104.8538 3: 104.8538 cents 104.8538 4: 87.1641 cents 87.1641 5: 118.5648 cents 118.5648 6: 79.7096 cents 79.7096 7: 114.3938 cents 114.3938 8: 95.3138 cents 95.3138 9: 95.3138 cents 95.3138 10: 114.3938 cents 114.3938 11: 79.7096 cents 79.7096 12: 118.5648 cents 118.5648
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 79.70961 cents 1: 2 87.16412 cents 1: 2 95.31380 cents 1: 2 104.85381 cents 1: 2 114.39382 cents 1: 2 118.56484 cents 2: 1 190.62760 cents 2: 2 192.01793 cents 2: 2 194.10343 cents 2: 2 198.27445 cents 2: 2 205.72896 cents 2: 3 209.70762 cents 3: 2 285.43857 cents 3: 2 289.41723 cents 3: 2 296.87174 cents 3: 2 305.02142 cents 3: 2 310.58277 cents 3: 2 312.66827 cents 4: 1 384.03586 cents 4: 2 384.73103 cents 4: 2 390.29238 cents 4: 2 399.83239 cents 4: 2 407.98207 cents 4: 2 415.43658 cents 4: 1 419.41524 cents 5: 4 495.14619 cents 5: 2 499.12485 cents 5: 2 502.60070 cents 5: 2 503.29587 cents 5: 2 504.68620 cents 6: 1 578.83447 cents 6: 2 582.31031 cents 6: 2 590.45999 cents 6: 2 600.00000 cents 6: 2 609.54001 cents 6: 2 617.68969 cents 6: 1 621.16553 cents 7: 2 695.31380 cents 7: 2 696.70413 cents 7: 2 697.39931 cents 7: 2 700.87515 cents 7: 4 704.85381 cents 8: 1 780.58476 cents 8: 2 784.56343 cents 8: 2 792.01793 cents 8: 2 800.16761 cents 8: 2 809.70762 cents 8: 2 815.26897 cents 8: 1 815.96414 cents 9: 2 887.33173 cents 9: 2 889.41724 cents 9: 2 894.97858 cents 9: 2 903.12826 cents 9: 2 910.58277 cents 9: 2 914.56143 cents 10: 3 990.29238 cents 10: 2 994.27105 cents 10: 2 1001.72555 cents 10: 2 1005.89657 cents 10: 2 1007.98207 cents 10: 1 1009.37240 cents 11: 2 1081.43517 cents 11: 2 1085.60618 cents 11: 2 1095.14619 cents 11: 2 1104.68620 cents 11: 2 1112.83588 cents 11: 2 1120.29039 cents Highest number of different intervals for one interval class: 7 Average number of different intervals per interval class: 6.09091 = 67/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 87.2 192.0 296.9 384.0 502.6 582.3 696.7 792.0 887.3 1001.7 1081.4 1200.0 87.2 : 104.9 209.7 296.9 415.4 495.1 609.5 704.9 800.2 914.6 994.3 1112.8 1200.0 192.0 : 104.9 192.0 310.6 390.3 504.7 600.0 695.3 809.7 889.4 1008.0 1095.1 1200.0 296.9 : 87.2 205.7 285.4 399.8 495.1 590.5 704.9 784.6 903.1 990.3 1095.1 1200.0 384.0 : 118.6 198.3 312.7 408.0 503.3 617.7 697.4 816.0 903.1 1008.0 1112.8 1200.0 502.6 : 79.7 194.1 289.4 384.7 499.1 578.8 697.4 784.6 889.4 994.3 1081.4 1200.0 582.3 : 114.4 209.7 305.0 419.4 499.1 617.7 704.9 809.7 914.6 1001.7 1120.3 1200.0 696.7 : 95.3 190.6 305.0 384.7 503.3 590.5 695.3 800.2 887.3 1005.9 1085.6 1200.0 792.0 : 95.3 209.7 289.4 408.0 495.1 600.0 704.9 792.0 910.6 990.3 1104.7 1200.0 887.3 : 114.4 194.1 312.7 399.8 504.7 609.5 696.7 815.3 895.0 1009.4 1104.7 1200.0 1001.7: 79.7 198.3 285.4 390.3 495.1 582.3 700.9 780.6 895.0 990.3 1085.6 1200.0 1081.4: 118.6 205.7 310.6 415.4 502.6 621.2 700.9 815.3 910.6 1005.9 1120.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 79.7096 .. 118.5648 cents 38.85523 cents 79.70961 cents 2: 190.6276 .. 209.7076 cents 19.08002 cents 72.06276 cents 3: 285.4386 .. 312.6683 cents 27.22970 cents 75.73095 cents 4: 384.0359 .. 419.4152 cents 35.37938 cents 71.36759 cents 5: 495.1462 .. 504.6862 cents 9.54001 cents 75.73095 cents 6: 578.8345 .. 621.1655 cents 42.33106 cents 74.14827 cents 7: 695.3138 .. 704.8538 cents 9.54001 cents 74.14827 cents 8: 780.5848 .. 815.9641 cents 35.37938 cents 75.73095 cents 9: 887.3317 .. 914.5614 cents 27.22970 cents 71.36759 cents 10: 990.2924 .. 1009.3724 cents 19.08002 cents 75.73095 cents 11: 1081.4352 .. 1120.2904 cents 38.85522 cents 72.06277 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 79.70961 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 118.56484 cents Average / Smallest step : 1.254554 Largest / Average step : 1.185648 Largest / Smallest step : 1.487460 Median interval of one step : 100.08380 cents, amount: 0 Least squares average step : 99.10296 cents, oct.: 1189.23553 cents Scale is strictly proper Step pattern alph. order: ABBACDEFFEDC Step pattern size order : BDDBFAECCEAF Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 71.36759 cents Scale diversity : 1.220860 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.747917 Rothenberg efficiency : 0.247938 redundancy: 0.752062 Efficiency x scale size : 2.975253 Number of different interval sizes : 67 = 6.09091 / class Number of one step interval sizes : 6 Highest interval variety : 7 Mean interval variety : 6.09091 = 67/11 Median interval variety : 5 Lowest interval variety : 5 Smallest interval difference : 0.69517 cents Most common intervals : 495.14619 cents & inv., amount: 4 Most common triad is 0.0 495.146 704.854 cents, amount: 3 Number of recognisable fifths : 12, average 700.0000 cents Best fifths form a closed circle Formal octave complements present : 1 = 8.3333% Limited transpositions with margin 38.8552 cents: 3 5 7 10 Limited inverse transpositions : 4 Inversional symmetry on degrees : 2 8 -- Rational properties --
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 10.7838 c. M:-18.5648 c. Twelve-tone Chromatic (1/11-comma) 31: 2 3 3 2 3 2 3 2 3 3 2 3 SP M ME SD: 7.2385 c. M: 17.8244 c. Meantone Chromatic (53/220-comma) 32: 2 3 3 2 3 3 3 2 3 3 2 3 SP D3 SD: 10.7395 c. M:-17.6897 c. 37: 3 3 3 3 3 3 3 3 3 4 2 4 SP D3 SD: 9.6897 c. M: 16.1142 c. 38: 3 3 3 3 4 2 4 3 3 4 2 4 SP D3 S SD: 7.1159 c. M: 13.8893 c. 41: 3 4 3 3 4 3 4 3 3 4 3 4 SP M ME SD: 5.7642 c. M:-12.8601 c. Schismic-12 43: 3 4 4 3 4 3 4 3 4 4 3 4 SP M ME SD: 5.6740 c. M: 10.6226 c. Meantone Chromatic (1/5-comma) 50: 4 4 4 4 5 3 5 4 4 5 3 5 SP D3 S SD: 4.4808 c. M: 8.8717 c. 54: 4 5 4 4 6 3 5 5 4 5 4 5 SP SD: 6.1121 c. M:-8.5104 c. 55: 4 5 5 4 5 4 5 4 5 5 4 5 SP M ME SD: 5.8123 c. M:-9.4739 c. Meantone Chromatic (3/17-comma) 57: 4 5 5 4 6 4 5 5 4 6 3 6 SP SD: 5.1551 c. M:-8.8008 c. 60: 4 6 5 4 6 4 6 5 4 6 4 6 SP T3 SD: 4.8869 c. M:-7.9821 c. 62: 5 5 5 5 6 4 6 5 5 6 4 6 SP D3 S SD: 3.9758 c. M:-9.6101 c. 74: 5 7 6 6 7 5 7 6 6 7 5 7 SP D3 SD: 3.6971 c. M: 6.0830 c. 80: 6 7 7 6 8 5 8 6 6 8 5 8 SP S SD: 4.2838 c. M:-8.2959 c. 81: 6 7 7 6 8 5 8 6 7 8 5 8 SP SD: 3.0039 c. M: 6.8327 c. 85: 6 8 7 6 9 5 8 7 7 8 6 8 SP SD: 3.5679 c. M:-5.6346 c. 88: 6 8 8 6 9 6 8 7 7 8 6 9 SP S SD: 3.2288 c. M: 6.2710 c. 93: 7 8 8 7 9 6 9 7 8 9 6 9 SP SD: 2.6817 c. M: 4.9212 c. 101: 7 9 9 7 10 7 10 8 8 9 7 10 SP SD: 3.0245 c. M:-4.2860 c. 105: 8 9 9 8 10 7 10 8 9 10 7 10 SP SD: 2.9918 c. M:-4.5356 c. 109: 8 9 10 8 11 7 10 9 9 10 7 11 SP SD: 2.5550 c. M: 4.8620 c. 112: 8 10 10 8 11 7 11 9 9 10 8 11 SP SD: 2.3001 c. M: 5.2970 c. 122: 9 11 10 9 12 8 12 10 9 12 8 12 SP SD: 2.3117 c. M:-4.7034 c. 124: 9 11 11 9 12 8 12 10 10 12 8 12 SP S SD: 2.3135 c. M:-4.7261 c. 126: 9 11 11 9 13 8 12 10 10 12 8 13 SP S SD: 2.2509 c. M: 5.2447 c. 127: 9 11 11 10 12 9 12 10 10 12 8 13 SP SD: 2.6541 c. M: 4.2698 c. 129: 9 12 11 9 13 9 12 10 10 13 8 13 SP SD: 2.4913 c. M:-3.7362 c. 132: 10 11 12 9 13 9 13 10 11 12 9 13 SP SD: 2.3270 c. M:-3.7450 c. 134: 10 11 12 10 13 9 13 10 11 13 9 13 SP SD: 2.0665 c. M: 3.9582 c. 138: 10 12 12 10 14 9 13 11 11 13 9 14 SP S SD: 1.3549 c. M: 3.1743 c. 150: 11 13 13 11 15 10 14 12 12 14 10 15 SP S SD: 1.0105 c. M: 1.7255 c. 250: 18 22 22 18 25 16 24 20 20 24 16 25 SP S SD: 0.9368 c. M: 1.5103 c. 262: 19 23 23 19 26 17 25 21 21 25 17 26 SP S SD: 0.7700 c. M:-1.3279 c. 274: 20 24 24 20 27 18 26 22 22 26 18 27 SP S SD: 0.8943 c. M:-1.7194 c. 291: 21 26 25 21 29 19 28 23 23 28 19 29 SP SD: 0.7435 c. M:-1.7965 c. 303: 22 26 27 22 30 20 29 24 24 29 20 30 SP SD: 0.5884 c. M: 1.9189 c. 315: 23 27 28 23 31 21 30 25 25 30 21 31 SP SD: 0.5922 c. M: 1.5417 c. 327: 24 28 29 24 32 22 31 26 26 31 22 32 SP SD: 0.8166 c. M:-1.2852 c. 332: 24 29 29 24 33 22 32 26 26 32 22 33 SP S SD: 0.7407 c. M: 1.7896 c. 344: 25 30 30 25 34 23 33 27 27 33 23 34 SP S SD: 0.5267 c. M: 1.2852 c. 356: 26 31 31 26 35 24 34 28 28 34 24 35 SP S SD: 0.5524 c. M:-1.0487 c. 437: 32 38 38 32 43 29 42 34 35 42 29 43 SP SD: 0.5402 c. M: 1.1712 c. 449: 33 39 39 33 44 30 43 35 36 43 30 44 SP SD: 0.6351 c. M:-1.0319 c. 453: 33 39 40 33 45 30 43 36 36 43 30 45 SP SD: 0.5060 c. M: 1.2895 c. 456: 33 40 40 33 45 30 44 36 36 44 30 45 SP S SD: 0.4523 c. M:-0.9060 c. 468: 34 41 41 34 46 31 45 37 37 45 31 46 SP S SD: 0.4594 c. M:-0.8386 c. 526: 38 46 46 38 52 35 50 42 42 50 35 52 SP S SD: 0.4865 c. M: 0.8866 c. 537: 39 47 47 39 53 36 51 42 43 51 36 53 SP SD: 0.4770 c. M: 0.9565 c. 538: 39 47 47 39 53 36 51 43 43 51 36 53 SP S SD: 0.3929 c. M: 0.7933 c. 550: 40 48 48 40 54 37 52 44 44 52 37 54 SP S SD: 0.4355 c. M: 0.7825 c. 556: 40 49 49 40 55 37 53 44 44 53 37 55 SP S SD: 0.4354 c. M:-0.9700 c. 575: 42 50 50 42 57 38 55 46 45 55 38 57 SP SD: 0.4188 c. M:-1.0255 c. 577: 42 50 51 42 57 38 55 46 46 55 38 57 SP SD: 0.4908 c. M:-0.7128 c. 587: 43 51 51 43 58 39 56 46 47 56 39 58 SP SD: 0.4064 c. M: 0.8765 c. 594: 43 52 52 43 59 39 57 47 47 57 39 59 SP S SD: 0.3351 c. M: 0.6271 c. 606: 44 53 53 44 60 40 58 48 48 58 40 60 SP S SD: 0.1991 c. M:-0.3696 c. 1132: 82 99 99 82 112 75 108 90 90 108 75 112 SP S SD: 0.1825 c. M: 0.3315 c. 1144: 83 100 100 83 113 76 109 91 91 109 76 113 SP S SD: 0.1018 c. M: 0.2006 c. 1750: 127 153 153 127 173 116 167 139 139 167 116 173 SP S SD: 0.0614 c. M: 0.1389 c. 1762: 128 154 154 128 174 117 168 140 140 168 117 174 SP S SD: 0.0496 c. M:-0.0906 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: 2.6254 cents 3 x x x x 4 x x x 5 x x 6 S SD: 1.8133 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 8.3485 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 6.5536 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 7.4271 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 12.5026 cents 8 x 9 x 10 x 11:12 x 13:14:15:16 S SD: 8.3427 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 8.2878 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.7708 cents 11 x 12:13:14:15 x 16:17:18:20:21:22 S SD: 10.7752 cents 12:13 x 14:15:16:17:18:19:20:21:22:24 S SD: 7.1332 cents 13:14 x 15:16:17:18:19:21:22:23:24:26 SD: 9.4082 cents 14:15:16:17 x 19:20:21:22:23:25:26:28 SD: 7.6218 cents 15:16:17:18:19:20:21:22:24:25:27:28:30 SD: 5.3257 cents 16:17:18:19:20:21:22:24:25:27:29:30:32 SD: 5.2744 cents 17:18:19:20:21:23:24:25:27:28:30:32:34 SD: 4.8256 cents 20:21:22:24:25:27:28:30:32:33:36:37:40 SD: 4.4214 cents 23:24:26:27:29:31:32:34:36:38:41:43:46 SD: 4.2283 cents 24:25:27:28:30:32:34:36:38:40:43:45:48 SD: 3.6558 cents 27:28:30:32:34:36:38:40:43:45:48:50:54 SD: 3.4624 cents 32:34:36:38:40:43:45:48:51:53:57:60:64 SD: 2.7614 cents 33:35:37:39:41:44:46:49:52:55:59:62:66 SD: 2.2992 cents 36:38:40:43:45:48:50:54:57:60:64:67:72 SD: 2.0018 cents 42:44:47:50:52:56:59:63:66:70:75:78:84 SD: 1.9792 cents 52:55:58:62:65:70:73:78:82:87:93:97:104 SD: 1.7074 cents 53:56:59:63:66:71:74:79:84:88:95:99:106 SD: 1.6414 cents 57:60:64:68:71:76:80:85:90:95:102:106:114 SD: 1.5072 cents 59:62:66:70:74:79:83:88:93:99:105:110:118 SD: 1.4237 cents 60:63:67:71:75:80:84:90:95:100:107:112:120 SD: 0.8883 cents 93:98:104:110:116:124:130:139:147:155:166:174:186 SD: 0.8491 cents 97:102:108:115:121:130:136:145:153:162:173:181:194 SD: 0.7542 cents 112:118:125:133:140:150:157:167:177:187:200:209:224 SD: 0.6842 cents 121:127:135:144:151:162:169:181:191:202:216:226:242 SD: 0.6699 cents 133:140:149:158:166:178:186:199:210:222:237:248:266 SD: 0.5600 cents 145:152:162:172:181:194:203:217:229:242:259:271:290 SD: 0.5595 cents 154:162:172:183:192:206:216:230:243:257:275:288:308 SD: 0.5450 cents 157:165:175:186:196:210:220:235:248:262:280:293:314 SD: 0.5199 cents 160:168:179:190:200:214:224:239:253:267:285:299:320 SD: 0.4643 cents 172:181:192:204:215:230:241:257:272:287:307:321:344 SD: 0.3937 cents 198:208:221:235:247:265:277:296:313:331:353:370:396 SD: 0.3801 cents 212:223:237:252:265:283:297:317:335:354:378:396:424 SD: 0.3780 cents 214:225:239:254:267:286:300:320:338:357:382:400:428 SD: 0.3196 cents 250:263:279:297:312:334:350:374:395:417:446:467:500 SD: 0.2764 cents 273:287:305:324:341:365:382:408:431:456:487:510:546 SD: 0.2113 cents 315:331:352:374:393:421:441:471:498:526:562:588:630 SD: 0.1953 cents 333:350:372:395:416:445:466:498:526:556:594:622:666 SD: 0.1926 cents 407:428:455:483:508:544:570:609:643:679:726:760:814 SD: 0.1830 cents 443:466:495:526:553:592:620:662:700:740:790:827:886 SD: 0.1685 cents 460:484:514:546:574:615:644:688:727:768:820:859:920 SD: 0.1327 cents 545:573:609:647:680:729:763:815:861:910:972:1018:1090 SD: 0.1264 cents 588:618:657:698:734:786:823:879:929:982:1049:1098:1176 SD: 0.1213 cents