Scala analysis: werck4
Andreas Werckmeister's temperament IV
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 82.406 cents 82.406000 2: 196.090 cents 196.090000 3: 32/27 294.134997 Pythagorean minor third 4: 392.180 cents 392.180000 5: 4/3 498.044999 perfect fourth 6: 1024/729 588.269995 Pythagorean diminished fifth 7: 694.135 cents 694.135000 8: 784.361 cents 784.361000 9: 890.225 cents 890.225000 10: 1003.910 cents 1003.910000 11: 4096/2187 1086.314994 Pythagorean diminished octave 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 2187/2048 113.6850 apotome 1: 82.4060 cents 82.4060 2: 113.6840 cents 113.6840 3: 98.0450 cents 98.0450 4: 98.0450 cents 98.0450 5: 105.8650 cents 105.8650 6: 256/243 90.2250 limma, Pythagorean minor second 7: 105.8650 cents 105.8650 8: 90.2260 cents 90.2260 9: 105.8640 cents 105.8640 10: 113.6850 cents 113.6850 11: 82.4050 cents 82.4050 12: 2187/2048 113.6850 apotome
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 82.40499 cents 1: 1 82.40600 cents 1: 1 256/243 90.225 cents limma, Pythagorean minor second 1: 1 90.22600 cents 1: 2 98.04500 cents 1: 1 105.86400 cents 1: 2 105.86500 cents 1: 1 113.68400 cents 1: 2 113.68500 cents 2: 7 196.09000 cents 2: 2 196.09101 cents 2: 1 9/8 203.910 cents major whole tone 2: 1 211.72900 cents 2: 1 219.54900 cents 3: 1 278.49600 cents 3: 1 286.31600 cents 3: 2 32/27 294.135 cents Pythagorean minor third 3: 1 301.95399 cents 3: 3 301.95500 cents 3: 1 309.77400 cents 3: 3 309.77500 cents 4: 5 392.18000 cents 4: 2 392.18100 cents 4: 1 400.00000 cents 4: 1 81/64 407.820 cents Pythagorean major third 4: 2 415.63900 cents 4: 1 415.64001 cents 5: 1 490.22500 cents 5: 1 490.22600 cents 5: 5 4/3 498.045 cents perfect fourth 5: 1 505.86399 cents 5: 4 505.86500 cents 6: 3 1024/729 588.270 cents Pythagorean diminished fifth 6: 2 588.27100 cents 6: 1 596.09000 cents 6: 1 603.91000 cents 6: 2 611.72900 cents 6: 3 611.73000 cents 7: 4 694.13500 cents 7: 1 694.13601 cents 7: 5 701.95500 cents 7: 1 709.77400 cents 7: 1 709.77500 cents 8: 1 784.35999 cents 8: 2 784.36100 cents 8: 1 128/81 792.180 cents Pythagorean minor sixth 8: 1 800.00000 cents 8: 2 807.81900 cents 8: 5 807.82000 cents 9: 3 890.22500 cents 9: 1 890.22600 cents 9: 3 898.04500 cents 9: 1 898.04601 cents 9: 2 27/16 905.865 cents Pythagorean major sixth 9: 1 913.68400 cents 9: 1 921.50400 cents 10: 1 980.45100 cents 10: 1 988.27100 cents 10: 1 16/9 996.090 cents Pythagorean minor seventh 10: 2 1003.90899 cents 10: 7 1003.91000 cents 11: 2 4096/2187 1086.315 cents Pythagorean diminished octave 11: 1 1086.31600 cents 11: 2 1094.13500 cents 11: 1 1094.13600 cents 11: 2 1101.95500 cents 11: 1 1109.77400 cents 11: 1 243/128 1109.775 cents Pythagorean major seventh 11: 1 1117.59400 cents 11: 1 1117.59501 cents Highest number of different intervals for one interval class: 9 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 : 82.4 196.1 294.1 392.2 498.0 588.3 694.1 784.4 890.2 1003.9 1086.3 1200.0 82.4 : 113.7 211.7 309.8 415.6 505.9 611.7 702.0 807.8 921.5 1003.9 1117.6 1200.0 196.1 : 98.0 196.1 302.0 392.2 498.0 588.3 694.1 807.8 890.2 1003.9 1086.3 1200.0 294.1 : 98.0 203.9 294.1 400.0 490.2 596.1 709.8 792.2 905.9 988.3 1102.0 1200.0 392.2 : 105.9 196.1 302.0 392.2 498.0 611.7 694.1 807.8 890.2 1003.9 1102.0 1200.0 498.0 : 90.2 196.1 286.3 392.2 505.9 588.3 702.0 784.4 898.0 996.1 1094.1 1200.0 588.3 : 105.9 196.1 302.0 415.6 498.0 611.7 694.1 807.8 905.9 1003.9 1109.8 1200.0 694.1 : 90.2 196.1 309.8 392.2 505.9 588.3 702.0 800.0 898.0 1003.9 1094.1 1200.0 784.4 : 105.9 219.5 302.0 415.6 498.0 611.7 709.8 807.8 913.7 1003.9 1109.8 1200.0 890.2 : 113.7 196.1 309.8 392.2 505.9 603.9 702.0 807.8 898.0 1003.9 1094.1 1200.0 1003.9: 82.4 196.1 278.5 392.2 490.2 588.3 694.1 784.4 890.2 980.5 1086.3 1200.0 1086.3: 113.7 196.1 309.8 407.8 505.9 611.7 702.0 807.8 898.0 1003.9 1117.6 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 82.4050 .. 113.6850 cents 31.28001 cents 82.40499 cents 2: 196.0900 .. 219.5490 cents 23.45900 cents 82.40500 cents 3: 278.4960 .. 309.7750 cents 31.27900 cents 58.94700 cents 4: 392.1800 .. 415.6400 cents 23.46001 cents 82.40500 cents 5: 490.2250 .. 505.8650 cents 15.64000 cents 74.58499 cents 6: 588.2700 .. 611.7300 cents 23.46001 cents 82.40499 cents 7: 694.1350 .. 709.7750 cents 15.64000 cents 82.40500 cents 8: 784.3600 .. 807.8200 cents 23.46001 cents 74.58499 cents 9: 890.2250 .. 921.5040 cents 31.27900 cents 82.40500 cents 10: 980.4510 .. 1003.9100 cents 23.45900 cents 58.94700 cents 11: 1086.3150 .. 1117.5950 cents 31.28001 cents 82.40499 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 82.40499 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 113.68500 cents Average / Smallest step : 1.213519 Largest / Average step : 1.136850 Largest / Smallest step : 1.379589 Median interval of one step : 101.95450 cents, amount: 0 Least squares average step : 99.19997 cents, oct.: 1190.39961 cents Scale is strictly proper Scale is a mode of a 1995-tone equal temperament with octave 2/1 degrees: 137 326 489 652 828 978 1154 1304 1480 1669 1806 1995 Least number of segments generator : 998 of 600.301 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 1832 of 1101.955 cents and inv. generated superset size: 677 = 665 more = 5641.667% Number of contiguous 1-step segments: 0 Step pattern alph. order: ABCCDEDFGHIH Step pattern size order : BHEEGCGDFIAI Scale is a Constant Structure, by a margin of 58.94700 cents Scale diversity : 1.191620 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.771920 Rothenberg efficiency : 0.271233 redundancy: 0.728767 Efficiency x scale size : 3.254798 Number of different interval sizes : 70 = 6.36364 / class Number of one step interval sizes : 9 Highest interval variety : 9 Mean interval variety : 6.36364 = 70/11 Median interval variety : 5 Lowest interval variety : 5 Smallest interval difference : 0.00100 cents Most common intervals : 196.09000 cents & inv., amount: 7 Most common triad is 0.0 196.090 1003.910 cents, amount: 5 Number of recognisable fifths : 12, average 700.0000 cents Best fifths form a closed circle Formal octave complements present : 3 = 25.0000% Limited transpositions with margin 31.2800 cents: 3 5 7 Limited inverse transpositions with margin 31.2800 cents: 1 3 6 11 -- Rational properties -- Prime limit : 3 (not all pitches rational)
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 9.7421 c. M:-17.5940 c. Twelve-tone Chromatic (1/11-comma) 31: 2 3 3 2 3 2 3 2 3 3 2 3 SP M ME SD: 6.4968 c. M:-15.5424 c. Meantone Chromatic (53/220-comma) 38: 3 3 3 3 4 3 3 3 3 4 2 4 SP D3 SD: 8.7802 c. M: 13.2326 c. 41: 3 4 3 3 4 3 4 3 3 4 3 4 SP M ME SD: 7.0380 c. M: 12.1762 c. Schismic-12 43: 3 4 4 3 4 3 4 3 4 4 3 4 SP M ME SD: 4.3454 c. M:-12.8417 c. Meantone Chromatic (1/5-comma) 45: 3 4 4 4 4 3 4 3 4 5 3 4 SP D3 SD: 7.0601 c. M: 11.0277 c. 49: 3 5 4 4 4 4 4 4 4 5 3 5 SP D3 SD: 5.5560 c. M: 8.9366 c. 53: 4 5 4 4 5 4 5 4 4 5 4 5 SP M ME SD: 5.8844 c. M:-8.1600 c. Schismic-12 55: 4 5 5 4 5 4 5 4 5 5 4 5 SP M ME SD: 4.3255 c. M:-11.3195 c. Meantone Chromatic (3/17-comma) 61: 4 6 5 5 5 5 5 5 5 6 4 6 SP D3 SD: 3.4302 c. M: 6.2417 c. 73: 5 7 6 6 6 6 6 6 6 7 5 7 SP D3 SD: 2.7664 c. M: 4.8943 c. 86: 6 8 7 7 8 6 8 6 8 8 6 8 SP D3 S SD: 2.2897 c. M:-4.2806 c. 98: 7 9 8 8 9 7 9 7 9 9 7 9 SP D3 S SD: 2.3800 c. M:-3.9958 c. 102: 7 10 8 8 9 8 9 8 9 9 7 10 SP SD: 2.9920 c. M: 3.9621 c. 104: 7 10 8 9 9 8 9 8 9 10 7 10 SP SD: 1.9981 c. M: 5.6735 c. 116: 8 11 9 10 10 9 10 9 10 11 8 11 SP SD: 1.5924 c. M: 4.4798 c. 128: 9 12 10 11 11 10 11 10 11 12 9 12 SP SD: 1.7883 c. M: 3.5100 c. 135: 9 13 11 11 12 10 12 10 12 13 9 13 SP SD: 1.3319 c. M: 2.4060 c. 147: 10 14 12 12 13 11 13 11 13 14 10 14 SP SD: 0.4279 c. M: 0.7733 c. 159: 11 15 13 13 14 12 14 12 14 15 11 15 SP SD: 0.3396 c. M:-0.6129 c. 306: 21 29 25 25 27 23 27 23 27 29 21 29 SP SD: 0.0291 c. M: 0.0531 c. 1077: 74 102 88 88 95 81 95 81 95 102 74 102 SP SD: 0.0253 c. M:-0.0453 c. 1383: 95 131 113 113 122 104 122 104 122 131 95 131 SP SD: 0.0133 c. M:-0.0235 c. 1689: 116 160 138 138 149 127 149 127 149 160 116 160 SP SD: 0.0056 c. M:-0.0096 c. 1995: 137 189 163 163 176 150 176 150 176 189 137 189 SP SD: 0.0004 c. M:-0.0008 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: 3.9100 cents 3 x x x x 4 x x x 5 x x 6 S SD: 1.9554 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 9.1051 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 7.8561 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 7.8251 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 11.0501 cents 8 x 9 x 10 x 11:12 x 13:14:15:16 S SD: 9.0370 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 8.0277 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.6931 cents 11 x 12:13:14:15 x 16:17:18:20:21:22 S SD: 10.3618 cents 12:13 x 14:15:16:17:18:19:20:21:22:24 S SD: 7.5879 cents 13:14 x 15:16:17:18:19:20:22:23:24:26 S SD: 9.6422 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 8.0039 cents 15:16:17:18:19:20:21:22:24:25:27:28:30 SD: 5.3698 cents 16:17:18:19:20:21:22:24:25:27:29:30:32 SD: 5.1575 cents 18:19:20:21:23:24:25:27:28:30:32:34:36 SD: 4.7331 cents 22:23:25:26:28:29:31:33:35:37:39:41:44 SD: 4.1870 cents 23:24:26:27:29:31:32:34:36:38:41:43:46 SD: 3.8718 cents 24:25:27:28:30:32:34:36:38:40:43:45:48 SD: 3.1735 cents 32:34:36:38:40:43:45:48:50:54:57:60:64 SD: 2.9671 cents 33:35:37:39:41:44:46:49:52:55:59:62:66 SD: 2.6656 cents 37:39:41:44:46:49:52:55:58:62:66:69:74 SD: 2.6211 cents 39:41:44:46:49:52:55:58:61:65:70:73:78 SD: 1.9833 cents 42:44:47:50:53:56:59:63:66:70:75:79:84 SD: 1.5148 cents 59:62:66:70:74:79:83:88:93:99:105:111:118 SD: 1.2540 cents 75:79:84:89:94:100:105:112:118:125:134:140:150 SD: 1.0694 cents 82:86:92:97:103:109:115:122:129:137:146:154:164 SD: 1.0369 cents 87:91:97:103:109:116:122:130:137:145:155:163:174 SD: 1.0283 cents 98:103:110:116:123:131:138:146:154:164:175:184:196 SD: 0.8854 cents 99:104:111:117:124:132:139:148:156:166:177:185:198 SD: 0.8132 cents 101:106:113:120:127:135:142:151:159:169:180:189:202 SD: 0.7661 cents 103:108:115:122:129:137:145:154:162:172:184:193:206 SD: 0.7464 cents 117:123:131:139:147:156:164:175:184:196:209:219:234 SD: 0.7258 cents 118:124:132:140:148:157:166:176:186:197:211:221:236 SD: 0.6926 cents 125:131:140:148:157:167:176:187:197:209:223:234:250 SD: 0.6634 cents 126:132:141:149:158:168:177:188:198:211:225:236:252 SD: 0.4671 cents 141:148:158:167:177:188:198:211:222:236:252:264:282 SD: 0.4353 cents 150:157:168:178:188:200:211:224:236:251:268:281:300 SD: 0.4265 cents 168:176:188:199:211:224:236:251:264:281:300:315:336 SD: 0.3741 cents 201:211:225:238:252:268:282:300:316:336:359:376:402 SD: 0.3483 cents 225:236:252:267:282:300:316:336:354:376:402:421:450 SD: 0.2812 cents 243:255:272:288:305:324:341:363:382:406:434:455:486 SD: 0.2769 cents 284:298:318:337:356:379:399:424:447:475:507:532:568 SD: 0.2655 cents 291:305:326:345:365:388:409:435:458:487:520:545:582 SD: 0.2593 cents 307:322:344:364:385:409:431:458:483:513:548:575:614 SD: 0.2549 cents 323:339:362:383:405:431:454:482:508:540:577:605:646 SD: 0.2542 cents 324:340:363:384:406:432:455:484:510:542:579:607:648 SD: 0.2323 cents 351:368:393:416:440:468:493:524:552:587:627:657:702 SD: 0.1457 cents 393:412:440:466:493:524:552:587:618:657:702:736:786 SD: 0.1366 cents 492:516:551:583:617:656:691:735:774:823:879:921:984 SD: 0.1268 cents 534:560:598:633:670:712:750:797:840:893:954:1000:1068 SD: 0.1102 cents