Scala analysis: werckmeisterIV_variant
Werckmeister IV with 1/3 syntonic comma temperings
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 85.010 cents 85.009950 2: 196.741 cents 196.741240 3: 32/27 294.134997 Pythagorean minor third 4: 393.482 cents 393.482480 5: 4/3 498.044999 perfect fourth 6: 45/32 590.223716 diatonic tritone 7: 694.786 cents 694.786240 8: 785.011 cents 785.011230 9: 891.527 cents 891.527480 10: 1003.259 cents 1003.258760 11: 15/8 1088.268715 classic major seventh 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 16/15 111.7313 minor diatonic semitone 1: 85.0099 cents 85.0099 2: 111.7313 cents 111.7313 3: 97.3938 cents 97.3938 4: 99.3475 cents 99.3475 5: 104.5625 cents 104.5625 6: 135/128 92.1787 major chroma, major limma 7: 104.5625 cents 104.5625 8: 90.2250 cents 90.2250 9: 106.5162 cents 106.5162 10: 111.7313 cents 111.7313 11: 85.0100 cents 85.0100 12: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 85.00995 cents 1: 1 90.22499 cents 1: 1 135/128 92.179 cents major chroma, major limma 1: 1 97.39376 cents 1: 1 99.34748 cents 1: 2 104.56252 cents 1: 1 106.51625 cents 1: 3 111.73129 cents 2: 1 194.78751 cents 2: 8 196.74124 cents 2: 1 9/8 203.910 cents major whole tone 2: 1 209.12505 cents 2: 1 218.24753 cents 3: 1 281.75119 cents 3: 1 286.96623 cents 3: 1 32/27 294.135 cents Pythagorean minor third 3: 1 1215/1024 296.089 cents wide augmented second 3: 3 301.30376 cents 3: 1 303.25748 cents 3: 4 308.47253 cents 4: 1 391.52875 cents 4: 6 393.48248 cents 4: 1 400.65124 cents 4: 1 512/405 405.866 cents narrow diminished fourth 4: 2 413.03505 cents 4: 1 414.98877 cents 5: 2 490.87623 cents 5: 4 4/3 498.045 cents perfect fourth 5: 1 499.99872 cents 5: 5 505.21377 cents 6: 1 588.26999 cents 6: 4 45/32 590.224 cents diatonic tritone 6: 1 597.39248 cents 6: 1 602.60752 cents 6: 4 609.77629 cents 6: 1 611.73001 cents 7: 5 694.78624 cents 7: 1 700.00128 cents 7: 4 3/2 701.955 cents perfect fifth 7: 2 709.12376 cents 8: 1 785.01123 cents 8: 2 786.96495 cents 8: 1 405/256 794.134 cents wide augmented fifth 8: 1 799.34876 cents 8: 6 806.51753 cents 8: 1 808.47125 cents 9: 4 891.52748 cents 9: 1 896.74252 cents 9: 3 898.69624 cents 9: 1 2048/1215 903.911 cents narrow diminished seventh 9: 1 27/16 905.865 cents Pythagorean major sixth 9: 1 913.03377 cents 9: 1 918.24881 cents 10: 1 981.75247 cents 10: 1 990.87495 cents 10: 1 16/9 996.090 cents Pythagorean minor seventh 10: 8 1003.25876 cents 10: 1 1005.21249 cents 11: 3 15/8 1088.269 cents classic major seventh 11: 1 1093.48375 cents 11: 2 1095.43748 cents 11: 1 1100.65252 cents 11: 1 1102.60624 cents 11: 1 256/135 1107.821 cents octave - major chroma 11: 1 1109.77501 cents 11: 2 1114.99005 cents Highest number of different intervals for one interval class: 8 Average number of different intervals per interval class: 6.00000 = 6
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 85.0 196.7 294.1 393.5 498.0 590.2 694.8 785.0 891.5 1003.3 1088.3 1200.0 85.0 : 111.7 209.1 308.5 413.0 505.2 609.8 700.0 806.5 918.2 1003.3 1115.0 1200.0 196.7 : 97.4 196.7 301.3 393.5 498.0 588.3 694.8 806.5 891.5 1003.3 1088.3 1200.0 294.1 : 99.3 203.9 296.1 400.7 490.9 597.4 709.1 794.1 905.9 990.9 1102.6 1200.0 393.5 : 104.6 196.7 301.3 391.5 498.0 609.8 694.8 806.5 891.5 1003.3 1100.7 1200.0 498.0 : 92.2 196.7 287.0 393.5 505.2 590.2 702.0 787.0 898.7 996.1 1095.4 1200.0 590.2 : 104.6 194.8 301.3 413.0 498.0 609.8 694.8 806.5 903.9 1003.3 1107.8 1200.0 694.8 : 90.2 196.7 308.5 393.5 505.2 590.2 702.0 799.3 898.7 1003.3 1095.4 1200.0 785.0 : 106.5 218.2 303.3 415.0 500.0 611.7 709.1 808.5 913.0 1005.2 1109.8 1200.0 891.5 : 111.7 196.7 308.5 393.5 505.2 602.6 702.0 806.5 898.7 1003.3 1093.5 1200.0 1003.3: 85.0 196.7 281.8 393.5 490.9 590.2 694.8 787.0 891.5 981.8 1088.3 1200.0 1088.3: 111.7 196.7 308.5 405.9 505.2 609.8 702.0 806.5 896.7 1003.3 1115.0 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 85.0099 .. 111.7313 cents 26.72134 cents 85.00995 cents 2: 194.7875 .. 218.2475 cents 23.46002 cents 83.05622 cents 3: 281.7512 .. 308.4725 cents 26.72134 cents 63.50366 cents 4: 391.5287 .. 414.9888 cents 23.46002 cents 83.05622 cents 5: 490.8762 .. 505.2138 cents 14.33753 cents 75.88746 cents 6: 588.2700 .. 611.7300 cents 23.46002 cents 83.05622 cents 7: 694.7862 .. 709.1238 cents 14.33752 cents 83.05623 cents 8: 785.0112 .. 808.4713 cents 23.46002 cents 75.88747 cents 9: 891.5275 .. 918.2488 cents 26.72133 cents 83.05623 cents 10: 981.7525 .. 1005.2125 cents 23.46002 cents 63.50366 cents 11: 1088.2687 .. 1114.9901 cents 26.72134 cents 83.05623 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 85.00995 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 111.73129 cents Average / Smallest step : 1.176333 Largest / Average step : 1.117313 Largest / Smallest step : 1.314332 Median interval of one step : 101.95500 cents, amount: 0 Most common interval of one step : 111.73129 cents, amount: 3 Least squares average step : 99.28812 cents, oct.: 1191.45746 cents Scale is strictly proper Step pattern alph. order: ABCDEFEGHBAB Step pattern size order : AHDEFCFBGHAH Scale is a Constant Structure, by a margin of 63.50366 cents Scale diversity : 1.179299 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.789283 Rothenberg efficiency : 0.301004 redundancy: 0.698996 Efficiency x scale size : 3.612049 Number of different interval sizes : 66 = 6.00000 / class Number of one step interval sizes : 8 Highest interval variety : 8 Mean interval variety : 6.00000 = 6 Median interval variety : 4 Lowest interval variety : 4 Smallest interval difference : 1.95372 cents Most common intervals : 196.74124 cents & inv., amount: 8 Most common triad is 0.0 196.741 1003.259 cents, amount: 6 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 26.7213 cents: 3 5 7 Limited inverse transpositions with margin 26.7213 cents: 1 3 6 11 -- Rational properties -- Prime limit : 5 (not all pitches rational)
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 8.5803 c. M:-14.9901 c. Twelve-tone Chromatic (1/11-comma) 31: 2 3 3 2 3 2 3 2 3 3 2 3 SP M ME SD: 7.1920 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: 8.8687 c. M: 14.5845 c. 40: 3 4 3 3 4 3 3 3 4 3 3 4 SP D3 SD: 8.4147 c. M:-13.2588 c. 41: 3 4 3 3 4 3 4 3 3 4 3 4 SP M ME SD: 7.2525 c. M: 13.4787 c. Schismic-12 43: 3 4 4 3 4 3 4 3 4 4 3 4 SP M ME SD: 4.4501 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: 7.3893 c. M: 11.6779 c. 49: 3 5 4 4 4 4 4 4 4 5 3 5 SP D3 SD: 6.5085 c. M: 11.5406 c. 50: 4 4 4 4 5 4 4 4 4 5 3 5 SP D3 SD: 6.8036 c. M:-10.9901 c. 53: 4 5 4 4 5 4 5 4 4 5 4 5 SP M ME SD: 5.6726 c. M: 8.5768 c. Schismic-12 55: 4 5 4 5 5 4 5 4 5 5 4 5 SP M ME SD: 3.6460 c. M: 10.4986 c. 57: 4 5 5 5 5 4 5 4 5 6 4 5 SP D3 SD: 5.2751 c. M: 7.3170 c. 61: 4 6 5 5 5 5 5 5 5 6 4 6 SP D3 SD: 4.0999 c. M: 6.3214 c. 67: 5 6 6 5 6 5 6 5 6 6 5 6 SP M ME SD: 4.0613 c. M:-10.3426 c. Meantone Chromatic (4/25-comma) 73: 5 7 6 6 6 6 6 6 6 7 5 7 SP D3 SD: 2.8927 c. M: 4.8943 c. 85: 6 8 7 7 7 7 7 7 7 8 6 8 SP D3 SD: 2.5731 c. M:-5.5770 c. 86: 6 8 7 7 8 6 8 6 8 8 6 8 SP D3 S SD: 2.4827 c. M:-4.2806 c. 98: 7 9 8 8 9 7 9 7 9 9 7 9 SP D3 S SD: 1.9663 c. M:-3.9958 c. 110: 8 10 9 9 10 8 10 8 10 10 8 10 SP D3 S SD: 2.0304 c. M:-3.7732 c. 116: 8 11 9 10 10 9 10 9 10 11 8 11 SP SD: 1.8240 c. M: 4.4798 c. 118: 8 11 10 10 10 9 10 9 11 11 8 11 SP SD: 2.4955 c. M: 3.6540 c. 122: 9 11 10 10 11 9 11 9 11 11 9 11 SP D3 S SD: 2.3733 c. M:-3.5943 c. 128: 9 12 10 11 11 10 11 10 11 12 9 12 SP SD: 1.3855 c. M: 3.5100 c. 147: 10 14 12 12 13 11 13 11 13 14 10 14 SP SD: 1.6777 c. M: 3.3773 c. 152: 11 14 12 13 13 12 13 11 14 14 11 14 SP SD: 1.5112 c. M: 3.4323 c. 159: 11 15 13 13 14 12 14 12 14 15 11 15 SP SD: 0.9704 c. M: 1.9911 c. 171: 12 16 14 14 15 13 15 13 15 16 12 16 SP SD: 0.5230 c. M:-0.9537 c. 311: 22 29 25 26 27 24 27 23 28 29 22 29 SP SD: 0.5810 c. M: 1.7315 c. 323: 23 30 26 27 28 25 28 24 29 30 23 30 SP SD: 0.4439 c. M: 1.1103 c. 397: 28 37 32 33 35 30 35 30 35 37 28 37 SP SD: 0.5459 c. M: 0.9360 c. 408: 29 38 33 34 35 32 35 31 36 38 29 38 SP SD: 0.5178 c. M: 0.9862 c. 433: 31 40 35 36 38 33 38 32 39 40 31 40 SP SD: 0.5978 c. M:-0.9023 c. 439: 31 41 36 36 38 34 38 33 39 41 31 41 SP SD: 0.4461 c. M:-1.0814 c. 451: 32 42 37 37 39 35 39 34 40 42 32 42 SP SD: 0.4300 c. M:-1.2087 c. 482: 34 45 39 40 42 37 42 36 43 45 34 45 SP SD: 0.3053 c. M: 0.7789 c. 494: 35 46 40 41 43 38 43 37 44 46 35 46 SP SD: 0.1336 c. M: 0.3958 c. 506: 36 47 41 42 44 39 44 38 45 47 36 47 SP SD: 0.1796 c. M:-0.3655 c. 665: 47 62 54 55 58 51 58 50 59 62 47 62 SP SD: 0.0969 c. M: 0.1979 c. 677: 48 63 55 56 59 52 59 51 60 63 48 63 SP SD: 0.0788 c. M:-0.2177 c. 1159: 82 108 94 96 101 89 101 87 103 108 82 108 SP SD: 0.0819 c. M: 0.1967 c. 1171: 83 109 95 97 102 90 102 88 104 109 83 109 SP SD: 0.0270 c. M:-0.0456 c. 1836: 130 171 149 152 160 141 160 138 163 171 130 171 SP SD: 0.0246 c. M: 0.0439 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: 3.5844 cents 3 x x x x 4 x x x 5 x x 6 S SD: 2.3896 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 8.9736 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 7.8735 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 8.0962 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 10.7581 cents 8 x 9 x 10 x 11:12 x 13:14:15:16 S SD: 9.2318 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 7.9742 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.6096 cents 11 x 12:13:14:15 x 16:17:18:20:21:22 S SD: 10.4149 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 7.4270 cents 13:14 x 15:16:17:18:19:20:22:23:24:26 S SD: 9.6780 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 7.8658 cents 15:16:17:18:19:20:21:22:24:25:27:28:30 SD: 5.2890 cents 16:17:18:19:20:21:23:24:25:27:29:30:32 SD: 5.1412 cents 18:19:20:21:23:24:25:27:28:30:32:34:36 SD: 4.6846 cents 22:23:25:26:28:29:31:33:35:37:39:41:44 SD: 4.0894 cents 23:24:26:27:29:31:32:34:36:38:41:43:46 SD: 4.0313 cents 24:25:27:28:30:32:34:36:38:40:43:45:48 SD: 3.1936 cents 32:34:36:38:40:43:45:48:50:54:57:60:64 SD: 2.7866 cents 33:35:37:39:41:44:46:49:52:55:59:62:66 SD: 2.6859 cents 38:40:43:45:48:51:53:57:60:64:68:71:76 SD: 2.6680 cents 39:41:44:46:49:52:55:58:61:65:70:73:78 SD: 1.9819 cents 42:44:47:50:53:56:59:63:66:70:75:79:84 SD: 1.4716 cents 59:62:66:70:74:79:83:88:93:99:105:111:118 SD: 1.0703 cents 75:79:84:89:94:100:105:112:118:126:134:141:150 SD: 1.0646 cents 81:85:91:96:102:108:114:121:127:136:145:152:162 SD: 1.0428 cents 91:96:102:108:114:121:128:136:143:152:162:171:182 SD: 1.0339 cents 93:98:104:110:117:124:131:139:146:156:166:174:186 SD: 0.9384 cents 97:102:109:115:122:129:136:145:153:162:173:182:194 SD: 0.9288 cents 98:103:110:116:123:131:138:146:154:164:175:184:196 SD: 0.7240 cents 117:123:131:139:147:156:165:175:184:196:209:219:234 SD: 0.6834 cents 125:131:140:148:157:167:176:187:197:209:223:234:250 SD: 0.6473 cents 134:141:150:159:168:179:188:200:211:224:239:251:268 SD: 0.6413 cents 138:145:155:164:173:184:194:206:217:231:246:259:276 SD: 0.6403 cents 140:147:157:166:176:187:197:209:220:234:250:263:280 SD: 0.5611 cents 141:148:158:167:177:188:198:211:222:236:252:264:282 SD: 0.4490 cents 183:192:205:217:230:244:257:273:288:306:327:343:366 SD: 0.4229 cents 199:209:223:236:250:265:280:297:313:333:355:373:398 SD: 0.3163 cents 200:210:224:237:251:267:281:299:315:335:357:375:400 SD: 0.3103 cents 239:251:268:283:300:319:336:357:376:400:427:448:478 SD: 0.2755 cents 258:271:289:306:324:344:363:385:406:432:461:484:516 SD: 0.2719 cents 275:289:308:326:345:367:387:411:433:460:491:516:550 SD: 0.2665 cents 297:312:333:352:373:396:418:444:467:497:530:557:594 SD: 0.2542 cents 298:313:334:353:374:397:419:445:469:499:532:559:596 SD: 0.1938 cents 340:357:381:403:427:453:478:508:535:569:607:638:680 SD: 0.1908 cents 357:375:400:423:448:476:502:533:562:597:637:669:714 SD: 0.1874 cents 399:419:447:473:501:532:561:596:628:668:712:748:798 SD: 0.1101 cents 573:602:642:679:719:764:806:856:902:959:1023:1074:1146 SD: 0.1012 cents