Scala analysis: werck3
Andreas Werckmeister's temperament III (the most famous one, 1681)
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 256/243 90.224996 limma, Pythagorean minor second 2: 192.180 cents 192.180000 3: 32/27 294.134997 Pythagorean minor third 4: 390.225 cents 390.225000 5: 4/3 498.044999 perfect fourth 6: 1024/729 588.269995 Pythagorean diminished fifth 7: 696.090 cents 696.090000 8: 128/81 792.179997 Pythagorean minor sixth 9: 888.270 cents 888.269990 10: 16/9 996.089998 Pythagorean minor seventh 11: 1092.180 cents 1092.180000 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 107.8200 cents 107.8200 1: 256/243 90.2250 limma, Pythagorean minor second 2: 101.9550 cents 101.9550 3: 101.9550 cents 101.9550 4: 96.0900 cents 96.0900 5: 107.8200 cents 107.8200 6: 256/243 90.2250 limma, Pythagorean minor second 7: 107.8200 cents 107.8200 8: 96.0900 cents 96.0900 9: 96.0900 cents 96.0900 10: 107.8200 cents 107.8200 11: 96.0900 cents 96.0900 12: 107.8200 cents 107.8200
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 256/243 90.225 cents limma, Pythagorean minor second 1: 4 96.09000 cents 1: 2 101.95500 cents 1: 4 107.82000 cents 2: 2 192.18000 cents 2: 4 198.04500 cents 2: 6 9/8 203.910 cents major whole tone 3: 4 32/27 294.135 cents Pythagorean minor third 3: 5 300.00000 cents 3: 2 305.86500 cents 3: 1 311.73001 cents 4: 2 390.22500 cents 4: 3 396.08999 cents 4: 4 401.95500 cents 4: 3 81/64 407.820 cents Pythagorean major third 5: 8 4/3 498.045 cents perfect fourth 5: 4 503.91000 cents 6: 1 1024/729 588.270 cents Pythagorean diminished fifth 6: 4 594.13499 cents 6: 2 600.00000 cents 6: 4 605.86500 cents 6: 1 729/512 611.730 cents Pythagorean tritone 7: 4 696.09000 cents 7: 8 3/2 701.955 cents perfect fifth 8: 3 128/81 792.180 cents Pythagorean minor sixth 8: 4 798.04499 cents 8: 3 803.91000 cents 8: 2 809.77500 cents 9: 1 888.26999 cents 9: 2 894.13500 cents 9: 5 900.00000 cents 9: 4 27/16 905.865 cents Pythagorean major sixth 10: 6 16/9 996.090 cents Pythagorean minor seventh 10: 4 1001.95500 cents 10: 2 1007.82000 cents 11: 4 1092.18000 cents 11: 2 1098.04500 cents 11: 4 1103.91000 cents 11: 2 243/128 1109.775 cents Pythagorean major seventh Highest number of different intervals for one interval class: 5 Average number of different intervals per interval class: 3.54545 = 39/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 90.2 192.2 294.1 390.2 498.0 588.3 696.1 792.2 888.3 996.1 1092.2 1200.0 90.2 : 102.0 203.9 300.0 407.8 498.0 605.9 702.0 798.0 905.9 1002.0 1109.8 1200.0 192.2 : 102.0 198.0 305.9 396.1 503.9 600.0 696.1 803.9 900.0 1007.8 1098.0 1200.0 294.1 : 96.1 203.9 294.1 402.0 498.0 594.1 702.0 798.0 905.9 996.1 1098.0 1200.0 390.2 : 107.8 198.0 305.9 402.0 498.0 605.9 702.0 809.8 900.0 1002.0 1103.9 1200.0 498.0 : 90.2 198.0 294.1 390.2 498.0 594.1 702.0 792.2 894.1 996.1 1092.2 1200.0 588.3 : 107.8 203.9 300.0 407.8 503.9 611.7 702.0 803.9 905.9 1002.0 1109.8 1200.0 696.1 : 96.1 192.2 300.0 396.1 503.9 594.1 696.1 798.0 894.1 1002.0 1092.2 1200.0 792.2 : 96.1 203.9 300.0 407.8 498.0 600.0 702.0 798.0 905.9 996.1 1103.9 1200.0 888.3 : 107.8 203.9 311.7 402.0 503.9 605.9 702.0 809.8 900.0 1007.8 1103.9 1200.0 996.1 : 96.1 203.9 294.1 396.1 498.0 594.1 702.0 792.2 900.0 996.1 1092.2 1200.0 1092.2: 107.8 198.0 300.0 402.0 498.0 605.9 696.1 803.9 900.0 996.1 1103.9 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.2250 .. 107.8200 cents 17.59500 cents 256/243, 90.2250 cents 2: 192.1800 .. 203.9100 cents 11.73000 cents 84.36000 cents 3: 294.1350 .. 311.7300 cents 17.59501 cents 256/243, 90.2250 cents 4: 390.2250 .. 407.8200 cents 17.59500 cents 78.49499 cents 5: 498.0450 .. 503.9100 cents 5.86500 cents 256/243, 90.2250 cents 6: 588.2700 .. 611.7300 cents 531441/524288, 23.4600 cents 84.35999 cents 7: 696.0900 .. 701.9550 cents 5.86500 cents 84.35999 cents 8: 792.1800 .. 809.7750 cents 17.59500 cents 256/243, 90.2250 cents 9: 888.2700 .. 905.8650 cents 17.59501 cents 78.49499 cents 10: 996.0900 .. 1007.8200 cents 11.73000 cents 256/243, 90.2250 cents 11: 1092.1800 .. 1109.7750 cents 17.59500 cents 84.36000 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 256/243, 90.2250 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 107.82000 cents Average / Smallest step : 1.108340 Largest / Average step : 1.078200 Largest / Smallest step : 1.195013 Median interval of one step : 99.02250 cents, amount: 0 Least squares average step : 99.25710 cents, oct.: 1191.08520 cents Scale is strictly proper Scale is a mode of a 1636-tone equal temperament with octave 2/1 degrees: 123 262 401 532 679 802 949 1080 1211 1358 1489 1636 Least number of segments generator : 819 of 600.733 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 1011 of 741.565 cents and inv. generated superset size: 1026 = 1014 more = 8550.000% Number of contiguous 1-step segments: 0 Step pattern alph. order: ABBCDADCCDCD Step pattern size order : ACCBDADBBDBD Scale is a Constant Structure, by a margin of 78.49499 cents Scale diversity : 1.113116 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.863150 Rothenberg efficiency : 0.341703 redundancy: 0.658297 Efficiency x scale size : 4.100433 Number of different interval sizes : 39 = 3.54545 / class Number of one step interval sizes : 4 Highest interval variety : 5 Mean interval variety : 3.54545 = 39/11 Median interval variety : 2 Lowest interval variety : 2 Smallest interval difference : 5.86499 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 8 Most common triad is 0.0 498.045 701.955 cents, amount: 6 Number of recognisable fifths : 12, average 700.0000 cents Scale is a well-temperament. It has 2 different sizes of fifth Scale has monotonic third-sizes over circle of fifths Best fifths form a closed circle Formal octave complements present : 1 = 8.3333% Limited transpositions with margin 23.4600 cents: 2 5 7 10 Limited inverse transpositions with margin 17.5950 cents: 1 2 4 6 8 9 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: 7.7381 c. M:-11.7300 c. Twelve-tone Chromatic (1/11-comma) 41: 3 3 4 3 4 3 4 3 3 4 3 4 SP M ME SD: 7.1944 c. M: 16.5702 c. 43: 3 4 4 3 4 3 4 3 4 4 3 4 SP M ME SD: 6.2580 c. M:-12.8417 c. Meantone Chromatic (1/5-comma) 45: 3 4 4 4 4 3 4 4 3 4 4 4 SP D3 SD: 6.6791 c. M: 10.2250 c. 49: 4 4 4 4 4 4 4 4 4 5 4 4 SP M ME SD: 6.6454 c. M: 10.3757 c. 53: 4 4 5 4 5 4 5 4 4 5 4 5 SP M ME SD: 4.4812 c. M: 11.0479 c. 57: 4 5 5 5 5 4 5 5 4 5 5 5 SP D3 SD: 5.1860 c. M:-9.7750 c. 58: 4 5 5 5 5 4 6 4 5 5 5 5 SP D3 SD: 5.1805 c. M: 8.9596 c. 62: 5 5 5 5 6 4 6 5 5 5 5 6 SP D3 SD: 5.0870 c. M: 8.9932 c. 65: 5 5 6 5 6 5 6 5 5 6 5 6 SP M ME SD: 3.1923 c. M: 7.5646 c. 70: 5 6 6 6 6 5 7 5 6 6 6 6 SP D3 SD: 3.9155 c. M:-6.7671 c. 77: 6 6 7 6 7 6 7 6 6 7 6 7 SP M ME SD: 2.8139 c. M:-5.2087 c. 89: 7 7 8 7 8 7 8 7 7 8 7 8 SP M ME SD: 2.9318 c. M:-5.0336 c. 101: 8 8 9 8 9 8 9 8 8 9 8 9 SP M ME SD: 3.2306 c. M:-5.7894 c. 104: 8 9 9 8 9 8 9 9 8 9 9 9 SP D3 SD: 3.1978 c. M:-5.8650 c. 106: 8 9 9 8 10 8 9 9 8 10 8 10 SP D3 SD: 3.1079 c. M: 5.5240 c. 108: 8 9 9 9 10 8 10 8 9 10 8 10 SP T3 SD: 2.8687 c. M: 5.2461 c. 110: 8 10 9 9 10 8 10 9 8 10 9 10 SP SD: 2.9415 c. M: 4.6336 c. 111: 8 10 9 9 10 8 10 9 9 10 9 10 SP SD: 2.5604 c. M: 4.4862 c. 118: 9 10 10 9 11 9 10 10 9 11 9 11 SP D3 SD: 2.4354 c. M: 4.5646 c. 120: 9 10 10 10 11 9 11 9 10 11 9 11 SP T3 SD: 2.4468 c. M: 4.1350 c. 123: 9 11 10 10 11 9 11 10 10 11 10 11 SP SD: 1.8720 c. M: 3.4071 c. 135: 10 12 11 11 12 10 12 11 11 12 11 12 SP SD: 1.5273 c. M:-3.3756 c. 142: 11 12 12 11 13 11 12 12 11 13 11 13 SP D3 SD: 2.0219 c. M:-3.2793 c. 147: 11 13 12 12 13 11 13 12 12 13 12 13 SP SD: 1.5096 c. M:-3.7384 c. 155: 12 13 13 12 14 12 14 12 13 14 12 14 SP SD: 1.9047 c. M: 3.1282 c. 157: 12 13 13 13 14 12 14 13 12 14 13 14 SP SD: 1.7308 c. M: 3.6891 c. 159: 12 13 14 13 14 12 14 13 13 14 13 14 SP SD: 1.5980 c. M: 3.5008 c. 164: 12 14 14 13 15 12 15 13 13 15 13 15 SP SD: 1.9158 c. M: 2.9041 c. 167: 13 14 14 13 15 13 15 13 14 15 13 15 SP SD: 1.9006 c. M:-3.1882 c. 169: 13 14 14 14 15 13 15 14 13 15 14 15 SP SD: 1.6252 c. M:-3.0863 c. 176: 13 15 15 14 16 13 16 14 14 16 14 16 SP SD: 1.2576 c. M: 1.9064 c. 188: 14 16 16 15 17 14 17 15 15 17 15 17 SP SD: 0.6834 c. M: 1.0360 c. 200: 15 17 17 16 18 15 18 16 16 18 16 18 SP SD: 0.1781 c. M: 0.2700 c. 412: 31 35 35 33 37 31 37 33 33 37 33 37 SP SD: 0.0525 c. M:-0.0795 c. 612: 46 52 52 49 55 46 55 49 49 55 49 55 SP SD: 0.0229 c. M: 0.0347 c. 1024: 77 87 87 82 92 77 92 82 82 92 82 92 SP SD: 0.0074 c. M:-0.0113 c. 1636: 123 139 139 131 147 123 147 131 131 147 131 147 SP SD: 0.0039 c. M: 0.0059 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: 2.9325 cents 3 x x x x 4 x x x 5 x x 6 S SD: 1.3038 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 7.0402 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 7.5385 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 8.5413 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 11.8770 cents 8 x 9 x 10 x 11:12 x 13:14:15:16 S SD: 8.4693 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 7.9377 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.3478 cents 11 x 12:13:14:15 x 16:17:18:20:21:22 S SD: 10.6170 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 6.6077 cents 13:14 x 15:16:17:18:19:21:22:23:24:26 SD: 9.4469 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 8.0711 cents 15:16:17:18:19:20:21:22:24:25:27:28:30 SD: 5.3279 cents 16:17:18:19:20:21:22:24:25:27:28:30:32 SD: 5.2540 cents 17:18:19:20:21:23:24:25:27:28:30:32:34 SD: 4.7144 cents 22:23:25:26:28:29:31:33:35:37:39:41:44 SD: 4.3741 cents 24:25:27:28:30:32:34:36:38:40:43:45:48 SD: 3.4895 cents 27:28:30:32:34:36:38:40:43:45:48:51:54 SD: 3.1721 cents 32:34:36:38:40:43:45:48:51:53:57:60:64 SD: 2.7166 cents 33:35:37:39:41:44:46:49:52:55:59:62:66 SD: 2.4325 cents 36:38:40:43:45:48:51:54:57:60:64:68:72 SD: 2.1417 cents 42:44:47:50:53:56:59:63:66:70:75:79:84 SD: 1.9031 cents 51:54:57:60:64:68:72:76:81:85:91:96:102 SD: 1.8525 cents 52:55:58:62:65:69:73:78:82:87:92:98:104 SD: 1.6664 cents 54:57:60:64:68:72:76:81:85:90:96:101:108 SD: 1.5805 cents 59:62:66:70:74:79:83:88:93:99:105:111:118 SD: 1.1565 cents 75:79:84:89:94:100:105:112:119:125:133:141:150 SD: 0.9923 cents 76:80:85:90:95:101:107:114:120:127:135:143:152 SD: 0.8886 cents 91:96:102:108:114:121:128:136:144:152:162:171:182 SD: 0.7458 cents 119:125:133:141:149:159:167:178:188:199:212:224:238 SD: 0.6924 cents 126:133:141:149:158:168:177:188:199:210:224:237:252 SD: 0.6723 cents 129:136:144:153:162:172:181:193:204:215:229:242:258 SD: 0.6517 cents 131:138:146:155:164:175:184:196:207:219:233:246:262 SD: 0.5739 cents 146:154:163:173:183:195:205:218:231:244:260:274:292 SD: 0.5303 cents 162:171:181:192:203:216:228:242:256:271:288:304:324 SD: 0.5068 cents 171:180:191:203:214:228:240:256:270:286:304:321:342 SD: 0.4758 cents 186:196:208:220:233:248:261:278:294:311:331:350:372 SD: 0.4469 cents 188:198:210:223:236:251:264:281:297:314:334:353:376 SD: 0.4073 cents 205:216:229:243:257:273:288:306:324:342:364:385:410 SD: 0.4057 cents 206:217:230:244:258:275:289:308:326:344:366:387:412 SD: 0.3768 cents 210:221:235:249:263:280:295:314:332:351:373:395:420 SD: 0.3487 cents 222:234:248:263:278:296:312:332:351:371:395:417:444 SD: 0.2305 cents 297:313:332:352:372:396:417:444:469:496:528:558:594 SD: 0.1553 cents 486:512:543:576:609:648:683:727:768:812:864:913:972 SD: 0.1339 cents 519:547:580:615:650:692:729:776:820:867:923:975:1038 SD: 0.1163 cents 546:575:610:647:684:728:767:816:863:912:971:1026:1092 SD: 0.0969 cents