Scala analysis: werckmeister3_eb
Harmonic equal-beating meta-version of the famous Well Temperament
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: 21995/19683 192.270569 3: 32/27 294.134997 Pythagorean minor third 4: 2740/2187 390.267207 5: 4/3 498.044999 perfect fourth 6: 1024/729 588.269995 Pythagorean diminished fifth 7: 9808/6561 696.051910 8: 128/81 792.179997 Pythagorean minor sixth 9: 10960/6561 888.312206 10: 16/9 996.089998 Pythagorean minor seventh 11: 1370/729 1092.222208 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 729/685 107.7778 1: 256/243 90.2250 limma, Pythagorean minor second 2: 21995/20736 102.0456 3: 23328/21995 101.8644 4: 685/648 96.1322 5: 729/685 107.7778 6: 256/243 90.2250 limma, Pythagorean minor second 7: 613/576 107.7819 8: 648/613 96.1281 9: 685/648 96.1322 10: 729/685 107.7778 11: 685/648 96.1322 12: 729/685 107.7778
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 256/243 90.225 cents limma, Pythagorean minor second 1: 1 648/613 96.128 cents 1: 3 685/648 96.132 cents 1: 1 23328/21995 101.864 cents 1: 1 21995/20736 102.046 cents 1: 3 729/685 107.778 cents 1: 1 613/576 107.782 cents 2: 1 685/613 192.260 cents 2: 1 21995/19683 192.271 cents 2: 1 4932/4399 197.997 cents 2: 2 768/685 198.003 cents 2: 1 2452/2187 198.007 cents 2: 6 9/8 203.910 cents major whole tone 3: 4 32/27 294.135 cents Pythagorean minor third 3: 1 729/613 300.038 cents 3: 3 685/576 300.042 cents 3: 1 4399/3699 300.048 cents 3: 1 26244/21995 305.774 cents 3: 1 2452/2055 305.785 cents 3: 1 6561/5480 311.688 cents 4: 2 2740/2187 390.267 cents 4: 1 27648/21995 395.999 cents 4: 1 6165/4904 396.170 cents 4: 1 21995/17496 396.181 cents 4: 3 864/685 401.913 cents 4: 1 613/486 401.917 cents 4: 3 81/64 407.820 cents Pythagorean major third 5: 8 4/3 498.045 cents perfect fourth 5: 1 29424/21995 503.781 cents 5: 1 6561/4904 503.948 cents 5: 1 685/512 503.952 cents 5: 1 4399/3288 503.958 cents 6: 1 1024/729 588.270 cents Pythagorean diminished fifth 6: 1 864/613 594.173 cents 6: 3 685/486 594.177 cents 6: 1 31104/21995 599.909 cents 6: 1 21995/15552 600.091 cents 6: 3 972/685 605.823 cents 6: 1 613/432 605.827 cents 6: 1 729/512 611.730 cents Pythagorean tritone 7: 1 6576/4399 696.042 cents 7: 1 1024/685 696.048 cents 7: 1 9808/6561 696.052 cents 7: 1 21995/14712 696.219 cents 7: 8 3/2 701.955 cents perfect fifth 8: 3 128/81 792.180 cents Pythagorean minor sixth 8: 1 972/613 798.083 cents 8: 3 685/432 798.087 cents 8: 1 34992/21995 803.819 cents 8: 1 9808/6165 803.830 cents 8: 1 21995/13824 804.001 cents 8: 2 2187/1370 809.733 cents 9: 1 10960/6561 888.312 cents 9: 1 2055/1226 894.215 cents 9: 1 21995/13122 894.226 cents 9: 1 7398/4399 899.952 cents 9: 3 1152/685 899.958 cents 9: 1 1226/729 899.962 cents 9: 4 27/16 905.865 cents Pythagorean major sixth 10: 6 16/9 996.090 cents Pythagorean minor seventh 10: 1 2187/1226 1001.993 cents 10: 2 685/384 1001.997 cents 10: 1 4399/2466 1002.003 cents 10: 1 39366/21995 1007.729 cents 10: 1 1226/685 1007.740 cents 11: 1 1152/613 1092.218 cents 11: 3 1370/729 1092.222 cents 11: 1 41472/21995 1097.954 cents 11: 1 21995/11664 1098.136 cents 11: 3 1296/685 1103.868 cents 11: 1 613/324 1103.872 cents 11: 2 243/128 1109.775 cents Pythagorean major seventh Highest number of different intervals for one interval class: 8 Average number of different intervals per interval class: 6.54545 = 72/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 90.2 192.3 294.1 390.3 498.0 588.3 696.1 792.2 888.3 996.1 1092.2 1200.0 90.2 : 102.0 203.9 300.0 407.8 498.0 605.8 702.0 798.1 905.9 1002.0 1109.8 1200.0 192.3 : 101.9 198.0 305.8 396.0 503.8 599.9 696.0 803.8 900.0 1007.7 1098.0 1200.0 294.1 : 96.1 203.9 294.1 401.9 498.0 594.2 702.0 798.1 905.9 996.1 1098.1 1200.0 390.3 : 107.8 198.0 305.8 401.9 498.0 605.8 702.0 809.7 900.0 1002.0 1103.9 1200.0 498.0 : 90.2 198.0 294.1 390.3 498.0 594.2 702.0 792.2 894.2 996.1 1092.2 1200.0 588.3 : 107.8 203.9 300.0 407.8 504.0 611.7 702.0 804.0 905.9 1002.0 1109.8 1200.0 696.1 : 96.1 192.3 300.0 396.2 503.9 594.2 696.2 798.1 894.2 1002.0 1092.2 1200.0 792.2 : 96.1 203.9 300.0 407.8 498.0 600.1 702.0 798.1 905.9 996.1 1103.9 1200.0 888.3 : 107.8 203.9 311.7 401.9 504.0 605.8 702.0 809.7 900.0 1007.7 1103.9 1200.0 996.1 : 96.1 203.9 294.1 396.2 498.0 594.2 702.0 792.2 900.0 996.1 1092.2 1200.0 1092.2: 107.8 198.0 300.0 401.9 498.0 605.8 696.0 803.8 900.0 996.1 1103.9 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.2250 .. 107.7819 cents 16551/16384, 17.5569 cents 256/243, 90.2250 cents 2: 192.2603 .. 203.9100 cents 5517/5480, 11.6497 cents 394560/375769, 84.4784 cents 3: 294.1350 .. 311.6878 cents 177147/175360, 17.5528 cents 256/243, 90.2250 cents 4: 390.2672 .. 407.8200 cents 177147/175360, 17.5528 cents 15015200/14348907, 78.5794 cents 5: 498.0450 .. 503.9584 cents 4399/4384, 5.9134 cents 256/243, 90.2250 cents 6: 588.2700 .. 611.7300 cents 531441/524288, 23.4600 cents 1122304/1068957, 84.3116 cents 7: 696.0416 .. 701.9550 cents 4399/4384, 5.9134 cents 1122304/1068957, 84.3116 cents 8: 792.1800 .. 809.7328 cents 177147/175360, 17.5528 cents 256/243, 90.2250 cents 9: 888.3122 .. 905.8650 cents 177147/175360, 17.5528 cents 15015200/14348907, 78.5794 cents 10: 996.0900 .. 1007.7397 cents 5517/5480, 11.6497 cents 256/243, 90.2250 cents 11: 1092.2181 .. 1109.7750 cents 16551/16384, 17.5569 cents 394560/375769, 84.4784 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 : 613/576, 107.7819 cents Average / Smallest step : 1.108340 Largest / Average step : 1.077819 Largest / Smallest step : 1.194590 Median interval of one step : 98.99832 cents, amount: 0 Least squares average step : 99.25853 cents, oct.: 1191.10232 cents Scale is strictly proper Step pattern alph. order: ABCDEAFGDEDE Step pattern size order : AEDCFAGBCFCF Scale is a Constant Structure, by a margin of 78.57941 cents Scale diversity : 1.112979 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.863407 Rothenberg efficiency : 0.294102 redundancy: 0.705898 Efficiency x scale size : 3.529221 Number of different interval sizes : 72 = 6.54545 / class Number of one step interval sizes : 7 Highest interval variety : 8 Mean interval variety : 6.54545 = 72/11 Median interval variety : 5 Lowest interval variety : 5 Smallest interval difference : 419905/419904, 0.0041 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 5 different sizes of fifth 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 17.5569 cents: 1 2 4 6 8 9 11 -- Rational properties -- Prime limit : 613 Odd number limit : 21995 (O: 21995 U: 21995) Highest odd numerator or denominator: 21995 Scale harmonicity : 0.024752 Fundamental : 1/19683, -14.2647 octaves, 0.0133 Hz. Guide tone : too high to compute Exponens Consonantiae : 3.723028E+16, 55.04733 octaves Euler's gradus suavitatis : 915 Sum of Mann's harmonic distance : 42574.5, average 3547.87500 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 28.19839, average 2.34987 Sum of Tenney's harmonic distance : 58.23645, average 4.85304 Vogel's harmonic complexity : 114.58333 Wille's k value : 10997 Wilson's harmonic complexity : 918 Rectangular lattice diameter : 14 Triangular lattice diameter : 9 Lattice compactness : 450.10262, average 5.77055 Lattice compactness (without 2's) : 297.13436, average 3.80941 Number of different primes : 7 Prime exponents' range, average, count, tones@limit: 2: 0 .. 10 4.00000 48 1 3: -9 .. 0 -4.91667 59 6 5: 0 .. 1 0.33333 4 53: 0 .. 1 0.08333 1 83: 0 .. 1 0.08333 1 1 137: 0 .. 1 0.25000 3 3 613: 0 .. 1 0.08333 1 1 Average exponent except 2's :-49 / 12 =-4.08333 Average absolute exponent except 2's: 69 / 12 = 5.75000 Scale is JI-epimorphic with val: <12 19 27 68 78 86 111| Scale is JI-epimorphic with val: <12 19 27 69 77 86 111| Scale is JI-epimorphic with val: <12 19 27 70 76 86 111| Scale is JI-epimorphic with val: <12 19 28 68 77 85 111| Scale is JI-epimorphic with val: <12 19 28 69 76 85 111| Scale is JI-epimorphic with val: <12 19 29 68 76 84 111|
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 7.7188 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: 7.2289 c. M: 16.6608 c. 43: 3 4 4 3 4 3 4 3 4 4 3 4 SP M ME SD: 6.2543 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: 6.6827 c. M: 10.2250 c. 49: 4 4 4 4 4 4 4 4 4 5 4 4 SP M ME SD: 6.6338 c. M: 10.3376 c. 53: 4 4 5 4 5 4 5 4 4 5 4 5 SP M ME SD: 4.5165 c. M: 11.1385 c. 57: 4 5 5 5 5 4 5 5 4 5 5 5 SP D3 SD: 5.1836 c. M:-9.7328 c. 58: 4 5 5 5 5 4 6 4 5 5 5 5 SP D3 SD: 5.1880 c. M: 8.9596 c. 62: 5 5 5 5 6 4 6 5 5 5 5 6 SP D3 SD: 5.0920 c. M: 8.9932 c. 65: 5 5 6 5 6 5 6 5 5 6 5 6 SP M ME SD: 3.2240 c. M: 7.6552 c. 70: 5 6 6 6 6 5 7 5 6 6 6 6 SP D3 SD: 3.9172 c. M:-6.8052 c. 77: 6 6 7 6 7 6 7 6 6 7 6 7 SP M ME SD: 2.8361 c. M: 5.2576 c. 89: 7 7 8 7 8 7 8 7 7 8 7 8 SP M ME SD: 2.9434 c. M:-5.0717 c. 101: 8 8 9 8 9 8 9 8 8 9 8 9 SP M ME SD: 3.2344 c. M:-5.7894 c. 104: 8 9 9 8 9 8 9 9 8 9 9 9 SP D3 SD: 3.1779 c. M:-5.8650 c. 106: 8 9 9 8 10 8 9 9 8 10 8 10 SP D3 SD: 3.1198 c. M: 5.4859 c. 108: 8 9 9 9 10 8 10 8 9 10 8 10 SP T3 SD: 2.8867 c. M: 5.2461 c. 110: 8 10 9 9 10 8 10 9 8 10 9 10 SP SD: 2.9373 c. M: 4.6758 c. 111: 8 10 9 9 10 8 10 9 9 10 9 10 SP SD: 2.5525 c. M: 4.4862 c. 118: 9 10 10 9 11 9 10 10 9 11 9 11 SP D3 SD: 2.4428 c. M: 4.5265 c. 120: 9 10 10 10 11 9 11 9 10 11 9 11 SP T3 SD: 2.4598 c. M: 4.1350 c. 123: 9 11 10 10 11 9 11 10 10 11 10 11 SP SD: 1.8545 c. M: 3.3690 c. 135: 10 12 11 11 12 10 12 11 11 12 11 12 SP SD: 1.4989 c. M:-3.2850 c. 142: 11 12 12 11 13 11 12 12 11 13 11 13 SP D3 SD: 2.0169 c. M:-3.2793 c. 147: 11 13 12 12 13 11 13 12 12 13 12 13 SP SD: 1.4751 c. M:-3.6478 c. 155: 12 13 13 12 14 12 14 12 13 14 12 14 SP SD: 1.9038 c. M: 3.1704 c. 157: 12 13 13 13 14 12 14 13 12 14 13 14 SP SD: 1.7374 c. M: 3.6891 c. 159: 12 14 13 13 14 12 14 13 13 14 13 14 SP SD: 1.6669 c. M:-3.9558 c. 164: 12 14 14 13 15 12 15 13 13 15 13 15 SP SD: 1.9354 c. M: 2.9464 c. 167: 13 14 14 13 15 13 15 13 14 15 13 15 SP SD: 1.8941 c. M:-3.1882 c. 169: 13 14 14 14 15 13 15 14 13 15 14 15 SP SD: 1.6253 c. M:-3.0863 c. 171: 13 14 15 14 15 13 15 14 14 15 14 15 SP SD: 1.7179 c. M:-2.9159 c. 176: 13 15 15 14 16 13 16 14 14 16 14 16 SP SD: 1.2773 c. M: 1.9486 c. 188: 14 16 16 15 17 14 17 15 15 17 15 17 SP SD: 0.7034 c. M: 1.0782 c. 200: 15 17 17 16 18 15 18 16 16 18 16 18 SP SD: 0.1997 c. M: 0.3122 c. 412: 31 35 35 33 37 31 37 33 33 37 33 37 SP SD: 0.0444 c. M:-0.0795 c. 1024: 77 87 87 82 92 77 92 82 82 92 82 92 SP SD: 0.0319 c. M: 0.0831 c. 1436: 108 122 122 115 129 108 129 115 115 129 115 129 SP SD: 0.0296 c. M: 0.0700 c. 1848: 139 157 157 148 166 139 166 148 148 166 148 166 SP SD: 0.0307 c. M: 0.0628 c. 2260: 170 192 192 181 203 170 203 181 181 203 181 203 SP SD: 0.0322 c. M: 0.0582 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: 2.9515 cents 3 x x x x 4 x x x 5 x x 6 S SD: 1.3178 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 7.0436 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: 8.5484 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 11.8659 cents 8 x 9 x 10 x 11:12 x 13:14:15:16 S SD: 8.4721 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 7.9416 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.3486 cents 11 x 12:13:14:15 x 16:17:18:20:21:22 S SD: 10.6181 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 6.6066 cents 13:14 x 15:16:17:18:19:21:22:23:24:26 SD: 9.4470 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 8.0684 cents 15:16:17:18:19:20:21:22:24:25:27:28:30 SD: 5.3232 cents 16:17:18:19:20:21:22:24:25:27:28:30:32 SD: 5.2523 cents 17:18:19:20:21:23:24:25:27:28:30:32:34 SD: 4.7156 cents 22:23:25:26:28:29:31:33:35:37:39:41:44 SD: 4.3686 cents 24:25:27:28:30:32:34:36:38:40:43:45:48 SD: 3.4888 cents 27:28:30:32:34:36:38:40:43:45:48:51:54 SD: 3.1715 cents 32:34:36:38:40:43:45:48:51:53:57:60:64 SD: 2.7169 cents 33:35:37:39:41:44:46:49:52:55:59:62:66 SD: 2.4319 cents 36:38:40:43:45:48:51:54:57:60:64:68:72 SD: 2.1451 cents 42:44:47:50:53:56:59:63:66:70:75:79:84 SD: 1.9015 cents 51:54:57:60:64:68:72:76:81:85:91:96:102 SD: 1.8514 cents 52:55:58:62:65:69:73:78:82:87:92:98:104 SD: 1.6679 cents 54:57:60:64:68:72:76:81:85:90:96:101:108 SD: 1.5860 cents 59:62:66:70:74:79:83:88:93:99:105:111:118 SD: 1.1516 cents 75:79:84:89:94:100:105:112:119:125:133:141:150 SD: 0.9900 cents 76:80:85:90:95:101:107:114:120:127:135:143:152 SD: 0.8896 cents 91:96:102:108:114:121:128:136:144:152:162:171:182 SD: 0.7413 cents 119:125:133:141:149:159:167:178:188:199:212:224:238 SD: 0.6909 cents 126:133:141:149:158:168:177:188:199:210:224:237:252 SD: 0.6690 cents 129:136:144:153:162:172:181:193:204:215:229:242:258 SD: 0.6552 cents 131:138:146:155:164:175:184:196:207:219:233:246:262 SD: 0.5801 cents 146:154:163:173:183:195:205:218:231:244:260:274:292 SD: 0.5315 cents 162:171:181:192:203:216:228:242:256:271:288:304:324 SD: 0.5063 cents 171:180:191:203:214:228:240:256:270:286:304:321:342 SD: 0.4792 cents 186:196:208:220:233:248:261:278:294:311:331:350:372 SD: 0.4424 cents 188:198:210:223:236:251:264:281:297:314:334:353:376 SD: 0.4069 cents 205:216:229:243:257:273:288:306:324:342:364:385:410 SD: 0.4065 cents 206:217:230:244:258:275:289:308:326:344:366:387:412 SD: 0.3807 cents 210:221:235:249:263:280:295:314:332:351:373:395:420 SD: 0.3428 cents 222:234:248:263:278:296:312:332:351:371:395:417:444 SD: 0.2337 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: 0.1370 cents 519:547:580:615:650:692:729:776:820:867:923:975:1038 SD: 0.1190 cents 546:575:610:647:684:728:767:816:863:912:971:1026:1092 SD: 0.0990 cents