Scala analysis: werckmeister3_eb89-l
Harmonic equal-beating 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: 2225/2112 90.234602 2: 295/264 192.212430 3: 6675/5632 294.144603 4: 441/352 390.243872 5: 4/3 498.044999 perfect fourth 6: 2225/1584 588.279601 7: 148/99 696.116095 8: 2225/1408 792.189603 9: 147/88 888.288871 10: 16/9 996.089998 Pythagorean minor seventh 11: 1323/704 1092.198873 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 1408/1323 107.8011 1: 2225/2112 90.2346 2: 472/445 101.9778 3: 4005/3776 101.9322 4: 2352/2225 96.0993 5: 1408/1323 107.8011 6: 2225/2112 90.2346 7: 2368/2225 107.8365 8: 20025/18944 96.0735 9: 2352/2225 96.0993 10: 1408/1323 107.8011 11: 11907/11264 96.1089 12: 1408/1323 107.8011
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 2225/2112 90.235 cents 1: 1 20025/18944 96.074 cents 1: 2 2352/2225 96.099 cents 1: 1 11907/11264 96.109 cents 1: 1 4005/3776 101.932 cents 1: 1 472/445 101.978 cents 1: 3 1408/1323 107.801 cents 1: 1 2368/2225 107.836 cents 2: 1 1323/1184 192.173 cents 2: 1 295/264 192.212 cents 2: 1 1323/1180 198.031 cents 2: 2 4450/3969 198.036 cents 2: 1 37/33 198.071 cents 2: 2 22528/20025 203.900 cents 2: 4 9/8 203.910 cents major whole tone 3: 1 32/27 294.135 cents Pythagorean minor third 3: 3 6675/5632 294.145 cents 3: 1 44/37 299.974 cents 3: 3 2646/2225 300.009 cents 3: 1 4720/3969 300.014 cents 3: 1 352/295 305.833 cents 3: 1 4736/3969 305.872 cents 3: 1 176/147 311.711 cents 4: 2 441/352 390.244 cents 4: 1 445/354 396.067 cents 4: 1 11907/9472 396.083 cents 4: 1 885/704 396.122 cents 4: 3 2225/1764 401.946 cents 4: 1 75776/60075 401.971 cents 4: 3 2816/2225 407.810 cents 5: 7 4/3 498.045 cents perfect fourth 5: 1 60075/45056 498.055 cents 5: 1 99/74 503.884 cents 5: 1 1184/885 503.904 cents 5: 1 11907/8900 503.919 cents 5: 1 590/441 503.924 cents 6: 1 2225/1584 588.280 cents 6: 1 6675/4736 594.119 cents 6: 1 3136/2225 594.144 cents 6: 2 3969/2816 594.154 cents 6: 1 1335/944 599.977 cents 6: 1 1888/1335 600.023 cents 6: 2 5632/3969 605.846 cents 6: 1 2225/1568 605.856 cents 6: 1 9472/6675 605.881 cents 6: 1 3168/2225 611.720 cents 7: 1 441/295 696.076 cents 7: 1 17800/11907 696.081 cents 7: 1 885/592 696.096 cents 7: 1 148/99 696.116 cents 7: 1 90112/60075 701.945 cents 7: 7 3/2 701.955 cents perfect fifth 8: 3 2225/1408 792.190 cents 8: 1 60075/37888 798.029 cents 8: 3 3528/2225 798.054 cents 8: 1 1408/885 803.878 cents 8: 1 18944/11907 803.917 cents 8: 1 708/445 803.933 cents 8: 2 704/441 809.756 cents 9: 1 147/88 888.289 cents 9: 1 3969/2368 894.128 cents 9: 1 295/176 894.167 cents 9: 1 3969/2360 899.986 cents 9: 3 2225/1323 899.991 cents 9: 1 37/22 900.026 cents 9: 3 11264/6675 905.855 cents 9: 1 27/16 905.865 cents Pythagorean major sixth 10: 4 16/9 996.090 cents Pythagorean minor seventh 10: 2 20025/11264 996.100 cents 10: 1 66/37 1001.929 cents 10: 2 3969/2225 1001.964 cents 10: 1 2360/1323 1001.969 cents 10: 1 528/295 1007.788 cents 10: 1 2368/1323 1007.827 cents 11: 1 2225/1184 1092.164 cents 11: 3 1323/704 1092.199 cents 11: 1 445/236 1098.022 cents 11: 1 7552/4005 1098.068 cents 11: 1 22528/11907 1103.891 cents 11: 2 2225/1176 1103.901 cents 11: 1 37888/20025 1103.926 cents 11: 2 4224/2225 1109.765 cents Highest number of different intervals for one interval class: 10 Average number of different intervals per interval class: 7.45455 = 82/11
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 : 102.0 203.9 300.0 407.8 498.0 605.9 702.0 798.1 905.9 1002.0 1109.8 1200.0 192.2 : 101.9 198.0 305.8 396.1 503.9 600.0 696.1 803.9 900.0 1007.8 1098.0 1200.0 294.1 : 96.1 203.9 294.1 402.0 498.0 594.1 701.9 798.1 905.9 996.1 1098.1 1200.0 390.2 : 107.8 198.0 305.9 401.9 498.0 605.8 702.0 809.8 900.0 1002.0 1103.9 1200.0 498.0 : 90.2 198.1 294.1 390.2 498.0 594.2 702.0 792.2 894.2 996.1 1092.2 1200.0 588.3 : 107.8 203.9 300.0 407.8 503.9 611.7 702.0 803.9 905.9 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 1001.9 1092.2 1200.0 792.2 : 96.1 203.9 300.0 407.8 498.0 600.0 702.0 798.1 905.9 996.1 1103.9 1200.0 888.3 : 107.8 203.9 311.7 401.9 503.9 605.9 702.0 809.8 900.0 1007.8 1103.9 1200.0 996.1 : 96.1 203.9 294.1 396.1 498.1 594.2 702.0 792.2 900.0 996.1 1092.2 1200.0 1092.2: 107.8 198.0 300.0 401.9 498.0 605.8 696.1 803.9 900.0 996.1 1103.9 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.2346 .. 107.8365 cents 5001216/4950625, 17.6019 cents 2225/2112, 90.2346 cents 2: 192.1728 .. 203.9100 cents 148/147, 11.7372 cents 2943675/2803712, 84.3363 cents 3: 294.1350 .. 311.7111 cents 99/98, 17.5761 cents 256/243, 90.2250 cents 4: 390.2439 .. 407.8104 cents 991232/981225, 17.5665 cents 64827/61952, 78.5327 cents 5: 498.0450 .. 503.9236 cents 295/294, 5.8786 cents 2225/2112, 90.2346 cents 6: 588.2796 .. 611.7204 cents 5018112/4950625, 23.4408 cents 21805/20768, 84.3560 cents 7: 696.0764 .. 701.9550 cents 295/294, 5.8786 cents 21805/20768, 84.3560 cents 8: 792.1896 .. 809.7561 cents 991232/981225, 17.5665 cents 2225/2112, 90.2346 cents 9: 888.2889 .. 905.8650 cents 99/98, 17.5761 cents 64827/61952, 78.5327 cents 10: 996.0900 .. 1007.8272 cents 148/147, 11.7372 cents 256/243, 90.2250 cents 11: 1092.1635 .. 1109.7654 cents 5001216/4950625, 17.6019 cents 2943675/2803712, 84.3363 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 2225/2112, 90.2346 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 2368/2225, 107.8365 cents Average / Smallest step : 1.108222 Largest / Average step : 1.078365 Largest / Smallest step : 1.195068 Median interval of one step : 99.02052 cents, amount: 0 Most common interval of one step : 1408/1323, 107.8011 cents, amount: 3 Least squares average step : 99.25844 cents, oct.: 1191.10132 cents Scale is strictly proper Step pattern alph. order: ABCDEAFGDEHE Step pattern size order : AFECGAHBCGDG Scale is a Constant Structure, by a margin of 78.53274 cents Scale diversity : 1.113127 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.863199 Rothenberg efficiency : 0.257912 redundancy: 0.742088 Efficiency x scale size : 3.094949 Number of different interval sizes : 82 = 7.45455 / class Number of one step interval sizes : 8 Highest interval variety : 10 Mean interval variety : 7.45455 = 82/11 Median interval variety : 6 Lowest interval variety : 6 Smallest interval difference : 5251000/5250987, 0.0043 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 7 Most common triad is 0.0 498.045 701.955 cents, amount: 4 Number of recognisable fifths : 12, average 700.0000 cents Scale is a well-temperament. It has 6 different sizes of fifth Best fifths form a closed circle Formal octave complements present : 1 = 8.3333% Limited transpositions with margin 23.4408 cents: 2 5 7 10 Limited inverse transpositions with margin 17.6019 cents: 1 2 4 6 8 9 11 -- Rational properties -- Prime limit : 89 Odd number limit : 60075 (O: 60075 U: 60075) Highest odd numerator or denominator: 6675 Scale harmonicity : 0.012839 Fundamental : 1/50688, -15.6294 octaves, 0.0052 Hz. Guide tone : too high to compute Exponens Consonantiae : 5.211572E+15, 52.21064 octaves Euler's gradus suavitatis : 236 Sum of Mann's harmonic distance : 13991.0, average 1165.91667 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 27.69658, average 2.30805 Sum of Tenney's harmonic distance : 55.15797, average 4.59650 Vogel's harmonic complexity : 59.33333 Wille's k value : 30037 Wilson's harmonic complexity : 235 Rectangular lattice diameter : 13 Triangular lattice diameter : 7 Lattice compactness : 449.78955, average 5.76653 Lattice compactness (without 2's) : 209.90547, average 2.69110 Number of different primes : 8 Prime exponents' range, average, count, tones@limit: 2: -9 .. 4 -2.83333 52 1 3: -2 .. 3 -0.16667 16 2 5: 0 .. 2 0.75000 9 7: 0 .. 2 0.50000 6 11: -1 .. 0 -0.75000 9 3 37: 0 .. 1 0.08333 1 1 59: 0 .. 1 0.08333 1 1 89: 0 .. 1 0.33333 4 4 Average exponent except 2's : 10 / 12 = 0.83333 Average absolute exponent except 2's: 46 / 12 = 3.83333 Scale is weakly epimorphic with val: <12 19 27 32 40 65 72 78| Scale is JI-epimorphic with val: <12 19 27 33 40 61 70 78| Scale is weakly epimorphic with val: <12 19 27 34 40 62 71 79| Scale is weakly epimorphic with val: <12 19 28 32 41 66 71 78| Scale is weakly epimorphic with val: <12 19 28 34 40 62 70 77| Scale is JI-epimorphic with val: <12 19 28 34 42 63 71 78| = patent Scale is weakly epimorphic with val: <12 19 28 35 42 64 72 79| Scale is weakly epimorphic with val: <12 19 29 35 42 64 71 77| Scale is JI-epimorphic with val: <12 19 29 35 44 65 72 78| Scale is weakly epimorphic with val: <12 19 29 36 42 63 70 79|
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 7.7247 c. M:-11.7204 c. Twelve-tone Chromatic (1/11-comma) 41: 3 3 4 3 4 3 4 3 3 4 3 4 SP M ME SD: 7.2061 c. M: 16.6027 c. 43: 3 4 4 3 4 3 4 3 4 4 3 4 SP M ME SD: 6.2566 c. M:-12.8321 c. Meantone Chromatic (1/5-comma) 45: 3 4 4 4 4 3 4 4 3 4 4 4 SP D3 SD: 6.6822 c. M: 10.2346 c. 49: 4 4 4 4 4 4 4 4 4 5 4 4 SP M ME SD: 6.6464 c. M: 10.4018 c. 53: 4 4 5 4 5 4 5 4 4 5 4 5 SP M ME SD: 4.4905 c. M: 11.0804 c. 57: 4 5 5 5 5 4 5 5 4 5 5 5 SP D3 SD: 5.1850 c. M:-9.7561 c. 58: 4 5 5 5 5 4 6 4 5 5 5 5 SP D3 SD: 5.1821 c. M: 8.9693 c. 62: 5 5 5 5 6 4 6 5 5 5 5 6 SP D3 SD: 5.0895 c. M: 8.9932 c. 65: 5 5 6 5 6 5 6 5 5 6 5 6 SP M ME SD: 3.1969 c. M: 7.5970 c. 70: 5 6 6 6 6 5 7 5 6 6 6 6 SP D3 SD: 3.9127 c. M:-6.7410 c. 77: 6 6 7 6 7 6 7 6 6 7 6 7 SP M ME SD: 2.8126 c. M: 5.1994 c. 89: 7 7 8 7 8 7 8 7 7 8 7 8 SP M ME SD: 2.9259 c. M:-5.0075 c. 101: 8 8 9 8 9 8 9 8 8 9 8 9 SP M ME SD: 3.2221 c. M:-5.7798 c. 104: 8 9 9 8 9 8 9 9 8 9 9 9 SP D3 SD: 3.1909 c. M:-5.8554 c. 106: 8 9 9 8 10 8 9 9 8 10 8 10 SP D3 SD: 3.1193 c. M: 5.5501 c. 108: 8 9 9 9 10 8 10 8 9 10 8 10 SP T3 SD: 2.8736 c. M: 5.2557 c. 110: 8 10 9 9 10 8 10 9 8 10 9 10 SP SD: 2.9373 c. M: 4.6525 c. 111: 8 10 9 9 10 8 10 9 9 10 9 10 SP SD: 2.5675 c. M: 4.4958 c. 118: 9 10 10 9 11 9 10 10 9 11 9 11 SP D3 SD: 2.4441 c. M: 4.5907 c. 120: 9 10 10 10 11 9 11 9 10 11 9 11 SP T3 SD: 2.4478 c. M: 4.1446 c. 123: 9 11 10 10 11 9 11 10 10 11 10 11 SP SD: 1.8754 c. M: 3.4332 c. 135: 10 12 11 11 12 10 12 11 11 12 11 12 SP SD: 1.5251 c. M:-3.3431 c. 142: 11 12 12 11 13 11 12 12 11 13 11 13 SP D3 SD: 2.0220 c. M:-3.2697 c. 147: 11 13 12 12 13 11 13 12 12 13 12 13 SP SD: 1.5020 c. M:-3.7059 c. 155: 12 13 13 12 14 12 14 12 13 14 12 14 SP SD: 1.9033 c. M: 3.1471 c. 157: 12 13 13 13 14 12 14 13 12 14 13 14 SP SD: 1.7340 c. M: 3.6987 c. 159: 12 13 14 13 14 12 14 13 13 14 13 14 SP SD: 1.5991 c. M: 3.5332 c. 164: 12 14 14 13 15 12 15 13 13 15 13 15 SP SD: 1.9293 c. M: 2.9230 c. 167: 13 14 14 13 15 13 15 13 14 15 13 15 SP SD: 1.8954 c. M:-3.1786 c. 169: 13 14 14 14 15 13 15 14 13 15 14 15 SP SD: 1.6239 c. M:-3.0767 c. 176: 13 15 15 14 16 13 16 14 14 16 14 16 SP SD: 1.2711 c. M: 1.9252 c. 188: 14 16 16 15 17 14 17 15 15 17 15 17 SP SD: 0.6969 c. M: 1.0548 c. 200: 15 17 17 16 18 15 18 16 16 18 16 18 SP SD: 0.1918 c. M: 0.2889 c. 412: 31 35 35 33 37 31 37 33 33 37 33 37 SP SD: 0.0401 c. M:-0.0699 c. 612: 46 52 52 49 55 46 55 49 49 55 49 55 SP SD: 0.0375 c. M: 0.0556 c. 1024: 77 87 87 82 92 77 92 82 82 92 82 92 SP SD: 0.0110 c. M: 0.0249 c. 1436: 108 122 122 115 129 108 129 115 115 129 115 129 SP SD: 0.0115 c. M:-0.0212 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: 2.9195 cents 3 x x x x 4 x x x 5 x x 6 S SD: 1.3101 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 7.0395 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 7.5366 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 8.5445 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 11.8723 cents 8 x 9 x 10 x 11:12 x 13:14:15:16 S SD: 8.4709 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 7.9390 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.3470 cents 11 x 12:13:14:15 x 16:17:18:20:21:22 S SD: 10.6195 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 6.6064 cents 13:14 x 15:16:17:18:19:21:22:23:24:26 SD: 9.4484 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 8.0692 cents 15:16:17:18:19:20:21:22:24:25:27:28:30 SD: 5.3271 cents 16:17:18:19:20:21:22:24:25:27:28:30:32 SD: 5.2535 cents 17:18:19:20:21:23:24:25:27:28:30:32:34 SD: 4.7166 cents 22:23:25:26:28:29:31:33:35:37:39:41:44 SD: 4.3716 cents 24:25:27:28:30:32:34:36:38:40:43:45:48 SD: 3.4894 cents 27:28:30:32:34:36:38:40:43:45:48:51:54 SD: 3.1734 cents 32:34:36:38:40:43:45:48:51:53:57:60:64 SD: 2.7155 cents 33:35:37:39:41:44:46:49:52:55:59:62:66 SD: 2.4341 cents 36:38:40:43:45:48:51:54:57:60:64:68:72 SD: 2.1410 cents 42:44:47:50:53:56:59:63:66:70:75:79:84 SD: 1.9020 cents 51:54:57:60:64:68:72:76:81:85:91:96:102 SD: 1.8524 cents 52:55:58:62:65:69:73:78:82:87:92:98:104 SD: 1.6654 cents 54:57:60:64:68:72:76:81:85:90:96:101:108 SD: 1.5815 cents 59:62:66:70:74:79:83:88:93:99:105:111:118 SD: 1.1556 cents 75:79:84:89:94:100:105:112:119:125:133:141:150 SD: 0.9918 cents 76:80:85:90:95:101:107:114:120:127:135:143:152 SD: 0.8872 cents 91:96:102:108:114:121:128:136:144:152:162:171:182 SD: 0.7436 cents 119:125:133:141:149:159:167:178:188:199:212:224:238 SD: 0.6920 cents 126:133:141:149:158:168:177:188:199:210:224:237:252 SD: 0.6727 cents 129:136:144:153:162:172:181:193:204:215:229:242:258 SD: 0.6523 cents 131:138:146:155:164:175:184:196:207:219:233:246:262 SD: 0.5758 cents 146:154:163:173:183:195:205:218:231:244:260:274:292 SD: 0.5314 cents 162:171:181:192:203:216:228:242:256:271:288:304:324 SD: 0.5064 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: 0.4459 cents 188:198:210:223:236:251:264:281:297:314:334:353:376 SD: 0.4073 cents 206:217:230:244:258:275:289:308:326:344:366:387:412 SD: 0.3783 cents 210:221:235:249:263:280:295:314:332:351:373:395:420 SD: 0.3461 cents 222:234:248:263:278:296:312:332:351:371:395:417:444 SD: 0.2303 cents 297:313:332:352:372:396:417:444:469:496:528:558:594 SD: 0.1560 cents 486:512:543:576:609:648:683:727:768:812:864:913:972 SD: 0.1322 cents 519:547:580:615:650:692:729:776:820:867:923:975:1038 SD: 0.1167 cents 546:575:610:647:684:728:767:816:863:912:971:1026:1092 SD: 0.0993 cents