Scala analysis: Sp5LimDodek
Sparschuh's 5-limit dodecatonics with two Kirnberger 5ths: C-G & A-E
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: 262144/234375 193.849002 3: 32/27 294.134997 Pythagorean minor third 4: 5/4 386.313714 major third 5: 4/3 498.044999 perfect fourth 6: 1024/729 588.269995 Pythagorean diminished fifth 7: 16384/10935 700.001280 fifth - schisma, 5-limit approximation to ET fifth 8: 128/81 792.179997 Pythagorean minor sixth 9: 78125/46656 892.465992 10: 16/9 996.089998 Pythagorean minor seventh 11: 4096/2187 1086.314994 Pythagorean diminished octave 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 2187/2048 113.6850 apotome 1: 256/243 90.2250 limma, Pythagorean minor second 2: 82944/78125 103.6240 3: 78125/73728 100.2860 Woolhouse semitone 4: 135/128 92.1787 major chroma, major limma 5: 16/15 111.7313 minor diatonic semitone 6: 256/243 90.2250 limma, Pythagorean minor second 7: 16/15 111.7313 minor diatonic semitone 8: 135/128 92.1787 major chroma, major limma 9: 78125/73728 100.2860 Woolhouse semitone 10: 82944/78125 103.6240 11: 256/243 90.2250 limma, Pythagorean minor second 12: 2187/2048 113.6850 apotome
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 3 256/243 90.225 cents limma, Pythagorean minor second 1: 2 135/128 92.179 cents major chroma, major limma 1: 2 78125/73728 100.286 cents Woolhouse semitone 1: 2 82944/78125 103.624 cents 1: 2 16/15 111.731 cents minor diatonic semitone 1: 1 2187/2048 113.685 cents apotome 2: 2 1171875/1048576 192.465 cents 2: 2 262144/234375 193.849 cents 2: 2 4096/3645 201.956 cents narrow diminished third 2: 6 9/8 203.910 cents major whole tone 3: 5 32/27 294.135 cents Pythagorean minor third 3: 2 1215/1024 296.089 cents wide augmented second 3: 2 78125/65536 304.196 cents 3: 2 93312/78125 307.534 cents 3: 1 65536/54675 313.688 cents 4: 2 5/4 386.314 cents major third 4: 2 78125/62208 394.421 cents 4: 2 98304/78125 397.759 cents 4: 2 512/405 405.866 cents narrow diminished fourth 4: 4 81/64 407.820 cents Pythagorean major third 5: 7 4/3 498.045 cents perfect fourth 5: 2 10935/8192 499.999 cents fourth + schisma, 5-limit approximation to ET fourth 5: 1 501.38301 cents 5: 2 15625/11664 506.152 cents two (semi-augmented whole tone) 6: 2 1024/729 588.270 cents Pythagorean diminished fifth 6: 2 45/32 590.224 cents diatonic tritone 6: 2 78125/55296 598.331 cents 6: 2 110592/78125 601.669 cents 6: 2 64/45 609.776 cents 2nd tritone 6: 2 729/512 611.730 cents Pythagorean tritone 7: 2 23328/15625 693.848 cents 7: 1 698.61699 cents 7: 2 16384/10935 700.001 cents fifth - schisma, 5-limit approximation to ET fifth 7: 7 3/2 701.955 cents perfect fifth 8: 4 128/81 792.180 cents Pythagorean minor sixth 8: 2 405/256 794.134 cents wide augmented fifth 8: 2 78125/49152 802.241 cents 8: 2 124416/78125 805.579 cents 8: 2 8/5 813.686 cents minor sixth 9: 1 54675/32768 886.312 cents 9: 2 78125/46656 892.466 cents 9: 2 131072/78125 895.804 cents 9: 2 2048/1215 903.911 cents narrow diminished seventh 9: 5 27/16 905.865 cents Pythagorean major sixth 10: 6 16/9 996.090 cents Pythagorean minor seventh 10: 2 3645/2048 998.044 cents wide augmented sixth 10: 2 234375/131072 1006.151 cents 10: 2 2097152/1171875 1007.535 cents 11: 1 4096/2187 1086.315 cents Pythagorean diminished octave 11: 2 15/8 1088.269 cents classic major seventh 11: 2 78125/41472 1096.376 cents 11: 2 147456/78125 1099.714 cents Woolhouse major seventh 11: 2 256/135 1107.821 cents octave - major chroma 11: 3 243/128 1109.775 cents Pythagorean major seventh Highest number of different intervals for one interval class: 6 Average number of different intervals per interval class: 4.90909 = 54/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 90.2 193.8 294.1 386.3 498.0 588.3 700.0 792.2 892.5 996.1 1086.3 1200.0 90.2 : 103.6 203.9 296.1 407.8 498.0 609.8 702.0 802.2 905.9 996.1 1109.8 1200.0 193.8 : 100.3 192.5 304.2 394.4 506.2 598.3 698.6 802.2 892.5 1006.2 1096.4 1200.0 294.1 : 92.2 203.9 294.1 405.9 498.0 598.3 702.0 792.2 905.9 996.1 1099.7 1200.0 386.3 : 111.7 202.0 313.7 405.9 506.2 609.8 700.0 813.7 903.9 1007.5 1107.8 1200.0 498.0 : 90.2 202.0 294.1 394.4 498.0 588.3 702.0 792.2 895.8 996.1 1088.3 1200.0 588.3 : 111.7 203.9 304.2 407.8 498.0 611.7 702.0 805.6 905.9 998.0 1109.8 1200.0 700.0 : 92.2 192.5 296.1 386.3 500.0 590.2 693.8 794.1 886.3 998.0 1088.3 1200.0 792.2 : 100.3 203.9 294.1 407.8 498.0 601.7 702.0 794.1 905.9 996.1 1107.8 1200.0 892.5 : 103.6 193.8 307.5 397.8 501.4 601.7 693.8 805.6 895.8 1007.5 1099.7 1200.0 996.1 : 90.2 203.9 294.1 397.8 498.0 590.2 702.0 792.2 903.9 996.1 1096.4 1200.0 1086.3: 113.7 203.9 307.5 407.8 500.0 611.7 702.0 813.7 905.9 1006.2 1109.8 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.2250 .. 113.6850 cents 531441/524288, 23.4600 cents 256/243, 90.2250 cents 2: 192.4647 .. 203.9100 cents 393216/390625, 11.4453 cents 390625/373248, 78.7797 cents 3: 294.1350 .. 313.6876 cents 2048/2025, 19.5526 cents 256/243, 90.2250 cents 4: 386.3137 .. 407.8200 cents 81/80, 21.5063 cents 273375/262144, 72.6261 cents 5: 498.0450 .. 506.1523 cents 15625/15552, 8.1073 cents 256/243, 90.2250 cents 6: 588.2700 .. 611.7300 cents 531441/524288, 23.4600 cents 16384/15625, 82.1177 cents 7: 693.8477 .. 701.9550 cents 15625/15552, 8.1073 cents 16384/15625, 82.1177 cents 8: 792.1800 .. 813.6863 cents 81/80, 21.5063 cents 256/243, 90.2250 cents 9: 886.3124 .. 905.8650 cents 2048/2025, 19.5526 cents 273375/262144, 72.6261 cents 10: 996.0900 .. 1007.5353 cents 393216/390625, 11.4453 cents 256/243, 90.2250 cents 11: 1086.3150 .. 1109.7750 cents 531441/524288, 23.4600 cents 390625/373248, 78.7797 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 : 2187/2048, 113.6850 cents Average / Smallest step : 1.108340 Largest / Average step : 1.136850 Largest / Smallest step : 1.260017 Median interval of one step : 78125/73728, 100.2860 cents, amount: 2 Most common interval of one step : 256/243, 90.2250 cents, amount: 3 Least squares average step : 99.23913 cents, oct.: 1190.86958 cents Scale is strictly proper Step pattern alph. order: ABCDEAEDCBAF Step pattern size order : ADCBEAEBCDAF Scale is a Constant Structure, by a margin of 72.62615 cents Scale diversity : 1.131864 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.840331 Rothenberg efficiency : 0.279582 redundancy: 0.720418 Efficiency x scale size : 3.354978 Number of different interval sizes : 54 = 4.90909 / class Number of one step interval sizes : 6 Highest interval variety : 6 Mean interval variety : 4.90909 = 54/11 Median interval variety : 4 Lowest interval variety : 4 Smallest interval difference : 1.38429 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 7 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 4 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 9 10 Limited inverse transpositions : 11 Inversional symmetry on intervals : 5-6 11-12 -- Rational properties -- Prime limit : 5 Odd number limit : too high to compute Highest odd numerator or denominator: 234375 Scale harmonicity : 0.007560 Average absolute harmonicity : 0.145649 Fundamental : too low to compute Guide tone : too high to compute Exponens Consonantiae : 2.239488E+20, 67.60173 octaves Euler's gradus suavitatis : 95 Sum of Mann's harmonic distance : 328733.0, average 27394.41667 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 28.35989, average 2.36332 Sum of Tenney's harmonic distance : 58.01428, average 4.83452 Vogel's harmonic complexity : too high to compute Wille's k value : too high to compute Wilson's harmonic complexity : 91 Rectangular lattice diameter : 21 Triangular lattice diameter : 14 Lattice compactness : 742.46890, average 9.51883 Lattice compactness (without 2's) : 375.82426, average 4.81826 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -6 .. 18 6.08333 89 1 3: -7 .. 0 -3.50000 42 7 5: -7 .. 7 0.00000 16 4 Average exponent except 2's :-42 / 12 =-3.50000 Average absolute exponent except 2's: 58 / 12 = 4.83333
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 8.2512 c. M:-13.6863 c. Twelve-tone Chromatic (1/11-comma) 41: 3 4 3 3 4 3 4 3 3 4 3 4 SP M ME SD: 5.7892 c. M: 14.4172 c. Schismic-12 43: 3 4 4 3 4 3 4 3 4 4 3 4 SP M ME SD: 6.1340 c. M:-12.8417 c. Meantone Chromatic (1/5-comma) 49: 4 4 4 4 4 4 5 3 4 5 3 5 SP D3 SD: 7.0562 c. M: 10.8333 c. 50: 4 4 4 4 5 4 4 4 4 5 3 5 SP D3 SD: 6.2619 c. M:-11.9100 c. 53: 4 5 4 4 5 4 5 4 4 5 4 5 SP M ME SD: 4.0201 c. M:-9.9246 c. Schismic-12 61: 5 5 5 5 5 5 6 4 5 6 4 6 SP D3 SD: 5.6761 c. M:-8.1954 c. 62: 5 5 5 5 6 4 6 5 5 5 5 6 SP D3 SD: 4.5354 c. M: 8.9932 c. 65: 5 6 5 5 6 5 6 5 5 6 5 6 SP M ME SD: 3.5771 c. M:-9.2279 c. 68: 5 6 6 5 6 5 7 5 6 5 6 6 SP D3 SD: 5.0846 c. M: 7.8547 c. 70: 5 6 6 5 7 5 7 5 6 6 5 7 SP D3 SD: 4.4265 c. M: 9.1709 c. 74: 6 6 6 6 7 5 7 6 6 6 6 7 SP D3 SD: 3.7284 c. M:-7.0723 c. 77: 6 7 6 6 7 6 7 6 6 7 6 7 SP M ME SD: 3.7266 c. M:-8.7484 c. 82: 6 7 7 6 8 6 8 6 7 7 6 8 SP D3 SD: 2.6728 c. M: 5.8259 c. 93: 7 8 8 7 9 7 8 7 8 8 7 9 SP SD: 3.1046 c. M:-5.2784 c. 94: 7 8 8 7 9 7 9 7 8 8 7 9 SP D3 SD: 1.4919 c. M: 3.3350 c. 106: 8 9 9 8 10 8 10 8 9 9 8 10 SP D3 SD: 0.9853 c. M:-1.8855 c. 253: 19 22 21 19 24 19 24 19 21 22 19 24 SP SD: 0.8871 c. M: 2.1240 c. 261: 20 22 22 20 24 20 24 20 22 23 19 25 SP SD: 1.0611 c. M:-1.7290 c. 265: 20 23 22 20 25 20 25 20 22 23 20 25 SP SD: 0.7689 c. M:-1.8855 c. 277: 21 24 23 21 26 21 26 21 23 24 21 26 SP SD: 0.8254 c. M:-1.8038 c. 289: 22 25 24 22 27 22 27 22 24 25 22 27 SP SD: 0.9947 c. M:-1.7288 c. 298: 22 26 25 23 28 22 28 23 25 25 23 28 SP SD: 0.9902 c. M: 1.6344 c. 302: 23 26 25 23 28 23 28 23 26 26 22 29 SP SD: 1.0707 c. M:-1.5737 c. 304: 23 26 25 24 28 23 28 24 25 26 23 29 SP SD: 0.9673 c. M: 2.0297 c. 316: 24 27 26 25 29 24 29 25 26 27 24 30 SP SD: 0.9391 c. M: 1.7299 c. 317: 24 27 27 24 30 23 30 24 27 27 24 30 SP SD: 0.8924 c. M:-1.6395 c. 323: 24 28 27 25 30 24 30 25 27 28 24 31 SP SD: 0.9048 c. M: 1.5493 c. 329: 25 28 28 25 31 24 31 25 28 28 25 31 SP SD: 0.8757 c. M:-1.6510 c. 335: 25 29 28 26 31 25 31 26 28 29 25 32 SP SD: 0.6782 c. M: 1.4938 c. 347: 26 30 29 27 32 26 32 27 29 30 26 33 SP SD: 0.5577 c. M: 1.4422 c. 351: 26 31 29 27 33 26 33 27 29 30 27 33 SP SD: 0.8196 c. M: 1.3361 c. 359: 27 31 30 28 33 27 33 28 30 31 27 34 SP SD: 0.5768 c. M:-1.4300 c. 363: 27 32 30 28 34 27 34 28 30 31 28 34 SP SD: 0.8559 c. M:-1.2883 c. 367: 28 31 31 28 34 28 34 28 31 32 27 35 SP SD: 0.7610 c. M:-1.3281 c. 371: 28 32 31 28 35 28 34 29 31 32 28 35 SP SD: 0.6170 c. M: 1.4081 c. 401: 30 35 33 31 37 31 37 31 33 35 30 38 SP SD: 0.7149 c. M: 1.2869 c. 410: 31 35 34 32 38 31 38 32 34 35 31 39 SP SD: 0.6757 c. M: 1.4521 c. 412: 31 36 34 32 38 31 38 32 34 36 31 39 SP SD: 0.6604 c. M:-1.2966 c. 416: 31 36 35 32 39 31 39 32 34 36 32 39 SP SD: 0.7977 c. M:-1.1850 c. 417: 31 36 35 32 39 31 39 32 35 36 31 40 SP SD: 0.8180 c. M: 1.4229 c. 420: 32 36 35 32 39 32 39 32 35 37 31 40 SP SD: 0.7092 c. M:-1.2036 c. 422: 32 36 35 33 39 32 39 33 35 36 32 40 SP SD: 0.6676 c. M: 1.2440 c. 429: 32 37 36 33 40 32 40 33 36 37 32 41 SP SD: 0.5897 c. M: 1.0003 c. 441: 33 38 37 34 41 33 41 34 37 38 33 42 SP SD: 0.4010 c. M: 0.6816 c. 453: 34 39 38 35 42 34 42 35 38 39 34 43 SP SD: 0.2973 c. M: 0.6635 c. 559: 42 48 47 43 52 42 52 43 47 48 42 53 SP SD: 0.2570 c. M: 0.6469 c. 588: 44 51 49 45 55 44 55 45 49 51 44 56 SP SD: 0.3859 c. M: 0.6293 c. 600: 45 52 50 46 56 45 56 46 50 52 45 57 SP SD: 0.2276 c. M: 0.4660 c. 612: 46 53 51 47 57 46 57 47 51 53 46 58 SP SD: 0.1204 c. M: 0.3091 c. 718: 54 62 60 55 67 54 67 55 60 62 54 68 SP SD: 0.1078 c. M:-0.2773 c. 730: 55 63 61 56 68 55 68 56 61 63 55 69 SP SD: 0.1587 c. M:-0.2727 c. 1065: 80 92 89 82 99 80 99 82 89 92 80 101 SP SD: 0.1134 c. M: 0.2830 c. 1159: 87 100 97 89 108 87 108 89 97 100 87 110 SP SD: 0.1287 c. M: 0.2338 c. 1171: 88 101 98 90 109 88 109 90 98 101 88 111 SP SD: 0.0701 c. M: 0.1684 c. 1330: 100 115 111 102 124 100 124 102 111 115 100 126 SP SD: 0.0821 c. M:-0.1491 c. 1771: 133 153 148 136 165 133 165 136 148 153 133 168 SP SD: 0.0849 c. M: 0.1490 c. 1783: 134 154 149 137 166 134 166 137 149 154 134 169 SP SD: 0.0341 c. M: 0.0574 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: 0.9769 cents 3 x x x x 4 x x x 5 x x 6 S SD: 2.7024 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 6.8335 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: 7.7788 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 11.5937 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 8.4365 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 7.8632 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.4041 cents 11 x 12:13:14:15 x 16:17:18:20:21:22 S SD: 11.2694 cents 12:13 x 14:15:16:17:18:19:20:21:22:24 S SD: 6.7344 cents 13:14 x 15:16:17:18:19:21:22:23:24:26 SD: 9.2695 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 8.1830 cents 15:16:17:18:19:20:21:22:24:25:27:28:30 SD: 5.5245 cents 16:17:18:19:20:21:22:24:25:27:28:30:32 SD: 5.1039 cents 17:18:19:20:21:23:24:25:27:28:30:32:34 SD: 4.9682 cents 22:23:25:26:28:29:31:33:35:37:39:41:44 SD: 4.2840 cents 24:25:27:28:30:32:34:36:38:40:43:45:48 SD: 3.4461 cents 31:33:35:37:39:41:44:46:49:52:55:58:62 SD: 3.4232 cents 32:34:36:38:40:43:45:48:51:54:57:60:64 SD: 2.5420 cents 33:35:37:39:41:44:46:49:52:55:59:62:66 SD: 2.5336 cents 36:38:40:43:45:48:51:54:57:60:64:67:72 SD: 2.2685 cents 40:42:45:47:50:53:56:60:63:67:71:75:80 SD: 1.9667 cents 52:55:58:62:65:69:73:78:82:87:92:97:104 SD: 1.6366 cents 59:62:66:70:74:79:83:88:93:99:105:111:118 SD: 1.3823 cents 60:63:67:71:75:80:84:90:95:100:107:112:120 SD: 1.2495 cents 72:76:81:85:90:96:101:108:114:121:128:135:144 SD: 1.2422 cents 76:80:85:90:95:101:107:114:120:127:135:142:152 SD: 0.7907 cents 108:114:121:128:135:144:152:162:171:181:192:202:216 SD: 0.6135 cents 129:136:144:153:161:172:181:193:204:216:229:242:258 SD: 0.5783 cents 135:142:151:160:169:180:190:202:213:226:240:253:270 SD: 0.5234 cents 151:159:169:179:189:201:212:226:239:253:268:283:302 SD: 0.5153 cents 153:161:171:181:191:204:215:229:242:256:272:287:306 SD: 0.5024 cents 167:176:187:198:209:223:235:250:264:280:297:313:334 SD: 0.4935 cents 168:177:188:199:210:224:236:252:265:281:299:315:336 SD: 0.4307 cents 188:198:210:223:235:251:264:282:297:315:334:352:376 SD: 0.3707 cents 205:216:229:243:256:273:288:307:324:343:364:384:410 SD: 0.3679 cents 221:233:247:262:276:295:310:331:349:370:393:414:442 SD: 0.3532 cents 227:239:254:269:284:303:319:340:359:380:404:425:454 SD: 0.3190 cents 243:256:272:288:304:324:341:364:384:407:432:455:486 SD: 0.2236 cents 297:313:332:352:371:396:417:445:469:497:528:556:594 SD: 0.2152 cents 319:336:357:378:399:425:448:478:504:534:567:597:638 SD: 0.2138 cents 339:357:379:402:424:452:476:508:536:568:603:635:678 SD: 0.2121 cents 356:375:398:422:445:475:500:533:563:596:633:667:712 SD: 0.2098 cents 372:392:416:441:465:496:523:557:588:623:661:697:744 SD: 0.1989 cents 388:409:434:460:485:517:545:581:613:650:690:727:776 SD: 0.1927 cents 396:417:443:469:495:528:556:593:626:663:704:742:792 SD: 0.1823 cents 405:427:453:480:506:540:569:607:640:678:720:759:810 SD: 0.1738 cents 415:437:464:492:519:553:583:622:656:695:738:777:830 SD: 0.1736 cents 432:455:483:512:540:576:607:647:683:723:768:809:864 SD: 0.1440 cents 464:489:519:550:580:619:652:695:733:777:825:869:928 SD: 0.1283 cents 584:615:653:692:730:779:820:875:923:978:1038:1094:1168 SD: 0.1251 cents 600:632:671:711:750:800:843:899:948:1005:1067:1124:1200 SD: 0.0908 cents