Scala analysis: meanqratapprox
A very close approximation of quarter-comma meantone
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 5375/5144 76.048934 2: 6470/5787 193.140082 3: 3846/3215 310.248030 4: 5/4 386.313714 major third 5: 860/643 503.421507 6: 16175/11574 579.453796 7: 1923/1286 696.561744 8: 25/16 772.627428 classic augmented fifth 9: 1075/643 889.735221 10: 10352/5787 1006.826368 11: 9615/5144 1082.875458 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 10288/9615 117.1245 1: 5375/5144 76.0489 2: 10352/9675 117.0911 3: 17307/16175 117.1079 4: 16075/15384 76.0657 5: 688/643 117.1078 6: 3235/3096 76.0323 7: 17307/16175 117.1079 8: 16075/15384 76.0657 9: 688/643 117.1078 10: 10352/9675 117.0911 11: 86535/82816 76.0491 12: 10288/9615 117.1245
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 3235/3096 76.032 cents 1: 1 5375/5144 76.049 cents 1: 1 86535/82816 76.049 cents 1: 2 16075/15384 76.066 cents 1: 2 10352/9675 117.091 cents 1: 2 688/643 117.108 cents 1: 2 17307/16175 117.108 cents 1: 1 10288/9615 117.125 cents 2: 2 6470/5787 193.140 cents 2: 2 1923/1720 193.140 cents 2: 3 2150/1923 193.173 cents 2: 3 5787/5176 193.174 cents 2: 1 165632/144675 234.199 cents 2: 1 30768/26875 234.199 cents 3: 1 80875/69228 269.206 cents 3: 1 3215/2752 269.206 cents 3: 1 48375/41408 269.223 cents 3: 3 3846/3215 310.248 cents 3: 2 20704/17307 310.265 cents 3: 2 1286/1075 310.265 cents 3: 2 774/647 310.281 cents 4: 8 5/4 386.314 cents major third 4: 4 32/25 427.373 cents classic diminished fourth 5: 1 80375/61536 462.379 cents 5: 3 2588/1935 503.405 cents 5: 3 860/643 503.422 cents 5: 3 17307/12940 503.422 cents 5: 2 2572/1923 503.438 cents 6: 1 16175/11574 579.454 cents 6: 1 1923/1376 579.454 cents 6: 2 5375/3846 579.487 cents 6: 2 28935/20704 579.487 cents 6: 2 41408/28935 620.513 cents 6: 2 7692/5375 620.513 cents 6: 1 2752/1923 620.546 cents 6: 1 23148/16175 620.546 cents 7: 2 1923/1286 696.562 cents 7: 3 25880/17307 696.578 cents 7: 3 643/430 696.578 cents 7: 3 1935/1294 696.595 cents 7: 1 123072/80375 737.621 cents 8: 4 25/16 772.627 cents classic augmented fifth 8: 8 8/5 813.686 cents minor sixth 9: 2 647/387 889.719 cents 9: 2 1075/643 889.735 cents 9: 2 17307/10352 889.735 cents 9: 3 3215/1923 889.752 cents 9: 1 82816/48375 930.777 cents 9: 1 5504/3215 930.794 cents 9: 1 138456/80875 930.794 cents 10: 1 26875/15384 965.801 cents 10: 1 144675/82816 965.801 cents 10: 3 10352/5787 1006.826 cents 10: 3 1923/1075 1006.827 cents 10: 2 3440/1923 1006.860 cents 10: 2 5787/3235 1006.860 cents 11: 1 9615/5144 1082.875 cents 11: 2 32350/17307 1082.892 cents 11: 2 643/344 1082.892 cents 11: 2 9675/5176 1082.909 cents 11: 2 30768/16075 1123.934 cents 11: 1 165632/86535 1123.951 cents 11: 1 10288/5375 1123.951 cents 11: 1 6192/3235 1123.968 cents Highest number of different intervals for one interval class: 8 Average number of different intervals per interval class: 5.81818 = 64/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 76.0 193.1 310.2 386.3 503.4 579.5 696.6 772.6 889.7 1006.8 1082.9 1200.0 76.0 : 117.1 234.2 310.3 427.4 503.4 620.5 696.6 813.7 930.8 1006.8 1124.0 1200.0 193.1 : 117.1 193.2 310.3 386.3 503.4 579.5 696.6 813.7 889.7 1006.9 1082.9 1200.0 310.2 : 76.1 193.2 269.2 386.3 462.4 579.5 696.6 772.6 889.8 965.8 1082.9 1200.0 386.3 : 117.1 193.1 310.2 386.3 503.4 620.5 696.6 813.7 889.7 1006.8 1123.9 1200.0 503.4 : 76.0 193.1 269.2 386.3 503.4 579.5 696.6 772.6 889.7 1006.8 1082.9 1200.0 579.5 : 117.1 193.2 310.3 427.4 503.4 620.5 696.6 813.7 930.8 1006.9 1124.0 1200.0 696.6 : 76.1 193.2 310.3 386.3 503.4 579.5 696.6 813.7 889.8 1006.9 1082.9 1200.0 772.6 : 117.1 234.2 310.2 427.4 503.4 620.5 737.6 813.7 930.8 1006.8 1123.9 1200.0 889.7 : 117.1 193.1 310.3 386.3 503.4 620.5 696.6 813.7 889.7 1006.8 1082.9 1200.0 1006.8: 76.0 193.2 269.2 386.3 503.4 579.5 696.6 772.6 889.7 965.8 1082.9 1200.0 1082.9: 117.1 193.2 310.3 427.4 503.4 620.5 696.6 813.7 889.8 1006.9 1124.0 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 76.0323 .. 117.1245 cents 10617216/10368175, 41.0923 cents 3235/3096, 76.0323 cents 2: 193.1401 .. 234.1991 cents 89027208/86940625, 41.0590 cents 10368175/9922776, 76.0155 cents 3: 269.2058 .. 310.2814 cents 53582472/52326125, 41.0757 cents 35.00667 cents 4: 386.3137 .. 427.3726 cents 128/125, 41.0589 cents 3235/3096, 76.0323 cents 5: 462.3794 .. 503.4383 cents 128/125, 41.0589 cents 2009375/1969152, 35.0068 cents 6: 579.4538 .. 620.5462 cents 267914952/261630625, 41.0924 cents 10368175/9922776, 76.0155 cents 7: 696.5617 .. 737.6206 cents 128/125, 41.0589 cents 10368175/9922776, 76.0155 cents 8: 772.6274 .. 813.6863 cents 128/125, 41.0589 cents 2009375/1969152, 35.0068 cents 9: 889.7186 .. 930.7942 cents 53582472/52326125, 41.0757 cents 3235/3096, 76.0323 cents 10: 965.8009 .. 1006.8599 cents 89027208/86940625, 41.0590 cents 35.00667 cents 11: 1082.8755 .. 1123.9677 cents 10617216/10368175, 41.0923 cents 10368175/9922776, 76.0155 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 3235/3096, 76.0323 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 10288/9615, 117.1245 cents Average / Smallest step : 1.315231 Largest / Average step : 1.171245 Largest / Smallest step : 1.540458 Median interval of one step : 10352/9675, 117.0911 cents, amount: 2 Least squares average step : 99.04095 cents, oct.: 1188.49146 cents Scale is strictly proper Step pattern alph. order: ABCDEFCDEBGH Step pattern size order : BDECEAECEDBF Scale is a Constant Structure, by a margin of 35.00667 cents Scale diversity : 1.218877 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.623515 Rothenberg efficiency : 0.319529 redundancy: 0.680471 Efficiency x scale size : 3.834343 Number of different interval sizes : 64 = 5.81818 / class Number of one step interval sizes : 8 Highest interval variety : 8 Mean interval variety : 5.81818 = 64/11 Median interval variety : 2 Lowest interval variety : 2 Smallest interval difference : 11128401/11128400, 0.0002 cents Most common intervals : 5/4, 386.3137 cents & inv., amount: 8 Most common triad is 0.0 386.314 813.686 cents, amount: 4 Number of recognisable fifths : 11, average 696.5799 cents Best fifths form a closed circle Formal octave complements present : 1 = 8.3333% Limited inverse transpositions : 11 Inversional symmetry on intervals : 5-6 11-12 -- Rational properties -- Prime limit : 647 Odd number limit : 144675 (O: 144675 U: 144675) Highest odd numerator or denominator: 16175 Scale harmonicity : 0.024038 Fundamental : 1/462960, -18.8205 octaves, 0.0006 Hz. Guide tone : too high to compute Exponens Consonantiae : 4.953652E+16, 55.45934 octaves Euler's gradus suavitatis : 2001 Sum of Mann's harmonic distance : 47483.5, average 3956.95833 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 34.17477, average 2.84790 Sum of Tenney's harmonic distance : 68.03095, average 5.66925 Vogel's harmonic complexity : 512.33333 Wille's k value : 361687 Wilson's harmonic complexity : 2003 Rectangular lattice diameter : 11 Triangular lattice diameter : 6 Lattice compactness : 276.24944, average 3.54166 Lattice compactness (without 2's) : 156.07914, average 2.00101 Number of different primes : 7 Prime exponents' range, average, count, tones@limit: 2: -4 .. 4 -0.41667 23 1 3: -2 .. 1 -0.25000 9 5: -1 .. 3 1.00000 14 2 43: 0 .. 1 0.25000 3 641: 0 .. 1 0.25000 3 643: -1 .. 0 -0.75000 9 6 647: 0 .. 1 0.25000 3 3 Average exponent except 2's : 9 / 12 = 0.75000 Average absolute exponent except 2's: 41 / 12 = 3.41667 Scale is weakly epimorphic with val: <12 18 28 62 113 113 112| Scale is weakly epimorphic with val: <12 18 28 63 112 112 113| Scale is weakly epimorphic with val: <12 18 28 66 113 113 112| Scale is weakly epimorphic with val: <12 18 28 67 112 112 113| Scale is weakly epimorphic with val: <12 19 28 62 113 113 113| Scale is weakly epimorphic with val: <12 19 28 63 113 113 112| Scale is weakly epimorphic with val: <12 19 28 64 111 113 113| Scale is weakly epimorphic with val: <12 19 28 65 111 112 113| Scale is JI-epimorphic with val: <12 19 28 65 112 112 112| = patent Scale is JI-epimorphic with val: <12 19 28 66 113 113 113| Scale is weakly epimorphic with val: <12 19 28 67 113 113 112| Scale is weakly epimorphic with val: <12 19 28 68 111 113 113| Scale is weakly epimorphic with val: <12 20 28 62 111 112 113| Scale is weakly epimorphic with val: <12 20 28 66 111 112 113| Scale is weakly epimorphic with val: <12 20 29 65 112 112 113|
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 14.5865 c. M:-27.3726 c. Twelve-tone Chromatic (1/11-comma) 19: 1 2 2 1 2 1 2 1 2 2 1 2 P M ME SD: 7.8463 c. M: 14.7327 c. Meantone Chromatic (1/3-comma) 31: 2 3 3 2 3 2 3 2 3 3 2 3 SP M ME SD: 0.8374 c. M:-1.5661 c. Meantone Chromatic (53/220-comma) 112: 7 11 11 7 11 7 11 7 11 11 7 11 SP M DE SD: 0.6359 c. M: 1.1989 c. Meantone Chromatic (7/27-comma), Meanpop-12 143: 9 14 14 9 14 9 14 9 14 14 9 14 SP M DE SD: 0.3166 c. M: 0.5995 c. Meantone Chromatic (19/75-comma) 174: 11 17 17 11 17 11 17 11 17 17 11 17 SP M DE SD: 0.1115 c. M: 0.2136 c. Meantone Chromatic (1/4-comma), Milliet de Chales-12 205: 13 20 20 13 20 13 20 13 20 20 13 20 SP M DE SD: 0.0345 c. M:-0.0584 c. 584: 37 57 57 37 57 37 57 37 57 57 37 57 SP M DE SD: 0.0153 c. M:-0.0259 c. 789: 50 77 77 50 77 50 77 50 77 77 50 77 SP M DE SD: 0.0115 c. M:-0.0181 c. 994: 63 97 97 63 97 63 97 63 97 97 63 97 SP M DE SD: 0.0137 c. M:-0.0231 c. 1373: 87 134 134 87 134 87 134 87 134 134 87 134 SP M DE SD: 0.0121 c. M:-0.0215 c. 1783: 113 174 174 113 174 113 174 113 174 174 113 174 SP M DE SD: 0.0123 c. M:-0.0190 c. 2162: 137 211 211 137 211 137 211 137 211 211 137 211 SP M DE SD: 0.0116 c. M:-0.0202 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: 2.6966 cents 3 x x x x 4 x x x 5 x x 6 S SD: 2.5345 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 9.5953 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 8.5797 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 9.2670 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 12.2699 cents 8 x 9 x 10 x 11:12 x 13:14:15:16 S SD: 8.6884 cents 9 x 10:11 x 12:13 x 14:15:16:17:18 S SD: 8.1092 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 8.1977 cents 11 x 12:13:14:15 x 16:17:18:20:21:22 S SD: 10.2524 cents 12 x 13:14:15:16:17:18:19:20:21:22:24 S SD: 8.4274 cents 13:14:15 x 16:17:18:19:20:22:23:24:26 S SD: 9.8604 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 8.1822 cents 15:16:17:18:19:20:21:22:23:25:27:28:30 SD: 5.7484 cents 16:17:18:19:20:21:22:24:25:27:29:30:32 SD: 5.1416 cents 20:21:22:24:25:27:28:30:31:33:36:37:40 SD: 4.0283 cents 24:25:27:29:30:32:34:36:38:40:43:45:48 SD: 3.3855 cents 25:26:28:30:31:33:35:37:39:42:45:47:50 SD: 3.1616 cents 32:33:36:38:40:43:45:48:50:54:57:60:64 SD: 3.0629 cents 35:37:39:42:44:47:49:52:55:59:63:65:70 SD: 2.9945 cents 36:38:40:43:45:48:50:54:56:60:64:67:72 SD: 2.4484 cents 40:42:45:48:50:54:56:60:63:67:72:75:80 SD: 2.4446 cents 41:43:46:49:51:55:57:61:64:69:73:77:82 SD: 2.0473 cents 45:47:50:54:56:60:63:67:70:75:80:84:90 SD: 1.8775 cents 48:50:54:57:60:64:67:72:75:80:86:90:96 SD: 1.7380 cents 51:53:57:61:64:68:71:76:80:85:91:95:102 SD: 1.6119 cents 60:63:67:72:75:80:84:90:94:100:107:112:120 SD: 1.3326 cents 67:70:75:80:84:90:94:100:105:112:120:125:134 SD: 1.1657 cents 68:71:76:81:85:91:95:102:106:114:122:127:136 SD: 1.0356 cents 83:87:93:99:104:111:116:124:130:139:148:155:166 SD: 1.0161 cents 85:89:95:102:106:114:119:127:133:142:152:159:170 SD: 0.8443 cents 91:95:102:109:114:122:127:136:142:152:163:170:182 SD: 0.7395 cents 113:118:126:135:141:151:158:169:177:189:202:211:226 SD: 0.6658 cents 128:134:143:153:160:171:179:191:200:214:229:239:256 SD: 0.5013 cents 153:160:171:183:191:205:214:229:239:256:274:286:306 SD: 0.4386 cents 204:213:228:244:255:273:285:305:319:341:365:381:408 SD: 0.2288 cents 332:347:371:397:415:444:464:496:519:555:594:621:664 SD: 0.2082 cents 397:415:444:475:496:531:555:594:620:664:710:742:794 SD: 0.1773 cents 448:468:501:536:560:599:626:670:700:749:801:837:896 SD: 0.1249 cents 551:576:616:659:689:737:770:824:861:921:986:1030:1102 SD: 0.1095 cents