Scala analysis: locomotive
A 2.9.11.13 subgroup scale
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 88/81 143.497939 2nd undecimal neutral second 2: 9/8 203.910002 major whole tone 3: 11/9 347.407941 undecimal neutral third 4: 16/13 359.472338 tridecimal neutral third 5: 11/8 551.317942 undecimal semi-augmented fourth 6: 13/9 636.617660 tridecimal diminished fifth 7: 16/11 648.682058 undecimal semi-diminished fifth 8: 13/8 840.527662 tridecimal neutral sixth 9: 18/11 852.592059 undecimal neutral sixth 10: 16/9 996.089998 Pythagorean minor seventh 11: 81/44 1056.502061 2nd undecimal neutral seventh 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 88/81 143.4979 2nd undecimal neutral second 1: 88/81 143.4979 2nd undecimal neutral second 2: 729/704 60.4121 undecimal major diesis 3: 88/81 143.4979 2nd undecimal neutral second 4: 144/143 12.0644 Grossma 5: 143/128 191.8456 6: 104/99 85.2997 7: 144/143 12.0644 Grossma 8: 143/128 191.8456 9: 144/143 12.0644 Grossma 10: 88/81 143.4979 2nd undecimal neutral second 11: 729/704 60.4121 undecimal major diesis 12: 88/81 143.4979 2nd undecimal neutral second
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 3 144/143 12.064 cents Grossma 1: 2 729/704 60.412 cents undecimal major diesis 1: 1 104/99 85.300 cents 1: 4 88/81 143.498 cents 2nd undecimal neutral second 1: 2 143/128 191.846 cents 2: 1 128/121 97.364 cents undecimal semitone 2: 2 128/117 155.562 cents 2: 7 9/8 203.910 cents major whole tone 2: 1 169/144 277.145 cents two (tridecimal 2/3-tone) 2: 1 7744/6561 286.996 cents two (2nd undecimal neutral second) 3: 3 162/143 215.974 cents 3: 3 13/11 289.210 cents tridecimal minor third 3: 6 11/9 347.408 cents undecimal neutral third 4: 2 144/121 301.274 cents two (undecimal neutral second, 3/4-tone) 4: 3 16/13 359.472 cents tridecimal neutral third 4: 3 81/64 407.820 cents Pythagorean major third 4: 1 104/81 432.708 cents 4: 1 169/128 481.055 cents 4: 2 968/729 490.906 cents 5: 1 729/572 419.884 cents 5: 2 128/99 444.772 cents 5: 3 117/88 493.120 cents tridecimal gentle fourth 5: 2 1408/1053 502.970 cents 5: 4 11/8 551.318 cents undecimal semi-augmented fourth 6: 3 162/121 505.184 cents 6: 3 18/13 563.382 cents tridecimal augmented fourth 6: 3 13/9 636.618 cents tridecimal diminished fifth 6: 3 121/81 694.816 cents two (undecimal neutral third) 7: 4 16/11 648.682 cents undecimal semi-diminished fifth 7: 2 1053/704 697.030 cents 7: 3 176/117 706.880 cents 7: 2 99/64 755.228 cents 7: 1 1144/729 780.116 cents 8: 2 729/484 709.094 cents two (neutral third, Zalzal wosta of al-Farabi) 8: 1 256/169 718.945 cents two (tridecimal neutral third) 8: 1 81/52 767.292 cents 8: 3 128/81 792.180 cents Pythagorean minor sixth 8: 3 13/8 840.528 cents tridecimal neutral sixth 8: 2 121/72 898.726 cents 9: 6 18/11 852.592 cents undecimal neutral sixth 9: 3 22/13 910.790 cents tridecimal major sixth 9: 3 143/81 984.026 cents 10: 1 6561/3872 913.004 cents 10: 1 288/169 922.855 cents 10: 7 16/9 996.090 cents Pythagorean minor seventh 10: 2 117/64 1044.438 cents 10: 1 121/64 1102.636 cents two (undecimal semi-augmented fourth) 11: 2 256/143 1008.154 cents 11: 4 81/44 1056.502 cents 2nd undecimal neutral seventh 11: 1 99/52 1114.700 cents 11: 2 1408/729 1139.588 cents 11: 3 143/72 1187.936 cents Highest number of different intervals for one interval class: 6 Average number of different intervals per interval class: 4.72727 = 52/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 143.5 203.9 347.4 359.5 551.3 636.6 648.7 840.5 852.6 996.1 1056.5 1200.0 143.5 : 60.4 203.9 216.0 407.8 493.1 505.2 697.0 709.1 852.6 913.0 1056.5 1200.0 203.9 : 143.5 155.6 347.4 432.7 444.8 636.6 648.7 792.2 852.6 996.1 1139.6 1200.0 347.4 : 12.1 203.9 289.2 301.3 493.1 505.2 648.7 709.1 852.6 996.1 1056.5 1200.0 359.5 : 191.8 277.1 289.2 481.1 493.1 636.6 697.0 840.5 984.0 1044.4 1187.9 1200.0 551.3 : 85.3 97.4 289.2 301.3 444.8 505.2 648.7 792.2 852.6 996.1 1008.2 1200.0 636.6 : 12.1 203.9 216.0 359.5 419.9 563.4 706.9 767.3 910.8 922.9 1114.7 1200.0 648.7 : 191.8 203.9 347.4 407.8 551.3 694.8 755.2 898.7 910.8 1102.6 1187.9 1200.0 840.5 : 12.1 155.6 216.0 359.5 503.0 563.4 706.9 718.9 910.8 996.1 1008.2 1200.0 852.6 : 143.5 203.9 347.4 490.9 551.3 694.8 706.9 898.7 984.0 996.1 1187.9 1200.0 996.1 : 60.4 203.9 347.4 407.8 551.3 563.4 755.2 840.5 852.6 1044.4 1056.5 1200.0 1056.5: 143.5 287.0 347.4 490.9 503.0 694.8 780.1 792.2 984.0 996.1 1139.6 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 12.0644 .. 191.8456 cents 20449/18432, 179.7812 cents 144/143, 12.0644 cents 2: 97.3641 .. 286.9959 cents 14641/13122, 189.6318 cents 16384/17303,-94.4815 cents 3: 215.9744 .. 347.4079 cents 1573/1458, 131.4335 cents 531441/553696,-71.0215 cents 4: 301.2741 .. 490.9059 cents 14641/13122, 189.6318 cents 1296/1331,-46.1338 cents 5: 419.8844 .. 551.3179 cents 1573/1458, 131.4335 cents 531441/553696,-71.0215 cents 6: 505.1841 .. 694.8159 cents 14641/13122, 189.6318 cents 1296/1331,-46.1338 cents 7: 648.6821 .. 780.1156 cents 1573/1458, 131.4335 cents 1296/1331,-46.1338 cents 8: 709.0941 .. 898.7259 cents 14641/13122, 189.6318 cents 531441/553696,-71.0215 cents 9: 852.5921 .. 984.0256 cents 1573/1458, 131.4335 cents 1296/1331,-46.1338 cents 10: 913.0041 .. 1102.6359 cents 14641/13122, 189.6318 cents 531441/553696,-71.0215 cents 11: 1008.1544 .. 1187.9356 cents 20449/18432, 179.7812 cents 16384/17303,-94.4815 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 144/143, 12.0644 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 143/128, 191.8456 cents Average / Smallest step : 8.288851 Largest / Average step : 1.918456 Largest / Smallest step : 15.901797 Median interval of one step : 114.39883 cents, amount: 0 Most common interval of one step : 88/81, 143.4979 cents, amount: 4 Least squares average step : 99.27457 cents, oct.: 1191.29490 cents Scale is not proper Step pattern alph. order: ABACDECDCABA Step pattern size order : DBDAECAEADBD Scale is a Constant Structure, by a margin of 2.21384 cents Scale diversity : 1.391834 Lumma stability : 0.020107 Lumma impropriety factor : 0.547987 Rothenberg efficiency : 0.304052 redundancy: 0.695948 Efficiency x scale size : 3.648629 Number of different interval sizes : 52 = 4.72727 / class Number of one step interval sizes : 5 Highest interval variety : 6 Mean interval variety : 4.72727 = 52/11 Median interval variety : 3 Lowest interval variety : 3 Smallest interval difference : 85293/85184, 2.2138 cents Most common intervals : 9/8, 203.9100 cents & inv., amount: 7 Most common triad is 0.0 203.910 996.090 cents, amount: 3 Number of recognisable fifths : 11, average 703.2983 cents Number of appreciable fifths : 10, average 701.7336 cents Scale contains a complete diamond : 1 9 11 Formal octave complements present : 11 = 91.6667% Scale is n-1 differentially coherent in interval classes 2 and 3 combined -- Rational properties -- Prime limit : 13 Odd number limit : 6561 (O: 6561 U: 6561) Highest odd numerator or denominator: 81 Scale harmonicity : 0.009174 Average absolute harmonicity : 0.131744 Specific harmonicity : 0.065755 Fundamental : 1/92664, -16.4997 octaves, 0.0028 Hz. Guide tone : 185328, 17.4997 octaves, 48486542.766 Hz. Exponens Consonantiae : 1.717323E+10, 33.99944 octaves Euler's gradus suavitatis : 68 Sum of Mann's harmonic distance : 253.0, average 21.08333 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 13.23286, average 1.10274 Sum of Tenney's harmonic distance : 26.60721, average 2.21727 Vogel's harmonic complexity : 40.08333 Wille's k value : 75289 Wilson's harmonic complexity : 72 Rectangular lattice diameter : 12 Triangular lattice diameter : 8 Lattice compactness : 356.47859, average 4.57024 Lattice compactness (without 2's) : 216.19648, average 2.77175 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -3 .. 4 0.50000 28 1 3: -4 .. 4 -0.16667 18 2 11: -1 .. 1 0.00000 6 6 13: -1 .. 1 0.08333 3 3 Average exponent except 2's :-1 / 12 =-0.08333 Average absolute exponent except 2's: 27 / 12 = 2.25000 Scale is JI-epimorphic with val: <12 19 41 44| Scale is weakly epimorphic with val: <12 20 39 46| Scale is weakly epimorphic with val: <12 20 41 46| Scale is weakly epimorphic with val: <12 20 43 46| Scale is weakly epimorphic with val: <12 20 45 46|
FIT/MODE
12: 1 1 1 0 2 1 0 2 0 2 0 2 N T3 SD: 42.4778 c. M: 59.4723 c. 13: 2 0 2 0 2 1 0 2 0 2 0 2 N SD: 21.1591 c. M:-41.1174 c. 14: 2 0 2 0 2 1 1 2 0 2 0 2 N D3 I SD: 26.3747 c. M: 37.0322 c. 16: 2 1 2 0 2 1 1 2 0 2 1 2 N I SD: 21.8043 c. M: 36.6177 c. 17: 2 1 2 0 3 1 0 3 0 2 1 2 N SD: 7.3083 c. M: 13.3879 c. 24: 3 1 3 0 4 2 0 4 0 3 1 3 N SD: 6.3939 c. M:-13.3823 c. 41: 5 2 5 0 7 3 0 7 0 5 2 5 N SD: 4.8483 c. M: 8.2528 c. 53: 6 3 6 1 8 4 1 8 1 6 3 6 N SD: 5.6746 c. M: 7.9217 c. 58: 7 3 7 0 10 4 0 10 0 7 3 7 N SD: 5.0665 c. M: 7.7482 c. 59: 7 3 7 1 9 4 1 9 1 7 3 7 N SD: 3.4531 c. M: 6.6294 c. 66: 8 3 8 1 10 5 1 10 1 8 3 8 N SD: 3.5281 c. M: 5.8634 c. 76: 9 4 9 1 12 5 1 12 1 9 4 9 N SD: 2.3013 c. M: 5.0387 c. 83: 10 4 10 1 13 6 1 13 1 10 4 10 N SD: 1.3717 c. M: 1.9734 c. 100: 12 5 12 1 16 7 1 16 1 12 5 12 N SD: 0.5075 c. M:-0.6821 c. 294: 35 15 35 3 47 21 3 47 3 35 15 35 N SD: 0.3739 c. M:-0.6408 c. 377: 45 19 45 4 60 27 4 60 4 45 19 45 N SD: 0.3628 c. M: 0.6548 c. 394: 47 20 47 4 63 28 4 63 4 47 20 47 N SD: 0.1813 c. M:-0.3507 c. 494: 59 25 59 5 79 35 5 79 5 59 25 59 N SD: 0.1157 c. M: 0.1804 c. 971: 116 49 116 10 155 69 10 155 10 116 49 116 N SD: 0.1251 c. M: 0.1604 c. 995: 119 50 119 10 159 71 10 159 10 119 50 119 N SD: 0.0996 c. M:-0.1663 c. 1012: 121 51 121 10 162 72 10 162 10 121 51 121 N SD: 0.0920 c. M: 0.1838 c. 1095: 131 55 131 11 175 78 11 175 11 131 55 131 N SD: 0.0603 c. M:-0.0947 c. 1195: 143 60 143 12 191 85 12 191 12 143 60 143 N SD: 0.0533 c. M:-0.1004 c. 1489: 178 75 178 15 238 106 15 238 15 178 75 178 N SD: 0.0489 c. M: 0.0755 c. 1589: 190 80 190 16 254 113 16 254 16 190 80 190 N SD: 0.0153 c. M:-0.0278 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: 26.6365 cents 3 x x x x 4 x x x 5 x x 6 S SD: 20.6751 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 16.3970 cents 5 x x 6 x 7 x x 8 x 9 x 10 S SD: 11.2520 cents 6 x 7 x x 8 x 9 x 10 x 11:12 S SD: 17.2403 cents 7 x 8 x 9 x 10 x 11 x 12:13:14 S SD: 17.1367 cents 8 x 9 x 10:11 x 12:13 x 14:15:16 S SD: 9.1095 cents 9 x 10:11 x 12:13 x x 15:16:17:18 SD: 9.9167 cents 10:11 x 12 x 14 x 15:16 x 18 x 20 S SD: 11.4871 cents 11:12 x 13:14:15 x 16 x 18 x 20:22 S SD: 10.8073 cents 12:13:14 x 15 x 17 x x 20:21:22:24 SD: 10.9276 cents 13:14:15 x 16:18 x 19:21 x 23:24:26 SD: 5.6285 cents 14:15:16:17 x 19:20 x x 23:25:26:28 SD: 5.7357 cents 15:16:17:18 x 21 x 22:24:25:27:28:30 SD: 7.8305 cents 16:17:18 x 20:22:23 x 26 x 28:29:32 SD: 6.7986 cents 17:18:19 x 21:23 x 25 x 28:30:31:34 SD: 6.9015 cents 18 x 20:22 x 25:26 x 29 x 32:33:36 SD: 4.0254 cents 19:21 x 23 x 26:27:28 x 31:34:35:38 SD: 5.7727 cents 20:22:23:24:25:28 x 29 x 33:36:37:40 SD: 7.3376 cents 21:23:24 x 26:29:30:31:34 x 37:39:42 SD: 5.1314 cents 22:24:25 x 27:30 x 32 x 36:39:41:44 SD: 3.6173 cents 23:25:26:28 x 32:33 x 37:38:41:42:46 SD: 3.8323 cents 24:26:27:29:30:33 x 35:39 x 43:44:48 SD: 3.7269 cents 25:27:28 x 31:34:36 x x 41:44:46:50 SD: 3.5822 cents 26:28:29 x 32:36 x 38:42:43:46:48:52 SD: 3.4703 cents 27:29:30:33 x 37:39 x 44 x 48:50:54 SD: 3.5445 cents 28:30:32:34 x 39:40:41 x 46:50:52:56 SD: 5.2775 cents 29:32:33:35:36:40:42 x 47 x 52:53:58 SD: 4.7592 cents 30:33:34 x 37:41:43:44 x 49:53:55:60 SD: 3.6245 cents 31:34:35:38 x 43 x 45:50:51:55:57:62 SD: 2.9084 cents 32:35:36:39 x 44:46:47:52 x 57:59:64 SD: 2.3043 cents 33:36:37:40:41:45 x 48 x 54:59:61:66 SD: 3.0181 cents 34:37:38 x 42:47:49 x 55:56:60:63:68 SD: 2.7523 cents 35:38:39 x 43:48 x 51:57 x 62:64:70 SD: 2.4931 cents 36:39:41:44 x 50:52 x x 59:64:66:72 SD: 3.2133 cents 37:40:42:45:46:51:53:54:60:61:66:68:74 SD: 2.8572 cents 38:41:43:46:47:52:55 x x 62:68:70:76 SD: 2.8835 cents 39:42:44 x 48:54:56:57:63:64:69:72:78 SD: 2.5698 cents 40:43:45:49 x 55 x 58:65 x 71:74:80 SD: 2.3819 cents 41:45:46:50 x 56:59:60 x 67:73:75:82 SD: 2.7722 cents 42:46:47:51:52:58 x 61:68:69:75:77:84 SD: 2.5281 cents 43:47:48 x 53:59:62:63:70 x 76:79:86 SD: 2.4529 cents 44:48:50 x 54:61 x 64 x 72:78:81:88 SD: 2.7336 cents 45:49:51:55 x 62:65 x 73:74:80:83:90 SD: 1.6836 cents 46:50:52:56:57:63:66:67:75 x 82:85:92 SD: 2.1124 cents 47:51:53:57:58:65:68 x 76:77:84:87:94 SD: 2.1955 cents 48:52:54 x 59:66:69:70:78:79:85:88:96 SD: 1.6158 cents 49:53:55:60 x 67:71 x x 80:87:90:98 SD: 1.7313 cents 51:55:57:62:63:70 x 74:83 x 91:94:102 SD: 2.2311 cents 52:56:59 x 64:72:75:76 x 85:92:96:104 SD: 2.7685 cents 53:58:60:65 x 73 x 77:86:87:94:98:106 SD: 2.0859 cents 54:59:61:66 x 74:78:79:88 x 96:99:108 SD: 1.8865 cents 55:60:62:67:68:76:79:80:89:90:98:101:110 SD: 1.6917 cents 56:61:63:68:69:77:81 x 91:92:100:103:112 SD: 1.4832 cents 57:62:64 x 70:78:82:83 x 93:101:105:114 SD: 1.4543 cents 58:63:65:71 x 80:84 x 94:95:103:107:116 SD: 1.1956 cents 60:65:68:73:74:83 x 87 x 98:107:110:120 SD: 2.2179 cents 61:66:69 x 75:84:88:89:99:100:108:112:122 SD: 1.4894 cents 62:67:70:76 x 85 x 90:101 x 110:114:124 SD: 1.6561 cents 63:68:71:77:78:87:91:92:102:103:112:116:126 SD: 1.6411 cents 64:70:72:78:79:88:92:93:104:105:114:118:128 SD: 1.4355 cents 65:71:73:79:80:89:94:95 x 106:116:120:130 SD: 1.8639 cents 66:72:74 x 81:91:95:96:107:108:117:122:132 SD: 1.4656 cents 67:73:75:82 x 92:97 x 109:110:119:123:134 SD: 1.3733 cents 68:74:77:83:84:94:98:99 x 111:121:125:136 SD: 1.6041 cents 69:75:78:84:85:95:100 x 112:113:123:127:138 SD: 1.2407 cents 70:76:79 x 86:96:101:102:114:115:124:129:140 SD: 1.2114 cents 71:77:80:87 x 98 x 103:115:116:126:131:142 SD: 1.2844 cents 72:78:81:88:89:99:104:105:117:118:128:133:144 SD: 0.9965 cents 73:79:82:89:90:100:105:106:119 x 130:134:146 SD: 1.4307 cents 74:80:83:90:91:102:107:108:120:121:132:136:148 SD: 1.4123 cents 75:81:84 x 92:103:108:109:122:123:133:138:150 SD: 1.4953 cents 76:83:86:93:94:105:110:111 x 124:135:140:152 SD: 1.8641 cents 77:84:87:94:95:106:111:112:125:126:137:142:154 SD: 1.0516 cents 78:85:88:95:96:107:113 x 127:128:139:144:156 SD: 1.3140 cents 79:86:89 x 97:109:114:115:128:129:140:145:158 SD: 1.1885 cents 80:87:90:98 x 110 x 116:130:131:142:147:160 SD: 0.8175 cents 84:91:95:103 x 116:121:122 x 137:149:155:168 SD: 1.6996 cents 85:92:96:104:105:117:123:124:138:139:151:156:170 SD: 1.1910 cents 86:93:97:105:106:118:124:125:140:141:153:158:172 SD: 1.0314 cents 88:96:99 x 108:121:127:128:143:144:156:162:176 SD: 0.9171 cents 89:97:100:109:110:122:129 x 145:146:158:164:178 SD: 1.3237 cents 90:98:101:110:111:124:130:131:146:147:160:166:180 SD: 0.7928 cents 93:101:105:114 x 128:134:135:151:152:165:171:186 SD: 0.9994 cents 94:102:106:115:116:129:136:137:153:154:167:173:188 SD: 0.7902 cents 97:105:109 x 119:133:140:141:158:159:172:179:194 SD: 1.1674 cents 99:108:111:121:122:136:143:144:161:162:176:182:198 SD: 0.8326 cents 104:113:117:127:128:143:150:151:169:170:185:191:208 SD: 0.5277 cents 106:115:119:130 x 146:153:154:172:173:188:195:212 SD: 0.9701 cents 107:116:120:131:132:147:155:156:174:175:190:197:214 SD: 0.9029 cents 112:122:126:137:138:154:162:163:182:183:199:206:224 SD: 0.5464 cents 126:137:142:154:155:173:182:183:205:206:224:232:252 SD: 0.4701 cents 152:165:171:186:187:209:220:221:247:249:270:280:304 SD: 0.4288 cents 176:191:198:215:217:242:254:256:286:288:313:324:352 SD: 0.3387 cents 207:225:233:253:255:285:299:301:336:339:368:381:414 SD: 0.3258 cents 234:254:263:286:288:322:338:340:380:383:416:431:468 SD: 0.2950 cents 256:278:288:313:315:352:370:372:416:419:455:471:512 SD: 0.2111 cents 346:376:389:423:426:476:500:503:562:566:615:637:692 SD: 0.1929 cents 360:391:405:440:443:495:520:524:585:589:640:663:720 SD: 0.1279 cents 464:504:522:567:571:638:670:675:754:759:825:854:928 SD: 0.0930 cents