Scala analysis: SpBruckner
Sparschuh's view of Bruckner's "dissonant 5th"D-A=40/27 in C-major key
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 2227/2112 91.790068 2: 9/8 203.910002 major whole tone 3: 313/264 294.749673 4: 5/4 386.313714 major third 5: 4/3 498.044999 perfect fourth 6: 45/32 590.223716 diatonic tritone 7: 3/2 701.955001 perfect fifth 8: 835/528 793.485922 9: 5/3 884.358713 major sixth, BP sixth 10: 313/176 996.704674 11: 15/8 1088.268715 classic major seventh 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 16/15 111.7313 minor diatonic semitone 1: 2227/2112 91.7901 2: 2376/2227 112.1199 3: 313/297 90.8397 4: 330/313 91.5640 5: 16/15 111.7313 minor diatonic semitone 6: 135/128 92.1787 major chroma, major limma 7: 16/15 111.7313 minor diatonic semitone 8: 835/792 91.5309 9: 176/167 90.8728 10: 939/880 112.3460 11: 330/313 91.5640 12: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 313/297 90.840 cents 1: 1 176/167 90.873 cents 1: 1 835/792 91.531 cents 1: 2 330/313 91.564 cents 1: 1 2227/2112 91.790 cents 1: 1 135/128 92.179 cents major chroma, major limma 1: 3 16/15 111.731 cents minor diatonic semitone 1: 1 2376/2227 112.120 cents 1: 1 939/880 112.346 cents 2: 2 10/9 182.404 cents minor whole tone 2: 1 2504/2227 202.960 cents 2: 1 939/835 203.219 cents 2: 1 334/297 203.262 cents 2: 2 352/313 203.295 cents 2: 1 2227/1980 203.521 cents 2: 4 9/8 203.910 cents major whole tone 3: 2 32/27 294.135 cents Pythagorean minor third 3: 1 2640/2227 294.524 cents 3: 2 313/264 294.750 cents 3: 1 198/167 294.783 cents 3: 1 2227/1878 295.085 cents 3: 1 835/704 295.441 cents 3: 1 1485/1252 295.474 cents 3: 3 6/5 315.641 cents minor third 4: 4 5/4 386.314 cents major third 4: 1 2816/2227 406.255 cents 4: 2 626/495 406.481 cents 4: 1 1056/835 406.514 cents 4: 1 167/132 407.172 cents 4: 2 396/313 407.205 cents 4: 1 2227/1760 407.431 cents 5: 7 4/3 498.045 cents perfect fourth 5: 1 2227/1670 498.304 cents 5: 1 2970/2227 498.434 cents 5: 1 939/704 498.660 cents 5: 1 835/626 498.736 cents 5: 1 27/20 519.551 cents acute fourth 6: 1 835/594 589.576 cents 6: 2 440/313 589.609 cents 6: 1 2227/1584 589.835 cents 6: 2 45/32 590.224 cents diatonic tritone 6: 2 64/45 609.776 cents 2nd tritone 6: 1 3168/2227 610.165 cents 6: 2 313/220 610.391 cents 6: 1 1188/835 610.424 cents 7: 1 40/27 680.449 cents grave fifth 7: 1 1252/835 701.264 cents 7: 1 1408/939 701.340 cents 7: 1 2227/1485 701.566 cents 7: 1 3340/2227 701.696 cents 7: 7 3/2 701.955 cents perfect fifth 8: 1 3520/2227 792.569 cents 8: 2 313/198 792.795 cents 8: 1 264/167 792.828 cents 8: 1 835/528 793.486 cents 8: 2 495/313 793.519 cents 8: 1 2227/1408 793.745 cents 8: 4 8/5 813.686 cents minor sixth 9: 3 5/3 884.359 cents major sixth, BP sixth 9: 1 2504/1485 904.526 cents 9: 1 1408/835 904.559 cents 9: 1 3756/2227 904.915 cents 9: 1 167/99 905.217 cents 9: 2 528/313 905.250 cents 9: 1 2227/1320 905.476 cents 9: 2 27/16 905.865 cents Pythagorean major sixth 10: 4 16/9 996.090 cents Pythagorean minor seventh 10: 1 3960/2227 996.479 cents 10: 2 313/176 996.705 cents 10: 1 297/167 996.738 cents 10: 1 1670/939 996.781 cents 10: 1 2227/1252 997.040 cents 10: 2 9/5 1017.596 cents just minor seventh, BP seventh 11: 1 1760/939 1087.654 cents 11: 1 2227/1188 1087.880 cents 11: 3 15/8 1088.269 cents classic major seventh 11: 1 256/135 1107.821 cents octave - major chroma 11: 1 4224/2227 1108.210 cents 11: 2 313/165 1108.436 cents 11: 1 1584/835 1108.469 cents 11: 1 167/88 1109.127 cents 11: 1 594/313 1109.160 cents Highest number of different intervals for one interval class: 9 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 : 91.8 203.9 294.7 386.3 498.0 590.2 702.0 793.5 884.4 996.7 1088.3 1200.0 91.8 : 112.1 203.0 294.5 406.3 498.4 610.2 701.7 792.6 904.9 996.5 1108.2 1200.0 203.9 : 90.8 182.4 294.1 386.3 498.0 589.6 680.4 792.8 884.4 996.1 1087.9 1200.0 294.7 : 91.6 203.3 295.5 407.2 498.7 589.6 702.0 793.5 905.3 997.0 1109.2 1200.0 386.3 : 111.7 203.9 315.6 407.2 498.0 610.4 702.0 813.7 905.5 1017.6 1108.4 1200.0 498.0 : 92.2 203.9 295.4 386.3 498.7 590.2 702.0 793.7 905.9 996.7 1088.3 1200.0 590.2 : 111.7 203.3 294.1 406.5 498.0 609.8 701.6 813.7 904.5 996.1 1107.8 1200.0 702.0 : 91.5 182.4 294.7 386.3 498.0 589.8 702.0 792.8 884.4 996.1 1088.3 1200.0 793.5 : 90.9 203.2 294.8 406.5 498.3 610.4 701.3 792.8 904.6 996.7 1108.5 1200.0 884.4 : 112.3 203.9 315.6 407.4 519.6 610.4 702.0 813.7 905.9 1017.6 1109.1 1200.0 996.7 : 91.6 203.3 295.1 407.2 498.0 589.6 701.3 793.5 905.3 996.8 1087.7 1200.0 1088.3: 111.7 203.5 315.6 406.5 498.0 609.8 702.0 813.7 905.2 996.1 1108.4 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.8397 .. 112.3460 cents 81/80, 21.5063 cents 313/297, 90.8397 cents 2: 182.4037 .. 203.9100 cents 81/80, 21.5063 cents 8800/8451, 70.0578 cents 3: 294.1350 .. 315.6413 cents 81/80, 21.5063 cents 256/243, 90.2250 cents 4: 386.3137 .. 407.4314 cents 2227/2200, 21.1176 cents 25/24, 70.6724 cents 5: 498.0450 .. 519.5513 cents 81/80, 21.5063 cents 7040/6681, 90.6136 cents 6: 589.5759 .. 610.4241 cents 705672/697225, 20.8482 cents 8350/8019, 70.0246 cents 7: 680.4487 .. 701.9550 cents 81/80, 21.5063 cents 8350/8019, 70.0246 cents 8: 792.5686 .. 813.6863 cents 2227/2200, 21.1176 cents 7040/6681, 90.6136 cents 9: 884.3587 .. 905.8650 cents 81/80, 21.5063 cents 25/24, 70.6724 cents 10: 996.0900 .. 1017.5963 cents 81/80, 21.5063 cents 256/243, 90.2250 cents 11: 1087.6540 .. 1109.1603 cents 81/80, 21.5063 cents 8800/8451, 70.0578 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 313/297, 90.8397 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 939/880, 112.3460 cents Average / Smallest step : 1.100841 Largest / Average step : 1.123460 Largest / Smallest step : 1.236750 Median interval of one step : 91.98439 cents, amount: 0 Most common interval of one step : 16/15, 111.7313 cents, amount: 3 Least squares average step : 99.26075 cents, oct.: 1191.12895 cents Scale is strictly proper Step pattern alph. order: ABCDEFEGHIDE Step pattern size order : EHADGFGCBIDG Scale is a Constant Structure, by a margin of 70.02463 cents Scale diversity : 1.146949 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.804055 Rothenberg efficiency : 0.260269 redundancy: 0.739731 Efficiency x scale size : 3.123232 Number of different interval sizes : 82 = 7.45455 / class Number of one step interval sizes : 9 Highest interval variety : 9 Mean interval variety : 7.45455 = 82/11 Median interval variety : 6 Lowest interval variety : 6 Smallest interval difference : 52272/52271, 0.0331 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 : 11, average 701.7774 cents Best fifths form a closed circle Scale contains a complete diamond : 1 3 Formal octave complements present : 3 = 25.0000% Limited inverse transpositions with margin 21.5063 cents: 11 Inversional symmetry on intervals : 5-6 11-12 -- Rational properties -- Prime limit : 313 Odd number limit : 2227 (O: 2227 U: 2227) Highest odd numerator or denominator: 2227 Scale harmonicity : 0.014144 Fundamental : 1/2112, -11.0444 octaves, 0.1239 Hz. Guide tone : too high to compute Exponens Consonantiae : 4.425348E+13, 45.33086 octaves Euler's gradus suavitatis : 653 Sum of Mann's harmonic distance : 3458.5, average 288.20833 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 17.39622, average 1.44969 Sum of Tenney's harmonic distance : 33.70530, average 2.80877 Vogel's harmonic complexity : 94.75000 Wille's k value : 1113 Wilson's harmonic complexity : 653 Rectangular lattice diameter : 9 Triangular lattice diameter : 5 Lattice compactness : 276.52314, average 3.54517 Lattice compactness (without 2's) : 136.53602, average 1.75046 Number of different primes : 8 Prime exponents' range, average, count, tones@limit: 2: -6 .. 2 -2.33333 34 1 3: -1 .. 2 0.08333 11 3 5: 0 .. 1 0.41667 5 4 11: -1 .. 0 -0.33333 4 17: 0 .. 1 0.08333 1 131: 0 .. 1 0.08333 1 1 167: 0 .. 1 0.08333 1 1 313: 0 .. 1 0.16667 2 2 Average exponent except 2's : 7 / 12 = 0.58333 Average absolute exponent except 2's: 25 / 12 = 2.08333 Scale is JI-epimorphic with val: <12 19 28 41 48 85 88 99| Scale is JI-epimorphic with val: <12 19 28 41 49 84 88 99| Scale is JI-epimorphic with val: <12 19 28 41 50 83 88 99| Scale is JI-epimorphic with val: <12 19 28 42 49 85 89 100| Scale is JI-epimorphic with val: <12 19 28 42 50 84 89 100| Scale is JI-epimorphic with val: <12 19 28 42 51 83 89 100|
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 8.3484 c. M:-15.6413 c. Twelve-tone Chromatic (1/11-comma) 29: 2 3 2 2 3 2 3 2 2 3 2 3 SP M ME SD: 8.6122 c. M: 15.3932 c. Schismic-12 40: 3 4 3 3 4 3 3 3 3 4 3 4 SP D3 SD: 8.9640 c. M: 14.3587 c. 41: 3 4 3 3 4 3 4 3 3 4 3 4 SP M ME SD: 3.6568 c. M: 6.3099 c. Schismic-12 53: 4 5 4 4 5 4 5 4 4 5 4 5 SP M ME SD: 0.9718 c. M: 1.5445 c. Schismic-12 65: 5 6 5 5 6 5 6 5 5 6 5 6 SP M ME SD: 0.8284 c. M:-1.7951 c. 118: 9 11 9 9 11 9 11 9 9 11 9 11 SP M DE SD: 0.2770 c. M: 0.5202 c. Schismic-12 289: 22 27 22 22 27 22 27 22 22 27 22 27 SP M DE SD: 0.3204 c. M: 0.6043 c. 407: 31 38 31 31 38 31 38 31 31 38 31 38 SP M DE SD: 0.2983 c. M: 0.5431 c. 525: 40 49 40 40 49 40 49 40 40 49 40 49 SP M DE SD: 0.2891 c. M: 0.5094 c. 643: 49 60 49 49 60 49 60 49 49 60 49 60 SP M DE SD: 0.2846 c. M: 0.4885 c. 696: 53 65 53 53 65 53 65 53 53 65 53 65 SP M DE SD: 0.3066 c. M: 0.5685 c. 730: 56 68 55 56 68 56 68 56 55 68 56 68 SP D3 SD: 0.2694 c. M: 0.5403 c. 783: 60 73 59 60 73 60 73 60 59 73 60 73 SP D3 SD: 0.2565 c. M: 0.5361 c. 795: 61 74 60 61 74 61 74 61 60 74 61 74 SP D3 SD: 0.2541 c. M: 0.4783 c. 836: 64 78 63 64 78 64 78 64 63 78 64 78 SP D3 SD: 0.2601 c. M: 0.5324 c. 848: 65 79 64 65 79 65 79 65 64 79 65 79 SP D3 SD: 0.2297 c. M: 0.4783 c. 860: 66 80 65 66 80 66 80 66 65 80 66 80 SP D3 SD: 0.2566 c. M:-0.4676 c. 889: 68 83 67 68 83 68 83 68 67 83 68 83 SP D3 SD: 0.2745 c. M: 0.5292 c. 901: 69 84 68 69 84 69 84 69 68 84 69 84 SP D3 SD: 0.2232 c. M: 0.4783 c. 913: 70 85 69 70 85 70 85 70 69 85 70 85 SP D3 SD: 0.2228 c. M: 0.4287 c. 954: 73 89 72 73 89 73 89 73 72 89 73 89 SP D3 SD: 0.2301 c. M: 0.4783 c. 966: 74 90 73 74 90 74 90 74 73 90 74 90 SP D3 SD: 0.2054 c. M: 0.4314 c. 978: 75 91 74 75 91 75 91 75 74 91 75 91 SP D3 SD: 0.2300 c. M: 0.3857 c. 1007: 77 94 76 77 94 77 94 77 76 94 77 94 SP D3 SD: 0.2458 c. M: 0.4783 c. 1010: 77 95 76 77 94 78 94 77 76 95 77 94 SP SD: 0.2439 c. M:-0.4464 c. 1019: 78 95 77 78 95 78 95 78 77 95 78 95 SP D3 SD: 0.2027 c. M: 0.4338 c. 1072: 82 100 81 82 100 82 100 82 81 100 82 100 SP D3 SD: 0.2112 c. M: 0.4360 c. 1075: 82 101 81 82 100 83 100 82 81 101 82 100 SP SD: 0.2033 c. M:-0.3691 c. 1084: 83 101 82 83 101 83 101 83 82 101 83 101 SP D3 SD: 0.1911 c. M: 0.3947 c. 1096: 84 102 83 84 102 84 102 84 83 102 84 102 SP D3 SD: 0.2133 c. M: 0.3543 c. 1125: 86 105 85 86 105 86 105 86 85 105 86 105 SP D3 SD: 0.2269 c. M: 0.4380 c. 1128: 86 106 85 86 105 87 105 86 85 106 86 105 SP SD: 0.1967 c. M:-0.3453 c. 1137: 87 106 86 87 106 87 106 87 86 106 87 106 SP D3 SD: 0.1908 c. M: 0.3986 c. 1140: 87 107 86 87 106 88 106 87 86 107 87 106 SP SD: 0.1768 c. M:-0.3026 c. 1152: 88 108 87 88 107 89 107 88 87 108 88 107 SP SD: 0.1982 c. M:-0.4013 c. 1193: 91 112 90 91 111 92 111 91 90 112 91 111 SP SD: 0.1625 c. M:-0.2811 c. 1205: 92 113 91 92 112 93 112 92 91 113 92 112 SP SD: 0.1645 c. M:-0.3157 c. 1246: 95 117 94 95 116 96 116 95 94 117 95 116 SP SD: 0.1602 c. M: 0.2973 c. 1258: 96 118 95 96 117 97 117 96 95 118 96 117 SP SD: 0.1417 c. M:-0.2373 c. 1270: 97 119 96 97 118 98 118 97 96 119 97 118 SP SD: 0.1647 c. M:-0.3275 c. 1311: 100 123 99 100 122 101 122 100 99 123 100 122 SP SD: 0.1311 c. M: 0.2569 c. 1323: 101 124 100 101 123 102 123 101 100 124 101 123 SP SD: 0.1347 c. M:-0.2525 c. 1364: 104 128 103 104 127 105 127 104 103 128 104 127 SP SD: 0.1325 c. M: 0.2945 c. 1376: 105 129 104 105 128 106 128 105 104 129 105 128 SP SD: 0.1150 c. M: 0.2203 c. 1429: 109 134 108 109 133 110 133 109 108 134 109 133 SP SD: 0.1076 c. M: 0.2575 c. 1441: 110 135 109 110 134 111 134 110 109 135 110 134 SP SD: 0.1126 c. M:-0.1996 c. 1482: 113 139 112 113 138 114 138 113 112 139 113 138 SP SD: 0.1120 c. M: 0.2921 c. 1494: 114 140 113 114 139 115 139 114 113 140 114 139 SP SD: 0.0958 c. M: 0.2238 c. 1547: 118 145 117 118 144 119 144 118 117 145 118 144 SP SD: 0.0912 c. M: 0.2581 c. 1559: 119 146 118 119 145 120 145 119 118 146 119 145 SP SD: 0.0974 c. M: 0.1929 c. 1612: 123 151 122 123 150 124 150 123 122 151 123 150 SP SD: 0.0833 c. M: 0.2268 c. 1624: 124 152 123 124 151 125 151 124 123 152 124 151 SP SD: 0.1093 c. M:-0.1704 c. 1665: 127 156 126 127 155 128 155 127 126 156 127 155 SP SD: 0.0812 c. M: 0.2585 c. 1677: 128 157 127 128 156 129 156 128 127 157 128 156 SP SD: 0.0883 c. M: 0.1979 c. 1730: 132 162 131 132 161 133 161 132 131 162 132 161 SP SD: 0.0769 c. M: 0.2294 c. 1783: 136 167 135 136 166 137 166 136 135 167 136 166 SP SD: 0.0769 c. M: 0.2589 c. 1848: 141 173 140 141 172 142 172 141 140 173 141 172 SP SD: 0.0757 c. M: 0.2316 c. 1901: 145 178 144 145 177 146 177 145 144 178 145 177 SP SD: 0.0770 c. M: 0.2593 c. 2330: 178 218 176 178 217 179 217 178 176 218 178 217 SP SD: 0.0883 c. M:-0.1621 c. 2342: 179 219 177 179 218 180 218 179 177 219 179 218 SP SD: 0.0809 c. M:-0.1947 c. 2395: 183 224 181 183 223 184 223 183 181 224 183 223 SP SD: 0.0783 c. M:-0.1675 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: 0.0000 cents 3 x x x x 4 x x x 5 x x 6 S SD: 0.0000 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 6.9697 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 7.3224 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 7.9772 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 11.6401 cents 8 x 9 x 10 x 11:12 x 13:14:15:16 S SD: 8.1125 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 7.9415 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 8.0476 cents 11 x 12:13:14:15 x x 17:18:20:21:22 SD: 11.3267 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 6.6547 cents 13:14:15 x 16:17:18:20:21:22:23:24:26 S SD: 9.3585 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 7.6136 cents 15:16:17:18:19:20:21:23:24:25:27:28:30 SD: 5.2271 cents 17:18:19:20:21:23:24:26:27:28:30:32:34 SD: 4.7829 cents 22:23:25:26:28:29:31:33:35:37:39:41:44 SD: 4.1028 cents 24:25:27:28:30:32:34:36:38:40:43:45:48 SD: 3.2872 cents 31:33:35:37:39:41:44:47:49:52:55:58:62 SD: 3.2480 cents 32:34:36:38:40:43:45:48:51:53:57:60:64 SD: 2.1540 cents 39:41:44:46:49:52:55:59:62:65:69:73:78 SD: 2.0794 cents 40:42:45:47:50:53:56:60:63:67:71:75:80 SD: 2.0555 cents 48:51:54:57:60:64:68:72:76:80:85:90:96 S SD: 1.6732 cents 54:57:61:64:68:72:76:81:85:90:96:101:108 SD: 1.4518 cents 64:67:72:76:80:85:90:96:101:107:114:120:128 SD: 1.3421 cents 71:75:80:84:89:95:100:107:112:118:126:133:142 SD: 1.2447 cents 72:76:81:85:90:96:101:108:114:120:128:135:144 SD: 0.7495 cents 96:101:108:114:120:128:135:144:152:160:171:180:192 SD: 0.4909 cents 144:152:162:171:180:192:203:216:228:240:256:270:288 SD: 0.4846 cents 162:171:182:192:203:216:228:243:256:270:288:304:324 SD: 0.4836 cents 167:176:188:198:209:223:235:251:264:278:297:313:334 SD: 0.4687 cents 168:177:189:199:210:224:236:252:266:280:299:315:336 SD: 0.3115 cents 216:228:243:256:270:288:304:324:342:360:384:405:432 SD: 0.2683 cents 280:295:315:332:350:373:394:420:443:467:498:525:560 SD: 0.2336 cents 296:312:333:351:370:395:416:444:468:493:526:555:592 SD: 0.2215 cents 312:329:351:370:390:416:439:468:493:520:555:585:624 SD: 0.1537 cents 384:405:432:455:480:512:540:576:607:640:683:720:768 SD: 0.1144 cents 480:506:540:569:600:640:675:720:759:800:854:900:960 SD: 0.0782 cents