Scala analysis: bicycle
13-limit harmonic bicycle, George Secor, 1963
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 13/12 138.572661 tridecimal 2/3-tone 2: 9/8 203.910002 major whole tone 3: 7/6 266.870906 septimal minor third 4: 5/4 386.313714 major third 5: 4/3 498.044999 perfect fourth 6: 11/8 551.317942 undecimal semi-augmented fourth 7: 3/2 701.955001 perfect fifth 8: 13/8 840.527662 tridecimal neutral sixth 9: 5/3 884.358713 major sixth, BP sixth 10: 7/4 968.825906 harmonic seventh 11: 11/6 1049.362941 undecimal neutral seventh, 21/4-tone 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 12/11 150.6371 undecimal neutral second, 3/4-tone 1: 13/12 138.5727 tridecimal 2/3-tone 2: 27/26 65.3373 tridecimal comma 3: 28/27 62.9609 Archytas' 1/3-tone 4: 15/14 119.4428 major diatonic semitone 5: 16/15 111.7313 minor diatonic semitone 6: 33/32 53.2729 undecimal comma, al-Farabi's 1/4-tone 7: 12/11 150.6371 undecimal neutral second, 3/4-tone 8: 13/12 138.5727 tridecimal 2/3-tone 9: 40/39 43.8311 tridecimal minor diesis 10: 21/20 84.4672 minor semitone 11: 22/21 80.5370 undecimal minor semitone 12: 12/11 150.6371 undecimal neutral second, 3/4-tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 40/39 43.831 cents tridecimal minor diesis 1: 1 33/32 53.273 cents undecimal comma, al-Farabi's 1/4-tone 1: 1 28/27 62.961 cents Archytas' 1/3-tone 1: 1 27/26 65.337 cents tridecimal comma 1: 1 22/21 80.537 cents undecimal minor semitone 1: 1 21/20 84.467 cents minor semitone 1: 1 16/15 111.731 cents minor diatonic semitone 1: 1 15/14 119.443 cents major diatonic semitone 1: 2 13/12 138.573 cents tridecimal 2/3-tone 1: 2 12/11 150.637 cents undecimal neutral second, 3/4-tone 2: 2 14/13 128.298 cents 2/3-tone 2: 2 11/10 165.004 cents Ptolemy's second, 4/5-tone 2: 2 10/9 182.404 cents minor whole tone 2: 2 9/8 203.910 cents major whole tone 2: 2 8/7 231.174 cents septimal whole tone 2: 2 13/11 289.210 cents tridecimal minor third 3: 1 44/39 208.835 cents 3: 1 15/13 247.741 cents tridecimal 5/4-tone 3: 2 7/6 266.871 cents septimal minor third 3: 1 33/28 284.447 cents undecimal minor third 3: 1 32/27 294.135 cents Pythagorean minor third 3: 2 6/5 315.641 cents minor third 3: 1 40/33 333.041 cents 3: 1 39/32 342.483 cents 39th harmonic, Zalzal wosta of Ibn Sina 3: 1 27/22 354.547 cents neutral third, Zalzal wosta of al-Farabi 3: 1 26/21 369.747 cents 4: 2 11/9 347.408 cents undecimal neutral third 4: 2 16/13 359.472 cents tridecimal neutral third 4: 2 5/4 386.314 cents major third 4: 2 14/11 417.508 cents undecimal diminished fourth or major third 4: 2 9/7 435.084 cents septimal major third, BP third 4: 2 13/10 454.214 cents tridecimal semi-diminished fourth 5: 1 33/26 412.745 cents tridecimal major third 5: 1 21/16 470.781 cents narrow fourth 5: 7 4/3 498.045 cents perfect fourth 5: 1 27/20 519.551 cents acute fourth 5: 1 15/11 536.951 cents undecimal augmented fourth 5: 1 39/28 573.657 cents 6: 2 11/8 551.318 cents undecimal semi-augmented fourth 6: 2 18/13 563.382 cents tridecimal augmented fourth 6: 2 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 6: 2 10/7 617.488 cents Euler's tritone 6: 2 13/9 636.618 cents tridecimal diminished fifth 6: 2 16/11 648.682 cents undecimal semi-diminished fifth 7: 1 56/39 626.343 cents 7: 1 22/15 663.049 cents undecimal diminished fifth 7: 1 40/27 680.449 cents grave fifth 7: 7 3/2 701.955 cents perfect fifth 7: 1 32/21 729.219 cents wide fifth 7: 1 52/33 787.255 cents tridecimal minor sixth 8: 2 20/13 745.786 cents tridecimal semi-augmented fifth 8: 2 14/9 764.916 cents septimal minor sixth 8: 2 11/7 782.492 cents undecimal augmented fifth 8: 2 8/5 813.686 cents minor sixth 8: 2 13/8 840.528 cents tridecimal neutral sixth 8: 2 18/11 852.592 cents undecimal neutral sixth 9: 1 21/13 830.253 cents 9: 1 44/27 845.453 cents neutral sixth 9: 1 64/39 857.517 cents 39th subharmonic 9: 1 33/20 866.959 cents 9: 2 5/3 884.359 cents major sixth, BP sixth 9: 1 27/16 905.865 cents Pythagorean major sixth 9: 1 56/33 915.553 cents 9: 2 12/7 933.129 cents septimal major sixth 9: 1 26/15 952.259 cents tridecimal semi-augmented sixth 9: 1 39/22 991.165 cents 10: 2 22/13 910.790 cents tridecimal major sixth 10: 2 7/4 968.826 cents harmonic seventh 10: 2 16/9 996.090 cents Pythagorean minor seventh 10: 2 9/5 1017.596 cents just minor seventh, BP seventh 10: 2 20/11 1034.996 cents large minor seventh 10: 2 13/7 1071.702 cents 16/3-tone 11: 2 11/6 1049.363 cents undecimal neutral seventh, 21/4-tone 11: 2 24/13 1061.427 cents tridecimal neutral seventh 11: 1 28/15 1080.557 cents grave major seventh 11: 1 15/8 1088.269 cents classic major seventh 11: 1 40/21 1115.533 cents acute major seventh 11: 1 21/11 1119.463 cents undecimal major seventh 11: 1 52/27 1134.663 cents 11: 1 27/14 1137.039 cents septimal major seventh 11: 1 64/33 1146.727 cents 33rd subharmonic 11: 1 39/20 1156.169 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 : 138.6 203.9 266.9 386.3 498.0 551.3 702.0 840.5 884.4 968.8 1049.4 1200.0 138.6 : 65.3 128.3 247.7 359.5 412.7 563.4 702.0 745.8 830.3 910.8 1061.4 1200.0 203.9 : 63.0 182.4 294.1 347.4 498.0 636.6 680.4 764.9 845.5 996.1 1134.7 1200.0 266.9 : 119.4 231.2 284.4 435.1 573.7 617.5 702.0 782.5 933.1 1071.7 1137.0 1200.0 386.3 : 111.7 165.0 315.6 454.2 498.0 582.5 663.0 813.7 952.3 1017.6 1080.6 1200.0 498.0 : 53.3 203.9 342.5 386.3 470.8 551.3 702.0 840.5 905.9 968.8 1088.3 1200.0 551.3 : 150.6 289.2 333.0 417.5 498.0 648.7 787.3 852.6 915.6 1035.0 1146.7 1200.0 702.0 : 138.6 182.4 266.9 347.4 498.0 636.6 702.0 764.9 884.4 996.1 1049.4 1200.0 840.5 : 43.8 128.3 208.8 359.5 498.0 563.4 626.3 745.8 857.5 910.8 1061.4 1200.0 884.4 : 84.5 165.0 315.6 454.2 519.6 582.5 702.0 813.7 867.0 1017.6 1156.2 1200.0 968.8 : 80.5 231.2 369.7 435.1 498.0 617.5 729.2 782.5 933.1 1071.7 1115.5 1200.0 1049.4: 150.6 289.2 354.5 417.5 537.0 648.7 702.0 852.6 991.2 1035.0 1119.5 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 43.8311 .. 150.6371 cents 117/110, 106.8060 cents 40/39, 43.8311 cents 2: 128.2982 .. 289.2097 cents 169/154, 160.9115 cents 77/78,-22.3388 cents 3: 208.8353 .. 369.7468 cents 169/154, 160.9115 cents 484/507,-80.3744 cents 4: 347.4079 .. 454.2139 cents 117/110, 106.8060 cents 77/78,-22.3388 cents 5: 412.7453 .. 573.6568 cents 169/154, 160.9115 cents 165/169,-41.4687 cents 6: 551.3179 .. 648.6821 cents 128/121, 97.3641 cents 77/78,-22.3388 cents 7: 626.3432 .. 787.2547 cents 169/154, 160.9115 cents 77/78,-22.3388 cents 8: 745.7861 .. 852.5921 cents 117/110, 106.8060 cents 165/169,-41.4687 cents 9: 830.2532 .. 991.1647 cents 169/154, 160.9115 cents 77/78,-22.3388 cents 10: 910.7903 .. 1071.7018 cents 169/154, 160.9115 cents 484/507,-80.3744 cents 11: 1049.3629 .. 1156.1689 cents 117/110, 106.8060 cents 77/78,-22.3388 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 40/39, 43.8311 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 12/11, 150.6371 cents Average / Smallest step : 2.281488 Largest / Average step : 1.506371 Largest / Smallest step : 3.436766 Median interval of one step : 98.09924 cents, amount: 0 Least squares average step : 98.33658 cents, oct.: 1180.03900 cents Scale is not proper Step pattern alph. order: ABCDEFGAHIJG Step pattern size order : IDCHGBJIAFEJ Scale is a Constant Structure, by a margin of 4.76268 cents Scale diversity : 1.659922 Lumma stability : 0.073052 Lumma impropriety factor : 0.314766 Rothenberg efficiency : 0.267677 redundancy: 0.732323 Efficiency x scale size : 3.212121 Number of different interval sizes : 82 = 7.45455 / class Number of one step interval sizes : 10 Highest interval variety : 10 Mean interval variety : 7.45455 = 82/11 Median interval variety : 6 Lowest interval variety : 6 Smallest interval difference : 729/728, 2.3764 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 7 Most common triad is 0.0 498.045 701.955 cents, amount: 2 Number of recognisable fifths : 7, average 701.9550 cents Scale contains a complete diamond : 1 3 All steps are superparticular (epimoric) Length of longest harmonic part : 2 at degree 1 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 3 = 25.0000% Scale is differentially coherent in interval class 1 Scale is differentially coherent in interval class 2 Scale is differentially coherent in interval class 3 Scale is differentially coherent in interval class 4 Scale is differentially coherent in interval class 5 Scale is differentially coherent in interval class 6 Scale is differentially coherent in interval classes 7 and 8 combined Scale is differentially coherent in interval class 8 Scale is differentially coherent in interval classes 9 and 10 combined Scale is differentially coherent in interval class 10 -- Rational properties -- Prime limit : 13 Odd number limit : 39 (O: 39 U: 39) Highest odd numerator or denominator: 13 Scale harmonicity : 0.014281 Average absolute harmonicity : 0.179953 Specific harmonicity : 0.106035 Fundamental : 1/24, -4.5850 octaves, 10.9011 Hz. Guide tone : 180180, 17.4591 octaves, 47139694.356 Hz. Exponens Consonantiae : 4.324320E+06, 22.04404 octaves Euler's gradus suavitatis : 44 Sum of Mann's harmonic distance : 77.5, average 6.45833 Mersenne's string divisions : 581981400 Sum of van Prooijen's expressibility: 9.30729, average 0.77561 Sum of Tenney's harmonic distance : 17.53741, average 1.46145 Vogel's harmonic complexity : 10.25000 Wille's k value : 19 Wilson's harmonic complexity : 45 Rectangular lattice diameter : 7 Triangular lattice diameter : 3 Lattice compactness : 201.34362, average 2.58133 Lattice compactness (without 2's) : 120.92024, average 1.55026 Number of different primes : 6 Prime exponents' range, average, count, tones@limit: 2: -3 .. 2 -1.25000 21 1 3: -1 .. 2 -0.16667 8 3 5: 0 .. 1 0.16667 2 2 7: 0 .. 1 0.16667 2 2 11: 0 .. 1 0.16667 2 2 13: 0 .. 1 0.16667 2 2 Average exponent except 2's : 6 / 12 = 0.50000 Average absolute exponent except 2's: 16 / 12 = 1.33333 Scale is weakly epimorphic with val: <12 19 28 32 42 46| Scale is JI-epimorphic with val: <12 19 28 34 42 44| = patent
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 29.7050 c. M:-50.6371 c. Twelve-tone Chromatic (1/11-comma) 13: 1 1 1 1 1 1 1 2 0 1 1 2 N D3 SD: 33.6316 c. M: 55.8012 c. 14: 1 1 1 1 2 0 2 2 0 1 1 2 N SD: 28.6567 c. M: 52.8584 c. 15: 2 1 0 2 1 1 2 2 0 1 1 2 N SD: 20.6967 c. M:-39.4723 c. 16: 2 1 1 1 2 0 2 2 1 1 1 2 N T3 SD: 19.3024 c. M:-33.1291 c. 17: 2 1 1 2 1 1 2 2 1 1 1 2 N D3 SD: 17.1276 c. M:-37.2157 c. 19: 2 1 1 2 2 1 2 2 1 1 2 2 P D3 SD: 14.2701 c. M:-24.3213 c. 22: 3 1 1 2 2 1 3 2 1 2 1 3 N SD: 12.8938 c. M:-25.0637 c. 24: 3 1 1 3 2 1 3 3 1 1 2 3 N SD: 10.4737 c. M: 18.8259 c. 26: 3 1 2 2 3 1 3 3 1 2 2 3 N SD: 10.2332 c. M: 19.2946 c. 27: 3 2 1 3 2 1 4 3 1 2 2 3 N SD: 11.0177 c. M:-18.3122 c. 29: 3 2 1 3 3 1 4 3 1 2 2 4 N SD: 12.4441 c. M: 18.5950 c. 31: 4 1 2 3 3 1 4 4 1 2 2 4 N SD: 7.6830 c. M:-16.2660 c. 37: 4 2 2 4 3 2 5 4 1 3 2 5 N SD: 7.7898 c. M:-11.5585 c. 41: 5 2 2 4 4 2 5 5 1 3 3 5 N SD: 4.7055 c. M:-8.2528 c. 46: 5 3 2 5 4 2 6 5 2 3 3 6 N SD: 4.6646 c. M: 8.1379 c. 53: 6 3 3 5 5 2 7 6 2 4 3 7 N SD: 3.9728 c. M: 7.9217 c. 58: 7 3 3 6 5 3 7 7 2 4 4 7 N SD: 4.9247 c. M:-7.7482 c. 63: 7 4 3 6 6 3 8 7 2 5 4 8 N SD: 3.9485 c. M: 8.1682 c. 67: 8 3 4 7 6 3 8 8 2 5 5 8 N SD: 4.7910 c. M:-7.7161 c. 68: 8 4 3 7 6 3 9 8 2 5 4 9 N SD: 4.4967 c. M: 8.1865 c. 69: 8 4 3 7 7 3 8 8 3 5 4 9 N SD: 4.8283 c. M:-6.3028 c. 70: 8 4 4 7 6 3 9 8 3 5 4 9 N SD: 4.7059 c. M:-8.3170 c. 72: 8 4 4 7 7 3 9 8 3 5 5 9 N SD: 3.1480 c. M: 7.1943 c. 77: 9 4 4 8 7 3 10 9 3 5 5 10 N SD: 2.9573 c. M: 5.8634 c. 81: 9 5 4 8 8 3 10 10 3 5 6 10 N SD: 3.9950 c. M: 5.8629 c. 84: 10 4 5 8 8 4 10 10 3 6 6 10 N SD: 3.7740 c. M:-7.7799 c. 87: 10 5 4 9 8 4 11 10 3 6 6 11 N SD: 2.0905 c. M: 4.8019 c. 94: 11 5 5 9 9 4 12 11 3 7 6 12 N SD: 1.9735 c. M: 3.5076 c. 111: 13 6 6 11 10 5 14 13 4 8 7 14 N SD: 2.1730 c. M:-4.1471 c. 113: 13 6 6 11 11 5 14 13 4 8 8 14 N SD: 1.9796 c. M: 4.0128 c. 130: 15 7 7 13 12 6 16 15 5 9 9 16 N SD: 1.3649 c. M:-2.9448 c. 159: 18 9 8 16 15 7 20 18 6 11 11 20 N SD: 1.6937 c. M: 2.7918 c. 166: 19 9 9 16 16 7 21 19 6 12 11 21 N SD: 1.5873 c. M: 3.1812 c. 167: 19 9 9 17 15 8 21 19 6 12 11 21 N SD: 1.6146 c. M: 2.7124 c. 171: 20 9 9 17 16 8 21 20 6 12 12 21 N SD: 1.4815 c. M:-3.2686 c. 176: 20 10 9 18 16 8 22 20 7 12 12 22 N SD: 1.3271 c. M:-2.3226 c. 183: 21 10 10 18 17 8 23 21 7 13 12 23 N SD: 0.9508 c. M:-1.9816 c. 217: 25 12 11 22 20 10 27 25 8 15 15 27 N SD: 0.8872 c. M:-1.6774 c. 224: 26 12 12 22 21 10 28 26 8 16 15 28 N SD: 0.5616 c. M:-0.9862 c. 270: 31 15 14 27 25 12 34 31 10 19 18 34 N SD: 0.3856 c. M: 0.7949 c. 311: 36 17 16 31 29 14 39 36 11 22 21 39 N SD: 0.4620 c. M: 0.7574 c. 400: 46 22 21 40 37 18 50 46 15 28 27 50 N SD: 0.4490 c. M:-0.6863 c. 441: 51 24 23 44 41 20 55 51 16 31 30 55 N SD: 0.4393 c. M:-1.0630 c. 494: 57 27 26 49 46 22 62 57 18 35 33 62 N SD: 0.1728 c. M:-0.4049 c. 764: 88 42 40 76 71 34 96 88 28 54 51 96 N SD: 0.1870 c. M: 0.3528 c. 954: 110 52 50 95 89 42 120 110 35 67 64 120 N SD: 0.2095 c. M: 0.3745 c. 1012: 117 55 53 101 94 45 127 117 37 71 68 127 N SD: 0.1257 c. M:-0.2476 c. 1178: 136 64 62 117 110 52 148 136 43 83 79 148 N SD: 0.1309 c. M: 0.2356 c. 1236: 143 67 65 123 115 55 155 143 45 87 83 155 N SD: 0.1354 c. M:-0.2623 c. 1282: 148 70 67 128 119 57 161 148 47 90 86 161 N SD: 0.1159 c. M:-0.2697 c. 1376: 159 75 72 137 128 61 173 159 50 97 92 173 N SD: 0.1167 c. M: 0.2350 c. 1395: 161 76 73 139 130 62 175 161 51 98 94 175 N SD: 0.1062 c. M: 0.2238 c. 1448: 167 79 76 144 135 64 182 167 53 102 97 182 N SD: 0.1234 c. M: 0.2130 c. 1506: 174 82 79 150 140 67 189 174 55 106 101 189 N SD: 0.0800 c. M:-0.1405 c. 1889: 218 103 99 188 176 84 237 218 69 133 127 237 N SD: 0.0635 c. M: 0.0867 c. 2190: 253 119 115 218 204 97 275 253 80 154 147 275 N SD: 0.0468 c. M: 0.0851 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: 0.0000 cents 5 x x 6 x x 7 x 8 x x 9:10 S SD: 14.2565 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 0.0000 cents 7 x 8 x 9 x 10 x 11 x 12:13:14 S SD: 16.0622 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 4.8632 cents 9 x 10 x 11:12 x 14 x 15:16:17:18 S SD: 12.0899 cents 10:11 x 12 x 13:14:15:16:17 x 18:20 S SD: 10.4682 cents 11:12 x 13:14 x 15:17:18 x 19:20:22 SD: 8.1341 cents 12:13 x 14:15:16:17:18 x 20:21:22:24 S SD: 5.1682 cents 13:14 x 15:16:17:18:20:21:22:23:24:26 S SD: 6.8614 cents 14:15:16 x 18 x 19:21:23 x 25:26:28 SD: 8.6640 cents 15:16:17:18:19:20:21:23:24:25:26:28:30 SD: 7.5356 cents 16:17:18:19:20:21:22:24:26:27:28:29:32 SD: 5.0459 cents 17:18:19:20:21:23 x 26 x 28:30:31:34 SD: 6.9375 cents 18 x 20:21:23:24:25:27:29:30:32:33:36 SD: 5.1243 cents 19 x 21:22:24:25:26:29:31:32:33:35:38 SD: 5.4212 cents 20:22 x 23:25:27:28:30 x 33:35:37:40 SD: 5.7471 cents 21:23:24:25:26:28:29:32:34:35:37:39:42 SD: 5.3200 cents 22:24:25:26:28:29:30:33:36:37:39:40:44 SD: 5.0102 cents 23:25:26:27:29:31:32:35:37:38:40:42:46 SD: 4.1888 cents 24:26:27:28:30:32:33:36:39:40:42:44:48 S SD: 0.0000 cents