Scala analysis: badings1
Henk Badings, harmonic scale, Lydomixolydisch
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 9/8 203.910002 major whole tone 2: 5/4 386.313714 major third 3: 11/8 551.317942 undecimal semi-augmented fourth 4: 3/2 701.955001 perfect fifth 5: 13/8 840.527662 tridecimal neutral sixth 6: 7/4 968.825906 harmonic seventh 7: 2/1 1200.000000 octave 8: 9/4 1403.910002 major ninth 9: 5/2 1586.313714 major 10th
SHOW/INTERVAL
0: 10/9 182.4037 minor whole tone 1: 9/8 203.9100 major whole tone 2: 10/9 182.4037 minor whole tone 3: 11/10 165.0042 Ptolemy's second, 4/5-tone 4: 12/11 150.6371 undecimal neutral second, 3/4-tone 5: 13/12 138.5727 tridecimal 2/3-tone 6: 14/13 128.2982 2/3-tone 7: 8/7 231.1741 septimal whole tone 8: 9/8 203.9100 major whole tone 9: 10/9 182.4037 minor whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 14/13 128.298 cents 2/3-tone 1: 1 13/12 138.573 cents tridecimal 2/3-tone 1: 1 12/11 150.637 cents undecimal neutral second, 3/4-tone 1: 1 11/10 165.004 cents Ptolemy's second, 4/5-tone 1: 2 10/9 182.404 cents minor whole tone 1: 2 9/8 203.910 cents major whole tone 1: 1 8/7 231.174 cents septimal whole tone 2: 1 7/6 266.871 cents septimal minor third 2: 1 13/11 289.210 cents tridecimal minor third 2: 1 6/5 315.641 cents minor third 2: 1 11/9 347.408 cents undecimal neutral third 2: 1 16/13 359.472 cents tridecimal neutral third 2: 3 5/4 386.314 cents major third 2: 1 9/7 435.084 cents septimal major third, BP third 3: 1 14/11 417.508 cents undecimal diminished fourth or major third 3: 1 13/10 454.214 cents tridecimal semi-diminished fourth 3: 2 4/3 498.045 cents perfect fourth 3: 1 11/8 551.318 cents undecimal semi-augmented fourth 3: 1 18/13 563.382 cents tridecimal augmented fourth 3: 1 25/18 568.717 cents classic augmented fourth 3: 1 45/32 590.224 cents diatonic tritone 3: 1 10/7 617.488 cents Euler's tritone 4: 1 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 4: 1 13/9 636.618 cents tridecimal diminished fifth 4: 1 16/11 648.682 cents undecimal semi-diminished fifth 4: 2 3/2 701.955 cents perfect fifth 4: 1 55/36 733.722 cents undecimal semi-augmented fifth 4: 1 20/13 745.786 cents tridecimal semi-augmented fifth 4: 1 25/16 772.627 cents classic augmented fifth 4: 1 45/28 821.398 cents septimale wide minor sixth 5: 1 14/9 764.916 cents septimal minor sixth 5: 1 8/5 813.686 cents minor sixth 5: 1 13/8 840.528 cents tridecimal neutral sixth 5: 1 18/11 852.592 cents undecimal neutral sixth 5: 2 5/3 884.359 cents major sixth, BP sixth 5: 1 55/32 937.632 cents 5: 1 45/26 949.696 cents 5: 1 25/14 1003.802 cents middle minor seventh 6: 1 7/4 968.826 cents harmonic seventh 6: 1 16/9 996.090 cents Pythagorean minor seventh 6: 1 9/5 1017.596 cents just minor seventh, BP seventh 6: 1 65/36 1022.931 cents 6: 1 20/11 1034.996 cents large minor seventh 6: 2 15/8 1088.269 cents classic major seventh 6: 1 25/13 1132.100 cents 6: 1 55/28 1168.806 cents 7: 1 35/18 1151.230 cents septimal semi-diminished octave 7: 3 2/1 1200.000 cents octave 7: 1 65/32 1226.841 cents 13th-partial chroma +1 octave 7: 1 45/22 1238.906 cents 1/5-tone +1 octave 7: 1 25/12 1270.672 cents classic augmented octave 7: 1 55/26 1297.104 cents 7: 1 15/7 1319.443 cents septimal minor ninth, BP ninth 8: 1 35/16 1355.140 cents septimal neutral second +1 octave 8: 2 20/9 1382.404 cents small ninth 8: 2 9/4 1403.910 cents major ninth 8: 1 25/11 1421.309 cents undecimal acute whole tone +1 octave 8: 1 55/24 1435.677 cents undecimal semi-augmented whole tone +1 octave 8: 1 30/13 1447.741 cents tridecimal 5/4-tone +1 octave 8: 1 65/28 1458.015 cents Highest number of different intervals for one interval class: 8 Average number of different intervals per interval class: 7.50000 = 15/2
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 0.0 : 203.9 386.3 551.3 702.0 840.5 968.8 1200.0 1403.9 1586.3 203.9 : 182.4 347.4 498.0 636.6 764.9 996.1 1200.0 1382.4 1586.3 386.3 : 165.0 315.6 454.2 582.5 813.7 1017.6 1200.0 1403.9 1586.3 551.3 : 150.6 289.2 417.5 648.7 852.6 1035.0 1238.9 1421.3 1586.3 702.0 : 138.6 266.9 498.0 702.0 884.4 1088.3 1270.7 1435.7 1586.3 840.5 : 128.3 359.5 563.4 745.8 949.7 1132.1 1297.1 1447.7 1586.3 968.8 : 231.2 435.1 617.5 821.4 1003.8 1168.8 1319.4 1458.0 1586.3 1200.0: 203.9 386.3 590.2 772.6 937.6 1088.3 1226.8 1355.1 1586.3 1403.9: 182.4 386.3 568.7 733.7 884.4 1022.9 1151.2 1382.4 1586.3 1586.3
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 128.2982 .. 231.1741 cents 52/49, 102.8758 cents 14/13, 128.2982 cents 2: 266.8709 .. 435.0841 cents 54/49, 168.2132 cents 49/48, 35.6968 cents 3: 417.5080 .. 617.4878 cents 55/49, 199.9798 cents 98/99,-17.5761 cents 4: 582.5122 .. 821.3978 cents 225/196, 238.8856 cents 49/50,-34.9756 cents 5: 764.9159 .. 1003.8015 cents 225/196, 238.8856 cents 392/405,-56.4819 cents 6: 968.8259 .. 1168.8057 cents 55/49, 199.9798 cents 49/50,-34.9756 cents 7: 1151.2296 .. 1319.4428 cents 54/49, 168.2132 cents 98/99,-17.5761 cents 8: 1355.1396 .. 1458.0155 cents 52/49, 102.8758 cents 49/48, 35.6968 cents
SHOW DATA
Number of notes : 9 -- Interval properties -- Smallest interval : 14/13, 128.2982 cents, class 1 Average step (divided formal octave): 176.2571 cents Largest one step interval : 8/7, 231.1741 cents Average / Smallest step : 1.373807 Largest / Average step : 1.311573 Largest / Smallest step : 1.801849 Median interval of one step : 10/9, 182.4037 cents, amount: 2 Least squares average step : 173.20003 cents, oct.: 1558.80027 cents Scale is not proper Step pattern alph. order: ABCDEFGAB Step pattern size order : FEDCBAGFE Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 7.71152 cents Scale diversity : 0.769398 Lumma stability : 0.206762 Lumma impropriety factor : 0.101862 Rothenberg efficiency : 0.258818 redundancy: 0.741182 Efficiency x scale size : 2.329365 Number of different interval sizes : 60 = 7.50000 / class Number of one step interval sizes : 7 Highest interval variety : 8 Mean interval variety : 7.50000 = 15/2 Median interval variety : 7 Lowest interval variety : 7 Smallest interval difference : 325/324, 5.3351 cents Most common intervals : 5/4, 386.3137 cents & inv., amount: 3 Number of recognisable fifths : 2, average 701.9550 cents Number of recognisable fourths : 2, average 498.0450 cents All steps are superparticular (epimoric) Length of longest harmonic part : 6 at degree 0 Length of longest subharmonic part : 2 at degree 8 Formal octave complements present : 3 = 33.3333% 2/1 octave complements present : 1 = 11.1111% Scale is differentially coherent in interval class 1 Scale is n-1 differentially coherent in interval classes 6 and 7 combined -- Rational properties -- Prime limit : 13 Odd number limit : 65 (O: 65 U: 13) Highest odd numerator or denominator: 13 Scale harmonicity : 0.015071 Average absolute harmonicity : 0.216848 Specific harmonicity : 0.137616 Fundamental : 1/8, -3.0000 octaves, 32.7032 Hz. Guide tone : 90090, 16.4591 octaves, 23569847.178 Hz. Exponens Consonantiae : 7.207200E+05, 19.45908 octaves Euler's gradus suavitatis : 41 Sum of Mann's harmonic distance : 52.5, average 5.83333 Mersenne's string divisions : 14549535 Sum of van Prooijen's expressibility: 6.78398, average 0.75378 Sum of Tenney's harmonic distance : 12.20252, average 1.35584 Vogel's harmonic complexity : 7.44444 Wille's k value : 6 Wilson's harmonic complexity : 42 Rectangular lattice diameter : 6 Triangular lattice diameter : 2 Lattice compactness : 104.18664, average 2.31526 Lattice compactness (without 2's) : 64.15967, average 1.42577 Number of different primes : 6 Prime exponents' range, average, count, tones@limit: 2: -3 .. 1 -1.77778 18 1 3: 0 .. 2 0.55556 5 3 5: 0 .. 1 0.22222 2 2 7: 0 .. 1 0.11111 1 1 11: 0 .. 1 0.11111 1 1 13: 0 .. 1 0.11111 1 1 Average exponent except 2's : 10 / 9 = 1.11111 Average absolute exponent except 2's: 10 / 9 = 1.11111 Scale is JI-epimorphic with val: <7 11 16 20 24 26|
FIT/MODE
9: 1 1 1 1 1 0 2 1 1 N T3 SD: 37.9028 c. M: 87.5405 c. 11: 1 2 1 1 1 1 1 2 1 N D3 I SD: 37.4068 c. M: 59.6997 c. 13: 2 1 2 1 1 1 2 2 1 N D3 SD: 34.4573 c. M:-60.3796 c. 14: 2 1 2 1 1 2 2 1 2 P D3 SD: 37.0200 c. M:-50.9472 c. 15: 2 2 1 2 1 1 2 2 2 N SD: 25.5969 c. M:-38.3247 c. 16: 2 2 2 1 1 2 2 2 2 P D3 S SD: 24.0601 c. M: 47.3708 c. 17: 2 2 2 1 2 1 3 2 2 N SD: 22.0715 c. M: 48.7670 c. 20: 3 2 2 2 2 1 3 3 2 N SD: 19.4410 c. M:-34.0371 c. 21: 3 2 2 2 2 2 3 3 2 N SD: 17.9765 c. M:-31.3262 c. 23: 3 2 3 2 2 2 3 3 3 N SD: 19.4717 c. M: 41.4629 c. 24: 3 3 2 3 2 2 3 3 3 P SD: 16.6521 c. M:-25.1055 c. 25: 3 3 3 2 2 2 4 3 3 N SD: 11.7885 c. M:-19.7550 c. 32: 4 4 3 3 3 2 5 4 4 N SD: 12.1349 c. M: 26.9521 c. 34: 4 4 4 3 3 3 5 4 4 P SD: 9.7532 c. M: 17.2849 c. 36: 5 4 4 3 3 3 5 5 4 N SD: 10.5492 c. M:-21.5176 c. 37: 5 4 4 3 4 3 5 5 4 N SD: 11.0830 c. M:-17.2610 c. 38: 5 4 4 4 3 3 6 5 4 N SD: 8.9891 c. M:-15.4233 c. 41: 5 5 4 4 4 3 6 5 5 N SD: 7.2415 c. M: 11.0492 c. 45: 6 5 5 4 4 4 6 6 5 N SD: 8.3842 c. M:-18.2137 c. 46: 6 5 5 4 4 4 7 6 5 N SD: 7.6945 c. M: 12.8857 c. 49: 6 6 5 5 4 4 7 6 6 N SD: 6.2938 c. M: 11.8388 c. 54: 7 6 6 5 5 4 8 7 6 N SD: 5.4350 c. M:-11.3816 c. 61: 8 7 6 6 5 5 9 8 7 N SD: 4.5543 c. M: 8.3631 c. 70: 9 8 7 7 6 6 10 9 8 N SD: 3.2485 c. M: 7.4390 c. 77: 10 9 8 7 7 6 11 10 9 N SD: 3.4910 c. M: 5.1143 c. 95: 12 11 10 9 8 8 14 12 11 N SD: 2.5070 c. M: 5.6257 c. 110: 14 13 11 11 9 9 16 14 13 N SD: 3.4159 c. M: 5.0697 c. 111: 14 13 12 10 10 9 16 14 13 N SD: 3.0122 c. M:-6.0355 c. 115: 15 13 12 11 10 9 17 15 13 N SD: 1.8981 c. M: 3.2436 c. 147: 19 17 15 14 13 12 21 19 17 N SD: 1.4946 c. M:-2.3866 c. 172: 22 20 18 16 15 14 25 22 20 N SD: 1.2655 c. M: 2.0514 c. 210: 27 24 22 20 18 17 31 27 24 N SD: 1.1425 c. M: 2.0476 c. 242: 31 28 25 23 21 20 35 31 28 N SD: 0.8763 c. M: 1.4857 c. 287: 37 33 30 27 25 23 42 37 33 N SD: 0.7903 c. M: 1.5614 c. 296: 38 34 31 28 26 24 43 38 34 N SD: 0.5894 c. M:-1.1835 c. 357: 46 41 37 34 31 29 52 46 41 N SD: 0.3473 c. M: 0.7145 c. 506: 65 58 53 48 44 41 74 65 58 N SD: 0.4421 c. M:-0.7078 c. 583: 75 67 61 55 51 47 85 75 67 N SD: 0.3660 c. M:-1.0349 c. 653: 84 75 68 62 57 53 95 84 75 N SD: 0.1840 c. M:-0.4531 c. 904: 116 104 94 86 79 73 132 116 104 N SD: 0.2150 c. M: 0.3565 c. 981: 126 113 102 93 86 79 143 126 113 N SD: 0.2036 c. M:-0.3318 c. 1010: 130 116 105 96 88 82 147 130 116 N SD: 0.1692 c. M:-0.2690 c. 1166: 150 134 121 111 102 94 170 150 134 N SD: 0.1635 c. M: 0.3256 c. 1261: 162 145 131 120 110 102 184 162 145 N SD: 0.1545 c. M: 0.3224 c. 1338: 172 154 139 127 117 108 195 172 154 N SD: 0.1305 c. M: 0.2024 c. 1487: 191 171 155 141 130 120 217 191 171 N SD: 0.1288 c. M:-0.2114 c. 1557: 200 179 162 148 136 126 227 200 179 N SD: 0.1104 c. M:-0.1783 c. 1634: 210 188 170 155 143 132 238 210 188 N SD: 0.0942 c. M:-0.1992 c. 1819: 234 209 189 173 159 147 265 234 209 N SD: 0.1074 c. M: 0.1633 c. 1844: 237 212 192 175 161 149 269 237 212 N SD: 0.0778 c. M: 0.1766 c. 1991: 256 229 207 189 174 161 290 256 229 N SD: 0.0587 c. M: 0.1063 c.
FIT/HARMONIC
1 x x x x x x 2 x 3 S SD: 157.8206 cents 2 x x x 3 x x 4 x 5 S SD: 0.0000 cents 3 x x 4 x 5 x 6:7:8 S SD: 29.1253 cents 4 x 5 x 6 x 7:8:9:10 S SD: 0.0000 cents 5 x 6:7 x 8:9:10:11:13 SD: 17.6073 cents 6:7 x 8:9:10:11:12:14:15 S SD: 17.3086 cents 7:8:9:10 x 11:12:14:16:18 S SD: 15.4438 cents 8:9:10:11:12:13:14:16:18:20 S SD: 0.0000 cents