Scala analysis: badings2
Henk Badings, subharmonic scale, Dorophrygisch
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 10/9 182.403712 minor whole tone 2: 5/4 386.313714 major third 3: 10/7 617.487807 Euler's tritone 4: 20/13 745.786052 tridecimal semi-augmented fifth 5: 5/3 884.358713 major sixth, BP sixth 6: 20/11 1034.995772 large minor seventh 7: 2/1 1200.000000 octave 8: 20/9 1382.403712 small ninth 9: 5/2 1586.313714 major 10th
SHOW/INTERVAL
0: 9/8 203.9100 major whole tone 1: 10/9 182.4037 minor whole tone 2: 9/8 203.9100 major whole tone 3: 8/7 231.1741 septimal whole tone 4: 14/13 128.2982 2/3-tone 5: 13/12 138.5727 tridecimal 2/3-tone 6: 12/11 150.6371 undecimal neutral second, 3/4-tone 7: 11/10 165.0042 Ptolemy's second, 4/5-tone 8: 10/9 182.4037 minor whole tone 9: 9/8 203.9100 major 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 : 182.4 386.3 617.5 745.8 884.4 1035.0 1200.0 1382.4 1586.3 182.4 : 203.9 435.1 563.4 702.0 852.6 1017.6 1200.0 1403.9 1586.3 386.3 : 231.2 359.5 498.0 648.7 813.7 996.1 1200.0 1382.4 1586.3 617.5 : 128.3 266.9 417.5 582.5 764.9 968.8 1151.2 1355.1 1586.3 745.8 : 138.6 289.2 454.2 636.6 840.5 1022.9 1226.8 1458.0 1586.3 884.4 : 150.6 315.6 498.0 702.0 884.4 1088.3 1319.4 1447.7 1586.3 1035.0: 165.0 347.4 551.3 733.7 937.6 1168.8 1297.1 1435.7 1586.3 1200.0: 182.4 386.3 568.7 772.6 1003.8 1132.1 1270.7 1421.3 1586.3 1382.4: 203.9 386.3 590.2 821.4 949.7 1088.3 1238.9 1403.9 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 : 175.99460 cents, oct.: 1583.95137 cents Scale is not proper Step pattern alph. order: ABCDEFGAB Step pattern size order : EFGABCDEF 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 Scale contains two identical disjunct "tetrachords" of 3 notes All steps are superparticular (epimoric) Length of longest harmonic part : 2 at degree 8 Length of longest subharmonic part : 6 at degree 3 Formal octave complements present : 3 = 33.3333% 2/1 octave complements present : 1 = 11.1111% Scale is differentially coherent in interval class 1 -- 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 : 0.182524 Specific harmonicity : 0.137616 Fundamental : 1/36036, -15.1372 octaves, 0.0073 Hz. Guide tone : 20, 4.3219 octaves, 5232.511 Hz. Exponens Consonantiae : 7.207200E+05, 19.45908 octaves Euler's gradus suavitatis : 41 Sum of Mann's harmonic distance : 78.0, average 8.66667 Mersenne's string divisions : 1609850088 Sum of van Prooijen's expressibility: 7.00583, average 0.77843 Sum of Tenney's harmonic distance : 14.59016, average 1.62113 Vogel's harmonic complexity : 39.00000 Wille's k value : 22522 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: -2 .. 2 0.66667 12 1 3: -2 .. 0 -0.55556 5 5: 0 .. 1 0.88889 8 5 7: -1 .. 0 -0.11111 1 1 11: -1 .. 0 -0.11111 1 1 13: -1 .. 0 -0.11111 1 1 Average exponent except 2's : 0 / 9 = 0.00000 Average absolute exponent except 2's: 16 / 9 = 1.77778 Scale is JI-epimorphic with val: <7 11 16 20 24 26| Scale is weakly epimorphic with val: <7 11 17 20 23 26|
FIT/MODE
9: 1 1 2 0 1 1 1 1 1 N T3 SD: 37.9028 c. M:-87.5405 c. 12: 1 2 2 1 1 1 1 1 2 N SD: 37.5824 c. M: 60.4756 c. 13: 1 2 2 1 1 1 2 1 2 N D3 SD: 34.4573 c. M: 60.3796 c. 15: 2 2 2 1 1 2 1 2 2 N SD: 25.5969 c. M: 38.3247 c. 16: 2 2 2 2 1 1 2 2 2 P D3 S SD: 24.0601 c. M:-47.3708 c. 17: 2 2 3 1 1 2 2 2 2 N SD: 21.0567 c. M: 44.5456 c. 19: 2 3 2 2 2 1 2 3 2 N SD: 27.7286 c. M:-36.9296 c. 20: 2 3 3 1 2 2 2 2 3 N SD: 19.4410 c. M: 34.0371 c. 21: 2 3 3 2 2 2 2 2 3 N SD: 17.9765 c. M: 31.3262 c. 23: 3 3 3 2 2 2 2 3 3 N S SD: 16.4984 c. M:-27.5073 c. 24: 3 3 3 2 2 3 2 3 3 P SD: 16.6521 c. M: 25.1055 c. 25: 3 3 4 2 2 2 3 3 3 N SD: 11.7885 c. M: 19.7550 c. 32: 4 4 4 3 3 3 3 4 4 P S SD: 11.1083 c. M: 22.6202 c. 34: 4 4 5 3 3 3 4 4 4 P SD: 9.7532 c. M:-17.2849 c. 36: 4 5 5 3 3 3 4 4 5 N SD: 10.5492 c. M: 21.5176 c. 38: 4 5 6 3 3 4 4 4 5 N SD: 8.9891 c. M: 15.4233 c. 41: 5 5 6 3 4 4 4 5 5 N SD: 7.2415 c. M:-11.0492 c. 45: 5 6 6 4 4 4 5 5 6 N SD: 8.3842 c. M: 18.2137 c. 46: 5 6 7 4 4 4 5 5 6 N SD: 7.6945 c. M:-12.8857 c. 49: 6 6 7 4 4 5 5 6 6 N SD: 6.2938 c. M:-11.8388 c. 54: 6 7 8 4 5 5 6 6 7 N SD: 5.4350 c. M: 11.3816 c. 61: 7 8 9 5 5 6 6 7 8 N SD: 4.5543 c. M:-8.3631 c. 70: 8 9 10 6 6 7 7 8 9 N SD: 3.2485 c. M:-7.4390 c. 77: 9 10 11 6 7 7 8 9 10 N SD: 3.4910 c. M: 5.1143 c. 95: 11 12 14 8 8 9 10 11 12 N SD: 2.5070 c. M:-5.6257 c. 110: 13 14 16 9 9 11 11 13 14 N SD: 3.4159 c. M:-5.0697 c. 111: 13 14 16 9 10 10 12 13 14 N SD: 3.0122 c. M: 6.0355 c. 115: 13 15 17 9 10 11 12 13 15 N SD: 1.8981 c. M:-3.2436 c. 147: 17 19 21 12 13 14 15 17 19 N SD: 1.4946 c. M: 2.3866 c. 172: 20 22 25 14 15 16 18 20 22 N SD: 1.2655 c. M:-2.0514 c. 210: 24 27 31 17 18 20 22 24 27 N SD: 1.1425 c. M:-2.0476 c. 242: 28 31 35 20 21 23 25 28 31 N SD: 0.8763 c. M:-1.4857 c. 287: 33 37 42 23 25 27 30 33 37 N SD: 0.7903 c. M:-1.5614 c. 296: 34 38 43 24 26 28 31 34 38 N SD: 0.5894 c. M: 1.1835 c. 357: 41 46 52 29 31 34 37 41 46 N SD: 0.3473 c. M:-0.7145 c. 506: 58 65 74 41 44 48 53 58 65 N SD: 0.4421 c. M:-0.7078 c. 583: 67 75 85 47 51 55 61 67 75 N SD: 0.3660 c. M: 1.0349 c. 653: 75 84 95 53 57 62 68 75 84 N SD: 0.1840 c. M: 0.4531 c. 904: 104 116 132 73 79 86 94 104 116 N SD: 0.2150 c. M:-0.3565 c. 981: 113 126 143 79 86 93 102 113 126 N SD: 0.2036 c. M: 0.3318 c. 1010: 116 130 147 82 88 96 105 116 130 N SD: 0.1692 c. M: 0.2690 c. 1166: 134 150 170 94 102 111 121 134 150 N SD: 0.1635 c. M:-0.3256 c. 1261: 145 162 184 102 110 120 131 145 162 N SD: 0.1545 c. M:-0.3224 c. 1338: 154 172 195 108 117 127 139 154 172 N SD: 0.1305 c. M:-0.2024 c. 1487: 171 191 217 120 130 141 155 171 191 N SD: 0.1288 c. M: 0.2114 c. 1557: 179 200 227 126 136 148 162 179 200 N SD: 0.1104 c. M:-0.1783 c. 1634: 188 210 238 132 143 155 170 188 210 N SD: 0.0942 c. M: 0.1992 c. 1819: 209 234 265 147 159 173 189 209 234 N SD: 0.1074 c. M:-0.1633 c. 1844: 212 237 269 149 161 175 192 212 237 N SD: 0.0778 c. M:-0.1766 c. 1991: 229 256 290 161 174 189 207 229 256 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: 14.6104 cents 3 x 4 x x 5 x 6:7:8 S SD: 35.8343 cents 4 x 5 x 6 x 7:8:9:10 S SD: 13.7054 cents 5 x 6:7:8 x 9:10:11:13 SD: 18.0951 cents 6:7:8 x 9:10:11:12:13:15 SD: 19.2302 cents 7:8:9:10:11:12:13:14:16:18 S SD: 13.4198 cents 8:9:10:11:12:13:15:16:18:20 SD: 12.1634 cents 9:10:11:13:14:15:16:18:20:23 SD: 8.0178 cents 12:13:15:17:18:20:22:24:27:30 SD: 7.6345 cents 15:17:19:21:23:25:27:30:33:38 SD: 7.1052 cents 16:18:20:23:25:27:29:32:36:40 SD: 5.2748 cents 17:19:21:24:26:28:31:34:38:43 SD: 4.9761 cents 19:21:24:27:29:32:35:38:42:48 SD: 4.8898 cents 21:23:26:30:32:35:38:42:47:53 SD: 4.5254 cents 24:27:30:34:37:40:44:48:53:60 SD: 3.5305 cents 28:31:35:40:43:47:51:56:62:70 SD: 1.7486 cents 44:49:55:63:68:73:80:88:98:110 SD: 1.4495 cents 56:62:70:80:86:93:102:112:124:140 SD: 1.2872 cents 63:70:79:90:97:105:115:126:140:158 SD: 1.1607 cents 72:80:90:103:111:120:131:144:160:180 SD: 0.4994 cents 152:169:190:217:234:253:276:304:338:380 SD: 0.4386 cents 171:190:214:244:263:285:311:342:380:428 SD: 0.3976 cents 180:200:225:257:277:300:327:360:400:450 SD: 0.2000 cents 252:280:315:360:388:420:458:504:560:630 SD: 0.1706 cents 396:440:495:566:609:660:720:792:880:990 SD: 0.1214 cents 548:609:685:783:843:913:996:1096:1218:1370 SD: 0.1177 cents 576:640:720:823:886:960:1047:1152:1280:1440 SD: 0.0689 cents