Scala analysis: hirajoshi2
Japanese pentatonic koto scale, theoretical. Helmholz/Ellis p.519, nr.110
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: 6/5 315.641287 minor third 3: 3/2 701.955001 perfect fifth 4: 8/5 813.686286 minor sixth 5: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 5/4 386.3137 major third 1: 9/8 203.9100 major whole tone 2: 16/15 111.7313 minor diatonic semitone 3: 5/4 386.3137 major third 4: 16/15 111.7313 minor diatonic semitone 5: 5/4 386.3137 major third
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 16/15 111.731 cents minor diatonic semitone 1: 1 9/8 203.910 cents major whole tone 1: 2 5/4 386.314 cents major third 2: 1 6/5 315.641 cents minor third 2: 3 4/3 498.045 cents perfect fourth 2: 1 45/32 590.224 cents diatonic tritone 3: 1 64/45 609.776 cents 2nd tritone 3: 3 3/2 701.955 cents perfect fifth 3: 1 5/3 884.359 cents major sixth, BP sixth 4: 2 8/5 813.686 cents minor sixth 4: 1 16/9 996.090 cents Pythagorean minor seventh 4: 2 15/8 1088.269 cents classic major seventh Highest number of different intervals for one interval class: 3 Average number of different intervals per interval class: 3.00000 = 3
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 0.0 : 203.9 315.6 702.0 813.7 1200.0 203.9 : 111.7 498.0 609.8 996.1 1200.0 315.6 : 386.3 498.0 884.4 1088.3 1200.0 702.0 : 111.7 498.0 702.0 813.7 1200.0 813.7 : 386.3 590.2 702.0 1088.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 111.7313 .. 386.3137 cents 75/64, 274.5824 cents 16/15, 111.7313 cents 2: 315.6413 .. 590.2237 cents 75/64, 274.5824 cents 24/25,-70.6724 cents 3: 609.7763 .. 884.3587 cents 75/64, 274.5824 cents 2048/2025, 19.5526 cents 4: 813.6863 .. 1088.2687 cents 75/64, 274.5824 cents 24/25,-70.6724 cents
SHOW DATA
Number of notes : 5 -- Interval properties -- Smallest interval : 16/15, 111.7313 cents, class 1 Average step (divided formal octave): 240.0000 cents Largest one step interval : 5/4, 386.3137 cents Average / Smallest step : 2.148011 Largest / Average step : 1.609640 Largest / Smallest step : 3.457525 Median interval of one step : 9/8, 203.9100 cents, amount: 1 Least squares average step : 221.74187 cents, oct.: 1108.70934 cents Scale is not proper Scale has trivalence property Step pattern alph. order: ABCBC Step pattern size order : MSLSL Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 19.55257 cents Scale diversity : 1.547790 Lumma stability : 0.202513 Lumma impropriety factor : 0.117787 Rothenberg efficiency : 0.580000 redundancy: 0.420000 Efficiency x scale size : 2.900000 Number of different interval sizes : 12 = 3.00000 / class Number of one step interval sizes : 3 Highest interval variety : 3 Mean interval variety : 3.00000 = 3 Median interval variety : 3 Lowest interval variety : 3 Smallest interval difference : 2048/2025, 19.5526 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 3 Scale is a chain of 2 triads 0.0 111.731 498.045 cents Number of recognisable fifths : 3, average 701.9550 cents Best fifths form a closed circle All steps are superparticular (epimoric) Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 1 = 20.0000% Scale is differentially coherent in interval class 1 Scale is n-1 differentially coherent in interval class 2 Scale is n-1 differentially coherent in interval class 3 -- Rational properties -- Prime limit : 5 Odd number limit : 45 (O: 45 U: 45) Highest odd numerator or denominator: 9 Scale harmonicity : 0.056391 Average absolute harmonicity : 0.319689 Specific harmonicity : 0.286776 Fundamental : 1/40, -5.3219 octaves, 6.5406 Hz. Guide tone : 72, 6.1699 octaves, 18837.041 Hz. Exponens Consonantiae : 2.880000E+03, 11.49185 octaves Euler's gradus suavitatis : 15 Sum of Mann's harmonic distance : 24.5, average 4.90000 Mersenne's string divisions : 36465 Sum of van Prooijen's expressibility: 2.82930, average 0.56586 Sum of Tenney's harmonic distance : 6.01569, average 1.20314 Vogel's harmonic complexity : 8.60000 Wille's k value : 22 Wilson's harmonic complexity : 11 Rectangular lattice diameter : 3 Triangular lattice diameter : 3 Lattice compactness : 42.86919, average 2.85795 Lattice compactness (without 2's) : 18.89292, average 1.25953 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -3 .. 3 0.20000 9 1 3: 0 .. 2 0.80000 4 2 5: -1 .. 0 -0.40000 2 2 Average exponent except 2's : 2 / 5 = 0.40000 Average absolute exponent except 2's: 6 / 5 = 1.20000 Scale is JI-epimorphic with val: <5 8 11| Scale is weakly epimorphic with val: <5 9 13|
FIT/MODE
5: 1 0 2 0 2 N T3 SD: 56.7923 c. M: 93.6863 c. 7: 1 1 2 1 2 P M ME S SD: 28.0998 c. M:-43.4566 c. Thai mode 1 10: 2 1 3 1 3 P T3 SD: 29.2855 c. M:-44.3587 c. 12: 2 1 4 1 4 N T3 SD: 9.4982 c. M: 15.6413 c. Aeolian Pentatonic, Hira-joshi, Kata-kumoi, Yona Nuki Minor: Japan, Tizita Minor (Half tizita): Ethiopia 19: 3 2 6 2 6 N T3 SD: 7.9347 c. M: 14.4363 c. 22: 4 2 7 2 7 N T3 SD: 9.0566 c. M:-14.2718 c. 31: 5 3 10 3 10 N T3 SD: 5.8375 c. M: 10.3616 c. 34: 6 3 11 3 11 N T3 SD: 4.1191 c. M:-7.8547 c. 41: 7 4 13 4 13 N T3 SD: 3.8711 c. M:-6.3099 c. 46: 8 4 15 4 15 N T3 SD: 3.4723 c. M: 4.9906 c. 53: 9 5 17 5 17 N T3 SD: 0.8719 c. M:-1.4081 c. 118: 20 11 38 11 38 N T3 SD: 0.3175 c. M: 0.5202 c. 171: 29 16 55 16 55 N T3 SD: 0.2626 c. M: 0.4012 c. 289: 49 27 93 27 93 N T3 SD: 0.2374 c. M: 0.4498 c. 323: 55 30 104 30 104 N T3 SD: 0.2241 c. M:-0.4244 c. 376: 64 35 121 35 121 N T3 SD: 0.2322 c. M:-0.3453 c. 388: 66 36 125 36 125 N T3 SD: 0.1840 c. M: 0.2842 c. 441: 75 41 142 41 142 N T3 SD: 0.0931 c. M:-0.1716 c. 494: 84 46 159 46 159 N T3 SD: 0.1022 c. M:-0.1482 c. 559: 95 52 180 52 180 N T3 SD: 0.0549 c. M: 0.0906 c. 612: 104 57 197 57 197 N T3 SD: 0.0273 c. M:-0.0450 c. 1171: 199 109 377 109 377 N T3 SD: 0.0150 c. M: 0.0228 c. 1783: 303 166 574 166 574 N T3 SD: 0.0085 c. M:-0.0160 c.
FIT/HARMONIC
1 x x x x 2 S SD: 0.0000 cents 2 x x 3 x 4 S SD: 0.0000 cents 3 x 4 x 5:6 S SD: 65.2054 cents 4 x 5:6 x 8 S SD: 23.5575 cents 5 x 6 x 8:10 S SD: 0.0000 cents 6 x 7:9:10:12 SD: 21.4668 cents 7:8 x x 11:14 SD: 13.8099 cents 9:10:11 x 14:18 SD: 15.5125 cents 10:11:12:15:16:20 S SD: 7.7812 cents 12 x 14:18:19:24 SD: 13.0077 cents 14:16:17:21:22:28 SD: 9.2438 cents 15:17:18:23:24:30 SD: 8.0277 cents 16:18:19:24:26:32 SD: 6.4779 cents 24:27:29:36:38:48 SD: 4.3459 cents 30:34:36:45:48:60 SD: 2.5553 cents 40:45:48:60:64:80 S SD: 0.0000 cents