Scala analysis: 455_D15
Diatonic tetrachord 16/15 * 9/8 * 10/9, Ptolemy, Didymos
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 16/15 111.731285 minor diatonic semitone 2: 6/5 315.641287 minor third 3: 4/3 498.044999 perfect fourth
SHOW/INTERVAL
0: 10/9 182.4037 minor whole tone 1: 16/15 111.7313 minor diatonic semitone 2: 9/8 203.9100 major whole tone 3: 10/9 182.4037 minor whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 16/15 111.731 cents minor diatonic semitone 1: 1 10/9 182.404 cents minor whole tone 1: 1 9/8 203.910 cents major whole tone 2: 1 32/27 294.135 cents Pythagorean minor third 2: 1 6/5 315.641 cents minor third 2: 1 5/4 386.314 cents major third 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 0.0 : 111.7 315.6 498.0 111.7: 203.9 386.3 498.0 315.6: 182.4 294.1 498.0 498.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 111.7313 .. 203.9100 cents 135/128, 92.1787 cents 16/15, 111.7313 cents 2: 294.1350 .. 386.3137 cents 135/128, 92.1787 cents 256/243, 90.2250 cents
SHOW DATA
Number of notes : 3 -- Interval properties -- Smallest interval : 16/15, 111.7313 cents, class 1 Average step (divided formal octave): 166.0150 cents Largest one step interval : 9/8, 203.9100 cents Average / Smallest step : 1.485842 Largest / Average step : 1.228263 Largest / Smallest step : 1.825004 Median interval of one step : 10/9, 182.4037 cents, amount: 1 Least squares average step : 159.79635 cents, oct.: 479.38904 cents Scale is strictly proper Scale has trivalence property Scale is a mode of a 1783-tone equal temperament with octave 4/3 degrees: 400 1130 1783 Least number of segments generator : 892 of 249.162 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 1425 of 398.045 cents and inv. generated superset size: 44 = 41 more = 1466.667% Number of contiguous 1-step segments: 0 Step pattern alph. order: ABC Step pattern size order : SLM Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 90.22500 cents Scale diversity : 1.236030 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.629838 Rothenberg efficiency : 0.666667 redundancy: 0.333333 Efficiency x scale size : 2.000000 Number of different interval sizes : 6 = 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 : 81/80, 21.5063 cents Number of recognisable fifths : 0 Number of recognisable fourths : 0 Best fifths form a closed circle Best major thirds form a closed circle Best minor thirds form a closed circle 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 : 1 = 33.3333% 2/1 octave complements present : 0 = 0.0000% Scale is differentially coherent in interval class 1 Scale is n-1 differentially coherent in interval class 2 Limited inverse transpositions with margin 92.1787 cents: 1 -- Rational properties -- Prime limit : 5 Odd number limit : 27 (O: 9 U: 27) Highest odd numerator or denominator: 15 Scale harmonicity : 0.063559 Average absolute harmonicity : 0.130051 Specific harmonicity : 0.279394 Fundamental : 1/15, -3.9069 octaves, 17.4417 Hz. Guide tone : 48, 5.5850 octaves, 12558.027 Hz. Exponens Consonantiae : 7.200000E+02, 9.49185 octaves Euler's gradus suavitatis : 13 Sum of Mann's harmonic distance : 24.5, average 8.16667 Mersenne's string divisions : 2387 Sum of van Prooijen's expressibility: 2.35218, average 0.78406 Sum of Tenney's harmonic distance : 4.93651, average 1.64550 Vogel's harmonic complexity : 6.00000 Wille's k value : 4 Wilson's harmonic complexity : 11 Rectangular lattice diameter : 3 Triangular lattice diameter : 2 Lattice compactness : 16.50187, average 2.75031 Lattice compactness (without 2's) : 9.06450, average 1.51075 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: 1 .. 4 2.33333 7 3: -1 .. 1 -0.33333 3 1 5: -1 .. 0 -0.66667 2 2 Average exponent except 2's :-3 / 3 =-1.00000 Average absolute exponent except 2's: 5 / 3 = 1.66667 Scale is weakly epimorphic with val: <7 10 15| Scale is JI-epimorphic with val: <7 11 16| Scale is weakly epimorphic with val: <7 11 19| Scale is weakly epimorphic with val: <8 11 18|
FIT/MODE
3: 1 1 1 SP B ME I SD: 32.7379 c. M:-54.2837 c. 5: 1 2 2 SP M ME S SD: 11.9676 c. M: 16.8143 c. 8: 2 3 3 SP M ME S SD: 7.7967 c. M:-12.7800 c. 13: 3 5 5 SP M DE S SD: 5.5980 c. M: 9.1521 c. 14: 3 6 5 SP T3 SD: 3.8987 c. M: 5.0074 c. 22: 5 9 8 SP T3 SD: 1.1276 c. M:-1.4608 c. 49: 11 20 18 SP T3 SD: 0.3214 c. M: 0.5516 c. 71: 16 29 26 SP T3 SD: 0.2914 c. M:-0.5042 c. 120: 27 49 44 SP T3 SD: 0.2261 c. M:-0.3288 c. 134: 30 55 49 SP T3 SD: 0.2100 c. M:-0.2828 c. 156: 35 64 57 SP T3 SD: 0.2459 c. M:-0.4257 c. 161: 36 66 59 SP T3 SD: 0.2212 c. M: 0.3672 c. 183: 41 75 67 SP T3 SD: 0.0918 c. M: 0.1474 c. 232: 52 95 85 SP T3 SD: 0.0706 c. M: 0.1005 c. 254: 57 104 93 SP T3 SD: 0.0345 c. M:-0.0487 c. 303: 68 124 111 SP T3 SD: 0.0367 c. M: 0.0484 c. 486: 109 199 178 SP T3 SD: 0.0178 c. M: 0.0298 c. 740: 166 303 271 SP T3 SD: 0.0080 c. M:-0.0116 c. 1043: 234 427 382 SP T3 SD: 0.0051 c. M:-0.0065 c. 1529: 343 626 560 SP T3 SD: 0.0047 c. M: 0.0065 c. 1783: 400 730 653 SP T3 SD: 0.0009 c. M:-0.0014 c.
FIT/HARMONIC
1 x x 2 S SD: 701.9550 cents 2 x x 3 S SD: 203.9100 cents 3 x x 4 S SD: 0.0000 cents 4 x 5 x S SD: 70.6724 cents 5 x 6:7 S SD: 42.2336 cents 6 x 7:8 S SD: 24.3852 cents 7 x 8:9 S SD: 52.6754 cents 8:9:10:11 H SD: 42.5957 cents 9:10:11:12 H SD: 25.8279 cents 10:11:12:13 H SD: 22.9956 cents 11:12:13:15 SD: 20.3468 cents 12:13:14:16 S SD: 18.5562 cents 14:15:17:19 SD: 12.5529 cents 15:16:18:20 S SD: 0.0000 cents