Scala analysis: 419_D9
Diatonic tetrachord 27/25 * 25/23 * 92/81
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 27/25 133.237575 large limma, BP small semitone, Zarlino semitone 2: 27/23 277.590655 vicesimotertial minor third 3: 4/3 498.044999 perfect fourth
SHOW/INTERVAL
0: 92/81 220.4543 1: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 2: 25/23 144.3531 3: 92/81 220.4543
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 27/25 133.238 cents large limma, BP small semitone, Zarlino semitone 1: 1 25/23 144.353 cents 1: 1 92/81 220.454 cents 2: 1 27/23 277.591 cents vicesimotertial minor third 2: 1 92/75 353.692 cents 2: 1 100/81 364.807 cents grave 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 : 133.2 277.6 498.0 133.2: 144.4 364.8 498.0 277.6: 220.5 353.7 498.0 498.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 133.2376 .. 220.4543 cents 2300/2187, 87.2168 cents 27/25, 133.2376 cents 2: 277.5907 .. 364.8074 cents 2300/2187, 87.2168 cents 2187/2116, 57.1363 cents
SHOW DATA
Number of notes : 3 -- Interval properties -- Smallest interval : 27/25, 133.2376 cents, class 1 Average step (divided formal octave): 166.0150 cents Largest one step interval : 92/81, 220.4543 cents Average / Smallest step : 1.246007 Largest / Average step : 1.327918 Largest / Smallest step : 1.654596 Median interval of one step : 25/23, 144.3531 cents, amount: 1 Least squares average step : 155.89671 cents, oct.: 467.69012 cents Scale is strictly proper Scale has trivalence property Scale is a mode of a 1656-tone equal temperament with octave 4/3 degrees: 443 923 1656 Least number of segments generator : 829 of 249.323 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 1429 of 429.774 cents and inv. generated superset size: 72 = 69 more = 2400.000% Number of contiguous 1-step segments: 0 Step pattern alph. order: ABC Step pattern size order : SML Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 57.13631 cents Scale diversity : 1.325221 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.649763 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 : 625/621, 11.1155 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 Steps are strictly increasing Formal octave complements present : 1 = 33.3333% 2/1 octave complements present : 0 = 0.0000% Limited inverse transpositions with margin 76.1013 cents: 2 -- Rational properties -- Prime limit : 23 Odd number limit : 81 (O: 27 U: 81) Highest odd numerator or denominator: 27 Scale harmonicity : 0.014803 Average absolute harmonicity : 0.094109 Specific harmonicity : 0.075951 Fundamental : 1/1725, -10.7524 octaves, 0.1517 Hz. Guide tone : 108, 6.7549 octaves, 28255.561 Hz. Exponens Consonantiae : 1.863000E+05, 17.50727 octaves Euler's gradus suavitatis : 41 Sum of Mann's harmonic distance : 54.5, average 18.16667 Mersenne's string divisions : 9100 Sum of van Prooijen's expressibility: 3.33985, average 1.11328 Sum of Tenney's harmonic distance : 6.70158, average 2.23386 Vogel's harmonic complexity : 30.66667 Wille's k value : 1012 Wilson's harmonic complexity : 45 Rectangular lattice diameter : 7 Triangular lattice diameter : 4 Lattice compactness : 20.72152, average 3.45359 Lattice compactness (without 2's) : 18.59914, average 3.09986 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: 0 .. 2 0.66667 2 3: -1 .. 3 1.66667 7 1 5: -2 .. 0 -0.66667 2 1 23: -1 .. 0 -0.33333 1 1 Average exponent except 2's : 2 / 3 = 0.66667 Average absolute exponent except 2's: 10 / 3 = 3.33333 Scale is weakly epimorphic with val: <6 11 15 31| Scale is weakly epimorphic with val: <6 11 15 34| Scale is weakly epimorphic with val: <6 11 17 30| Scale is JI-epimorphic with val: <7 11 16 31| Scale is weakly epimorphic with val: <7 11 16 34| Scale is weakly epimorphic with val: <7 11 17 32| Scale is weakly epimorphic with val: <7 11 17 35| Scale is weakly epimorphic with val: <7 11 19 31| Scale is weakly epimorphic with val: <7 11 19 34| Scale is weakly epimorphic with val: <7 12 16 33| Scale is weakly epimorphic with val: <7 12 19 33| Scale is weakly epimorphic with val: <8 11 15 32| Scale is weakly epimorphic with val: <8 11 15 35| Scale is weakly epimorphic with val: <8 11 16 30| Scale is weakly epimorphic with val: <8 11 19 30| Scale is weakly epimorphic with val: <8 12 17 33|
FIT/MODE
3: 1 1 1 SP B ME I SD: 36.6879 c. M:-54.4393 c. 4: 1 1 2 P M ME S SD: 17.2461 c. M: 28.5682 c. 7: 2 2 3 SP M ME S SD: 6.6129 c. M:-9.0610 c. 11: 3 3 5 SP M DE S SD: 3.7365 c. M: 5.9297 c. 18: 5 5 8 SP M DE S SD: 2.9946 c. M:-5.1083 c. 23: 6 7 10 SP T3 SD: 2.9601 c. M:-3.9130 c. 27: 7 8 12 SP T3 SD: 2.4317 c. M: 4.1148 c. 34: 9 10 15 SP T3 SD: 0.9123 c. M: 1.4021 c. 41: 11 12 18 SP T3 SD: 1.0629 c. M:-1.8004 c. 45: 12 13 20 SP T3 SD: 0.5743 c. M: 0.8990 c. 52: 14 15 23 SP T3 SD: 0.5008 c. M:-0.8515 c. 79: 21 23 35 SP T3 SD: 0.5016 c. M: 0.8459 c. 86: 23 25 38 SP T3 SD: 0.2251 c. M:-0.3879 c. 97: 26 28 43 SP T3 SD: 0.2415 c. M: 0.3285 c. 131: 35 38 58 SP T3 SD: 0.1042 c. M: 0.1721 c. 183: 49 53 81 SP T3 SD: 0.0687 c. M:-0.1187 c. 314: 84 91 139 SP T3 SD: 0.0104 c. M: 0.0178 c. 1028: 275 298 455 SP T3 SD: 0.0099 c. M:-0.0161 c. 1211: 324 351 536 SP T3 SD: 0.0115 c. M:-0.0149 c. 1342: 359 389 594 SP T3 SD: 0.0055 c. M:-0.0082 c. 1656: 443 480 733 SP T3 SD: 0.0032 c. M: 0.0045 c. 1970: 527 571 872 SP T3 SD: 0.0024 c. M: 0.0042 c. 2284: 611 662 1011 SP T3 SD: 0.0027 c. M: 0.0040 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: 108.7231 cents 5 x 6:7 S SD: 46.3210 cents 6 x 7:8 S SD: 5.3599 cents 7 x 8:9 S SD: 39.1107 cents 8:9 x 11 SD: 44.2510 cents 9:10:11:12 H SD: 28.4639 cents 10:11:12:13 H SD: 22.0558 cents 11:12:13:15 SD: 14.7249 cents 12:13:14:16 S SD: 3.9913 cents 39:42:46:52 SD: 3.1912 cents 51:55:60:68 SD: 1.5102 cents 63:68:74:84 SD: 0.4798 cents 75:81:88:100 SD: 0.2850 cents 138:149:162:184 SD: 0.1549 cents 213:230:250:284 SD: 0.1419 cents 351:379:412:468 SD: 0.1362 cents