Scala analysis: xen15-chalmers-triadic-diamond-8192-6561-tetrachord
Upper tetrachord 2187/2048 * 134217728/129140163 * 19683/16384 of triadic diamond for M=8192/6561, D=3/2
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 2187/2048 113.685006 apotome 2: 65536/59049 180.449991 Pythagorean diminished third 3: 4/3 498.044999 perfect fourth
SHOW/INTERVAL
0: 19683/16384 317.5950 Pythagorean augmented second 1: 2187/2048 113.6850 apotome 2: 134217728/129140163 66.7650 Pythagorean double diminished third 3: 19683/16384 317.5950 Pythagorean augmented second
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 134217728/129140163 66.765 cents Pythagorean double diminished third 1: 1 2187/2048 113.685 cents apotome 1: 1 19683/16384 317.595 cents Pythagorean augmented second 2: 1 65536/59049 180.450 cents Pythagorean diminished third 2: 1 8192/6561 384.360 cents Pythagorean diminished fourth 2: 1 43046721/33554432 431.280 cents Pythagorean double augmented second 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 : 113.7 180.4 498.0 113.7: 66.8 384.4 498.0 180.4: 317.6 431.3 498.0 498.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 66.7650 .. 317.5950 cents 250.83002 cents 134217728/129140163, 66.7650 cents 2: 180.4500 .. 431.2800 cents 250.83002 cents 1073741824/1162261467,-137.1450 cents
SHOW DATA
Number of notes : 3 -- Interval properties -- Smallest interval : 134217728/129140163, 66.7650 cents, class 1 Average step (divided formal octave): 166.0150 cents Largest one step interval : 19683/16384, 317.5950 cents Average / Smallest step : 2.486558 Largest / Average step : 1.913050 Largest / Smallest step : 4.756910 Median interval of one step : 2187/2048, 113.6850 cents, amount: 1 Least squares average step : 140.62286 cents, oct.: 421.86857 cents Scale is not proper Scale has trivalence property Scale is a mode of a 276-tone equal temperament with octave 4/3 degrees: 63 100 276 Least number of segments generator : 139 of 250.827 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 163 of 294.135 cents and inv. generated superset size: 8 = 5 more = 266.667% Number of contiguous 1-step segments: 0 Step pattern alph. order: ABC Step pattern size order : MSL Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 66.76499 cents Scale diversity : 0.953493 Lumma stability : 0.268108 Lumma impropriety factor : 0.275367 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 : 46.92002 cents Number of recognisable fifths : 0 Number of recognisable fourths : 0 Formal octave complements present : 1 = 33.3333% 2/1 octave complements present : 0 = 0.0000% -- Rational properties -- Prime limit : 3 Odd number limit : 129140163 (O: 43046721 U: 129140163) Highest odd numerator or denominator: 59049 Scale harmonicity : 0.013825 Average absolute harmonicity : 0.090477 Specific harmonicity : 0.072180 Fundamental : 1/120932352, -26.8496 octaves, 0.0000 Hz. Guide tone : 143327232, 27.0947 octaves, 37498068095.028 Hz. Exponens Consonantiae : 1.733290E+16, 53.94436 octaves Euler's gradus suavitatis : 62 Sum of Mann's harmonic distance : 64413.5, average 21471.16667 Mersenne's string divisions : 105523495 Sum of van Prooijen's expressibility: 8.58818, average 2.86273 Sum of Tenney's harmonic distance : 17.31805, average 5.77268 Vogel's harmonic complexity : 39.33333 Wille's k value : 64570081 Wilson's harmonic complexity : 51 Rectangular lattice diameter : 17 Triangular lattice diameter : 17 Lattice compactness : 97.95620, average 16.32603 Lattice compactness (without 2's) : 52.00000, average 8.66667 Number of different primes : 2 Prime exponents' range, average, count, tones@limit: 2: -11 .. 16 2.33333 29 3: -10 .. 7 -1.33333 18 3 Average exponent except 2's :-4 / 3 =-1.33333 Average absolute exponent except 2's: 18 / 3 = 6.00000
FIT/MODE
3: 1 0 2 N G T3 SD: 31.3411 c. M:-52.3300 c. 5: 1 1 3 N M DE S SD: 13.5446 c. M:-18.7680 c. 8: 2 1 5 N T3 SD: 7.2367 c. M:-10.8262 c. 13: 3 2 8 N T3 SD: 6.4523 c. M:-11.1058 c. 14: 3 2 9 N T3 SD: 4.2855 c. M: 6.9611 c. 17: 4 2 11 N T3 SD: 3.3699 c. M: 4.6694 c. 22: 5 3 14 N T3 SD: 0.4744 c. M:-0.6573 c. 83: 19 11 53 N T3 SD: 0.3130 c. M: 0.4337 c. 105: 24 14 67 N T3 SD: 0.1480 c. M: 0.2051 c. 127: 29 17 81 N T3 SD: 0.0402 c. M: 0.0557 c. 149: 34 20 95 N T3 SD: 0.0358 c. M:-0.0495 c. 276: 63 37 176 N T3 SD: 0.0008 c. M:-0.0011 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 x 5 S SD: 111.7313 cents 5 x 6:7 S SD: 79.7048 cents 6 x 7:8 S SD: 43.2105 cents 7 x 8:9 S SD: 40.4259 cents 8 x 9:11 SD: 29.1049 cents 9 x 10:12 S SD: 0.9769 cents 10 x 11:13 SD: 23.2365 cents 11 x 12:15 S SD: 24.5075 cents 12:13 x 16 SD: 12.4438 cents 13:14 x 17 SD: 18.3280 cents 15:16:17:20 SD: 12.0964 cents 17:18:19:23 SD: 10.5533 cents 18:19:20:24 S SD: 6.7256 cents 27:29:30:36 SD: 3.4052 cents 45:48:50:60 S SD: 0.9210 cents 117:125:130:156 SD: 0.7061 cents 147:157:163:196 SD: 0.5344 cents 192:205:213:256 SD: 0.2661 cents 237:253:263:316 SD: 0.2105 cents 264:282:293:352 SD: 0.1680 cents 309:330:343:412 SD: 0.1029 cents 501:535:556:668 SD: 0.0405 cents