Scala analysis: 413_D8
Diatonic tetrachord 264/251 * 251/225 * 25/22
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 264/251 87.420678 2: 88/75 276.735514 3: 4/3 498.044999 perfect fourth
SHOW/INTERVAL
0: 25/22 221.3095 undecimal acute whole tone 1: 264/251 87.4207 2: 251/225 189.3148 3: 25/22 221.3095 undecimal acute whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 264/251 87.421 cents 1: 1 251/225 189.315 cents 1: 1 25/22 221.309 cents undecimal acute whole tone 2: 1 88/75 276.736 cents 2: 1 300/251 308.730 cents 2: 1 251/198 410.624 cents 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 : 87.4 276.7 498.0 87.4 : 189.3 410.6 498.0 276.7: 221.3 308.7 498.0 498.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 87.4207 .. 221.3095 cents 6275/5808, 133.8888 cents 264/251, 87.4207 cents 2: 276.7355 .. 410.6243 cents 6275/5808, 133.8888 cents 1936/1875, 55.4260 cents
SHOW DATA
Number of notes : 3 -- Interval properties -- Smallest interval : 264/251, 87.4207 cents, class 1 Average step (divided formal octave): 166.0150 cents Largest one step interval : 25/22, 221.3095 cents Average / Smallest step : 1.899036 Largest / Average step : 1.333069 Largest / Smallest step : 2.531546 Median interval of one step : 251/225, 189.3148 cents, amount: 1 Least squares average step : 152.50191 cents, oct.: 457.50572 cents Scale is strictly proper Scale has trivalence property Scale is a mode of a 1447-tone equal temperament with octave 4/3 degrees: 254 804 1447 Least number of segments generator : 724 of 249.195 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 1337 of 460.184 cents and inv. generated superset size: 25 = 22 more = 833.333% 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 55.42603 cents Scale diversity : 1.087453 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.462343 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 : 5625/5522, 31.9947 cents Number of recognisable fifths : 0 Number of recognisable fourths : 0 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% -- Rational properties -- Prime limit : 251 Odd number limit : 251 (O: 251 U: 251) Highest odd numerator or denominator: 251 Scale harmonicity : 0.025435 Fundamental : 1/18825, -14.2004 octaves, 0.0139 Hz. Guide tone : 264, 8.0444 octaves, 69069.149 Hz. Exponens Consonantiae : 4.969800E+06, 22.24476 octaves Euler's gradus suavitatis : 276 Sum of Mann's harmonic distance : 342.5, average 114.16667 Mersenne's string divisions : 587615 Sum of van Prooijen's expressibility: 4.75186, average 1.58395 Sum of Tenney's harmonic distance : 9.72000, average 3.24000 Vogel's harmonic complexity : 186.00000 Wille's k value : 3137 Wilson's harmonic complexity : 278 Rectangular lattice diameter : 6 Triangular lattice diameter : 4 Lattice compactness : 17.15308, average 2.85885 Lattice compactness (without 2's) : 12.16427, average 2.02738 Number of different primes : 5 Prime exponents' range, average, count, tones@limit: 2: 2 .. 3 2.66667 8 3: -1 .. 1 -0.33333 3 1 5: -2 .. 0 -0.66667 2 11: 0 .. 1 0.66667 2 1 251: -1 .. 0 -0.33333 1 1 Average exponent except 2's :-2 / 3 =-0.66667 Average absolute exponent except 2's: 8 / 3 = 2.66667 Scale is weakly epimorphic with val: <6 10 15 25 58| Scale is weakly epimorphic with val: <6 10 16 27 57| Scale is weakly epimorphic with val: <6 11 15 26 59| Scale is weakly epimorphic with val: <7 10 16 24 59| Scale is weakly epimorphic with val: <7 10 17 26 58| Scale is weakly epimorphic with val: <7 11 15 24 57| Scale is weakly epimorphic with val: <7 11 15 25 59| Scale is weakly epimorphic with val: <7 11 15 27 57| Scale is weakly epimorphic with val: <7 11 16 23 59| Scale is weakly epimorphic with val: <7 11 16 24 58| Scale is weakly epimorphic with val: <7 11 16 26 59| Scale is weakly epimorphic with val: <7 11 16 27 58| Scale is weakly epimorphic with val: <7 11 17 23 57| Scale is weakly epimorphic with val: <7 11 17 25 58| = patent Scale is JI-epimorphic with val: <7 11 17 26 57| Scale is weakly epimorphic with val: <7 11 18 24 57| Scale is weakly epimorphic with val: <7 11 18 25 59| Scale is weakly epimorphic with val: <7 11 18 27 57| Scale is weakly epimorphic with val: <7 11 19 23 59| Scale is weakly epimorphic with val: <7 11 19 24 58| Scale is weakly epimorphic with val: <7 11 19 26 59| Scale is weakly epimorphic with val: <7 11 19 27 58| Scale is weakly epimorphic with val: <7 12 15 24 59| Scale is weakly epimorphic with val: <7 12 16 26 58| Scale is weakly epimorphic with val: <7 12 16 27 57| Scale is weakly epimorphic with val: <7 12 19 27 57| Scale is weakly epimorphic with val: <8 11 15 27 59| Scale is weakly epimorphic with val: <8 11 16 26 58| Scale is weakly epimorphic with val: <8 11 17 24 58| Scale is weakly epimorphic with val: <8 11 17 25 57| Scale is weakly epimorphic with val: <8 11 18 26 57| Scale is weakly epimorphic with val: <8 11 18 27 59| Scale is weakly epimorphic with val: <8 11 19 26 58| Scale is weakly epimorphic with val: <8 12 15 26 59| Scale is weakly epimorphic with val: <8 12 16 23 57| Scale is weakly epimorphic with val: <8 12 16 25 58| Scale is weakly epimorphic with val: <8 12 17 24 57| Scale is weakly epimorphic with val: <8 12 17 25 59| Scale is weakly epimorphic with val: <8 12 18 26 59| Scale is weakly epimorphic with val: <8 12 18 27 58| Scale is weakly epimorphic with val: <8 12 19 25 58|
FIT/MODE
3: 0 1 2 N G T3 SD: 81.4482 c. M: 110.7205 c. 4: 1 1 2 P M ME S SD: 26.7315 c. M:-37.0906 c. 5: 1 2 2 SP M ME S SD: 14.5670 c. M:-22.0915 c. 7: 1 3 3 SP M DE S SD: 10.4333 c. M: 16.2714 c. 11: 2 4 5 SP T3 SD: 3.4432 c. M: 5.0746 c. 16: 3 6 7 SP T3 SD: 3.9672 c. M:-5.9628 c. 18: 3 7 8 SP T3 SD: 2.5481 c. M: 4.4132 c. 29: 5 11 13 SP T3 SD: 1.4394 c. M: 1.9521 c. 34: 6 13 15 SP T3 SD: 0.9537 c. M:-1.5838 c. 45: 8 17 20 SP T3 SD: 0.6475 c. M:-1.1207 c. 63: 11 24 28 SP T3 SD: 0.2670 c. M: 0.4604 c. 108: 19 41 48 SP T3 SD: 0.1173 c. M:-0.1983 c. 171: 30 65 76 SP T3 SD: 0.0360 c. M: 0.0444 c. 450: 79 171 200 SP T3 SD: 0.0266 c. M: 0.0438 c. 621: 109 236 276 SP T3 SD: 0.0253 c. M: 0.0438 c. 655: 115 249 291 SP T3 SD: 0.0268 c. M:-0.0406 c. 718: 126 273 319 SP T3 SD: 0.0224 c. M:-0.0332 c. 826: 145 314 367 SP T3 SD: 0.0142 c. M:-0.0231 c. 997: 175 379 443 SP T3 SD: 0.0067 c. M:-0.0117 c. 1168: 205 444 519 SP T3 SD: 0.0045 c. M: 0.0070 c. 1447: 254 550 643 SP T3 SD: 0.0040 c. M: 0.0056 c. 2273: 399 864 1010 SP T3 SD: 0.0043 c. M:-0.0056 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: 109.5782 cents 5 x 6:7 S SD: 46.4983 cents 6 x 7:8 S SD: 4.9323 cents 7 x 8:9 S SD: 38.8585 cents 8 x 9:11 SD: 45.1153 cents 9 x 11:12 SD: 35.3362 cents 10:11:12:13 H SD: 32.4106 cents 11:12:13:15 SD: 25.0900 cents 12:13:14:16 S SD: 17.3648 cents 15:16:18:20 S SD: 15.2922 cents 16:17:19:21 SD: 12.8344 cents 17:18:20:23 SD: 9.3878 cents 18:19:21:24 SD: 3.8806 cents 39:41:46:52 SD: 3.0317 cents 57:60:67:76 SD: 1.1322 cents 75:79:88:100 SD: 0.8446 cents 132:139:155:176 SD: 0.8125 cents 138:145:162:184 SD: 0.6520 cents 156:164:183:208 SD: 0.3074 cents 213:224:250:284 SD: 0.2021 cents 231:243:271:308 SD: 0.1204 cents 387:407:454:516 SD: 0.1192 cents 444:467:521:592 SD: 0.0446 cents