Scala analysis: 078_E8
Enharmonic tetrachord 40/39 * 416/405 * 81/64
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 40/39 43.831051 tridecimal minor diesis 2: 256/243 90.224996 limma, Pythagorean minor second 3: 4/3 498.044999 perfect fourth
SHOW/INTERVAL
0: 81/64 407.8200 Pythagorean major third 1: 40/39 43.8311 tridecimal minor diesis 2: 416/405 46.3939 3: 81/64 407.8200 Pythagorean major third
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 40/39 43.831 cents tridecimal minor diesis 1: 1 416/405 46.394 cents 1: 1 81/64 407.820 cents Pythagorean major third 2: 1 256/243 90.225 cents limma, Pythagorean minor second 2: 1 135/104 451.651 cents 2: 1 13/10 454.214 cents tridecimal semi-diminished fourth 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 : 43.8 90.2 498.0 43.8 : 46.4 454.2 498.0 90.2 : 407.8 451.7 498.0 498.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 43.8311 .. 407.8200 cents 3159/2560, 363.9890 cents 40/39, 43.8311 cents 2: 90.2250 .. 454.2139 cents 3159/2560, 363.9890 cents 16384/19683,-317.5950 cents
SHOW DATA
Number of notes : 3 -- Interval properties -- Smallest interval : 40/39, 43.8311 cents, class 1 Average step (divided formal octave): 166.0150 cents Largest one step interval : 81/64, 407.8200 cents Average / Smallest step : 3.787612 Largest / Average step : 2.456525 Largest / Smallest step : 9.304363 Median interval of one step : 416/405, 46.3939 cents, amount: 1 Least squares average step : 122.74400 cents, oct.: 368.23201 cents Scale is not proper Scale has trivalence property Scale is a mode of a 1761-tone equal temperament with octave 4/3 degrees: 155 319 1761 Least number of segments generator : 881 of 249.164 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 1615 of 456.753 cents and inv. generated superset size: 35 = 32 more = 1166.667% 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 43.83105 cents Scale diversity : 0.355212 Lumma stability : 0.176012 Lumma impropriety factor : 0.637683 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 : 676/675, 2.5629 cents Number of recognisable fifths : 0 Number of recognisable fourths : 0 Steps are strictly increasing Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 1 = 33.3333% 2/1 octave complements present : 0 = 0.0000% Limited inverse transpositions with margin 361.4261 cents: 2 -- Rational properties -- Prime limit : 13 Odd number limit : 405 (O: 135 U: 405) Highest odd numerator or denominator: 243 Scale harmonicity : 0.020045 Average absolute harmonicity : 0.096794 Specific harmonicity : 0.106198 Fundamental : 1/3159, -11.6253 octaves, 0.0828 Hz. Guide tone : 1280, 10.3219 octaves, 334880.724 Hz. Exponens Consonantiae : 4.043520E+06, 21.94718 octaves Euler's gradus suavitatis : 35 Sum of Mann's harmonic distance : 292.5, average 97.50000 Mersenne's string divisions : 275947 Sum of van Prooijen's expressibility: 4.45379, average 1.48460 Sum of Tenney's harmonic distance : 9.06615, average 3.02205 Vogel's harmonic complexity : 22.66667 Wille's k value : 526 Wilson's harmonic complexity : 33 Rectangular lattice diameter : 7 Triangular lattice diameter : 5 Lattice compactness : 30.63474, average 5.10579 Lattice compactness (without 2's) : 17.38891, average 2.89815 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: 2 .. 8 4.33333 13 3: -5 ..-1 -2.33333 7 2 5: 0 .. 1 0.33333 1 13: -1 .. 0 -0.33333 1 1 Average exponent except 2's :-7 / 3 =-2.33333 Average absolute exponent except 2's: 9 / 3 = 3.00000 Scale is weakly epimorphic with val: <7 11 15 26| Scale is weakly epimorphic with val: <7 11 15 29| Scale is weakly epimorphic with val: <7 11 16 27| Scale is weakly epimorphic with val: <7 11 16 30| Scale is weakly epimorphic with val: <7 11 17 25| Scale is weakly epimorphic with val: <7 11 17 28| Scale is weakly epimorphic with val: <7 11 18 26| Scale is weakly epimorphic with val: <7 11 18 29| Scale is weakly epimorphic with val: <7 11 19 27| Scale is weakly epimorphic with val: <7 11 19 30|
FIT/MODE
3: 0 0 3 N M DE S SD: 57.9129 c. M: 90.2250 c. 4: 0 1 3 N G T3 SD: 32.1284 c. M: 43.8311 c. 5: 0 1 4 N G T3 SD: 25.8793 c. M: 43.8311 c. 6: 0 1 5 N G T3 SD: 25.6467 c. M: 43.8311 c. 7: 0 1 6 N G T3 SD: 27.5986 c. M: 43.8311 c. 8: 1 0 7 N G T3 SD: 19.3369 c. M: 27.9694 c. 9: 1 1 7 N M DE S SD: 13.5485 c. M:-20.4517 c. 10: 1 1 8 N M DE S SD: 6.4224 c. M:-9.3840 c. 11: 1 1 9 N M DE S SD: 0.8560 c. M:-1.4458 c. 56: 5 5 46 N M DE S SD: 0.8299 c. M: 1.2884 c. 67: 6 6 55 N M DE S SD: 0.7392 c. M: 1.0229 c. 78: 7 7 64 N M DE S SD: 0.6932 c. M:-0.8653 c. 89: 8 8 73 N M DE S SD: 0.6714 c. M:-0.9370 c. 100: 9 9 82 N M DE S SD: 0.6630 c. M:-0.9930 c. 104: 9 10 85 N T3 SD: 0.6105 c. M:-0.7640 c. 105: 9 10 86 N T3 SD: 0.6617 c. M: 1.1415 c. 115: 10 11 94 N T3 SD: 0.5148 c. M:-0.7224 c. 116: 10 11 95 N T3 SD: 0.5186 c. M: 0.8961 c. 126: 11 12 103 N T3 SD: 0.4459 c. M:-0.6880 c. 127: 11 12 104 N T3 SD: 0.4006 c. M: 0.6933 c. 137: 12 13 112 N T3 SD: 0.3988 c. M:-0.6591 c. 138: 12 13 113 N T3 SD: 0.3018 c. M: 0.5228 c. 149: 13 14 122 N T3 SD: 0.2184 c. M: 0.3775 c. 160: 14 15 131 N T3 SD: 0.1479 c. M: 0.2521 c. 171: 15 16 140 N T3 SD: 0.0904 c. M: 0.1429 c. 182: 16 17 149 N T3 SD: 0.0535 c. M:-0.0799 c. 193: 17 18 158 N T3 SD: 0.0586 c. M:-0.0940 c. 375: 33 35 307 N T3 SD: 0.0504 c. M:-0.0872 c. 535: 47 50 438 N T3 SD: 0.0622 c. M: 0.0776 c. 557: 49 52 456 N T3 SD: 0.0500 c. M:-0.0848 c. 569: 50 53 466 N T3 SD: 0.0553 c. M: 0.0692 c. 580: 51 54 475 N T3 SD: 0.0417 c. M: 0.0617 c. 591: 52 55 484 N T3 SD: 0.0319 c. M: 0.0544 c. 602: 53 56 493 N T3 SD: 0.0290 c. M: 0.0474 c. 613: 54 57 502 N T3 SD: 0.0339 c. M:-0.0424 c. 762: 67 71 624 N T3 SD: 0.0280 c. M: 0.0397 c. 773: 68 72 633 N T3 SD: 0.0170 c. M: 0.0228 c. 784: 69 73 642 N T3 SD: 0.0104 c. M: 0.0179 c. 966: 85 90 791 N T3 SD: 0.0042 c. M: 0.0072 c. 1750: 154 163 1433 N T3 SD: 0.0048 c. M: 0.0077 c. 1761: 155 164 1442 N T3 SD: 0.0047 c. M:-0.0060 c. 1943: 171 181 1591 N T3 SD: 0.0015 c. M:-0.0024 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 x 7 SD: 84.4672 cents 6 x 7:8 S SD: 88.3230 cents 7 x 8:9 S SD: 77.1860 cents 8 x 9:11 SD: 62.7740 cents 9 x 10:12 S SD: 46.0894 cents 10 x 11:13 SD: 43.3391 cents 11 x 12:15 S SD: 35.9280 cents 12 x 13:16 SD: 24.1738 cents 13 x 14:17 SD: 25.3953 cents 14 x 15:19 SD: 21.1697 cents 15 x 16:20 S SD: 10.7531 cents 16 x 17:21 SD: 15.4945 cents 17 x 18:23 SD: 13.3695 cents 18 x 19:24 SD: 1.6890 cents 19 x 20:25 S SD: 11.4874 cents 20 x 21:27 SD: 11.1319 cents 21 x 22:28 SD: 4.8440 cents 22 x 23:29 SD: 11.9115 cents 23 x 24:31 SD: 12.4901 cents 24 x 25:32 SD: 9.7763 cents 25 x 26:33 SD: 14.1522 cents 26:27 x 35 SD: 13.5737 cents 27:28 x 36 SD: 9.5649 cents 30:31:32:40 S SD: 8.3656 cents 32:33:34:43 SD: 7.3609 cents 33:34:35:44 SD: 4.6806 cents 36:37:38:48 SD: 1.6463 cents 39:40:41:52 SD: 1.2151 cents 75:77:79:100 SD: 0.5838 cents 189:194:199:252 SD: 0.5598 cents 240:246:253:320 SD: 0.5141 cents 276:283:291:368 SD: 0.4911 cents 279:286:294:372 SD: 0.3427 cents 315:323:332:420 SD: 0.2919 cents 318:326:335:424 SD: 0.2731 cents 354:363:373:472 SD: 0.1552 cents 393:403:414:524 SD: 0.1154 cents 429:440:452:572 SD: 0.0631 cents 468:480:493:624 SD: 0.0434 cents