Scala analysis: xen15-chalmers-triadic-reversed-diamond-8192-6561
Triadic reversed 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: 8192/6561 384.359993 Pythagorean diminished fourth 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 6561/4096 815.640007 Pythagorean augmented fifth 6: 4096/2187 1086.314994 Pythagorean diminished octave 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 2187/2048 113.6850 apotome 1: 2187/2048 113.6850 apotome 2: 16777216/14348907 270.6750 Pythagorean double diminished fourth 3: 2187/2048 113.6850 apotome 4: 9/8 203.9100 major whole tone 5: 2187/2048 113.6850 apotome 6: 16777216/14348907 270.6750 Pythagorean double diminished fourth 7: 2187/2048 113.6850 apotome
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 4 2187/2048 113.685 cents apotome 1: 1 9/8 203.910 cents major whole tone 1: 2 16777216/14348907 270.675 cents Pythagorean double diminished fourth 2: 1 4782969/4194304 227.370 cents Pythagorean double augmented prime 2: 2 19683/16384 317.595 cents Pythagorean augmented second 2: 4 8192/6561 384.360 cents Pythagorean diminished fourth 3: 1 43046721/33554432 431.280 cents Pythagorean double augmented second 3: 4 4/3 498.045 cents perfect fourth 3: 2 1024/729 588.270 cents Pythagorean diminished fifth 4: 2 729/512 611.730 cents Pythagorean tritone 4: 4 3/2 701.955 cents perfect fifth 4: 1 67108864/43046721 768.720 cents Pythagorean double diminished seventh 5: 4 6561/4096 815.640 cents Pythagorean augmented fifth 5: 2 32768/19683 882.405 cents Pythagorean diminished seventh 5: 1 8388608/4782969 972.630 cents Pythagorean double diminished octave 6: 2 14348907/8388608 929.325 cents Pythagorean double augmented fifth 6: 1 16/9 996.090 cents Pythagorean minor seventh 6: 4 4096/2187 1086.315 cents Pythagorean diminished octave 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 4 5 6 7 0.0 : 113.7 384.4 498.0 702.0 815.6 1086.3 1200.0 113.7 : 270.7 384.4 588.3 702.0 972.6 1086.3 1200.0 384.4 : 113.7 317.6 431.3 702.0 815.6 929.3 1200.0 498.0 : 203.9 317.6 588.3 702.0 815.6 1086.3 1200.0 702.0 : 113.7 384.4 498.0 611.7 882.4 996.1 1200.0 815.6 : 270.7 384.4 498.0 768.7 882.4 1086.3 1200.0 1086.3: 113.7 227.4 498.0 611.7 815.6 929.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 113.6850 .. 270.6750 cents 156.98998 cents 2187/2048, 113.6850 cents 2: 227.3700 .. 384.3600 cents 156.98998 cents -43.30497 cents 3: 431.2800 .. 588.2700 cents 156.98998 cents 46.92002 cents 4: 611.7300 .. 768.7200 cents 156.98998 cents 531441/524288, 23.4600 cents 5: 815.6400 .. 972.6300 cents 156.98998 cents 46.92002 cents 6: 929.3250 .. 1086.3150 cents 156.98998 cents -43.30497 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 2187/2048, 113.6850 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 16777216/14348907, 270.6750 cents Average / Smallest step : 1.507926 Largest / Average step : 1.578937 Largest / Smallest step : 2.380921 Median interval of one step : 2187/2048, 113.6850 cents, amount: 4 Most common interval of one step : 2187/2048, 113.6850 cents, amount: 4 Least squares average step : 172.71750 cents, oct.: 1209.02250 cents Scale is not proper Scale has trivalence property Scale is a mode of a 665-tone equal temperament with octave 2/1 degrees: 63 213 276 389 452 602 665 Least number of segments generator : 333 of 600.902 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 389 of 701.955 cents and inv. generated superset size: 17 = 10 more = 242.857% Number of contiguous 1-step segments: 0 Step pattern alph. order: ABACABA Step pattern size order : SLSMSLS Scale is a Constant Structure, by a margin of 23.46001 cents Scale diversity : 1.422273 Lumma stability : 0.287225 Lumma impropriety factor : 0.072175 Rothenberg efficiency : 0.506122 redundancy: 0.493878 Efficiency x scale size : 3.542857 Number of different interval sizes : 18 = 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 : 531441/524288, 23.4600 cents Number of recognisable fifths : 4, average 701.9550 cents Scale is Dorian octave species of abstract tetrachord Scale contains two identical disjunct tetrachords Best fifths form a closed circle Scale is a complete diamond : 1 3 6561 Formal octave complements present : 7 = 100.0000% Scale is differentially coherent in interval classes 3 and 4 combined Inversional symmetry on degrees : 0 Inversional symmetry on intervals : 3-4 -- Rational properties -- Prime limit : 3 Odd number limit : 43046721 (O: 43046721 U: 43046721) Highest odd numerator or denominator: 6561 Scale harmonicity : 0.014778 Average absolute harmonicity : 0.230350 Specific harmonicity : 0.079866 Fundamental : 1/26873856, -24.6797 octaves, 0.0000 Hz. Guide tone : 53747712, 25.6797 octaves, 14061775535.635 Hz. Exponens Consonantiae : 1.444408E+15, 50.35940 octaves Euler's gradus suavitatis : 58 Sum of Mann's harmonic distance : 17971.5, average 2567.35714 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 15.26788, average 2.18113 Sum of Tenney's harmonic distance : 30.92144, average 4.41735 Vogel's harmonic complexity : 36.57143 Wille's k value : 21523360 Wilson's harmonic complexity : 48 Rectangular lattice diameter : 16 Triangular lattice diameter : 16 Lattice compactness : 354.39661, average 12.65702 Lattice compactness (without 2's) : 188.00000, average 6.71429 Number of different primes : 2 Prime exponents' range, average, count, tones@limit: 2: -12 .. 13 0.57143 52 1 3: -8 .. 8 0.00000 32 6 Average exponent except 2's : 0 / 7 = 0.00000 Average absolute exponent except 2's: 32 / 7 = 4.57143
FIT/MODE
7: 1 1 1 1 1 1 1 SP B ME I SD: 38.9893 c. M: 57.7436 c. Lebeng 9: 1 2 1 1 1 2 1 P M ME I SD: 23.1513 c. M:-35.2883 c. 10: 1 2 1 2 1 2 1 P M ME I SD: 16.5522 c. M:-24.3600 c. 12: 1 3 1 2 1 3 1 N T3 I SD: 11.1575 c. M: 15.6400 c. Double Harmonic Major, Major Gipsy, Bhairav That, Mela Mayamalavagaula, Raga Paraj, Kalingada, Gaulipantu, Lalitapancamam, Gandhakriya, Kalingda, Manjiri, Chromatic 2nd Byzantine Liturgical, Hitzazkiar: Greece, Maqam Zengule, Hijaz Kar, Suzidil 19: 2 4 2 3 2 4 2 P T3 I SD: 8.2969 c. M: 12.6308 c. 22: 2 5 2 4 2 5 2 N T3 I SD: 4.7355 c. M:-7.1359 c. Double Harmonic Major 31: 3 7 3 5 3 7 3 N T3 I SD: 3.3934 c. M: 5.1808 c. Double Harmonic Major, Major Gipsy 41: 4 9 4 7 4 9 4 N T3 I SD: 2.7624 c. M: 3.8722 c. 53: 5 12 5 9 5 12 5 N T3 I SD: 0.3893 c. M: 0.5457 c. 200: 19 45 19 34 19 45 19 N T3 I SD: 0.2568 c. M: 0.3600 c. 253: 24 57 24 43 24 57 24 N T3 I SD: 0.1215 c. M: 0.1703 c. 306: 29 69 29 52 29 69 29 N T3 I SD: 0.0330 c. M: 0.0463 c. 359: 34 81 34 61 34 81 34 N T3 I SD: 0.0293 c. M: 0.0411 c. 665: 63 150 63 113 63 150 63 N T3 I SD: 0.0006 c. M:-0.0009 c.
FIT/HARMONIC
1 x x x x x x 2 S SD: 0.0000 cents 2 x x x 3 x x 4 S SD: 0.0000 cents 3 x x 4 x 5 x 6 S SD: 22.9062 cents 4 x 5 x 6 x x 8 S SD: 0.6512 cents 5 x 6:7 x 8:9:10 S SD: 25.7550 cents 6 x x 8:9:10:11:12 S SD: 15.6048 cents 7:8:9 x x 11:13:14 SD: 26.6000 cents 8:9:10:11:12:13:15:16 SD: 15.3899 cents 9:10:11:12 x 14:17:18 SD: 15.7035 cents 10:11:12:13:15:16:19:20 SD: 14.2142 cents 11:12:14:15:17:18:21:22 SD: 13.6376 cents 12:13:15:16:18:19:22:24 SD: 6.9868 cents 15:16:19:20:23:24:28:30 SD: 6.5588 cents 16:17:20:21:24:26:30:32 SD: 5.4334 cents 24:26:30:32:36:38:45:48 SD: 4.5855 cents 28:30:35:37:42:45:52:56 SD: 3.2725 cents 30:32:37:40:45:48:56:60 SD: 3.1746 cents 32:34:40:43:48:51:60:64 SD: 2.6402 cents 40:43:50:53:60:64:75:80 SD: 2.3121 cents 45:48:56:60:68:72:84:90 SD: 2.2001 cents 48:51:60:64:72:77:90:96 SD: 1.3578 cents 60:64:75:80:90:96:112:120 SD: 0.9541 cents 72:77:90:96:108:115:135:144 SD: 0.8893 cents 88:94:110:117:132:141:165:176 SD: 0.8132 cents 116:124:145:155:174:186:217:232 SD: 0.7576 cents 120:128:150:160:180:192:225:240 SD: 0.5582 cents 148:158:185:197:222:237:277:296 SD: 0.5390 cents 174:186:217:232:261:279:326:348 SD: 0.4696 cents 204:218:255:272:306:327:382:408 SD: 0.3758 cents 222:237:277:296:333:356:416:444 SD: 0.3560 cents 234:250:292:312:351:375:438:468 SD: 0.2624 cents 282:301:352:376:423:452:528:564 SD: 0.2209 cents 294:314:367:392:441:471:551:588 SD: 0.1838 cents 354:378:442:472:531:567:663:708 SD: 0.0244 cents