Scala analysis: xen15-gilson-eratosthenes-diatonic
Eratosthenes' Diatonic (or Ptolemy's Diatonic Ditonaion)
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 256/243 90.224996 limma, Pythagorean minor second 2: 32/27 294.134997 Pythagorean minor third 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 128/81 792.179997 Pythagorean minor sixth 6: 16/9 996.089998 Pythagorean minor seventh 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 9/8 203.9100 major whole tone 1: 256/243 90.2250 limma, Pythagorean minor second 2: 9/8 203.9100 major whole tone 3: 9/8 203.9100 major whole tone 4: 9/8 203.9100 major whole tone 5: 256/243 90.2250 limma, Pythagorean minor second 6: 9/8 203.9100 major whole tone 7: 9/8 203.9100 major whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 256/243 90.225 cents limma, Pythagorean minor second 1: 5 9/8 203.910 cents major whole tone 2: 4 32/27 294.135 cents Pythagorean minor third 2: 3 81/64 407.820 cents Pythagorean major third 3: 6 4/3 498.045 cents perfect fourth 3: 1 729/512 611.730 cents Pythagorean tritone 4: 1 1024/729 588.270 cents Pythagorean diminished fifth 4: 6 3/2 701.955 cents perfect fifth 5: 3 128/81 792.180 cents Pythagorean minor sixth 5: 4 27/16 905.865 cents Pythagorean major sixth 6: 5 16/9 996.090 cents Pythagorean minor seventh 6: 2 243/128 1109.775 cents Pythagorean major seventh Highest number of different intervals for one interval class: 2 Average number of different intervals per interval class: 2.00000 = 2
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 0.0 : 90.2 294.1 498.0 702.0 792.2 996.1 1200.0 90.2 : 203.9 407.8 611.7 702.0 905.9 1109.8 1200.0 294.1 : 203.9 407.8 498.0 702.0 905.9 996.1 1200.0 498.0 : 203.9 294.1 498.0 702.0 792.2 996.1 1200.0 702.0 : 90.2 294.1 498.0 588.3 792.2 996.1 1200.0 792.2 : 203.9 407.8 498.0 702.0 905.9 1109.8 1200.0 996.1 : 203.9 294.1 498.0 702.0 905.9 996.1 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.2250 .. 203.9100 cents 2187/2048, 113.6850 cents 256/243, 90.2250 cents 2: 294.1350 .. 407.8200 cents 2187/2048, 113.6850 cents 256/243, 90.2250 cents 3: 498.0450 .. 611.7300 cents 2187/2048, 113.6850 cents 256/243, 90.2250 cents 4: 588.2700 .. 701.9550 cents 2187/2048, 113.6850 cents 524288/531441,-23.4600 cents 5: 792.1800 .. 905.8650 cents 2187/2048, 113.6850 cents 256/243, 90.2250 cents 6: 996.0900 .. 1109.7750 cents 2187/2048, 113.6850 cents 256/243, 90.2250 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 256/243, 90.2250 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 9/8, 203.9100 cents Average / Smallest step : 1.900012 Largest / Average step : 1.189475 Largest / Smallest step : 2.260017 Median interval of one step : 9/8, 203.9100 cents, amount: 5 Most common interval of one step : 9/8, 203.9100 cents, amount: 5 Least squares average step : 166.55636 cents, oct.: 1165.89450 cents Scale is not proper Scale has Myhill's property chromas: 2187/2048, 113.6850 cents and 531441/524288, 23.4600 cents, ratio: 4.845906 generators: 3 of 498.04500 cents and 4 of 701.95500 cents ET pair of linear temperament : 12&41 Scale is distributional even Scale is a mode of a 665-tone equal temperament with octave 2/1 degrees: 50 163 276 389 439 552 665 generators: 276 and 389 Number of contiguous 1-step segments: 0 Step pattern alph. order: ABBBABB Step pattern size order : SLLLSLL Scale is a Constant Structure, by a margin of 23.46001 cents Ratio Whole / Half tone : 2.260017 Scale diversity : 1.288369 Lumma stability : 0.451125 Lumma impropriety factor : 0.019550 Rothenberg efficiency : 0.761905 redundancy: 0.238095 Efficiency x scale size : 5.333333 Number of different interval sizes : 12 = 2.00000 / class Number of one step interval sizes : 2 Highest interval variety : 2 Mean interval variety : 2.00000 = 2 Median interval variety : 2 Lowest interval variety : 2 Smallest interval difference : 531441/524288, 23.4600 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 6 Scale is a chain of 3 triads 0.0 498.045 996.090 cents Most common triad is 0.0 498.045 701.955 cents, amount: 5 Number of recognisable fifths : 6, average 701.9550 cents Scale is Dorian octave species of abstract tetrachord Scale is Hypolydian octave species of abstract tetrachord Scale is Hypophrygian octave species of abstract tetrachord Scale contains two identical disjunct tetrachords Best major thirds form a closed circle Best minor thirds form a closed circle Scale is a complete genus musicum : 2^m 3^6 Scale contains a complete diamond : 1 3 Formal octave complements present : 3 = 42.8571% Scale is n-1 differentially coherent in interval classes 2 and 3 combined Scale is n-1 differentially coherent in interval class 3 Scale is n-1 differentially coherent in interval class 4 Limited inverse transpositions : 5 Inversional symmetry on degrees : 6 Inversional symmetry on intervals : 2-3 -- Rational properties -- Prime limit : 3 Odd number limit : 729 (O: 729 U: 729) Highest odd numerator or denominator: 243 Scale harmonicity : 0.040000 Average absolute harmonicity : 0.253508 Specific harmonicity : 0.210526 Fundamental : 1/486, -8.9248 octaves, 0.5383 Hz. Guide tone : 768, 9.5850 octaves, 200928.434 Hz. Exponens Consonantiae : 3.732480E+05, 18.50978 octaves Euler's gradus suavitatis : 22 Sum of Mann's harmonic distance : 403.5, average 57.64286 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 7.63394, average 1.09056 Sum of Tenney's harmonic distance : 16.06278, average 2.29468 Vogel's harmonic complexity : 13.71429 Wille's k value : 364 Wilson's harmonic complexity : 18 Rectangular lattice diameter : 6 Triangular lattice diameter : 6 Lattice compactness : 133.57638, average 4.77059 Lattice compactness (without 2's) : 72.00000, average 2.57143 Number of different primes : 2 Prime exponents' range, average, count, tones@limit: 2: -1 .. 8 3.71429 28 1 3: -5 .. 1 -2.00000 16 6 Average exponent except 2's :-14 / 7 =-2.00000 Average absolute exponent except 2's: 16 / 7 = 2.28571 Scale is weakly epimorphic with val: <7 10| Scale is JI-epimorphic with val: <7 11| = patent Scale is weakly epimorphic with val: <7 12|
FIT/MODE
7: 1 1 1 1 1 1 1 SP B ME I SD: 45.9357 c. M:-81.2036 c. Lebeng 8: 1 1 1 2 0 2 1 P D3 S SD: 42.9444 c. M:-59.7750 c. 9: 1 1 2 1 1 1 2 P M ME S SD: 36.0613 c. M: 62.7567 c. Pseudo Pelog, Pelogic-7, Antiionian 10: 1 1 2 2 1 1 2 P D3 S SD: 33.9277 c. M: 54.1350 c. 12: 1 2 2 2 1 2 2 P M ME S SD: 5.5296 c. M:-9.7750 c. G.Dorian, M.Phrygian, G.M.Hypoaeolian, Bhairavi That, Mela Hanumatodi, Raga Asavari (Asaveri), Bilashkhani Todi, Ghanta, Makam Kurd, Gregorian nr.3, In, Zokuso: Japan, Ousak: Greece, Major inverse 29: 2 5 5 5 2 5 5 N M DE S SD: 4.2236 c. M: 7.4664 c. 41: 3 7 7 7 3 7 7 N M DE S SD: 1.3690 c. M: 2.4201 c. 53: 4 9 9 9 4 9 9 N M DE S SD: 0.1929 c. M:-0.3410 c. Makam Ferahnuma, Askefza, Acem Kurdi 200: 15 34 34 34 15 34 34 N M DE S SD: 0.1273 c. M: 0.2250 c. 253: 19 43 43 43 19 43 43 N M DE S SD: 0.0602 c. M: 0.1064 c. 306: 23 52 52 52 23 52 52 N M DE S SD: 0.0164 c. M: 0.0289 c. 359: 27 61 61 61 27 61 61 N M DE S SD: 0.0145 c. M:-0.0257 c. 665: 50 113 113 113 50 113 113 N M DE S SD: 0.0003 c. M:-0.0006 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: 30.7262 cents 4 x 5 x 6 x 7:8 S SD: 24.0316 cents 5 x 6:7 x 8:9:10 S SD: 18.4632 cents 6 x 7:8:9 x 11:12 SD: 11.9689 cents 7 x 8:9 x 11:12:14 SD: 21.8962 cents 8 x 9:11:12:13:14:16 SD: 19.7620 cents 9:10:11:12 x 14:16:18 S SD: 18.3169 cents 10:11:12:13:15:16:18:20 SD: 13.4776 cents 11:12:13:15 x 17:20:22 SD: 15.0801 cents 12:13:14:16:18:19:21:24 SD: 8.8474 cents 14:15:17:19:21:22:25:28 SD: 8.7008 cents 15:16:18:20:23:24:27:30 SD: 8.2039 cents 16:17:19:21:24:25:28:32 SD: 6.5424 cents 18:19:21:24:27:28:32:36 SD: 5.5293 cents 21:22:25:28:32:33:37:42 SD: 5.0134 cents 22:23:26:29:33:35:39:44 SD: 3.9164 cents 32:34:38:43:48:51:57:64 SD: 3.6092 cents 33:35:39:44:50:52:59:66 SD: 3.4493 cents 36:38:43:48:54:57:64:72 SD: 2.0421 cents 54:57:64:72:81:85:96:108 SD: 1.0816 cents 76:80:90:101:114:120:135:152 SD: 0.9109 cents 96:101:114:128:144:152:171:192 SD: 0.8995 cents 108:114:128:144:162:171:192:216 SD: 0.6825 cents 114:120:135:152:171:180:203:228 SD: 0.5380 cents 150:158:178:200:225:237:267:300 SD: 0.4403 cents 188:198:223:251:282:297:334:376 SD: 0.4332 cents 210:221:249:280:315:332:373:420 SD: 0.3768 cents 222:234:263:296:333:351:395:444 SD: 0.2976 cents 264:278:313:352:396:417:469:528 SD: 0.2504 cents 324:341:384:432:486:512:576:648 SD: 0.2416 cents 336:354:398:448:504:531:597:672 SD: 0.1968 cents 372:392:441:496:558:588:661:744 SD: 0.1649 cents 432:455:512:576:648:683:768:864 SD: 0.1350 cents 486:512:576:648:729:768:864:972 SD: 0.0000 cents