Scala analysis: xen15-gilson-pythagorean-diatonic
Pythagorean Intonation Diatonic Scale (PIDS)
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 9/8 203.910002 major whole tone 2: 81/64 407.820003 Pythagorean major third 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 27/16 905.865003 Pythagorean major sixth 6: 243/128 1109.775004 Pythagorean major seventh 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 256/243 90.2250 limma, Pythagorean minor second 1: 9/8 203.9100 major whole tone 2: 9/8 203.9100 major whole tone 3: 256/243 90.2250 limma, Pythagorean minor second 4: 9/8 203.9100 major whole tone 5: 9/8 203.9100 major whole tone 6: 9/8 203.9100 major whole tone 7: 256/243 90.2250 limma, Pythagorean minor second
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 : 203.9 407.8 498.0 702.0 905.9 1109.8 1200.0 203.9 : 203.9 294.1 498.0 702.0 905.9 996.1 1200.0 407.8 : 90.2 294.1 498.0 702.0 792.2 996.1 1200.0 498.0 : 203.9 407.8 611.7 702.0 905.9 1109.8 1200.0 702.0 : 203.9 407.8 498.0 702.0 905.9 996.1 1200.0 905.9 : 203.9 294.1 498.0 702.0 792.2 996.1 1200.0 1109.8: 90.2 294.1 498.0 588.3 792.2 996.1 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 : 177.92486 cents, oct.: 1245.47400 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: 113 226 276 389 502 615 665 generators: 276 and 389 Number of contiguous 1-step segments: 0 Step pattern alph. order: AABAAAB Step pattern size order : LLSLLLS 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 Lydian octave species of abstract tetrachord Scale is Phrygian octave species of abstract tetrachord Scale is Dorian 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 Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 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 : 2 Inversional symmetry on degrees : 1 Inversional symmetry on intervals : 4-5 -- 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 : 0.257075 Specific harmonicity : 0.210526 Fundamental : 1/384, -8.5850 octaves, 0.6813 Hz. Guide tone : 972, 9.9248 octaves, 254300.049 Hz. Exponens Consonantiae : 3.732480E+05, 18.50978 octaves Euler's gradus suavitatis : 22 Sum of Mann's harmonic distance : 295.5, average 42.21429 Mersenne's string divisions : 117972435 Sum of van Prooijen's expressibility: 7.63394, average 1.09056 Sum of Tenney's harmonic distance : 14.85866, average 2.12267 Vogel's harmonic complexity : 13.42857 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: -7 .. 2 -2.57143 24 1 3: -1 .. 5 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 9: 2 1 1 1 2 1 1 P M ME S SD: 36.0613 c. M:-62.7567 c. 12: 2 2 1 2 2 2 1 P M ME S SD: 5.5296 c. M: 9.7750 c. G.Lydian, M.Ionian, M.Hypolydian, Major, Bilaval That, Mela Shankarabharanam, Raga Atana, Ghana Heptatonic, Peruvian Major, Matzore, Rast ascending: Greece, 4th plagal Byzantine, Ararai: Ethiopia, Makam Cargah, Ajam Ashiran, Dastgah-e Mahur, Dastgah-e Rast Panjgah, Xin: China, DS2, Heptatonia prima 29: 5 5 2 5 5 5 2 N M DE S SD: 4.2236 c. M:-7.4664 c. Twenty-nine tone Major 41: 7 7 3 7 7 7 3 N M DE S SD: 1.3690 c. M:-2.4201 c. Forty-one tone Major 53: 9 9 4 9 9 9 4 N M DE S SD: 0.1929 c. M: 0.3410 c. Makam Cargah II, Zavil, Dastgah-e Mahur, Fifty-three tone Major 200: 34 34 15 34 34 34 15 N M DE S SD: 0.1273 c. M:-0.2250 c. 253: 43 43 19 43 43 43 19 N M DE S SD: 0.0602 c. M:-0.1064 c. 306: 52 52 23 52 52 52 23 N M DE S SD: 0.0164 c. M:-0.0289 c. 359: 61 61 27 61 61 61 27 N M DE S SD: 0.0145 c. M: 0.0257 c. 665: 113 113 50 113 113 113 50 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: 7.1688 cents 4 x 5 x 6:7 x 8 S SD: 16.6332 cents 5 x 6:7 x 8 x 10 S SD: 38.8334 cents 6:7 x 8:9:10:11:12 S SD: 14.9780 cents 7:8:9 x 11:12:13:14 SD: 16.8042 cents 8:9:10:11:12:14:15:16 S SD: 12.5577 cents 10:11 x 13:15:17:19:20 SD: 10.0001 cents 14:16:18:19:21:24:27:28 SD: 8.9354 cents 15:17:19:20:23:25:28:30 SD: 7.7319 cents 16:18:20:21:24:27:30:32 S SD: 5.8351 cents 18:20:23:24:27:30:34:36 SD: 5.1009 cents 22:25:28:29:33:37:42:44 SD: 4.3237 cents 26:29:33:35:39:44:49:52 SD: 3.7900 cents 30:34:38:40:45:51:57:60 SD: 2.5973 cents 38:43:48:51:57:64:72:76 SD: 2.3256 cents 42:47:53:56:63:71:80:84 SD: 1.7658 cents 48:54:61:64:72:81:91:96 SD: 1.0709 cents 80:90:101:107:120:135:152:160 SD: 1.0053 cents 90:101:114:120:135:152:171:180 SD: 0.7057 cents 120:135:152:160:180:203:228:240 SD: 0.6744 cents 128:144:162:171:192:216:243:256 SD: 0.4826 cents 144:162:182:192:216:243:273:288 SD: 0.4801 cents 176:198:223:235:264:297:334:352 SD: 0.4569 cents 192:216:243:256:288:324:365:384 SD: 0.3390 cents 234:263:296:312:351:395:444:468 SD: 0.3089 cents 240:270:304:320:360:405:456:480 SD: 0.2878 cents 256:288:324:341:384:432:486:512 SD: 0.2416 cents 294:331:372:392:441:496:558:588 SD: 0.2158 cents 336:378:425:448:504:567:638:672 SD: 0.1533 cents 384:432:486:512:576:648:729:768 SD: 0.0000 cents