Scala analysis: pyth_17
17-tone Pythagorean scale
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: 2187/2048 113.685006 apotome 3: 9/8 203.910002 major whole tone 4: 32/27 294.134997 Pythagorean minor third 5: 19683/16384 317.595008 Pythagorean augmented second 6: 81/64 407.820003 Pythagorean major third 7: 4/3 498.044999 perfect fourth 8: 1024/729 588.269995 Pythagorean diminished fifth 9: 729/512 611.730005 Pythagorean tritone 10: 3/2 701.955001 perfect fifth 11: 128/81 792.179997 Pythagorean minor sixth 12: 6561/4096 815.640007 Pythagorean augmented fifth 13: 27/16 905.865003 Pythagorean major sixth 14: 16/9 996.089998 Pythagorean minor seventh 15: 59049/32768 1019.550009 Pythagorean augmented sixth 16: 243/128 1109.775004 Pythagorean major seventh 17: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 256/243 90.2250 limma, Pythagorean minor second 1: 256/243 90.2250 limma, Pythagorean minor second 2: 531441/524288 23.4600 Pythagorean comma, ditonic comma 3: 256/243 90.2250 limma, Pythagorean minor second 4: 256/243 90.2250 limma, Pythagorean minor second 5: 531441/524288 23.4600 Pythagorean comma, ditonic comma 6: 256/243 90.2250 limma, Pythagorean minor second 7: 256/243 90.2250 limma, Pythagorean minor second 8: 256/243 90.2250 limma, Pythagorean minor second 9: 531441/524288 23.4600 Pythagorean comma, ditonic comma 10: 256/243 90.2250 limma, Pythagorean minor second 11: 256/243 90.2250 limma, Pythagorean minor second 12: 531441/524288 23.4600 Pythagorean comma, ditonic comma 13: 256/243 90.2250 limma, Pythagorean minor second 14: 256/243 90.2250 limma, Pythagorean minor second 15: 531441/524288 23.4600 Pythagorean comma, ditonic comma 16: 256/243 90.2250 limma, Pythagorean minor second 17: 256/243 90.2250 limma, Pythagorean minor second
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 5 531441/524288 23.460 cents Pythagorean comma, ditonic comma 1: 12 256/243 90.225 cents limma, Pythagorean minor second 2: 10 2187/2048 113.685 cents apotome 2: 7 65536/59049 180.450 cents Pythagorean diminished third 3: 15 9/8 203.910 cents major whole tone 3: 2 16777216/14348907 270.675 cents Pythagorean double diminished fourth 4: 3 4782969/4194304 227.370 cents Pythagorean double augmented prime 4: 14 32/27 294.135 cents Pythagorean minor third 5: 8 19683/16384 317.595 cents Pythagorean augmented second 5: 9 8192/6561 384.360 cents Pythagorean diminished fourth 6: 13 81/64 407.820 cents Pythagorean major third 6: 4 2097152/1594323 474.585 cents Pythagorean double diminished fifth 7: 1 43046721/33554432 431.280 cents Pythagorean double augmented second 7: 16 4/3 498.045 cents perfect fourth 8: 6 177147/131072 521.505 cents Pythagorean augmented third 8: 11 1024/729 588.270 cents Pythagorean diminished fifth 9: 11 729/512 611.730 cents Pythagorean tritone 9: 6 262144/177147 678.495 cents Pythagorean diminished sixth 10: 16 3/2 701.955 cents perfect fifth 10: 1 67108864/43046721 768.720 cents Pythagorean double diminished seventh 11: 4 1594323/1048576 725.415 cents Pythagorean double augmented fourth 11: 13 128/81 792.180 cents Pythagorean minor sixth 12: 9 6561/4096 815.640 cents Pythagorean augmented fifth 12: 8 32768/19683 882.405 cents Pythagorean diminished seventh 13: 14 27/16 905.865 cents Pythagorean major sixth 13: 3 8388608/4782969 972.630 cents Pythagorean double diminished octave 14: 2 14348907/8388608 929.325 cents Pythagorean double augmented fifth 14: 15 16/9 996.090 cents Pythagorean minor seventh 15: 7 59049/32768 1019.550 cents Pythagorean augmented sixth 15: 10 4096/2187 1086.315 cents Pythagorean diminished octave 16: 12 243/128 1109.775 cents Pythagorean major seventh 16: 5 1048576/531441 1176.540 cents Pythagorean diminished ninth 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 8 9 10 11 12 13 14 15 16 17 0.0 : 90.2 113.7 203.9 294.1 317.6 407.8 498.0 588.3 611.7 702.0 792.2 815.6 905.9 996.1 1019.6 1109.8 1200.0 90.2 : 23.5 113.7 203.9 227.4 317.6 407.8 498.0 521.5 611.7 702.0 725.4 815.6 905.9 929.3 1019.6 1109.8 1200.0 113.7 : 90.2 180.4 203.9 294.1 384.4 474.6 498.0 588.3 678.5 702.0 792.2 882.4 905.9 996.1 1086.3 1176.5 1200.0 203.9 : 90.2 113.7 203.9 294.1 384.4 407.8 498.0 588.3 611.7 702.0 792.2 815.6 905.9 996.1 1086.3 1109.8 1200.0 294.1 : 23.5 113.7 203.9 294.1 317.6 407.8 498.0 521.5 611.7 702.0 725.4 815.6 905.9 996.1 1019.6 1109.8 1200.0 317.6 : 90.2 180.4 270.7 294.1 384.4 474.6 498.0 588.3 678.5 702.0 792.2 882.4 972.6 996.1 1086.3 1176.5 1200.0 407.8 : 90.2 180.4 203.9 294.1 384.4 407.8 498.0 588.3 611.7 702.0 792.2 882.4 905.9 996.1 1086.3 1109.8 1200.0 498.0 : 90.2 113.7 203.9 294.1 317.6 407.8 498.0 521.5 611.7 702.0 792.2 815.6 905.9 996.1 1019.6 1109.8 1200.0 588.3 : 23.5 113.7 203.9 227.4 317.6 407.8 431.3 521.5 611.7 702.0 725.4 815.6 905.9 929.3 1019.6 1109.8 1200.0 611.7 : 90.2 180.4 203.9 294.1 384.4 407.8 498.0 588.3 678.5 702.0 792.2 882.4 905.9 996.1 1086.3 1176.5 1200.0 702.0 : 90.2 113.7 203.9 294.1 317.6 407.8 498.0 588.3 611.7 702.0 792.2 815.6 905.9 996.1 1086.3 1109.8 1200.0 792.2 : 23.5 113.7 203.9 227.4 317.6 407.8 498.0 521.5 611.7 702.0 725.4 815.6 905.9 996.1 1019.6 1109.8 1200.0 815.6 : 90.2 180.4 203.9 294.1 384.4 474.6 498.0 588.3 678.5 702.0 792.2 882.4 972.6 996.1 1086.3 1176.5 1200.0 905.9 : 90.2 113.7 203.9 294.1 384.4 407.8 498.0 588.3 611.7 702.0 792.2 882.4 905.9 996.1 1086.3 1109.8 1200.0 996.1 : 23.5 113.7 203.9 294.1 317.6 407.8 498.0 521.5 611.7 702.0 792.2 815.6 905.9 996.1 1019.6 1109.8 1200.0 1019.6: 90.2 180.4 270.7 294.1 384.4 474.6 498.0 588.3 678.5 768.7 792.2 882.4 972.6 996.1 1086.3 1176.5 1200.0 1109.8: 90.2 180.4 203.9 294.1 384.4 407.8 498.0 588.3 678.5 702.0 792.2 882.4 905.9 996.1 1086.3 1109.8 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 23.4600 .. 90.2250 cents 134217728/129140163, 66.7650 cents 531441/524288, 23.4600 cents 2: 113.6850 .. 180.4500 cents 134217728/129140163, 66.7650 cents 531441/524288, 23.4600 cents 3: 203.9100 .. 270.6750 cents 134217728/129140163, 66.7650 cents 531441/524288, 23.4600 cents 4: 227.3700 .. 294.1350 cents 134217728/129140163, 66.7650 cents-43.30497 cents 5: 317.5950 .. 384.3600 cents 134217728/129140163, 66.7650 cents 531441/524288, 23.4600 cents 6: 407.8200 .. 474.5850 cents 134217728/129140163, 66.7650 cents 531441/524288, 23.4600 cents 7: 431.2800 .. 498.0450 cents 134217728/129140163, 66.7650 cents-43.30497 cents 8: 521.5050 .. 588.2700 cents 134217728/129140163, 66.7650 cents 531441/524288, 23.4600 cents 9: 611.7300 .. 678.4950 cents 134217728/129140163, 66.7650 cents 531441/524288, 23.4600 cents 10: 701.9550 .. 768.7200 cents 134217728/129140163, 66.7650 cents 531441/524288, 23.4600 cents 11: 725.4150 .. 792.1800 cents 134217728/129140163, 66.7650 cents-43.30497 cents 12: 815.6400 .. 882.4050 cents 134217728/129140163, 66.7650 cents 531441/524288, 23.4600 cents 13: 905.8650 .. 972.6300 cents 134217728/129140163, 66.7650 cents 531441/524288, 23.4600 cents 14: 929.3250 .. 996.0900 cents 134217728/129140163, 66.7650 cents-43.30497 cents 15: 1019.5500 .. 1086.3150 cents 134217728/129140163, 66.7650 cents 531441/524288, 23.4600 cents 16: 1109.7750 .. 1176.5400 cents 134217728/129140163, 66.7650 cents 531441/524288, 23.4600 cents
SHOW DATA
Number of notes : 17 -- Interval properties -- Smallest interval : 531441/524288, 23.4600 cents, class 1 Average step (divided formal octave): 70.5882 cents Largest one step interval : 256/243, 90.2250 cents Average / Smallest step : 3.008875 Largest / Average step : 1.278187 Largest / Smallest step : 3.845906 Median interval of one step : 256/243, 90.2250 cents, amount: 12 Most common interval of one step : 256/243, 90.2250 cents, amount: 12 Least squares average step : 69.76536 cents, oct.: 1186.01115 cents Scale is not proper Scale has Myhill's property chromas: 134217728/129140163, 66.7650 cents and 43.30497 cents, ratio: 1.541739 generators: 7 of 498.04500 cents and 10 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 63 113 163 176 226 276 326 339 389 439 452 502 552 565 615 665 generators: 276 and 389 Number of contiguous 1-step segments: 0 Step pattern alph. order: ABAABAAABAABAABAA Step pattern size order : LSLLSLLLSLLSLLSLL Scale is a Constant Structure, by a margin of 23.46001 cents Ratio Whole / Half tone : 2.260017 Scale diversity : 1.154229 Lumma stability : 0.254150 Lumma impropriety factor : 0.144350 Rothenberg efficiency : 0.705882 redundancy: 0.294118 Efficiency x scale size : 12.000000 Number of different interval sizes : 32 = 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: 16 Scale is a chain of 8 triads 0.0 498.045 996.090 cents Most common triad is 0.0 498.045 701.955 cents, amount: 15 Number of recognisable fifths : 16, average 701.9550 cents Scale contains two identical disjunct "tetrachords" of 8 notes Scale is a complete genus musicum : 2^m 3^16 Scale contains a complete diamond : 1 3 9 27 81 243 729 Formal octave complements present : 13 = 76.4706% Scale is n-1 differentially coherent in interval classes 6 and 7 combined Scale is n-1 differentially coherent in interval class 7 Scale is n-1 differentially coherent in interval classes 9 and 10 combined Scale is n-1 differentially coherent in interval class 10 Limited inverse transpositions : 6 Inversional symmetry on degrees : 3 Inversional symmetry on intervals : 11-12 -- Rational properties -- Prime limit : 3 Odd number limit : 43046721 (O: 43046721 U: 43046721) Highest odd numerator or denominator: 59049 Scale harmonicity : 0.014778 Average absolute harmonicity : 0.134091 Specific harmonicity : 0.085586 Fundamental : 1/23887872, -24.5098 octaves, 0.0000 Hz. Guide tone : 60466176, 25.8496 octaves, 15819497477.590 Hz. Exponens Consonantiae : 1.444408E+15, 50.35940 octaves Euler's gradus suavitatis : 58 Sum of Mann's harmonic distance : 73576.5, average 4328.02941 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 36.26122, average 2.13301 Sum of Tenney's harmonic distance : 72.08378, average 4.24022 Vogel's harmonic complexity : 36.35294 Wille's k value : 21523360 Wilson's harmonic complexity : 48 Rectangular lattice diameter : 16 Triangular lattice diameter : 16 Lattice compactness : 1666.85705, average 10.89449 Lattice compactness (without 2's) : 892.00000, average 5.83007 Number of different primes : 2 Prime exponents' range, average, count, tones@limit: 2: -15 .. 10 -2.64706 119 1 3: -6 .. 10 2.00000 76 16 Average exponent except 2's : 34 / 17 = 2.00000 Average absolute exponent except 2's: 76 / 17 = 4.47059 Scale is weakly epimorphic with val: <17 25| Scale is weakly epimorphic with val: <17 26| Scale is JI-epimorphic with val: <17 27| = patent Scale is weakly epimorphic with val: <17 28| Scale is weakly epimorphic with val: <17 29| Scale is weakly epimorphic with val: <17 30|
FIT/MODE
17: 1 1 1 1 0 2 1 1 1 1 1 1 1 1 0 2 1 N T3 SD: 19.9598 c. M: 35.2421 c. 18: 1 1 1 1 1 1 1 2 0 2 1 0 2 1 0 2 1 N D3 SD: 19.2152 c. M: 31.3783 c. 19: 1 1 1 2 0 1 2 1 1 1 2 0 1 2 0 2 1 N D3 SD: 18.2351 c. M: 28.8726 c. 20: 2 0 1 2 0 2 1 2 0 2 1 1 1 2 0 2 1 N T3 SD: 17.4533 c. M:-30.2250 c. 24: 2 0 2 2 0 2 2 2 0 2 2 0 2 2 0 2 2 N M DE S SD: 10.3449 c. M: 19.5500 c. 29: 2 1 2 2 1 2 2 2 1 2 2 1 2 2 1 2 2 P M ME S SD: 7.9017 c. M:-14.9327 c. 38: 3 1 2 3 1 3 3 3 0 3 3 1 3 3 0 3 3 N SD: 8.8187 c. M:-14.4363 c. 39: 3 1 3 3 0 3 3 3 1 3 3 1 2 3 1 3 3 N SD: 8.7008 c. M:-15.1292 c. 40: 3 1 3 3 1 3 3 3 0 3 3 1 3 3 1 3 3 N SD: 8.4942 c. M:-12.4050 c. 41: 3 1 3 3 1 3 3 3 1 3 3 1 3 3 1 3 3 N M DE S SD: 2.5612 c. M:-4.8402 c. 53: 4 1 4 4 1 4 4 4 1 4 4 1 4 4 1 4 4 N M DE S SD: 0.3609 c. M: 0.6821 c. 200: 15 4 15 15 4 15 15 15 4 15 15 4 15 15 4 15 15 N M DE S SD: 0.2381 c. M:-0.4500 c. 253: 19 5 19 19 5 19 19 19 5 19 19 5 19 19 5 19 19 N M DE S SD: 0.1126 c. M:-0.2128 c. 306: 23 6 23 23 6 23 23 23 6 23 23 6 23 23 6 23 23 N M DE S SD: 0.0306 c. M:-0.0578 c. 359: 27 7 27 27 7 27 27 27 7 27 27 7 27 27 7 27 27 N M DE S SD: 0.0272 c. M: 0.0514 c. 665: 50 13 50 50 13 50 50 50 13 50 50 13 50 50 13 50 50 N M DE S SD: 0.0006 c. M: 0.0011 c.
FIT/HARMONIC
1 .. 2 match: 1 H SD: 0.0000 cents 2 .. 4 match: 2 H SD: 0.0000 cents 3 .. 6 match: 3 H SD: 7.1688 cents 4 .. 8 match: 4 H SD: 8.6813 cents 5 .. 10 match: 5 H SD: 1.3357 cents 6 .. 12 match: 6 H SD: 7.6279 cents 7 .. 14 match: 7 H SD: 8.8139 cents 8 .. 16 match: 8 H SD: 7.5552 cents 9 .. 18 match: 9 H SD: 6.3370 cents 10 .. 20 match: 10 H SD: 6.0371 cents 11 .. 22 match: 11 H SD: 8.7067 cents 12 .. 24 match: 12 H SD: 5.4526 cents 13 .. 26 match: 11 SD: 8.3145 cents 14 .. 28 match: 12 SD: 5.5749 cents 15 .. 30 match: 12 SD: 4.5474 cents 16 .. 32 match: 15 S SD: 4.8262 cents 17 .. 34 match: 13 SD: 5.4004 cents 18 .. 36 match: 15 S SD: 4.6015 cents 19 .. 38 match: 12 SD: 4.3487 cents 20 .. 40 match: 12 SD: 4.2622 cents 21 .. 42 match: 14 SD: 4.3108 cents 22 .. 44 match: 13 SD: 3.0230 cents 23 .. 46 match: 16 SD: 4.0235 cents 24 .. 48 match: 14 SD: 4.0535 cents 25 .. 50 match: 15 SD: 3.7863 cents 26 .. 52 match: 15 SD: 2.9598 cents 27 .. 54 match: 14 SD: 3.5317 cents 28 .. 56 match: 15 SD: 3.3072 cents 29 .. 58 match: 13 SD: 3.3236 cents 30 .. 60 match: 15 SD: 1.9468 cents 31 .. 62 match: 14 SD: 2.4452 cents 32 .. 64 match: 14 SD: 2.5118 cents 33 .. 66 match: 15 SD: 2.4355 cents 34 .. 68 match: 14 SD: 2.4998 cents 35 .. 70 match: 16 SD: 2.5189 cents 36 .. 72 match: 14 SD: 2.4354 cents 37 .. 74 match: 16 SD: 2.6532 cents 38 .. 76 match: 16 SD: 2.0431 cents 39 .. 78 match: 16 SD: 2.1991 cents 40 .. 80 match: 17 SD: 1.7867 cents 41 .. 82 match: 15 SD: 1.9298 cents 42 .. 84 match: 16 SD: 1.5662 cents 44 .. 88 match: 16 SD: 2.1340 cents 45 .. 90 match: 17 SD: 1.6715 cents 46 .. 92 match: 15 SD: 2.0170 cents 47 .. 94 match: 15 SD: 1.9499 cents 48 .. 96 match: 16 SD: 1.5352 cents 49 .. 98 match: 16 SD: 1.6697 cents 50 .. 100 match: 16 SD: 1.3936 cents 51 .. 102 match: 16 SD: 2.0101 cents 52 .. 104 match: 16 SD: 1.8537 cents 54 .. 108 match: 17 SD: 1.3368 cents 57 .. 114 match: 16 SD: 1.2443 cents 60 .. 120 match: 17 SD: 1.1636 cents 66 .. 132 match: 16 SD: 1.3682 cents 69 .. 138 match: 17 SD: 1.2899 cents 70 .. 140 match: 17 SD: 1.1130 cents 71 .. 142 match: 17 SD: 0.9789 cents 76 .. 152 match: 17 SD: 0.9536 cents 80 .. 160 match: 17 SD: 0.9438 cents 86 .. 172 match: 17 SD: 0.8725 cents 90 .. 180 match: 17 SD: 0.6565 cents 114 .. 228 match: 17 SD: 0.6300 cents 118 .. 236 match: 17 SD: 0.6168 cents 128 .. 256 match: 17 SD: 0.5314 cents 135 .. 270 match: 17 SD: 0.5036 cents 146 .. 292 match: 17 SD: 0.4756 cents 150 .. 300 match: 17 SD: 0.4327 cents 188 .. 376 match: 17 SD: 0.3726 cents 207 .. 414 match: 17 SD: 0.3314 cents 208 .. 416 match: 17 SD: 0.3132 cents 222 .. 444 match: 17 SD: 0.2806 cents 238 .. 476 match: 17 SD: 0.2645 cents 248 .. 496 match: 17 SD: 0.2500 cents 264 .. 528 match: 17 SD: 0.1926 cents 336 .. 672 match: 17 SD: 0.1920 cents 354 .. 708 match: 17 SD: 0.1899 cents 368 .. 736 match: 17 SD: 0.1857 cents 384 .. 768 match: 17 SD: 0.1743 cents 410 .. 820 match: 17 SD: 0.1575 cents 432 .. 864 match: 17 SD: 0.1250 cents 486 .. 972 match: 17 SD: 0.1088 cents 576 .. 1152 match: 17 SD: 0.0917 cents