Scala analysis: xen07-rosenthal-helix
Scale for 'Helix Song'
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: 7/6 266.870906 septimal minor third 3: 5/4 386.313714 major third 4: 4/3 498.044999 perfect fourth 5: 11/8 551.317942 undecimal semi-augmented fourth 6: 3/2 701.955001 perfect fifth 7: 5/3 884.358713 major sixth, BP sixth 8: 7/4 968.825906 harmonic seventh 9: 11/6 1049.362941 undecimal neutral seventh, 21/4-tone 10: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 12/11 150.6371 undecimal neutral second, 3/4-tone 1: 9/8 203.9100 major whole tone 2: 28/27 62.9609 Archytas' 1/3-tone 3: 15/14 119.4428 major diatonic semitone 4: 16/15 111.7313 minor diatonic semitone 5: 33/32 53.2729 undecimal comma, al-Farabi's 1/4-tone 6: 12/11 150.6371 undecimal neutral second, 3/4-tone 7: 10/9 182.4037 minor whole tone 8: 21/20 84.4672 minor semitone 9: 22/21 80.5370 undecimal minor semitone 10: 12/11 150.6371 undecimal neutral second, 3/4-tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 33/32 53.273 cents undecimal comma, al-Farabi's 1/4-tone 1: 1 28/27 62.961 cents Archytas' 1/3-tone 1: 1 22/21 80.537 cents undecimal minor semitone 1: 1 21/20 84.467 cents minor semitone 1: 1 16/15 111.731 cents minor diatonic semitone 1: 1 15/14 119.443 cents major diatonic semitone 1: 2 12/11 150.637 cents undecimal neutral second, 3/4-tone 1: 1 10/9 182.404 cents minor whole tone 1: 1 9/8 203.910 cents major whole tone 2: 2 11/10 165.004 cents Ptolemy's second, 4/5-tone 2: 1 10/9 182.404 cents minor whole tone 2: 1 9/8 203.910 cents major whole tone 2: 2 8/7 231.174 cents septimal whole tone 2: 2 7/6 266.871 cents septimal minor third 2: 1 40/33 333.041 cents 2: 1 27/22 354.547 cents neutral third, Zalzal wosta of al-Farabi 3: 1 33/28 284.447 cents undecimal minor third 3: 1 32/27 294.135 cents Pythagorean minor third 3: 2 6/5 315.641 cents minor third 3: 1 11/9 347.408 cents undecimal neutral third 3: 2 5/4 386.314 cents major third 3: 2 14/11 417.508 cents undecimal diminished fourth or major third 3: 1 9/7 435.084 cents septimal major third, BP third 4: 1 11/9 347.408 cents undecimal neutral third 4: 1 9/7 435.084 cents septimal major third, BP third 4: 1 21/16 470.781 cents narrow fourth 4: 5 4/3 498.045 cents perfect fourth 4: 1 27/20 519.551 cents acute fourth 4: 1 15/11 536.951 cents undecimal augmented fourth 5: 1 4/3 498.045 cents perfect fourth 5: 2 11/8 551.318 cents undecimal semi-augmented fourth 5: 2 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 5: 2 10/7 617.488 cents Euler's tritone 5: 2 16/11 648.682 cents undecimal semi-diminished fifth 5: 1 3/2 701.955 cents perfect fifth 6: 1 22/15 663.049 cents undecimal diminished fifth 6: 1 40/27 680.449 cents grave fifth 6: 5 3/2 701.955 cents perfect fifth 6: 1 32/21 729.219 cents wide fifth 6: 1 14/9 764.916 cents septimal minor sixth 6: 1 18/11 852.592 cents undecimal neutral sixth 7: 1 14/9 764.916 cents septimal minor sixth 7: 2 11/7 782.492 cents undecimal augmented fifth 7: 2 8/5 813.686 cents minor sixth 7: 1 18/11 852.592 cents undecimal neutral sixth 7: 2 5/3 884.359 cents major sixth, BP sixth 7: 1 27/16 905.865 cents Pythagorean major sixth 7: 1 56/33 915.553 cents 8: 1 44/27 845.453 cents neutral sixth 8: 1 33/20 866.959 cents 8: 2 12/7 933.129 cents septimal major sixth 8: 2 7/4 968.826 cents harmonic seventh 8: 1 16/9 996.090 cents Pythagorean minor seventh 8: 1 9/5 1017.596 cents just minor seventh, BP seventh 8: 2 20/11 1034.996 cents large minor seventh 9: 1 16/9 996.090 cents Pythagorean minor seventh 9: 1 9/5 1017.596 cents just minor seventh, BP seventh 9: 2 11/6 1049.363 cents undecimal neutral seventh, 21/4-tone 9: 1 28/15 1080.557 cents grave major seventh 9: 1 15/8 1088.269 cents classic major seventh 9: 1 40/21 1115.533 cents acute major seventh 9: 1 21/11 1119.463 cents undecimal major seventh 9: 1 27/14 1137.039 cents septimal major seventh 9: 1 64/33 1146.727 cents 33rd subharmonic Highest number of different intervals for one interval class: 9 Average number of different intervals per interval class: 7.11111 = 64/9
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 0.0 : 203.9 266.9 386.3 498.0 551.3 702.0 884.4 968.8 1049.4 1200.0 203.9 : 63.0 182.4 294.1 347.4 498.0 680.4 764.9 845.5 996.1 1200.0 266.9 : 119.4 231.2 284.4 435.1 617.5 702.0 782.5 933.1 1137.0 1200.0 386.3 : 111.7 165.0 315.6 498.0 582.5 663.0 813.7 1017.6 1080.6 1200.0 498.0 : 53.3 203.9 386.3 470.8 551.3 702.0 905.9 968.8 1088.3 1200.0 551.3 : 150.6 333.0 417.5 498.0 648.7 852.6 915.6 1035.0 1146.7 1200.0 702.0 : 182.4 266.9 347.4 498.0 702.0 764.9 884.4 996.1 1049.4 1200.0 884.4 : 84.5 165.0 315.6 519.6 582.5 702.0 813.7 867.0 1017.6 1200.0 968.8 : 80.5 231.2 435.1 498.0 617.5 729.2 782.5 933.1 1115.5 1200.0 1049.4: 150.6 354.5 417.5 537.0 648.7 702.0 852.6 1035.0 1119.5 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 53.2729 .. 203.9100 cents 12/11, 150.6371 cents 33/32, 53.2729 cents 2: 165.0042 .. 354.5471 cents 135/121, 189.5428 cents 44/45,-38.9058 cents 3: 284.4470 .. 435.0841 cents 12/11, 150.6371 cents 121/126,-70.1000 cents 4: 347.4079 .. 536.9508 cents 135/121, 189.5428 cents 77/81,-87.6762 cents 5: 498.0450 .. 701.9550 cents 9/8, 203.9100 cents 44/45,-38.9058 cents 6: 663.0492 .. 852.5921 cents 135/121, 189.5428 cents 44/45,-38.9058 cents 7: 764.9159 .. 915.5530 cents 12/11, 150.6371 cents 77/81,-87.6762 cents 8: 845.4529 .. 1034.9958 cents 135/121, 189.5428 cents 121/126,-70.1000 cents 9: 996.0900 .. 1146.7271 cents 12/11, 150.6371 cents 44/45,-38.9058 cents
SHOW DATA
Number of notes : 10 -- Interval properties -- Smallest interval : 33/32, 53.2729 cents, class 1 Average step (divided formal octave): 120.0000 cents Largest one step interval : 9/8, 203.9100 cents Average / Smallest step : 2.252551 Largest / Average step : 1.699250 Largest / Smallest step : 3.827647 Median interval of one step : 115.58705 cents, amount: 0 Most common interval of one step : 12/11, 150.6371 cents, amount: 2 Least squares average step : 120.11033 cents, oct.: 1201.10331 cents Scale is not proper Step pattern alph. order: ABCDEFGHIF Step pattern size order : IBFEAGHDCG Scale is not a Constant Structure Scale diversity : 1.644827 Lumma stability : 0.088788 Lumma impropriety factor : 0.380748 Rothenberg efficiency : 0.324444 redundancy: 0.675556 Efficiency x scale size : 3.244444 Number of different interval sizes : 54 = 6.00000 / class Number of one step interval sizes : 9 Highest interval variety : 9 Mean interval variety : 7.11111 = 64/9 Median interval variety : 6 Lowest interval variety : 6 Smallest interval difference : 441/440, 3.9302 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 6 Most common triad is 0.0 498.045 701.955 cents, amount: 2 Number of recognisable fifths : 6, average 701.9550 cents Scale contains a complete diamond : 1 3 All steps are superparticular (epimoric) Length of longest harmonic part : 2 at degree 2 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 3 = 30.0000% Scale is differentially coherent in interval class 1 Scale is differentially coherent in interval class 2 Scale is differentially coherent in interval class 3 Scale is differentially coherent in interval class 4 Scale is differentially coherent in interval class 5 Scale is n-1 differentially coherent in interval class 6 Scale is n-1 differentially coherent in interval class 7 Scale is differentially coherent in interval classes 8 and 9 combined -- Rational properties -- Prime limit : 11 Odd number limit : 33 (O: 33 U: 33) Highest odd numerator or denominator: 11 Scale harmonicity : 0.020891 Average absolute harmonicity : 0.208240 Specific harmonicity : 0.137664 Fundamental : 1/24, -4.5850 octaves, 10.9011 Hz. Guide tone : 13860, 13.7586 octaves, 3626130.335 Hz. Exponens Consonantiae : 3.326400E+05, 18.34360 octaves Euler's gradus suavitatis : 32 Sum of Mann's harmonic distance : 54.5, average 5.45000 Mersenne's string divisions : 116396280 Sum of van Prooijen's expressibility: 7.07941, average 0.70794 Sum of Tenney's harmonic distance : 13.32725, average 1.33273 Vogel's harmonic complexity : 9.30000 Wille's k value : 16 Wilson's harmonic complexity : 32 Rectangular lattice diameter : 6 Triangular lattice diameter : 3 Lattice compactness : 142.09536, average 2.58355 Lattice compactness (without 2's) : 84.75139, average 1.54093 Number of different primes : 5 Prime exponents' range, average, count, tones@limit: 2: -3 .. 2 -1.00000 16 1 3: -1 .. 2 -0.10000 7 3 5: 0 .. 1 0.20000 2 2 7: 0 .. 1 0.20000 2 2 11: 0 .. 1 0.20000 2 2 Average exponent except 2's : 5 / 10 = 0.50000 Average absolute exponent except 2's: 13 / 10 = 1.30000 Scale is weakly epimorphic with val: <10 17 22 26 38| Scale is weakly epimorphic with val: <10 17 22 28 36|
FIT/MODE
10: 2 0 1 1 1 1 1 1 1 1 N T3 SD: 29.5336 c. M:-48.6821 c. 12: 2 1 1 1 1 1 2 1 1 1 P D3 SD: 27.3099 c. M:-50.6371 c. Minor Pentatonic with leading tones 14: 2 1 1 2 0 2 2 1 1 2 N T3 SD: 26.0469 c. M: 43.4566 c. 15: 3 0 2 1 1 2 2 1 1 2 N SD: 17.6721 c. M:-36.0900 c. 16: 3 1 1 2 0 2 3 1 1 2 N SD: 20.2467 c. M:-33.1291 c. 17: 3 1 2 1 1 2 3 1 1 2 N SD: 18.6301 c. M:-37.2157 c. 19: 3 1 2 2 1 2 3 1 2 2 N SD: 13.8352 c. M:-24.3213 c. 22: 4 1 2 2 1 3 3 2 1 3 N SD: 9.3139 c. M:-14.2718 c. 31: 5 2 3 3 1 4 5 2 2 4 N SD: 5.6645 c. M: 10.3616 c. 41: 7 2 4 4 2 5 6 3 3 5 N SD: 3.7046 c. M: 6.3099 c. 46: 8 2 5 4 2 6 7 3 3 6 N SD: 4.0233 c. M: 6.0013 c. 58: 10 3 6 5 3 7 9 4 4 7 N SD: 4.3803 c. M:-7.3027 c. 63: 11 3 6 6 3 8 9 5 4 8 N SD: 3.9208 c. M: 8.1682 c. 72: 12 4 7 7 3 9 11 5 5 9 N SD: 1.9928 c. M: 3.9100 c. 94: 16 5 9 9 4 12 14 7 6 12 N SD: 1.9799 c. M: 3.5076 c. 111: 19 6 11 10 5 14 17 8 7 14 N SD: 2.1311 c. M:-4.1471 c. 113: 19 6 11 11 5 14 17 8 8 14 N SD: 2.1031 c. M: 4.0128 c. 118: 20 6 12 11 5 15 18 8 8 15 N SD: 1.4961 c. M: 2.7242 c. 130: 22 7 13 12 6 16 20 9 9 16 N SD: 1.4855 c. M:-2.9448 c. 152: 26 8 15 14 7 19 23 11 10 19 N SD: 1.1198 c. M:-2.2267 c. 183: 31 10 18 17 8 23 28 13 12 23 N SD: 0.9325 c. M:-1.9816 c. 217: 37 11 22 20 10 27 33 15 15 27 N SD: 0.9664 c. M:-1.6774 c. 224: 38 12 22 21 10 28 34 16 15 28 N SD: 0.5460 c. M:-0.9862 c. 270: 46 14 27 25 12 34 41 19 18 34 N SD: 0.2956 c. M:-0.5344 c. 342: 58 18 34 32 15 43 52 24 23 43 N SD: 0.2903 c. M: 0.4407 c. 494: 84 26 49 46 22 62 75 35 33 62 N SD: 0.1855 c. M:-0.4049 c. 612: 104 32 61 57 27 77 93 43 41 77 N SD: 0.1780 c. M: 0.3433 c. 742: 126 39 74 69 33 93 113 52 50 93 N SD: 0.1546 c. M:-0.2774 c. 836: 142 44 83 78 37 105 127 59 56 105 N SD: 0.1039 c. M: 0.1893 c. 1106: 188 58 110 103 49 139 168 78 74 139 N SD: 0.0874 c. M: 0.1767 c. 1236: 210 65 123 115 55 155 188 87 83 155 N SD: 0.0944 c. M:-0.1516 c. 1354: 230 71 135 126 60 170 206 95 91 170 N SD: 0.0834 c. M: 0.1405 c. 1506: 256 79 150 140 67 189 229 106 101 189 N SD: 0.0771 c. M:-0.1405 c. 1578: 268 83 157 147 70 198 240 111 106 198 N SD: 0.0521 c. M: 0.1077 c. 1848: 314 97 184 172 82 232 281 130 124 232 N SD: 0.0259 c. M:-0.0569 c.
FIT/HARMONIC
1 x x x x x x x x x 2 S SD: 0.0000 cents 2 x x x x x 3 x x x 4 S SD: 0.0000 cents 3 x x x 4 x x 5 x x 6 S SD: 0.0000 cents 4 x x 5 x x 6 x 7 x 8 S SD: 0.0000 cents 5 x 6 x x 7 x 8 x 9:10 S SD: 19.3446 cents 6 x 7 x 8 x 9:10 x 11:12 S SD: 0.0000 cents 7:8 x 9 x 10:11 x 12:13:14 S SD: 17.9340 cents 8:9 x 10 x 11:12:13:14:15:16 S SD: 7.3259 cents 9:10 x 11:12 x 14:15:16:17:18 S SD: 12.0899 cents 10:11:12 x 13:14:15:17 x 18:20 SD: 11.8394 cents 11:12:13:14 x 15:17:18:19:20:22 SD: 10.4912 cents 12 x 14:15:16:17:18:20:21:22:24 S SD: 5.7425 cents 13 x 15:16:17:18:20:22:23:24:26 SD: 8.2293 cents 14:16 x 18 x 19:21:23:25:26:28 SD: 9.6638 cents 15:17:18:19:20:21:23:25:26:28:30 SD: 8.2074 cents 16:18:19:20:21:22:24:27:28:29:32 SD: 5.0361 cents 17:19:20:21:23 x 26:28:30:31:34 SD: 6.3277 cents 18:20:21:23:24:25:27:30:32:33:36 SD: 5.4373 cents 20 x 23:25:27:28:30:33:35:37:40 SD: 5.6703 cents 23:26:27:29:31:32:35:38:40:42:46 SD: 4.6781 cents 24:27:28:30:32:33:36:40:42:44:48 S SD: 0.0000 cents