Scala analysis: hahn15
Hahn-reduced 15 note scale
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 16/15 111.731285 minor diatonic semitone 2: 10/9 182.403712 minor whole tone 3: 7/6 266.870906 septimal minor third 4: 6/5 315.641287 minor third 5: 5/4 386.313714 major third 6: 4/3 498.044999 perfect fourth 7: 7/5 582.512193 septimal or Huygens' tritone, BP fourth 8: 10/7 617.487807 Euler's tritone 9: 3/2 701.955001 perfect fifth 10: 8/5 813.686286 minor sixth 11: 5/3 884.358713 major sixth, BP sixth 12: 7/4 968.825906 harmonic seventh 13: 9/5 1017.596288 just minor seventh, BP seventh 14: 15/8 1088.268715 classic major seventh 15: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 16/15 111.7313 minor diatonic semitone 1: 16/15 111.7313 minor diatonic semitone 2: 25/24 70.6724 classic chromatic semitone, minor chroma 3: 21/20 84.4672 minor semitone 4: 36/35 48.7704 septimal diesis, 1/4-tone 5: 25/24 70.6724 classic chromatic semitone, minor chroma 6: 16/15 111.7313 minor diatonic semitone 7: 21/20 84.4672 minor semitone 8: 50/49 34.9756 Erlich's decatonic comma, tritonic diesis 9: 21/20 84.4672 minor semitone 10: 16/15 111.7313 minor diatonic semitone 11: 25/24 70.6724 classic chromatic semitone, minor chroma 12: 21/20 84.4672 minor semitone 13: 36/35 48.7704 septimal diesis, 1/4-tone 14: 25/24 70.6724 classic chromatic semitone, minor chroma 15: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 50/49 34.976 cents Erlich's decatonic comma, tritonic diesis 1: 2 36/35 48.770 cents septimal diesis, 1/4-tone 1: 4 25/24 70.672 cents classic chromatic semitone, minor chroma 1: 4 21/20 84.467 cents minor semitone 1: 4 16/15 111.731 cents minor diatonic semitone 2: 4 15/14 119.443 cents major diatonic semitone 2: 2 27/25 133.238 cents large limma, BP small semitone, Zarlino semitone 2: 2 35/32 155.140 cents septimal neutral second 2: 4 10/9 182.404 cents minor whole tone 2: 2 28/25 196.198 cents middle second 2: 1 256/225 223.463 cents Neapolitan diminished third 3: 5 9/8 203.910 cents major whole tone 3: 4 8/7 231.174 cents septimal whole tone 3: 4 7/6 266.871 cents septimal minor third 3: 2 32/27 294.135 cents Pythagorean minor third 4: 2 75/64 274.582 cents classic augmented second 4: 2 25/21 301.847 cents BP second, quasi-equal minor third 4: 7 6/5 315.641 cents minor third 4: 1 128/105 342.905 cents septimal neutral third 4: 1 49/40 351.338 cents larger approximation to neutral third 4: 1 100/81 364.807 cents grave major third 4: 1 56/45 378.602 cents septimal narrow major third 5: 1 60/49 350.617 cents smaller approximation to neutral third 5: 7 5/4 386.314 cents major third 5: 2 63/50 400.108 cents quasi-equal major third 5: 1 80/63 413.578 cents wide major third 5: 3 32/25 427.373 cents classic diminished fourth 5: 1 35/27 449.275 cents septimal semi-diminished fourth, 9/4-tone 6: 3 9/7 435.084 cents septimal major third, BP third 6: 3 21/16 470.781 cents narrow fourth 6: 9 4/3 498.045 cents perfect fourth 7: 2 75/56 505.757 cents marvelous fourth 7: 2 27/20 519.551 cents acute fourth 7: 2 48/35 546.815 cents septimal semi-augmented fourth 7: 3 25/18 568.717 cents classic augmented fourth 7: 4 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 7: 2 64/45 609.776 cents 2nd tritone 8: 2 45/32 590.224 cents diatonic tritone 8: 4 10/7 617.488 cents Euler's tritone 8: 3 36/25 631.283 cents classic diminished fifth 8: 2 35/24 653.185 cents septimal semi-diminished fifth 8: 2 40/27 680.449 cents grave fifth 8: 2 112/75 694.243 cents marvelous fifth 9: 9 3/2 701.955 cents perfect fifth 9: 3 32/21 729.219 cents wide fifth 9: 3 14/9 764.916 cents septimal minor sixth 10: 1 54/35 750.725 cents septimal semi-augmented fifth 10: 3 25/16 772.627 cents classic augmented fifth 10: 1 63/40 786.422 cents narrow minor sixth 10: 2 100/63 799.892 cents quasi-equal minor sixth 10: 7 8/5 813.686 cents minor sixth 10: 1 49/30 849.383 cents larger approximation to neutral sixth 11: 1 45/28 821.398 cents septimale wide minor sixth 11: 1 81/50 835.193 cents acute minor sixth 11: 1 80/49 848.662 cents smaller approximation to neutral sixth 11: 1 105/64 857.095 cents septimal neutral sixth 11: 7 5/3 884.359 cents major sixth, BP sixth 11: 2 42/25 898.153 cents quasi-equal major sixth 11: 2 128/75 925.418 cents diminished seventh 12: 2 27/16 905.865 cents Pythagorean major sixth 12: 4 12/7 933.129 cents septimal major sixth 12: 4 7/4 968.826 cents harmonic seventh 12: 5 16/9 996.090 cents Pythagorean minor seventh 13: 1 225/128 976.537 cents augmented sixth 13: 2 25/14 1003.802 cents middle minor seventh 13: 4 9/5 1017.596 cents just minor seventh, BP seventh 13: 2 64/35 1044.860 cents septimal neutral seventh 13: 2 50/27 1066.762 cents grave major seventh 13: 4 28/15 1080.557 cents grave major seventh 14: 4 15/8 1088.269 cents classic major seventh 14: 4 40/21 1115.533 cents acute major seventh 14: 4 48/25 1129.328 cents classic diminished octave 14: 2 35/18 1151.230 cents septimal semi-diminished octave 14: 1 49/25 1165.024 cents BP eighth Highest number of different intervals for one interval class: 7 Average number of different intervals per interval class: 5.28571 = 37/7
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0.0 : 111.7 182.4 266.9 315.6 386.3 498.0 582.5 617.5 702.0 813.7 884.4 968.8 1017.6 1088.3 1200.0 111.7 : 70.7 155.1 203.9 274.6 386.3 470.8 505.8 590.2 702.0 772.6 857.1 905.9 976.5 1088.3 1200.0 182.4 : 84.5 133.2 203.9 315.6 400.1 435.1 519.6 631.3 702.0 786.4 835.2 905.9 1017.6 1129.3 1200.0 266.9 : 48.8 119.4 231.2 315.6 350.6 435.1 546.8 617.5 702.0 750.7 821.4 933.1 1044.9 1115.5 1200.0 315.6 : 70.7 182.4 266.9 301.8 386.3 498.0 568.7 653.2 702.0 772.6 884.4 996.1 1066.8 1151.2 1200.0 386.3 : 111.7 196.2 231.2 315.6 427.4 498.0 582.5 631.3 702.0 813.7 925.4 996.1 1080.6 1129.3 1200.0 498.0 : 84.5 119.4 203.9 315.6 386.3 470.8 519.6 590.2 702.0 813.7 884.4 968.8 1017.6 1088.3 1200.0 582.5 : 35.0 119.4 231.2 301.8 386.3 435.1 505.8 617.5 729.2 799.9 884.4 933.1 1003.8 1115.5 1200.0 617.5 : 84.5 196.2 266.9 351.3 400.1 470.8 582.5 694.2 764.9 849.4 898.2 968.8 1080.6 1165.0 1200.0 702.0 : 111.7 182.4 266.9 315.6 386.3 498.0 609.8 680.4 764.9 813.7 884.4 996.1 1080.6 1115.5 1200.0 813.7 : 70.7 155.1 203.9 274.6 386.3 498.0 568.7 653.2 702.0 772.6 884.4 968.8 1003.8 1088.3 1200.0 884.4 : 84.5 133.2 203.9 315.6 427.4 498.0 582.5 631.3 702.0 813.7 898.2 933.1 1017.6 1129.3 1200.0 968.8 : 48.8 119.4 231.2 342.9 413.6 498.0 546.8 617.5 729.2 813.7 848.7 933.1 1044.9 1115.5 1200.0 1017.6: 70.7 182.4 294.1 364.8 449.3 498.0 568.7 680.4 764.9 799.9 884.4 996.1 1066.8 1151.2 1200.0 1088.3: 111.7 223.5 294.1 378.6 427.4 498.0 609.8 694.2 729.2 813.7 925.4 996.1 1080.6 1129.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 34.9756 .. 111.7313 cents 392/375, 76.7557 cents 50/49, 34.9756 cents 2: 119.4428 .. 223.4626 cents 3584/3375, 104.0198 cents 225/224, 7.7115 cents 3: 203.9100 .. 294.1350 cents 256/243, 90.2250 cents 2025/2048,-19.5526 cents 4: 274.5824 .. 378.6022 cents 3584/3375, 104.0198 cents 2025/2048,-19.5526 cents 5: 350.6169 .. 449.2746 cents 343/324, 98.6577 cents 675/686,-27.9853 cents 6: 435.0841 .. 498.0450 cents 28/27, 62.9609 cents 243/245,-14.1905 cents 7: 505.7565 .. 609.7763 cents 3584/3375, 104.0198 cents 225/224, 7.7115 cents 8: 590.2237 .. 694.2435 cents 3584/3375, 104.0198 cents 2025/2048,-19.5526 cents 9: 701.9550 .. 764.9159 cents 28/27, 62.9609 cents 225/224, 7.7115 cents 10: 750.7254 .. 849.3831 cents 343/324, 98.6577 cents 243/245,-14.1905 cents 11: 821.3978 .. 925.4176 cents 3584/3375, 104.0198 cents 675/686,-27.9853 cents 12: 905.8650 .. 996.0900 cents 256/243, 90.2250 cents 2025/2048,-19.5526 cents 13: 976.5374 .. 1080.5572 cents 3584/3375, 104.0198 cents 2025/2048,-19.5526 cents 14: 1088.2687 .. 1165.0244 cents 392/375, 76.7557 cents 225/224, 7.7115 cents
SHOW DATA
Number of notes : 15 -- Interval properties -- Smallest interval : 50/49, 34.9756 cents, class 1 Average step (divided formal octave): 80.0000 cents Largest one step interval : 16/15, 111.7313 cents Average / Smallest step : 2.287308 Largest / Average step : 1.396641 Largest / Smallest step : 3.194548 Median interval of one step : 21/20, 84.4672 cents, amount: 4 Least squares average step : 79.63699 cents, oct.: 1194.55489 cents Scale is not proper Step pattern alph. order: ABCDBACECABCDBA Step pattern size order : ECDBCEDADECDBCE Scale is a Constant Structure, by a margin of 0.72120 cents Scale diversity : 1.507817 Lumma stability : 0.083998 Lumma impropriety factor : 0.151762 Number of different interval sizes : 74 = 5.28571 / class Number of one step interval sizes : 5 Highest interval variety : 7 Mean interval variety : 5.28571 = 37/7 Median interval variety : 3.5 Lowest interval variety : 3 Smallest interval difference : 2401/2400, 0.7212 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 9 Most common triad is 0.0 498.045 701.955 cents, amount: 5 Number of recognisable fifths : 11, average 700.5529 cents Scale contains two identical disjunct heptachords Scale contains a complete diamond : 1 3 5 15 All steps are superparticular (epimoric) Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 13 = 86.6667% Scale is differentially coherent in interval class 1 Scale is n-1 differentially coherent in interval classes 2 and 3 combined Scale is n-1 differentially coherent in interval class 3 -- Rational properties -- Prime limit : 7 Odd number limit : 225 (O: 225 U: 225) Highest odd numerator or denominator: 15 Scale harmonicity : 0.019593 Average absolute harmonicity : 0.167652 Specific harmonicity : 0.137038 Fundamental : 1/2520, -11.2992 octaves, 0.1038 Hz. Guide tone : 5040, 12.2992 octaves, 1318592.849 Hz. Exponens Consonantiae : 1.270080E+07, 23.59842 octaves Euler's gradus suavitatis : 36 Sum of Mann's harmonic distance : 98.0, average 6.53333 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 11.39118, average 0.75941 Sum of Tenney's harmonic distance : 22.24109, average 1.48274 Vogel's harmonic complexity : 22.00000 Wille's k value : 2362 Wilson's harmonic complexity : 36 Rectangular lattice diameter : 8 Triangular lattice diameter : 4 Lattice compactness : 360.36549, average 3.00305 Lattice compactness (without 2's) : 218.12650, average 1.81772 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -3 .. 4 0.26667 22 1 3: -2 .. 2 -0.06667 11 2 5: -1 .. 1 0.00000 10 8 7: -1 .. 1 0.13333 4 4 Average exponent except 2's : 1 / 15 = 0.06667 Average absolute exponent except 2's: 25 / 15 = 1.66667 Scale is weakly epimorphic with val: <15 24 32 42| Scale is weakly epimorphic with val: <15 24 34 42| Scale is JI-epimorphic with val: <15 24 35 42| = patent Scale is weakly epimorphic with val: <15 24 37 42| Scale is weakly epimorphic with val: <15 24 38 42|
FIT/MODE
15: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 19.8242 c. M:-31.7313 c. 19: 2 1 1 1 1 2 1 1 1 2 1 1 1 1 2 N D3 I SD: 10.9498 c. M: 21.4575 c. 22: 2 1 2 1 1 2 2 0 2 2 1 2 1 1 2 N D3 SD: 11.3899 c. M: 18.7673 c. 27: 3 1 2 1 2 2 2 1 2 2 2 2 1 1 3 N SD: 10.5970 c. M:-21.6020 c. 29: 3 1 2 2 1 3 2 1 2 3 1 2 2 1 3 N I SD: 12.6535 c. M: 18.5950 c. 31: 3 2 2 1 2 3 2 1 2 3 2 2 1 2 3 N SD: 5.4041 c. M: 11.1447 c. 41: 4 2 3 2 2 4 3 1 3 4 2 3 2 2 4 N SD: 4.7218 c. M: 6.7940 c. 53: 5 3 4 2 3 5 4 1 4 5 3 4 2 3 5 N SD: 3.0240 c. M:-6.1671 c. 68: 6 4 5 3 4 6 5 2 5 6 4 5 3 4 6 N SD: 3.5865 c. M: 5.9331 c. 72: 7 4 5 3 4 7 5 2 5 7 4 5 3 4 7 N SD: 2.3666 c. M: 4.9354 c. 84: 8 5 6 3 5 8 6 2 6 8 5 6 3 5 8 N SD: 2.5174 c. M:-4.5577 c. 87: 8 5 6 4 5 8 6 3 6 8 5 6 4 5 8 N SD: 2.4096 c. M: 4.8019 c. 99: 9 6 7 4 6 9 7 3 7 9 6 7 4 6 9 N SD: 1.2671 c. M:-2.6404 c. 130: 12 8 9 5 8 12 9 4 9 12 8 9 5 8 12 N SD: 1.2900 c. M: 2.2117 c. 140: 13 8 10 6 8 13 10 4 10 13 8 10 6 8 13 N SD: 1.1615 c. M: 2.4037 c. 152: 14 9 11 6 9 14 11 4 11 14 9 11 6 9 14 N SD: 1.1229 c. M:-2.2267 c. 171: 16 10 12 7 10 16 12 5 12 16 10 12 7 10 16 N SD: 0.2815 c. M:-0.5494 c. 441: 41 26 31 18 26 41 31 13 31 41 26 31 18 26 41 N SD: 0.1252 c. M: 0.2042 c. 612: 57 36 43 25 36 57 43 18 43 57 36 43 25 36 57 N SD: 0.0988 c. M: 0.2042 c. 935: 87 55 66 38 55 87 66 27 66 87 55 66 38 55 87 N SD: 0.1062 c. M:-0.1616 c. 1053: 98 62 74 43 62 98 74 31 74 98 62 74 43 62 98 N SD: 0.1000 c. M: 0.2042 c. 1106: 103 65 78 45 65 103 78 32 78 103 65 78 45 65 103 N SD: 0.0804 c. M:-0.1280 c. 1236: 115 73 87 50 73 115 87 36 87 115 73 87 50 73 115 N SD: 0.0852 c. M: 0.1206 c. 1277: 119 75 90 52 75 119 90 37 90 119 75 90 52 75 119 N SD: 0.0806 c. M:-0.1033 c. 1376: 128 81 97 56 81 128 97 40 97 128 81 97 56 81 128 N SD: 0.0765 c. M: 0.1363 c. 1407: 131 83 99 57 83 131 99 41 99 131 83 99 57 83 131 N SD: 0.0584 c. M: 0.1123 c. 1547: 144 91 109 63 91 144 109 45 109 144 91 109 63 91 144 N SD: 0.0557 c. M: 0.1154 c. 1578: 147 93 111 64 93 147 111 46 111 147 93 111 64 93 147 N SD: 0.0531 c. M: 0.1058 c. 1718: 160 101 121 70 101 160 121 50 121 160 101 121 70 101 160 N SD: 0.0533 c. M: 0.0987 c. 1848: 172 109 130 75 109 172 130 54 130 172 109 130 75 109 172 N SD: 0.0429 c. M: 0.0638 c. 2019: 188 119 142 82 119 188 142 59 142 188 119 142 82 119 188 N SD: 0.0345 c. M: 0.0629 c. 2190: 204 129 154 89 129 204 154 64 154 204 129 154 89 129 204 N SD: 0.0409 c. M: 0.0620 c.
FIT/HARMONIC
1 x x x x x x x x x x x x x x 2 S SD: 0.0000 cents 2 x x x x x x x x 3 x x x x x 4 S SD: 0.0000 cents 3 x x x x x 4 x x x x 5 x x x 6 S SD: 0.0000 cents 4 x x x x 5 x x x 6 x x 7 x x 8 S SD: 0.0000 cents 5 x x x 6 x x 7 x x 8 x x 9 x 10 S SD: 0.0000 cents 6 x x 7 x x 8 x x 9 x 10 x 11 x 12 S SD: 5.2944 cents 7 x x 8 x 9 x x 10 x 11 x 12 x 13:14 S SD: 11.2255 cents 8 x 9 x x 10 x 11 x 12:13 x 14 x 15:16 S SD: 5.8042 cents 9 x 10 x 11 x 12 x 13 x 14:15 x 16:17:18 S SD: 7.3530 cents 10 x 11 x 12 x 13:14 x 15:16:17 x 18:19:20 S SD: 6.2650 cents 11 x 12:13 x 14:15 x 16:17 x 18:19:20:21:22 S SD: 9.2870 cents 12:13 x 14 x 15:16 x 17:18:19:20:21:22:23:24 S SD: 5.0799 cents 13:14 x 15 x 16:17:18:19:20:21:22:23 x 24:26 S SD: 7.6329 cents 14:15 x 16:17:18:19 x 20:21:22:23 x 25:26:28 SD: 7.0363 cents 15:16:17 x 18:19:20:21 x 23:24:25:26:27:28:30 SD: 4.5397 cents 16:17:18 x 19:20:21:22:23:24:26:27:28:29:30:32 S SD: 4.5155 cents 17:18:19:20 x 21:23:24 x 26:27:28:30:31:32:34 SD: 4.9965 cents 18:19:20:21:22:23:24:25:26:27:29:30 x 32:34:36 SD: 4.5799 cents 19:20:21:22:23:24:25 x 27:29:30:32:33:34:36:38 SD: 4.6893 cents 20:21:22:23:24:25:27:28:29:30:32:33:35:36:38:40 SD: 4.0122 cents 21:22:23 x 25:26:28:29:30:32:34:35:37:38:39:42 SD: 4.6728 cents 22:23:24:26 x 28:29:31 x 33:35:37:39:40:41:44 SD: 5.6855 cents 23:25:26:27:28:29:31:32:33:35:37:38:40:41:43:46 SD: 4.6186 cents 24:26:27:28:29:30:32 x 34:36:38:40:42:43:45:48 SD: 3.1376 cents 27:29:30 x 32:34:36:38:39:41:43:45:47:49:51:54 SD: 3.3377 cents 28:30:31:33:34:35:37:39:40:42:45:47:49:50:53:56 SD: 2.8417 cents 29:31:32:34:35:36:39:41 x 44:46:48:51:52:54:58 SD: 3.1948 cents 30:32:33:35:36:38:40:42:43:45:48:50:53:54:56:60 SD: 2.3002 cents 35:37:39:41:42:44:47:49:50:53:56:58:61:63:66:70 SD: 2.1904 cents 40:43:44:47:48:50:53:56:57:60:64:67:70:72:75:80 SD: 1.9405 cents 44:47:49:51:53:55:59:62:63:66:70:73:77:79:83:88 SD: 1.7849 cents 45:48:50:53:54:56:60:63:64:68:72:75:79:81:84:90 SD: 1.6879 cents 48:51:53:56:58:60:64:67:69:72:77:80:84:86:90:96 SD: 1.5421 cents 55:59:61:64:66:69:73:77:79:83:88:92:96:99:103:110 SD: 1.4763 cents 56:60:62:65:67:70:75:78:80:84:90:93:98:101:105:112 SD: 1.4148 cents 60:64:67:70:72:75:80:84:86:90:96:100:105:108:113:120 SD: 0.8607 cents 72:77:80:84:86:90:96:101:103:108:115:120:126:130:135:144 SD: 0.7885 cents 90:96:100:105:108:113:120:126:129:135:144:150:158:162:169:180 SD: 0.7566 cents 100:107:111:117:120:125:133:140:143:150:160:167:175:180:188:200 SD: 0.7042 cents 108:115:120:126:130:135:144:151:154:162:173:180:189:194:203:216 SD: 0.6254 cents 120:128:133:140:144:150:160:168:171:180:192:200:210:216:225:240 SD: 0.4086 cents 168:179:187:196:202:210:224:235:240:252:269:280:294:302:315:336 SD: 0.3895 cents 180:192:200:210:216:225:240:252:257:270:288:300:315:324:338:360 SD: 0.1825 cents 240:256:267:280:288:300:320:336:343:360:384:400:420:432:450:480 SD: 0.1520 cents 360:384:400:420:432:450:480:504:514:540:576:600:630:648:675:720 SD: 0.0641 cents 600:640:667:700:720:750:800:840:857:900:960:1000:1050:1080:1125:1200 SD: 0.0608 cents