Scala analysis: rectoo
Hahn-reduced circle of fifths via <12 19 27 34| kernel
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 10/9 182.403712 minor whole tone 2: 8/7 231.174094 septimal whole tone 3: 6/5 315.641287 minor third 4: 5/4 386.313714 major third 5: 4/3 498.044999 perfect fourth 6: 3/2 701.955001 perfect fifth 7: 25/16 772.627428 classic augmented fifth 8: 8/5 813.686286 minor sixth 9: 5/3 884.358713 major sixth, BP sixth 10: 7/4 968.825906 harmonic seventh 11: 9/5 1017.596288 just minor seventh, BP seventh 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 10/9 182.4037 minor whole tone 1: 10/9 182.4037 minor whole tone 2: 36/35 48.7704 septimal diesis, 1/4-tone 3: 21/20 84.4672 minor semitone 4: 25/24 70.6724 classic chromatic semitone, minor chroma 5: 16/15 111.7313 minor diatonic semitone 6: 9/8 203.9100 major whole tone 7: 25/24 70.6724 classic chromatic semitone, minor chroma 8: 128/125 41.0589 minor diesis, diesis 9: 25/24 70.6724 classic chromatic semitone, minor chroma 10: 21/20 84.4672 minor semitone 11: 36/35 48.7704 septimal diesis, 1/4-tone 12: 10/9 182.4037 minor whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 128/125 41.059 cents minor diesis, diesis 1: 2 36/35 48.770 cents septimal diesis, 1/4-tone 1: 3 25/24 70.672 cents classic chromatic semitone, minor chroma 1: 2 21/20 84.467 cents minor semitone 1: 1 16/15 111.731 cents minor diatonic semitone 1: 2 10/9 182.404 cents minor whole tone 1: 1 9/8 203.910 cents major whole tone 2: 2 16/15 111.731 cents minor 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: 1 10/9 182.404 cents minor whole tone 2: 2 8/7 231.174 cents septimal whole tone 2: 1 75/64 274.582 cents classic augmented second 2: 1 6/5 315.641 cents minor third 2: 1 100/81 364.807 cents grave major third 3: 1 10/9 182.404 cents minor whole tone 3: 1 28/25 196.198 cents middle second 3: 2 9/8 203.910 cents major whole tone 3: 1 7/6 266.871 cents septimal minor third 3: 3 6/5 315.641 cents minor third 3: 2 5/4 386.314 cents major third 3: 2 80/63 413.578 cents wide major third 4: 1 144/125 244.969 cents classic diminished third 4: 1 7/6 266.871 cents septimal minor third 4: 1 6/5 315.641 cents minor third 4: 3 5/4 386.314 cents major third 4: 1 32/25 427.373 cents classic diminished fourth 4: 1 125/96 456.986 cents classic augmented third 4: 1 64/49 462.348 cents 2 septatones or septatonic major third 4: 1 21/16 470.781 cents narrow fourth 4: 2 4/3 498.045 cents perfect fourth 5: 1 6/5 315.641 cents minor third 5: 1 32/25 427.373 cents classic diminished fourth 5: 1 21/16 470.781 cents narrow fourth 5: 3 4/3 498.045 cents perfect fourth 5: 1 27/20 519.551 cents acute fourth 5: 1 175/128 541.453 cents 5: 2 48/35 546.815 cents septimal semi-augmented fourth 5: 2 25/18 568.717 cents classic augmented fourth 6: 1 4/3 498.045 cents perfect fourth 6: 1 27/20 519.551 cents acute fourth 6: 1 25/18 568.717 cents classic augmented fourth 6: 2 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 6: 1 45/32 590.224 cents diatonic tritone 6: 1 64/45 609.776 cents 2nd tritone 6: 2 10/7 617.488 cents Euler's tritone 6: 1 36/25 631.283 cents classic diminished fifth 6: 1 40/27 680.449 cents grave fifth 6: 1 3/2 701.955 cents perfect fifth 7: 2 36/25 631.283 cents classic diminished fifth 7: 2 35/24 653.185 cents septimal semi-diminished fifth 7: 1 256/175 658.547 cents 7: 1 40/27 680.449 cents grave fifth 7: 3 3/2 701.955 cents perfect fifth 7: 1 32/21 729.219 cents wide fifth 7: 1 25/16 772.627 cents classic augmented fifth 7: 1 5/3 884.359 cents major sixth, BP sixth 8: 2 3/2 701.955 cents perfect fifth 8: 1 32/21 729.219 cents wide fifth 8: 1 49/32 737.652 cents 8: 1 192/125 743.014 cents classic diminished sixth 8: 1 25/16 772.627 cents classic augmented fifth 8: 3 8/5 813.686 cents minor sixth 8: 1 5/3 884.359 cents major sixth, BP sixth 8: 1 12/7 933.129 cents septimal major sixth 8: 1 125/72 955.031 cents classic augmented sixth 9: 2 63/40 786.422 cents narrow minor sixth 9: 2 8/5 813.686 cents minor sixth 9: 3 5/3 884.359 cents major sixth, BP sixth 9: 1 12/7 933.129 cents septimal major sixth 9: 2 16/9 996.090 cents Pythagorean minor seventh 9: 1 25/14 1003.802 cents middle minor seventh 9: 1 9/5 1017.596 cents just minor seventh, BP seventh 10: 1 81/50 835.193 cents acute minor sixth 10: 1 5/3 884.359 cents major sixth, BP sixth 10: 1 128/75 925.418 cents diminished seventh 10: 2 7/4 968.826 cents harmonic seventh 10: 1 9/5 1017.596 cents just minor seventh, BP seventh 10: 2 64/35 1044.860 cents septimal neutral seventh 10: 2 50/27 1066.762 cents grave major seventh 10: 2 15/8 1088.269 cents classic major seventh 11: 1 16/9 996.090 cents Pythagorean minor seventh 11: 2 9/5 1017.596 cents just minor seventh, BP seventh 11: 1 15/8 1088.269 cents classic major seventh 11: 2 40/21 1115.533 cents acute major seventh 11: 3 48/25 1129.328 cents classic diminished octave 11: 2 35/18 1151.230 cents septimal semi-diminished octave 11: 1 125/64 1158.941 cents classic augmented seventh, octave - minor diesis Highest number of different intervals for one interval class: 10 Average number of different intervals per interval class: 8.00000 = 8
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 182.4 231.2 315.6 386.3 498.0 702.0 772.6 813.7 884.4 968.8 1017.6 1200.0 182.4 : 48.8 133.2 203.9 315.6 519.6 590.2 631.3 702.0 786.4 835.2 1017.6 1200.0 231.2 : 84.5 155.1 266.9 470.8 541.5 582.5 653.2 737.7 786.4 968.8 1151.2 1200.0 315.6 : 70.7 182.4 386.3 457.0 498.0 568.7 653.2 702.0 884.4 1066.8 1115.5 1200.0 386.3 : 111.7 315.6 386.3 427.4 498.0 582.5 631.3 813.7 996.1 1044.9 1129.3 1200.0 498.0 : 203.9 274.6 315.6 386.3 470.8 519.6 702.0 884.4 933.1 1017.6 1088.3 1200.0 702.0 : 70.7 111.7 182.4 266.9 315.6 498.0 680.4 729.2 813.7 884.4 996.1 1200.0 772.6 : 41.1 111.7 196.2 245.0 427.4 609.8 658.5 743.0 813.7 925.4 1129.3 1200.0 813.7 : 70.7 155.1 203.9 386.3 568.7 617.5 702.0 772.6 884.4 1088.3 1158.9 1200.0 884.4 : 84.5 133.2 315.6 498.0 546.8 631.3 702.0 813.7 1017.6 1088.3 1129.3 1200.0 968.8 : 48.8 231.2 413.6 462.3 546.8 617.5 729.2 933.1 1003.8 1044.9 1115.5 1200.0 1017.6: 182.4 364.8 413.6 498.0 568.7 680.4 884.4 955.0 996.1 1066.8 1151.2 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 41.0589 .. 203.9100 cents 1125/1024, 162.8511 cents 128/125, 41.0589 cents 2: 111.7313 .. 364.8074 cents 125/108, 253.0761 cents 128/135,-92.1787 cents 3: 182.4037 .. 413.5778 cents 8/7, 231.1741 cents 9/10,-182.4037 cents 4: 244.9689 .. 498.0450 cents 125/108, 253.0761 cents 567/625,-168.6089 cents 5: 315.6413 .. 568.7174 cents 125/108, 253.0761 cents 9/10,-182.4037 cents 6: 498.0450 .. 701.9550 cents 9/8, 203.9100 cents 24/25,-70.6724 cents 7: 631.2826 .. 884.3587 cents 125/108, 253.0761 cents 24/25,-70.6724 cents 8: 701.9550 .. 955.0311 cents 125/108, 253.0761 cents 9/10,-182.4037 cents 9: 786.4222 .. 1017.5963 cents 8/7, 231.1741 cents 567/625,-168.6089 cents 10: 835.1926 .. 1088.2687 cents 125/108, 253.0761 cents 9/10,-182.4037 cents 11: 996.0900 .. 1158.9411 cents 1125/1024, 162.8511 cents 128/135,-92.1787 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 128/125, 41.0589 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 9/8, 203.9100 cents Average / Smallest step : 2.435528 Largest / Average step : 2.039100 Largest / Smallest step : 4.966285 Median interval of one step : 77.56981 cents, amount: 0 Most common interval of one step : 25/24, 70.6724 cents, amount: 3 Least squares average step : 99.99664 cents, oct.: 1199.95965 cents Scale is not proper Step pattern alph. order: ABCDEFDGDCBA Step pattern size order : FBDCEGCACDBF Scale is not a Constant Structure Scale diversity : 1.540101 Lumma stability : 0.068431 Lumma impropriety factor : 0.761644 Rothenberg efficiency : 0.279780 redundancy: 0.720220 Efficiency x scale size : 3.357359 Number of different interval sizes : 58 = 5.27273 / class Number of one step interval sizes : 7 Highest interval variety : 10 Mean interval variety : 8.00000 = 8 Median interval variety : 7 Lowest interval variety : 7 Smallest interval difference : 6144/6125, 5.3620 cents Number of recognisable fifths : 6, average 701.9550 cents Scale contains a complete diamond : 1 3 5 Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 11 = 91.6667% 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 classes 3 and 4 combined -- Rational properties -- Prime limit : 7 Odd number limit : 175 (O: 175 U: 175) Highest odd numerator or denominator: 25 Scale harmonicity : 0.017410 Average absolute harmonicity : 0.191835 Specific harmonicity : 0.146439 Fundamental : 1/5040, -12.2992 octaves, 0.0519 Hz. Guide tone : 12600, 13.6211 octaves, 3296482.123 Hz. Exponens Consonantiae : 6.350400E+07, 25.92034 octaves Euler's gradus suavitatis : 40 Sum of Mann's harmonic distance : 78.0, average 6.50000 Mersenne's string divisions : 280720440 Sum of van Prooijen's expressibility: 8.74674, average 0.72890 Sum of Tenney's harmonic distance : 17.11953, average 1.42663 Vogel's harmonic complexity : 23.00000 Wille's k value : 3937 Wilson's harmonic complexity : 41 Rectangular lattice diameter : 9 Triangular lattice diameter : 4 Lattice compactness : 248.60209, average 3.18721 Lattice compactness (without 2's) : 141.92601, average 1.81956 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -4 .. 3 0.16667 20 1 3: -2 .. 2 0.00000 8 2 5: -1 .. 2 0.16667 8 7 7: -1 .. 1 0.00000 2 2 Average exponent except 2's : 2 / 12 = 0.16667 Average absolute exponent except 2's: 18 / 12 = 1.50000 Scale is weakly epimorphic with val: <12 19 27 34| Scale is weakly epimorphic with val: <12 20 27 34|
FIT/MODE
12: 2 0 1 1 1 2 1 0 1 1 0 2 N SD: 18.6720 c. M:-31.1741 c. 15: 2 1 1 1 1 3 1 0 1 1 1 2 N SD: 15.7395 c. M:-27.3726 c. 19: 3 1 1 1 2 3 1 1 1 1 1 3 N SD: 10.9949 c. M:-21.4575 c. 22: 3 1 2 1 2 4 1 1 1 2 1 3 N SD: 11.3126 c. M: 18.7673 c. 26: 4 1 2 1 3 4 2 1 1 2 1 4 N SD: 9.2835 c. M: 17.0829 c. 27: 4 1 2 2 2 5 1 1 2 2 1 4 N SD: 9.4774 c. M: 17.0719 c. 28: 4 1 2 2 3 4 2 1 2 2 1 4 N SD: 12.3502 c. M:-16.8884 c. 31: 5 1 2 2 3 5 2 1 2 2 1 5 N SD: 5.6218 c. M: 11.1447 c. 46: 7 2 3 3 4 8 3 1 3 3 2 7 N SD: 4.0877 c. M:-9.9813 c. 50: 8 2 3 3 5 8 3 2 3 3 2 8 N SD: 6.2552 c. M: 9.5963 c. 53: 8 2 4 3 5 9 3 2 3 4 2 8 N SD: 2.3099 c. M:-4.7590 c. 84: 13 3 6 5 8 14 5 3 5 6 3 13 N SD: 2.0195 c. M: 3.3106 c. 87: 13 4 6 5 8 15 5 3 5 6 4 13 N SD: 2.0551 c. M:-3.3087 c. 99: 15 4 7 6 9 17 6 3 6 7 4 15 N SD: 1.2810 c. M:-3.1301 c. 118: 18 5 8 7 11 20 7 4 7 8 5 18 N SD: 1.1623 c. M:-2.7242 c. 140: 21 6 10 8 13 24 8 5 8 10 6 21 N SD: 1.2903 c. M: 2.4037 c. 152: 23 6 11 9 14 26 9 5 9 11 6 23 N SD: 1.0767 c. M: 2.2267 c. 171: 26 7 12 10 16 29 10 6 10 12 7 26 N SD: 0.3146 c. M: 0.6976 c. 270: 41 11 19 16 25 46 16 9 16 19 11 41 N SD: 0.2856 c. M:-0.7059 c. 354: 54 14 25 21 33 60 21 12 21 25 14 54 N SD: 0.4335 c. M:-0.6656 c. 388: 59 16 27 23 36 66 23 13 23 27 16 59 N SD: 0.3886 c. M:-0.7847 c. 441: 67 18 31 26 41 75 26 15 26 31 18 67 N SD: 0.0906 c. M:-0.1617 c. 612: 93 25 43 36 57 104 36 21 36 43 25 93 N SD: 0.0900 c. M:-0.1985 c. 935: 142 38 66 55 87 159 55 32 55 66 38 142 N SD: 0.1019 c. M:-0.1581 c. 1053: 160 43 74 62 98 179 62 36 62 74 43 160 N SD: 0.0758 c. M:-0.1649 c. 1106: 168 45 78 65 103 188 65 38 65 78 45 168 N SD: 0.0816 c. M: 0.1252 c. 1376: 209 56 97 81 128 234 81 47 81 97 56 209 N SD: 0.0759 c. M: 0.1363 c. 1407: 214 57 99 83 131 239 83 48 83 99 57 214 N SD: 0.0665 c. M: 0.1123 c. 1547: 235 63 109 91 144 263 91 53 91 109 63 235 N SD: 0.0597 c. M: 0.1154 c. 1578: 240 64 111 93 147 268 93 54 93 111 64 240 N SD: 0.0530 c. M: 0.1058 c. 1718: 261 70 121 101 160 292 101 59 101 121 70 261 N SD: 0.0635 c. M: 0.1012 c. 1749: 266 71 123 103 163 297 103 60 103 123 71 266 N SD: 0.0600 c. M: 0.1006 c. 1848: 281 75 130 109 172 314 109 63 109 130 75 281 N SD: 0.0498 c. M:-0.0998 c. 2019: 307 82 142 119 188 343 119 69 119 142 82 307 N SD: 0.0358 c. M: 0.0629 c. 2190: 333 89 154 129 204 372 129 75 129 154 89 333 N SD: 0.0403 c. M: 0.0620 c.
FIT/HARMONIC
1 x x 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 x x 4 S SD: 0.0000 cents 3 x x x x 4 x x x 5 x x 6 S SD: 0.0000 cents 4 x x x 5 x 6 x x x 7 x 8 S SD: 0.0000 cents 5 x x 6 x 7 x x 8 x x 9:10 S SD: 16.8934 cents 6 x 7 x x 8:9 x x 10 x 11:12 S SD: 7.9641 cents 7 x 8 x 9 x x 11 x x 12:13:14 SD: 13.6194 cents 8:9 x x 10:11:12 x 13 x 14 x 16 S SD: 9.0587 cents 9:10 x 11 x 12 x 14 x 15 x 16:18 S SD: 5.5899 cents 10:11 x 12 x 13:15 x 16:17 x 18:20 SD: 7.2879 cents 11:12 x 13:14:15 x 17 x 18:19:20:22 SD: 8.8744 cents 12:13:14 x 15:16:18 x 19:20:21:22:24 S SD: 6.7329 cents 13:14:15 x 16:17 x 20:21:22:23 x 26 SD: 9.3650 cents 14 x 16:17:18:19:21:22 x 23 x 25:28 SD: 7.5730 cents 15 x 17:18:19:20 x 23:24:25:26:27:30 SD: 4.5545 cents 16:18 x 19:20:21:24:25:26:27:28:29:32 SD: 4.8626 cents 17:19 x 20:21:23:26 x 27:28:30:31:34 SD: 6.9033 cents 18:20:21:22:23:24:27:28:29:30 x 32:36 S SD: 6.0280 cents 19:21:22:23:24:25:29:30 x 32:33:34:38 SD: 5.3908 cents 20:22:23:24:25:27:30:31:32:33:35:36:40 SD: 3.0931 cents 23 x 26:28:29:31:35:36:37:38:40:41:46 SD: 4.8944 cents 24:27 x 29:30:32:36 x 38:40:42:43:48 SD: 3.1611 cents 28:31:32:34:35:37:42:44:45:47:49:50:56 SD: 2.8845 cents 35:39:40:42:44:47:53:55:56:58:61:63:70 SD: 2.3403 cents 36:40:41:43:45:48:54:56:58:60:63:65:72 SD: 1.5190 cents 55:61:63:66:69:73:83:86:88:92:96:99:110 SD: 1.4382 cents 60:67:69:72:75:80:90:94:96:100:105:108:120 SD: 1.2138 cents 64:71:73:77:80:85:96:100:102:107:112:115:128 SD: 1.0837 cents 80:89:91:96:100:107:120:125:128:133:140:144:160 SD: 0.9083 cents 84:93:96:101:105:112:126:131:134:140:147:151:168 SD: 0.8033 cents 96:107:110:115:120:128:144:150:154:160:168:173:192 SD: 0.7582 cents 100:111:114:120:125:133:150:156:160:167:175:180:200 SD: 0.6466 cents 120:133:137:144:150:160:180:188:192:200:210:216:240 SD: 0.5483 cents 135:150:154:162:169:180:203:211:216:225:236:243:270 SD: 0.5185 cents 144:160:165:173:180:192:216:225:230:240:252:259:288 SD: 0.4938 cents 156:173:178:187:195:208:234:244:250:260:273:281:312 SD: 0.4901 cents 160:178:183:192:200:213:240:250:256:267:280:288:320 SD: 0.3585 cents 180:200:206:216:225:240:270:281:288:300:315:324:360 SD: 0.2378 cents 240:267:274:288:300:320:360:375:384:400:420:432:480 SD: 0.2347 cents 260:289:297:312:325:347:390:406:416:433:455:468:520 SD: 0.2176 cents 300:333:343:360:375:400:450:469:480:500:525:540:600 SD: 0.1743 cents 384:427:439:461:480:512:576:600:614:640:672:691:768 SD: 0.1714 cents 405:450:463:486:506:540:608:633:648:675:709:729:810 SD: 0.1599 cents 420:467:480:504:525:560:630:656:672:700:735:756:840 SD: 0.1168 cents 540:600:617:648:675:720:810:844:864:900:945:972:1080 SD: 0.0542 cents