Scala analysis: Archytas3genera
All three Archytas's gerenra at once: diatonic+chromatic+enharmonic
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 28/27 62.960904 Archytas' 1/3-tone 2: 16/15 111.731285 minor diatonic semitone 3: 9/8 203.910002 major whole tone 4: 32/27 294.134997 Pythagorean minor third 5: 4/3 498.044999 perfect fourth 6: 3/2 701.955001 perfect fifth 7: 14/9 764.915905 septimal minor sixth 8: 8/5 813.686286 minor sixth 9: 27/16 905.865003 Pythagorean major sixth 10: 16/9 996.089998 Pythagorean minor seventh 11: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 9/8 203.9100 major whole tone 1: 28/27 62.9609 Archytas' 1/3-tone 2: 36/35 48.7704 septimal diesis, 1/4-tone 3: 135/128 92.1787 major chroma, major limma 4: 256/243 90.2250 limma, Pythagorean minor second 5: 9/8 203.9100 major whole tone 6: 9/8 203.9100 major whole tone 7: 28/27 62.9609 Archytas' 1/3-tone 8: 36/35 48.7704 septimal diesis, 1/4-tone 9: 135/128 92.1787 major chroma, major limma 10: 256/243 90.2250 limma, Pythagorean minor second 11: 9/8 203.9100 major whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 36/35 48.770 cents septimal diesis, 1/4-tone 1: 2 28/27 62.961 cents Archytas' 1/3-tone 1: 2 256/243 90.225 cents limma, Pythagorean minor second 1: 2 135/128 92.179 cents major chroma, major limma 1: 3 9/8 203.910 cents major whole tone 2: 2 16/15 111.731 cents minor diatonic semitone 2: 2 243/224 140.949 cents Archytas' 2/3-tone 2: 2 10/9 182.404 cents minor whole tone 2: 2 7/6 266.871 cents septimal minor third 2: 2 32/27 294.135 cents Pythagorean minor third 2: 1 81/64 407.820 cents Pythagorean major third 3: 2 9/8 203.910 cents major whole tone 3: 2 8/7 231.174 cents septimal whole tone 3: 2 6/5 315.641 cents minor third 3: 1 896/729 357.096 cents 3: 2 5/4 386.314 cents major third 3: 1 21/16 470.781 cents narrow fourth 3: 1 4/3 498.045 cents perfect fourth 4: 2 32/27 294.135 cents Pythagorean minor third 4: 1 512/405 405.866 cents narrow diminished fourth 4: 2 81/64 407.820 cents Pythagorean major third 4: 2 9/7 435.084 cents septimal major third, BP third 4: 1 35/27 449.275 cents septimal semi-diminished fourth, 9/4-tone 4: 1 27/20 519.551 cents acute fourth 4: 1 112/81 561.006 cents 4: 1 45/32 590.224 cents diatonic tritone 5: 7 4/3 498.045 cents perfect fourth 5: 1 64/45 609.776 cents 2nd tritone 5: 1 729/512 611.730 cents Pythagorean tritone 5: 1 81/56 638.994 cents 5: 1 35/24 653.185 cents septimal semi-diminished fifth 6: 1 48/35 546.815 cents septimal semi-augmented fourth 6: 1 112/81 561.006 cents 6: 1 1024/729 588.270 cents Pythagorean diminished fifth 6: 1 45/32 590.224 cents diatonic tritone 6: 7 3/2 701.955 cents perfect fifth 7: 1 64/45 609.776 cents 2nd tritone 7: 1 81/56 638.994 cents 7: 1 40/27 680.449 cents grave fifth 7: 1 54/35 750.725 cents septimal semi-augmented fifth 7: 2 14/9 764.916 cents septimal minor sixth 7: 2 128/81 792.180 cents Pythagorean minor sixth 7: 1 405/256 794.134 cents wide augmented fifth 7: 2 27/16 905.865 cents Pythagorean major sixth 8: 1 3/2 701.955 cents perfect fifth 8: 1 32/21 729.219 cents wide fifth 8: 2 8/5 813.686 cents minor sixth 8: 1 729/448 842.904 cents 8: 2 5/3 884.359 cents major sixth, BP sixth 8: 2 7/4 968.826 cents harmonic seventh 8: 2 16/9 996.090 cents Pythagorean minor seventh 9: 1 128/81 792.180 cents Pythagorean minor sixth 9: 2 27/16 905.865 cents Pythagorean major sixth 9: 2 12/7 933.129 cents septimal major sixth 9: 2 9/5 1017.596 cents just minor seventh, BP seventh 9: 2 448/243 1059.051 cents 9: 2 15/8 1088.269 cents classic major seventh 10: 3 16/9 996.090 cents Pythagorean minor seventh 10: 2 256/135 1107.821 cents octave - major chroma 10: 2 243/128 1109.775 cents Pythagorean major seventh 10: 2 27/14 1137.039 cents septimal major seventh 10: 2 35/18 1151.230 cents septimal semi-diminished octave Highest number of different intervals for one interval class: 8 Average number of different intervals per interval class: 6.20000 = 31/5
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 0.0 : 63.0 111.7 203.9 294.1 498.0 702.0 764.9 813.7 905.9 996.1 1200.0 63.0 : 48.8 140.9 231.2 435.1 639.0 702.0 750.7 842.9 933.1 1137.0 1200.0 111.7 : 92.2 182.4 386.3 590.2 653.2 702.0 794.1 884.4 1088.3 1151.2 1200.0 203.9 : 90.2 294.1 498.0 561.0 609.8 702.0 792.2 996.1 1059.1 1107.8 1200.0 294.1 : 203.9 407.8 470.8 519.6 611.7 702.0 905.9 968.8 1017.6 1109.8 1200.0 498.0 : 203.9 266.9 315.6 407.8 498.0 702.0 764.9 813.7 905.9 996.1 1200.0 702.0 : 63.0 111.7 203.9 294.1 498.0 561.0 609.8 702.0 792.2 996.1 1200.0 764.9 : 48.8 140.9 231.2 435.1 498.0 546.8 639.0 729.2 933.1 1137.0 1200.0 813.7 : 92.2 182.4 386.3 449.3 498.0 590.2 680.4 884.4 1088.3 1151.2 1200.0 905.9 : 90.2 294.1 357.1 405.9 498.0 588.3 792.2 996.1 1059.1 1107.8 1200.0 996.1 : 203.9 266.9 315.6 407.8 498.0 702.0 905.9 968.8 1017.6 1109.8 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 48.7704 .. 203.9100 cents 35/32, 155.1396 cents 36/35, 48.7704 cents 2: 111.7313 .. 407.8200 cents 1215/1024, 296.0887 cents 128/135,-92.1787 cents 3: 203.9100 .. 498.0450 cents 32/27, 294.1350 cents 8/9,-203.9100 cents 4: 294.1350 .. 590.2237 cents 1215/1024, 296.0887 cents 8/9,-203.9100 cents 5: 498.0450 .. 653.1846 cents 35/32, 155.1396 cents 128/135,-92.1787 cents 6: 546.8154 .. 701.9550 cents 35/32, 155.1396 cents 1152/1225,-106.3692 cents 7: 609.7763 .. 905.8650 cents 1215/1024, 296.0887 cents 128/135,-92.1787 cents 8: 701.9550 .. 996.0900 cents 32/27, 294.1350 cents 8/9,-203.9100 cents 9: 792.1800 .. 1088.2687 cents 1215/1024, 296.0887 cents 8/9,-203.9100 cents 10: 996.0900 .. 1151.2296 cents 35/32, 155.1396 cents 128/135,-92.1787 cents
SHOW DATA
Number of notes : 11 -- Interval properties -- Smallest interval : 36/35, 48.7704 cents, class 1 Average step (divided formal octave): 109.0909 cents Largest one step interval : 9/8, 203.9100 cents Average / Smallest step : 2.236827 Largest / Average step : 1.869175 Largest / Smallest step : 4.181021 Median interval of one step : 256/243, 90.2250 cents, amount: 2 Most common interval of one step : 9/8, 203.9100 cents, amount: 3 Least squares average step : 102.67635 cents, oct.: 1129.43989 cents Scale is not proper Step pattern alph. order: ABCDEEABCDE Step pattern size order : BADCEEBADCE Scale is not a Constant Structure Scale diversity : 1.551997 Lumma stability : 0.081284 Lumma impropriety factor : 0.813781 Rothenberg efficiency : 0.392916 redundancy: 0.607084 Efficiency x scale size : 4.322078 Number of different interval sizes : 50 = 5.00000 / class Number of one step interval sizes : 5 Highest interval variety : 8 Mean interval variety : 6.20000 = 31/5 Median interval variety : 5 Lowest interval variety : 5 Smallest interval difference : 32805/32768, 1.9537 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 8 Most common triad is 0.0 498.045 701.955 cents, amount: 5 Number of recognisable fifths : 8, average 701.9550 cents Scale contains two identical disjunct hexachords Scale contains a complete diamond : 1 3 9 27 Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 7 = 63.6364% Scale is differentially coherent in interval classes 5 and 6 combined -- Rational properties -- Prime limit : 7 Odd number limit : 729 (O: 729 U: 729) Highest odd numerator or denominator: 27 Scale harmonicity : 0.023989 Average absolute harmonicity : 0.196981 Specific harmonicity : 0.153346 Fundamental : 1/2160, -11.0768 octaves, 0.1211 Hz. Guide tone : 6048, 12.5622 octaves, 1582311.419 Hz. Exponens Consonantiae : 1.306368E+07, 23.63906 octaves Euler's gradus suavitatis : 32 Sum of Mann's harmonic distance : 140.5, average 12.77273 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 9.98612, average 0.90783 Sum of Tenney's harmonic distance : 20.70722, average 1.88247 Vogel's harmonic complexity : 18.00000 Wille's k value : 1822 Wilson's harmonic complexity : 30 Rectangular lattice diameter : 8 Triangular lattice diameter : 6 Lattice compactness : 277.75383, average 4.20839 Lattice compactness (without 2's) : 159.53762, average 2.41724 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -4 .. 5 1.27273 30 1 3: -3 .. 3 -0.54545 18 6 5: -1 .. 0 -0.18182 2 2 7: 0 .. 1 0.18182 2 2 Average exponent except 2's :-6 / 11 =-0.54545 Average absolute exponent except 2's: 22 / 11 = 2.00000 Scale is weakly epimorphic with val: <11 15 27 28| Scale is weakly epimorphic with val: <11 17 24 31| Scale is weakly epimorphic with val: <11 17 25 32| Scale is weakly epimorphic with val: <11 19 26 28|
FIT/MODE
11: 0 1 1 0 2 2 1 0 1 1 2 N SD: 42.4310 c. M: 75.9532 c. 12: 1 0 1 1 2 2 1 0 1 1 2 N SD: 16.6098 c. M:-37.0391 c. 13: 1 0 1 1 2 3 0 1 1 1 2 N SD: 23.8035 c. M:-36.5065 c. 16: 1 0 2 1 3 2 1 1 1 1 3 N SD: 19.6703 c. M: 36.7313 c. 17: 1 1 1 1 3 3 1 1 1 1 3 N D3 S SD: 15.3869 c. M:-33.3725 c. 18: 1 1 1 1 3 4 0 1 2 1 3 N SD: 21.6878 c. M: 31.5826 c. 19: 1 1 1 2 3 3 1 1 1 2 3 N SD: 12.7035 c. M:-21.6545 c. 22: 1 1 2 1 4 4 1 1 2 1 4 N SD: 11.7761 c. M:-21.4077 c. 24: 1 1 2 2 4 4 1 1 2 2 4 N SD: 8.6467 c. M: 14.9159 c. 33: 2 1 3 2 6 5 2 1 3 2 6 N SD: 9.3591 c. M:-14.2718 c. 34: 2 1 3 2 6 6 2 1 3 2 6 N SD: 7.7546 c. M: 11.7821 c. 36: 2 1 3 3 6 6 2 1 3 3 6 N SD: 6.3871 c. M: 13.6863 c. 41: 2 2 3 3 7 7 2 2 3 3 7 N SD: 3.0769 c. M:-5.8259 c. 53: 3 2 4 4 9 9 3 2 4 4 9 N SD: 2.1929 c. M:-4.9636 c. 77: 4 3 6 6 13 13 4 3 6 6 13 N SD: 1.7036 c. M: 3.2967 c. 94: 5 4 7 7 16 16 5 4 7 7 16 N SD: 1.4708 c. M:-3.3350 c. 99: 5 4 8 7 17 17 5 4 8 7 17 N SD: 2.1101 c. M:-3.2259 c. 111: 6 4 9 8 19 19 6 4 9 8 19 N SD: 2.0825 c. M: 3.6232 c. 118: 6 5 9 9 20 20 6 5 9 9 20 N SD: 0.9800 c. M: 2.2040 c. 130: 7 5 10 10 22 22 7 5 10 10 22 N SD: 1.0425 c. M:-1.6545 c. 171: 9 7 13 13 29 29 9 7 13 13 29 N SD: 0.3801 c. M: 0.6018 c. 323: 17 13 25 24 55 55 17 13 25 24 55 N SD: 0.3750 c. M:-0.6365 c. 400: 21 16 31 30 68 68 21 16 31 30 68 N SD: 0.3120 c. M: 0.7313 c. 441: 23 18 34 33 75 75 23 18 34 33 75 N SD: 0.2058 c. M: 0.3759 c. 494: 26 20 38 37 84 84 26 20 38 37 84 N SD: 0.1509 c. M:-0.2663 c. 612: 32 25 47 46 104 104 32 25 47 46 104 N SD: 0.0926 c. M: 0.2158 c. 665: 35 27 51 50 113 113 35 27 51 50 113 N SD: 0.1051 c. M:-0.1970 c. 783: 41 32 60 59 133 133 41 32 60 59 133 N SD: 0.1040 c. M: 0.1649 c. 1065: 56 43 82 80 181 181 56 43 82 80 181 N SD: 0.1005 c. M: 0.1820 c. 1106: 58 45 85 83 188 188 58 45 85 83 188 N SD: 0.0583 c. M: 0.1024 c. 1277: 67 52 98 96 217 217 67 52 98 96 217 N SD: 0.0406 c. M:-0.0960 c. 1771: 93 72 136 133 301 301 93 72 136 133 301 N SD: 0.0561 c. M:-0.0912 c. 1848: 97 75 142 139 314 314 97 75 142 139 314 N SD: 0.0248 c. M: 0.0499 c.
FIT/HARMONIC
1 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 4 S SD: 0.0000 cents 3 x x x x 4 x x x 5 x 6 S SD: 7.1688 cents 4 x x x 5 x 6 x x x 7:8 S SD: 24.0316 cents 5 x x x 6:7 x x 8 x 9:10 S SD: 17.9552 cents 6 x x x 7:8:9 x x 10:11:12 S SD: 10.5986 cents 7 x x 8 x 9 x 11 x 12 x 14 SD: 15.1785 cents 8 x x 9 x 11:12 x 13 x 14:16 SD: 10.9313 cents 9 x x 10:11:12 x 14 x 15:16:18 S SD: 8.7634 cents 10 x x 11:12:13:15 x 16:17:18:20 SD: 8.4068 cents 11 x 12 x 13:15 x 17:18:19:20:22 SD: 11.0628 cents 12 x 13 x 14:16:18 x 19:20:21:24 S SD: 6.8446 cents 13 x 14:15 x 17 x 20:21:22:23:26 SD: 7.9648 cents 14 x 15:16:17:19:21:22 x 24:25:28 SD: 7.5498 cents 15 x 16:17:18:20 x 23:24:25:27:30 SD: 5.1775 cents 16 x 17:18:19:21:24:25:26:27:28:32 SD: 4.8207 cents 17 x 18:19:20:23 x 26:27:29:30:34 SD: 5.7020 cents 18 x 19:20:21:24:27:28:29:30:32:36 S SD: 4.6266 cents 19 x 20:21:23:25:29 x 30:32:34:38 SD: 7.7815 cents 20:21 x 23:24:27:30:31:32:34:36:40 SD: 5.9157 cents 21:22 x 24:25:28:32:33:34:35:37:42 SD: 5.7335 cents 22:23 x 25:26:29:33:34:35:37:39:44 SD: 3.4601 cents 27:28:29:30:32:36:41:42:43:46:48:54 SD: 3.3961 cents 30:31:32:34:36:40:45:47:48:51:53:60 SD: 3.0110 cents 32:33:34:36:38:43:48:50:51:54:57:64 SD: 1.9274 cents 48:50:51:54:57:64:72:75:77:81:85:96 SD: 1.4155 cents 58:60:62:65:69:77:87:90:93:98:103:116 SD: 1.3335 cents 72:75:77:81:85:96:108:112:115:122:128:144 SD: 1.2373 cents 80:83:85:90:95:107:120:124:128:135:142:160 SD: 1.0476 cents 86:89:92:97:102:115:129:134:138:145:153:172 SD: 1.0054 cents 90:93:96:101:107:120:135:140:144:152:160:180 SD: 0.8523 cents 102:106:109:115:121:136:153:159:163:172:181:204 SD: 0.7577 cents 108:112:115:122:128:144:162:168:173:182:192:216 SD: 0.7565 cents 130:135:139:146:154:173:195:202:208:219:231:260 SD: 0.6839 cents 134:139:143:151:159:179:201:208:214:226:238:268 SD: 0.6506 cents 135:140:144:152:160:180:203:210:216:228:240:270 SD: 0.4292 cents 162:168:173:182:192:216:243:252:259:273:288:324 SD: 0.3758 cents 210:218:224:236:249:280:315:327:336:354:373:420 SD: 0.3630 cents 238:247:254:268:282:317:357:370:381:402:423:476 SD: 0.3323 cents 240:249:256:270:284:320:360:373:384:405:427:480 SD: 0.3168 cents 270:280:288:304:320:360:405:420:432:456:480:540 S SD: 0.1831 cents 378:392:403:425:448:504:567:588:605:638:672:756 SD: 0.1354 cents 432:448:461:486:512:576:648:672:691:729:768:864 SD: 0.0821 cents