Scala analysis: kirnberger2
Kirnberger 2: 1/2 synt. comma. "Die Kunst des reinen Satzes" (1774)
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 135/128 92.178716 major chroma, major limma 2: 9/8 203.910002 major whole tone 3: 32/27 294.134997 Pythagorean minor third 4: 5/4 386.313714 major third 5: 4/3 498.044999 perfect fourth 6: 45/32 590.223716 diatonic tritone 7: 3/2 701.955001 perfect fifth 8: 405/256 794.133717 wide augmented fifth 9: 895.112 cents 895.111860 10: 16/9 996.089998 Pythagorean minor seventh 11: 15/8 1088.268715 classic major seventh 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 16/15 111.7313 minor diatonic semitone 1: 135/128 92.1787 major chroma, major limma 2: 16/15 111.7313 minor diatonic semitone 3: 256/243 90.2250 limma, Pythagorean minor second 4: 135/128 92.1787 major chroma, major limma 5: 16/15 111.7313 minor diatonic semitone 6: 135/128 92.1787 major chroma, major limma 7: 16/15 111.7313 minor diatonic semitone 8: 135/128 92.1787 major chroma, major limma 9: 100.9781 cents 100.9781 10: 100.9781 cents 100.9781 11: 135/128 92.1787 major chroma, major limma 12: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 256/243 90.225 cents limma, Pythagorean minor second 1: 5 135/128 92.179 cents major chroma, major limma 1: 2 100.97814 cents 1: 4 16/15 111.731 cents minor diatonic semitone 2: 1 10/9 182.404 cents minor whole tone 2: 2 193.15686 cents 2: 2 4096/3645 201.956 cents narrow diminished third 2: 7 9/8 203.910 cents major whole tone 3: 5 32/27 294.135 cents Pythagorean minor third 3: 3 1215/1024 296.089 cents wide augmented second 3: 2 304.88814 cents 3: 2 6/5 315.641 cents minor third 4: 3 5/4 386.314 cents major third 4: 2 397.06686 cents 4: 4 512/405 405.866 cents narrow diminished fourth 4: 3 81/64 407.820 cents Pythagorean major third 5: 9 4/3 498.045 cents perfect fourth 5: 1 10935/8192 499.999 cents fourth + schisma, 5-limit approximation to ET fourth 5: 2 508.79815 cents 6: 5 45/32 590.224 cents diatonic tritone 6: 1 599.02314 cents 6: 1 600.97686 cents 6: 5 64/45 609.776 cents 2nd tritone 7: 2 691.20186 cents 7: 1 16384/10935 700.001 cents fifth - schisma, 5-limit approximation to ET fifth 7: 9 3/2 701.955 cents perfect fifth 8: 3 128/81 792.180 cents Pythagorean minor sixth 8: 4 405/256 794.134 cents wide augmented fifth 8: 2 802.93314 cents 8: 3 8/5 813.686 cents minor sixth 9: 2 5/3 884.359 cents major sixth, BP sixth 9: 2 895.11186 cents 9: 3 2048/1215 903.911 cents narrow diminished seventh 9: 5 27/16 905.865 cents Pythagorean major sixth 10: 7 16/9 996.090 cents Pythagorean minor seventh 10: 2 3645/2048 998.044 cents wide augmented sixth 10: 2 1006.84314 cents 10: 1 9/5 1017.596 cents just minor seventh, BP seventh 11: 4 15/8 1088.269 cents classic major seventh 11: 2 1099.02186 cents 11: 5 256/135 1107.821 cents octave - major chroma 11: 1 243/128 1109.775 cents Pythagorean major seventh Highest number of different intervals for one interval class: 4 Average number of different intervals per interval class: 3.81818 = 42/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 92.2 203.9 294.1 386.3 498.0 590.2 702.0 794.1 895.1 996.1 1088.3 1200.0 92.2 : 111.7 202.0 294.1 405.9 498.0 609.8 702.0 802.9 903.9 996.1 1107.8 1200.0 203.9 : 90.2 182.4 294.1 386.3 498.0 590.2 691.2 792.2 884.4 996.1 1088.3 1200.0 294.1 : 92.2 203.9 296.1 407.8 500.0 601.0 702.0 794.1 905.9 998.0 1109.8 1200.0 386.3 : 111.7 203.9 315.6 407.8 508.8 609.8 702.0 813.7 905.9 1017.6 1107.8 1200.0 498.0 : 92.2 203.9 296.1 397.1 498.0 590.2 702.0 794.1 905.9 996.1 1088.3 1200.0 590.2 : 111.7 203.9 304.9 405.9 498.0 609.8 702.0 813.7 903.9 996.1 1107.8 1200.0 702.0 : 92.2 193.2 294.1 386.3 498.0 590.2 702.0 792.2 884.4 996.1 1088.3 1200.0 794.1 : 101.0 202.0 294.1 405.9 498.0 609.8 700.0 792.2 903.9 996.1 1107.8 1200.0 895.1 : 101.0 193.2 304.9 397.1 508.8 599.0 691.2 802.9 895.1 1006.8 1099.0 1200.0 996.1 : 92.2 203.9 296.1 407.8 498.0 590.2 702.0 794.1 905.9 998.0 1099.0 1200.0 1088.3: 111.7 203.9 315.6 405.9 498.0 609.8 702.0 813.7 905.9 1006.8 1107.8 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.2250 .. 111.7313 cents 81/80, 21.5063 cents 256/243, 90.2250 cents 2: 182.4037 .. 203.9100 cents 81/80, 21.5063 cents 25/24, 70.6724 cents 3: 294.1350 .. 315.6413 cents 81/80, 21.5063 cents 256/243, 90.2250 cents 4: 386.3137 .. 407.8200 cents 81/80, 21.5063 cents 25/24, 70.6724 cents 5: 498.0450 .. 508.7981 cents 10.75315 cents 256/243, 90.2250 cents 6: 590.2237 .. 609.7763 cents 2048/2025, 19.5526 cents 81.42557 cents 7: 691.2019 .. 701.9550 cents 10.75314 cents 81.42557 cents 8: 792.1800 .. 813.6863 cents 81/80, 21.5063 cents 256/243, 90.2250 cents 9: 884.3587 .. 905.8650 cents 81/80, 21.5063 cents 25/24, 70.6724 cents 10: 996.0900 .. 1017.5963 cents 81/80, 21.5063 cents 256/243, 90.2250 cents 11: 1088.2687 .. 1109.7750 cents 81/80, 21.5063 cents 25/24, 70.6724 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 256/243, 90.2250 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 16/15, 111.7313 cents Average / Smallest step : 1.108340 Largest / Average step : 1.117313 Largest / Smallest step : 1.238363 Median interval of one step : 96.57843 cents, amount: 0 Most common interval of one step : 135/128, 92.1787 cents, amount: 5 Least squares average step : 99.40591 cents, oct.: 1192.87095 cents Scale is strictly proper Step pattern alph. order: ABCABABADDAB Step pattern size order : BDABDBDBCCBD Scale is a Constant Structure, by a margin of 70.67243 cents Scale diversity : 1.143641 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.822409 Rothenberg efficiency : 0.383691 redundancy: 0.616309 Efficiency x scale size : 4.604293 Number of different interval sizes : 42 = 3.81818 / class Number of one step interval sizes : 4 Highest interval variety : 4 Mean interval variety : 3.81818 = 42/11 Median interval variety : 3 Lowest interval variety : 3 Smallest interval difference : 32805/32768, 1.9537 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 9 Most common triad is 0.0 498.045 701.955 cents, amount: 7 Number of recognisable fifths : 12, average 700.0000 cents Scale is a well-temperament. It has 3 different sizes of fifth Best fifths form a closed circle Scale contains a complete diamond : 1 3 9 Formal octave complements present : 5 = 41.6667% Limited transpositions with margin 21.5063 cents: 5 7 9 Limited inverse transpositions with margin 21.5063 cents: 1 6 9 11 -- Rational properties -- Prime limit : 5 (not all pitches rational)
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 7.1450 c. M:-13.6863 c. Twelve-tone Chromatic (1/11-comma) 41: 3 4 3 3 4 3 4 3 4 3 3 4 SP M ME SD: 4.7763 c. M:-12.2052 c. 46: 4 4 3 4 4 4 4 3 4 4 4 4 SP D3 S SD: 7.2526 c. M:-12.1691 c. 49: 4 4 4 4 4 4 5 3 5 4 3 5 SP D3 SD: 7.5245 c. M:-11.0106 c. 51: 4 5 4 3 5 4 5 4 4 4 4 5 SP D3 SD: 6.2399 c. M:-11.7474 c. 52: 4 5 4 4 5 4 4 4 5 4 4 5 SP D3 SD: 6.5097 c. M:-9.7763 c. 53: 4 5 4 4 5 4 5 4 5 4 4 5 SP M ME SD: 3.2059 c. M:-10.5485 c. 63: 5 6 4 5 6 5 6 5 5 5 5 6 SP D3 SD: 4.3667 c. M: 8.4207 c. 65: 5 6 5 5 6 5 6 5 5 6 5 6 SP M ME SD: 2.6890 c. M: 8.9580 c. 69: 5 7 5 5 7 5 6 6 5 6 6 6 SP D3 SD: 5.2055 c. M: 8.1553 c. 70: 5 7 5 5 7 5 7 5 6 6 5 7 SP D3 S SD: 5.0703 c. M: 9.1709 c. 71: 5 7 5 6 6 6 6 6 6 6 5 7 SP D3 SD: 5.0203 c. M: 8.9973 c. 74: 6 7 5 6 7 5 7 6 6 6 6 7 SP SD: 4.4098 c. M:-6.9008 c. 75: 6 7 5 6 7 6 7 6 6 6 6 7 SP D3 SD: 3.3892 c. M: 6.1350 c. 77: 6 7 6 6 7 6 7 6 6 7 6 7 SP M ME SD: 2.5563 c. M: 6.8002 c. 78: 6 7 6 6 7 6 8 6 6 7 6 7 SP D3 SD: 4.0072 c. M:-5.8663 c. 82: 6 8 6 6 8 6 8 6 7 7 6 8 SP D3 S SD: 3.3001 c. M: 5.8259 c. 86: 7 8 6 7 8 6 8 7 7 7 7 8 SP SD: 3.7501 c. M:-5.4957 c. 87: 7 8 6 7 8 7 8 7 7 7 7 8 SP D3 SD: 3.0028 c. M:-5.8663 c. 89: 7 8 7 7 8 7 8 7 7 8 7 8 SP M ME SD: 2.7896 c. M: 5.2242 c. 94: 7 9 7 7 9 7 9 7 8 8 7 9 SP D3 S SD: 1.9956 c. M: 3.3350 c. 106: 8 10 8 8 10 8 10 8 9 9 8 10 SP D3 S SD: 1.0271 c. M: 1.6809 c. 118: 9 11 9 9 11 9 11 9 10 10 9 11 SP D3 S SD: 0.4807 c. M: 0.9134 c. 311: 24 29 23 24 29 24 29 24 26 26 24 29 SP SD: 0.4678 c. M: 0.8874 c. 323: 25 30 24 25 30 25 30 25 27 27 25 30 SP SD: 0.4617 c. M:-0.9127 c. 417: 32 39 31 32 39 32 39 32 35 35 32 39 SP SD: 0.3716 c. M: 0.7022 c. 429: 33 40 32 33 40 33 40 33 36 36 33 40 SP SD: 0.2223 c. M: 0.4287 c. 441: 34 41 33 34 41 34 41 34 37 37 34 41 SP SD: 0.2137 c. M:-0.4241 c. 547: 42 51 41 42 51 42 51 42 46 46 42 51 SP SD: 0.1076 c. M: 0.2077 c. 559: 43 52 42 43 52 43 52 43 47 47 43 52 SP SD: 0.0794 c. M:-0.1418 c. 677: 52 63 51 52 63 52 63 52 57 57 52 63 SP SD: 0.0570 c. M:-0.1043 c. 1106: 85 103 83 85 103 85 103 85 93 93 85 103 SP SD: 0.0536 c. M: 0.1024 c. 1224: 94 114 92 94 114 94 114 94 103 103 94 114 SP SD: 0.0204 c. M: 0.0392 c. 1783: 137 166 134 137 166 137 166 137 150 150 137 166 SP SD: 0.0169 c. M:-0.0334 c. 1901: 146 177 143 146 177 146 177 146 160 160 146 177 SP SD: 0.0145 c. M:-0.0260 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 x 3 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: 3.5844 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 6.8160 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 7.3941 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 8.1183 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 10.8049 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 8.3004 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 8.0593 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.6688 cents 11 x 12:13:14:15 x x 17:18:20:21:22 SD: 11.8027 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 6.6605 cents 13:14:15 x 16:17:18:20:21:22:23:24:26 S SD: 9.1175 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 7.9287 cents 15:16:17:18:19:20:21:23:24:25:27:28:30 SD: 5.3106 cents 16:17:18:19:20:21:23:24:25:27:28:30:32 SD: 5.0603 cents 21:22:24:25:26:28:30:32:33:35:37:39:42 SD: 4.9062 cents 22:23:25:26:28:29:31:33:35:37:39:41:44 SD: 3.8985 cents 24:25:27:28:30:32:34:36:38:40:43:45:48 SD: 3.4037 cents 31:33:35:37:39:41:44:47:49:52:55:58:62 SD: 3.1116 cents 32:34:36:38:40:43:45:48:51:54:57:60:64 SD: 2.1188 cents 40:42:45:47:50:53:56:60:63:67:71:75:80 SD: 1.9208 cents 48:51:54:57:60:64:68:72:76:81:85:90:96 SD: 1.8661 cents 54:57:61:64:68:72:76:81:85:91:96:101:108 SD: 1.6299 cents 55:58:62:65:69:73:77:83:87:92:98:103:110 SD: 1.5523 cents 56:59:63:66:70:75:79:84:89:94:100:105:112 SD: 1.4707 cents 64:68:72:76:80:85:90:96:101:107:114:120:128 SD: 1.3918 cents 71:75:80:84:89:95:100:107:112:119:126:133:142 SD: 1.1662 cents 72:76:81:85:90:96:101:108:114:121:128:135:144 SD: 0.7516 cents 96:101:108:114:120:128:135:144:152:161:171:180:192 SD: 0.5475 cents 152:160:171:180:190:203:214:228:240:255:270:285:304 SD: 0.5233 cents 160:169:180:190:200:213:225:240:253:268:284:300:320 SD: 0.5127 cents 167:176:188:198:209:223:235:251:264:280:297:313:334 SD: 0.4600 cents 168:177:189:199:210:224:236:252:266:282:299:315:336 SD: 0.3320 cents 184:194:207:218:230:245:259:276:291:309:327:345:368 SD: 0.3256 cents 216:228:243:256:270:288:304:324:342:362:384:405:432 SD: 0.2272 cents 288:304:324:341:360:384:405:432:456:483:512:540:576 SD: 0.2192 cents 312:329:351:370:390:416:439:468:494:523:555:585:624 SD: 0.2025 cents 328:346:369:389:410:437:461:492:519:550:583:615:656 SD: 0.1731 cents 384:405:432:455:480:512:540:576:608:644:683:720:768 SD: 0.1425 cents 480:506:540:569:600:640:675:720:759:805:853:900:960 SD: 0.1189 cents 512:540:576:607:640:683:720:768:810:859:910:960:1024 SD: 0.1077 cents 584:616:657:692:730:779:821:876:924:979:1038:1095:1168 SD: 0.1073 cents 600:633:675:711:750:800:844:900:949:1006:1067:1125:1200 SD: 0.0916 cents