Scala analysis: kirnberger3
Kirnberger 3: 1/4 synt. comma (1744)
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: 193.157 cents 193.156860 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: 696.578 cents 696.578430 8: 405/256 794.133717 wide augmented fifth 9: 889.735 cents 889.735290 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: 100.9781 cents 100.9781 3: 100.9781 cents 100.9781 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: 106.3547 cents 106.3547 8: 97.5553 cents 97.5553 9: 95.6016 cents 95.6016 10: 106.3547 cents 106.3547 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: 4 135/128 92.179 cents major chroma, major limma 1: 1 95.60157 cents 1: 1 97.55529 cents 1: 2 100.97814 cents 1: 2 106.35471 cents 1: 2 16/15 111.731 cents minor diatonic semitone 2: 3 193.15686 cents 2: 2 198.53343 cents 2: 2 4096/3645 201.956 cents narrow diminished third 2: 5 9/8 203.910 cents major whole tone 3: 3 32/27 294.135 cents Pythagorean minor third 3: 3 1215/1024 296.089 cents wide augmented second 3: 2 299.51157 cents 3: 2 304.88814 cents 3: 2 310.26472 cents 4: 1 5/4 386.314 cents major third 4: 2 391.69029 cents 4: 2 397.06686 cents 4: 2 402.44343 cents 4: 4 512/405 405.866 cents narrow diminished fourth 4: 1 81/64 407.820 cents Pythagorean major third 5: 7 4/3 498.045 cents perfect fourth 5: 1 10935/8192 499.999 cents fourth + schisma, 5-limit approximation to ET fourth 5: 4 503.42157 cents 6: 3 45/32 590.224 cents diatonic tritone 6: 2 595.60029 cents 6: 1 599.02314 cents 6: 1 600.97686 cents 6: 2 604.39971 cents 6: 3 64/45 609.776 cents 2nd tritone 7: 4 696.57843 cents 7: 1 16384/10935 700.001 cents fifth - schisma, 5-limit approximation to ET fifth 7: 7 3/2 701.955 cents perfect fifth 8: 1 128/81 792.180 cents Pythagorean minor sixth 8: 4 405/256 794.134 cents wide augmented fifth 8: 2 797.55657 cents 8: 2 802.93314 cents 8: 2 808.30971 cents 8: 1 8/5 813.686 cents minor sixth 9: 2 889.73529 cents 9: 2 895.11185 cents 9: 2 900.48843 cents 9: 3 2048/1215 903.911 cents narrow diminished seventh 9: 3 27/16 905.865 cents Pythagorean major sixth 10: 5 16/9 996.090 cents Pythagorean minor seventh 10: 2 3645/2048 998.044 cents wide augmented sixth 10: 2 1001.46657 cents 10: 3 1006.84314 cents 11: 2 15/8 1088.269 cents classic major seventh 11: 2 1093.64529 cents 11: 2 1099.02186 cents 11: 1 1102.44471 cents 11: 1 1104.39843 cents 11: 4 256/135 1107.821 cents octave - major chroma Highest number of different intervals for one interval class: 6 Average number of different intervals per interval class: 4.90909 = 54/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 92.2 193.2 294.1 386.3 498.0 590.2 696.6 794.1 889.7 996.1 1088.3 1200.0 92.2 : 101.0 202.0 294.1 405.9 498.0 604.4 702.0 797.6 903.9 996.1 1107.8 1200.0 193.2 : 101.0 193.2 304.9 397.1 503.4 601.0 696.6 802.9 895.1 1006.8 1099.0 1200.0 294.1 : 92.2 203.9 296.1 402.4 500.0 595.6 702.0 794.1 905.9 998.0 1099.0 1200.0 386.3 : 111.7 203.9 310.3 407.8 503.4 609.8 702.0 813.7 905.9 1006.8 1107.8 1200.0 498.0 : 92.2 198.5 296.1 391.7 498.0 590.2 702.0 794.1 895.1 996.1 1088.3 1200.0 590.2 : 106.4 203.9 299.5 405.9 498.0 609.8 702.0 802.9 903.9 996.1 1107.8 1200.0 696.6 : 97.6 193.2 299.5 391.7 503.4 595.6 696.6 797.6 889.7 1001.5 1093.6 1200.0 794.1 : 95.6 202.0 294.1 405.9 498.0 599.0 700.0 792.2 903.9 996.1 1102.4 1200.0 889.7 : 106.4 198.5 310.3 402.4 503.4 604.4 696.6 808.3 900.5 1006.8 1104.4 1200.0 996.1 : 92.2 203.9 296.1 397.1 498.0 590.2 702.0 794.1 900.5 998.0 1093.6 1200.0 1088.3: 111.7 203.9 304.9 405.9 498.0 609.8 702.0 808.3 905.9 1001.5 1107.8 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 92.1787 .. 111.7313 cents 2048/2025, 19.5526 cents 135/128, 92.1787 cents 2: 193.1569 .. 203.9100 cents 10.75314 cents 81.42557 cents 3: 294.1350 .. 310.2647 cents 16.12972 cents 256/243, 90.2250 cents 4: 386.3137 .. 407.8200 cents 81/80, 21.5063 cents 76.04900 cents 5: 498.0450 .. 503.4216 cents 5.37657 cents 256/243, 90.2250 cents 6: 590.2237 .. 609.7763 cents 2048/2025, 19.5526 cents 86.80215 cents 7: 696.5784 .. 701.9550 cents 5.37657 cents 86.80215 cents 8: 792.1800 .. 813.6863 cents 81/80, 21.5063 cents 256/243, 90.2250 cents 9: 889.7353 .. 905.8650 cents 16.12971 cents 76.04900 cents 10: 996.0900 .. 1006.8431 cents 10.75314 cents 256/243, 90.2250 cents 11: 1088.2687 .. 1107.8213 cents 2048/2025, 19.5526 cents 81.42557 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 135/128, 92.1787 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 16/15, 111.7313 cents Average / Smallest step : 1.084849 Largest / Average step : 1.117313 Largest / Smallest step : 1.212116 Median interval of one step : 99.26672 cents, amount: 0 Most common interval of one step : 135/128, 92.1787 cents, amount: 4 Least squares average step : 99.24048 cents, oct.: 1190.88576 cents Scale is strictly proper Step pattern alph. order: ABBACADEFDAC Step pattern size order : ADDAFAECBEAF Scale is a Constant Structure, by a margin of 76.04900 cents Scale diversity : 1.114973 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.861509 Rothenberg efficiency : 0.285795 redundancy: 0.714205 Efficiency x scale size : 3.429545 Number of different interval sizes : 54 = 4.90909 / class Number of one step interval sizes : 6 Highest interval variety : 6 Mean interval variety : 4.90909 = 54/11 Median interval variety : 3 Lowest interval variety : 3 Smallest interval difference : 1.95371 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 7 Most common triad is 0.0 498.045 701.955 cents, amount: 5 Number of recognisable fifths : 12, average 700.0000 cents Scale is a well-temperament. It has 3 different sizes of fifth Scale has monotonic third-sizes over circle of fifths Best fifths form a closed circle Formal octave complements present : 1 = 8.3333% Limited transpositions with margin 21.5063 cents: 2 3 5 7 9 10 Limited inverse transpositions with margin 21.5063 cents: 1 2 4 6 7 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: 7.8183 c. M:-13.6863 c. Twelve-tone Chromatic (1/11-comma) 41: 3 3 4 3 4 3 4 3 3 4 3 4 SP M ME SD: 7.0854 c. M: 17.5471 c. 43: 3 4 4 3 4 3 4 3 4 4 3 4 SP M ME SD: 6.7319 c. M:-12.8417 c. Meantone Chromatic (1/5-comma) 49: 4 4 4 4 4 4 4 4 4 5 3 5 SP D3 SD: 7.1739 c. M: 10.8641 c. 50: 4 4 4 4 5 4 4 4 4 5 3 5 SP D3 SD: 5.8316 c. M:-11.9100 c. 53: 4 5 4 4 5 4 5 4 4 5 4 5 SP M ME SD: 4.0625 c. M:-10.6167 c. Schismic-12 54: 4 5 4 4 5 5 4 5 4 5 4 5 SP D3 SD: 6.1059 c. M:-9.7763 c. 55: 4 5 4 5 5 4 5 4 5 5 4 5 SP M ME SD: 5.5241 c. M: 10.4986 c. 57: 4 5 5 4 6 4 5 5 4 5 5 5 SP D3 SD: 5.3095 c. M: 7.9682 c. 62: 5 5 5 5 6 4 6 5 5 5 5 6 SP D3 SD: 4.6184 c. M: 9.5786 c. 65: 5 5 6 5 6 5 6 5 5 6 5 6 SP M ME SD: 3.1096 c. M: 8.5415 c. 69: 5 6 6 5 7 5 6 6 5 6 6 6 SP D3 SD: 4.1637 c. M:-7.3835 c. 74: 6 6 6 6 7 5 7 6 6 6 6 7 SP D3 SD: 3.6619 c. M: 6.9008 c. 77: 6 6 7 6 7 6 7 6 6 7 6 7 SP M ME SD: 2.7700 c. M: 6.1439 c. 86: 7 7 7 7 8 6 8 7 7 7 7 8 SP D3 SD: 3.3393 c. M:-5.4957 c. 89: 7 7 8 7 8 7 8 7 7 8 7 8 SP M ME SD: 2.9243 c. M:-4.6975 c. 101: 8 8 9 8 9 8 9 8 8 9 8 9 SP M ME SD: 3.2475 c. M:-5.7655 c. 105: 8 9 9 8 10 8 9 9 8 9 8 10 SP SD: 2.9300 c. M:-5.8663 c. 106: 8 9 9 8 10 8 10 8 9 9 8 10 SP D3 SD: 2.2714 c. M:-5.3084 c. 116: 9 10 9 9 11 9 10 10 9 10 9 11 SP SD: 2.5616 c. M: 4.4798 c. 117: 9 10 10 9 11 9 10 10 9 10 9 11 SP SD: 3.0562 c. M:-5.8663 c. 118: 9 10 10 9 11 9 10 10 9 11 9 11 SP D3 SD: 2.0982 c. M: 5.0530 c. 120: 9 10 10 10 11 9 11 9 10 11 9 11 SP T3 SD: 2.8346 c. M: 4.1350 c. 124: 9 11 10 10 11 10 11 10 10 11 9 12 SP SD: 2.6123 c. M: 5.0819 c. 127: 10 10 11 10 12 9 12 10 10 11 10 12 SP SD: 2.5870 c. M: 4.3969 c. 128: 10 11 10 10 12 10 11 11 10 11 10 12 SP SD: 2.1655 c. M:-3.7181 c. 130: 10 11 11 10 12 10 11 11 10 12 10 12 SP D3 SD: 1.7605 c. M: 4.2707 c. 139: 11 11 12 11 13 10 13 11 11 12 11 13 SP SD: 2.2199 c. M: 3.2842 c. 140: 11 12 11 11 13 11 12 12 11 12 11 13 SP SD: 2.0576 c. M:-3.9860 c. 142: 11 12 12 11 13 11 12 12 11 13 11 13 SP D3 SD: 1.7402 c. M: 3.6207 c. 143: 11 12 12 11 13 11 13 12 11 13 11 13 SP SD: 1.8169 c. M:-3.0691 c. 159: 12 13 14 12 15 12 14 13 13 14 12 15 SP SD: 1.7746 c. M: 4.4776 c. 169: 13 14 14 13 16 13 15 14 13 15 13 16 SP SD: 1.7142 c. M: 3.0107 c. 171: 13 15 14 13 16 13 15 14 14 15 13 16 SP SD: 1.3076 c. M:-3.3344 c. 174: 13 15 15 13 16 14 15 14 14 15 14 16 SP S SD: 1.6994 c. M: 2.9865 c. 181: 14 15 15 14 17 14 16 15 14 16 14 17 SP SD: 1.2440 c. M: 2.4223 c. 183: 14 15 16 14 17 14 16 15 15 16 14 17 SP SD: 1.2208 c. M: 2.9929 c. 193: 15 16 16 15 18 15 17 16 15 17 15 18 SP SD: 0.9704 c. M: 1.9070 c. 205: 16 17 17 16 19 16 18 17 16 18 16 19 SP SD: 0.9390 c. M:-1.9638 c. 212: 16 18 18 16 20 16 19 17 17 19 16 20 SP S SD: 1.0713 c. M: 1.6809 c. 224: 17 19 19 17 21 17 20 18 18 20 17 21 SP S SD: 0.6709 c. M: 1.2766 c. 236: 18 20 20 18 22 18 21 19 19 21 18 22 SP S SD: 0.4482 c. M: 0.9134 c. 248: 19 21 21 19 23 19 22 20 20 22 19 23 SP S SD: 0.5318 c. M:-1.0263 c. 405: 31 34 34 31 38 31 36 33 32 36 31 38 SP SD: 0.6187 c. M: 1.1285 c. 417: 32 35 35 32 39 32 37 34 33 37 32 39 SP SD: 0.3937 c. M: 0.7022 c. 429: 33 36 36 33 40 33 38 35 34 38 33 40 SP SD: 0.2179 c. M: 0.4287 c. 441: 34 37 37 34 41 34 39 36 35 39 34 41 SP SD: 0.2044 c. M:-0.4241 c. 653: 50 55 55 50 61 50 58 53 52 58 50 61 SP SD: 0.2450 c. M: 0.4025 c. 665: 51 56 56 51 62 51 59 54 53 59 51 62 SP SD: 0.1039 c. M: 0.1488 c. 677: 52 57 57 52 63 52 60 55 54 60 52 63 SP SD: 0.0584 c. M:-0.1043 c. 1106: 85 93 93 85 103 85 98 90 88 98 85 103 SP SD: 0.0512 c. M: 0.1024 c. 1342: 103 113 113 103 125 103 119 109 107 119 103 125 SP SD: 0.0464 c. M: 0.0950 c. 1771: 136 149 149 136 165 136 157 144 141 157 136 165 SP SD: 0.0500 c. M: 0.0912 c. 1783: 137 150 150 137 166 137 158 145 142 158 137 166 SP SD: 0.0160 c. M:-0.0334 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: 2.6883 cents 3 x x x x 4 x x x 5 x x 6 S SD: 1.7922 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 6.9473 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: 8.0187 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 11.8323 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 8.4353 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 7.7200 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.3965 cents 11 x 12:13:14:15 x 16:17:18:20:21:22 S SD: 11.0088 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 6.6245 cents 13:14 x 15:16:17:18:19:21:22:23:24:26 SD: 9.1515 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 8.0918 cents 15:16:17:18:19:20:21:22:24:25:27:28:30 SD: 5.2975 cents 16:17:18:19:20:21:23:24:25:27:28:30:32 SD: 5.2549 cents 17:18:19:20:21:23:24:25:27:28:30:32:34 SD: 4.6200 cents 22:23:25:26:28:29:31:33:35:37:39:41:44 SD: 4.4009 cents 24:25:27:28:30:32:34:36:38:40:43:45:48 SD: 3.4622 cents 27:28:30:32:34:36:38:40:43:45:48:51:54 SD: 3.4557 cents 31:33:35:37:39:41:44:46:49:52:55:58:62 SD: 3.2394 cents 32:34:36:38:40:43:45:48:51:54:57:60:64 SD: 2.5489 cents 33:35:37:39:41:44:46:49:52:55:59:62:66 SD: 2.3837 cents 36:38:40:43:45:48:51:54:57:60:64:68:72 SD: 2.1818 cents 40:42:45:47:50:53:56:60:63:67:71:75:80 SD: 2.1762 cents 48:51:54:57:60:64:68:72:76:80:85:90:96 S SD: 1.9708 cents 49:52:55:58:61:65:69:73:78:82:87:92:98 SD: 1.8049 cents 52:55:58:62:65:69:73:78:82:87:92:98:104 SD: 1.7421 cents 54:57:60:64:68:72:76:81:85:90:96:101:108 SD: 1.7386 cents 55:58:61:65:69:73:77:82:87:92:98:103:110 SD: 1.7200 cents 57:60:64:68:71:76:80:85:90:95:101:107:114 SD: 1.5431 cents 59:62:66:70:74:79:83:88:93:99:105:111:118 SD: 1.3683 cents 69:73:77:82:86:92:97:103:109:115:123:129:138 SD: 1.1144 cents 76:80:85:90:95:101:107:114:120:127:135:143:152 SD: 0.9510 cents 91:96:102:108:114:121:128:136:144:152:162:171:182 SD: 0.7737 cents 103:109:115:122:129:137:145:154:163:172:183:193:206 SD: 0.7458 cents 113:119:126:134:141:151:159:169:179:189:201:212:226 SD: 0.6555 cents 125:132:140:148:156:167:176:187:198:209:222:234:250 SD: 0.6296 cents 129:136:144:153:161:172:181:193:204:216:229:242:258 SD: 0.5727 cents 144:152:161:171:180:192:203:215:228:241:256:270:288 SD: 0.5532 cents 145:153:162:172:181:193:204:217:229:242:258:272:290 SD: 0.5417 cents 153:161:171:181:191:204:215:229:242:256:272:287:306 SD: 0.5136 cents 162:171:181:192:203:216:228:242:256:271:288:304:324 SD: 0.4833 cents 167:176:187:198:209:223:235:250:264:279:297:313:334 SD: 0.4518 cents 168:177:188:199:210:224:236:251:266:281:299:315:336 SD: 0.3611 cents 204:215:228:242:255:272:287:305:323:341:363:383:408 SD: 0.3215 cents 237:250:265:281:296:316:333:354:375:396:421:444:474 SD: 0.3085 cents 256:270:286:303:320:341:360:383:405:428:455:480:512 SD: 0.2753 cents 271:286:303:321:339:361:381:405:429:453:482:508:542 SD: 0.2592 cents 296:312:331:351:370:395:416:443:468:495:526:555:592 SD: 0.2533 cents 297:313:332:352:371:396:418:444:470:497:528:557:594 SD: 0.2424 cents 313:330:350:371:391:417:440:468:495:523:556:587:626 SD: 0.2262 cents 345:364:386:409:431:460:485:516:546:577:613:647:690 SD: 0.1940 cents 347:366:388:411:434:463:488:519:549:580:617:651:694 SD: 0.1901 cents 384:405:429:455:480:512:540:574:608:642:683:720:768 SD: 0.1874 cents 399:421:446:473:499:532:561:597:631:667:709:748:798 SD: 0.1652 cents 492:519:550:583:615:656:692:736:778:823:875:923:984 SD: 0.1596 cents 507:535:567:601:634:676:713:758:802:848:901:951:1014 SD: 0.1486 cents 549:579:614:651:686:732:772:821:869:918:976:1029:1098 SD: 0.1410 cents 551:581:616:653:689:735:775:824:872:921:980:1033:1102 SD: 0.1310 cents 567:598:634:672:709:756:797:848:897:948:1008:1063:1134 SD: 0.0876 cents