Scala analysis: kirnberger1
Kirnberger's temperament 1 (1766)
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 256/243 90.224996 limma, Pythagorean minor second 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: 128/81 792.179997 Pythagorean minor sixth 9: 895.112 cents 895.112000 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: 256/243 90.2250 limma, Pythagorean minor second 2: 2187/2048 113.6850 apotome 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: 256/243 90.2250 limma, Pythagorean minor second 9: 102.9320 cents 102.9320 10: 100.9780 cents 100.9780 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: 3 256/243 90.225 cents limma, Pythagorean minor second 1: 3 135/128 92.179 cents major chroma, major limma 1: 1 100.97800 cents 1: 1 102.93200 cents 1: 3 16/15 111.731 cents minor diatonic semitone 1: 1 2187/2048 113.685 cents apotome 2: 1 10/9 182.404 cents minor whole tone 2: 1 193.15671 cents 2: 1 193.15700 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: 1 304.88800 cents 3: 1 304.88828 cents 3: 2 6/5 315.641 cents minor third 4: 3 5/4 386.314 cents major third 4: 1 395.11300 cents 4: 1 397.06700 cents 4: 3 512/405 405.866 cents narrow diminished fourth 4: 4 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: 1 508.79800 cents 5: 1 508.79829 cents 6: 2 1024/729 588.270 cents Pythagorean diminished fifth 6: 3 45/32 590.224 cents diatonic tritone 6: 1 599.02300 cents 6: 1 600.97700 cents 6: 3 64/45 609.776 cents 2nd tritone 6: 2 729/512 611.730 cents Pythagorean tritone 7: 1 691.20171 cents 7: 1 691.20200 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: 4 128/81 792.180 cents Pythagorean minor sixth 8: 3 405/256 794.134 cents wide augmented fifth 8: 1 802.93300 cents 8: 1 804.88700 cents 8: 3 8/5 813.686 cents minor sixth 9: 2 5/3 884.359 cents major sixth, BP sixth 9: 1 895.11172 cents 9: 1 895.11200 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: 1 1006.84300 cents 10: 1 1006.84329 cents 10: 1 9/5 1017.596 cents just minor seventh, BP seventh 11: 1 4096/2187 1086.315 cents Pythagorean diminished octave 11: 3 15/8 1088.269 cents classic major seventh 11: 1 1097.06800 cents 11: 1 1099.02200 cents 11: 3 256/135 1107.821 cents octave - major chroma 11: 3 243/128 1109.775 cents Pythagorean major seventh Highest number of different intervals for one interval class: 6 Average number of different intervals per interval class: 5.09091 = 56/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 90.2 203.9 294.1 386.3 498.0 590.2 702.0 792.2 895.1 996.1 1088.3 1200.0 90.2 : 113.7 203.9 296.1 407.8 500.0 611.7 702.0 804.9 905.9 998.0 1109.8 1200.0 203.9 : 90.2 182.4 294.1 386.3 498.0 588.3 691.2 792.2 884.4 996.1 1086.3 1200.0 294.1 : 92.2 203.9 296.1 407.8 498.0 601.0 702.0 794.1 905.9 996.1 1109.8 1200.0 386.3 : 111.7 203.9 315.6 405.9 508.8 609.8 702.0 813.7 903.9 1017.6 1107.8 1200.0 498.0 : 92.2 203.9 294.1 397.1 498.0 590.2 702.0 792.2 905.9 996.1 1088.3 1200.0 590.2 : 111.7 202.0 304.9 405.9 498.0 609.8 700.0 813.7 903.9 996.1 1107.8 1200.0 702.0 : 90.2 193.2 294.1 386.3 498.0 588.3 702.0 792.2 884.4 996.1 1088.3 1200.0 792.2 : 102.9 203.9 296.1 407.8 498.0 611.7 702.0 794.1 905.9 998.0 1109.8 1200.0 895.1 : 101.0 193.2 304.9 395.1 508.8 599.0 691.2 802.9 895.1 1006.8 1097.1 1200.0 996.1 : 92.2 203.9 294.1 407.8 498.0 590.2 702.0 794.1 905.9 996.1 1099.0 1200.0 1088.3: 111.7 202.0 315.6 405.9 498.0 609.8 702.0 813.7 903.9 1006.8 1107.8 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.2250 .. 113.6850 cents 531441/524288, 23.4600 cents 256/243, 90.2250 cents 2: 182.4037 .. 203.9100 cents 81/80, 21.5063 cents 20480/19683, 68.7187 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 .. 508.7983 cents 10.75329 cents 256/243, 90.2250 cents 6: 588.2700 .. 611.7300 cents 531441/524288, 23.4600 cents 79.47171 cents 7: 691.2017 .. 701.9550 cents 10.75329 cents 79.47171 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: 1086.3150 .. 1109.7750 cents 531441/524288, 23.4600 cents 20480/19683, 68.7187 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 : 2187/2048, 113.6850 cents Average / Smallest step : 1.108340 Largest / Average step : 1.136850 Largest / Smallest step : 1.260017 Median interval of one step : 96.57836 cents, amount: 0 Least squares average step : 99.37886 cents, oct.: 1192.54636 cents Scale is strictly proper Step pattern alph. order: ABACDCDAEFCD Step pattern size order : AFABEBEADCBE Scale is a Constant Structure, by a margin of 68.71871 cents Scale diversity : 1.149899 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.815896 Rothenberg efficiency : 0.369697 redundancy: 0.630303 Efficiency x scale size : 4.436364 Number of different interval sizes : 56 = 5.09091 / class Number of one step interval sizes : 6 Highest interval variety : 6 Mean interval variety : 5.09091 = 56/11 Median interval variety : 4 Lowest interval variety : 4 Smallest interval difference : 0.00028 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 4 different sizes of fifth Scale has monotonic third-sizes over circle of fifths 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 23.4600 cents: 5 7 9 Limited inverse transpositions with margin 23.4600 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: 7.4929 c. M:-13.6863 c. Twelve-tone Chromatic (1/11-comma) 40: 3 4 3 3 4 3 3 3 4 3 3 4 SP D3 SD: 7.8753 c. M: 12.1800 c. 41: 3 4 3 3 4 3 4 3 4 3 3 4 SP M ME SD: 4.5561 c. M:-12.2051 c. 46: 3 5 3 4 4 4 4 3 4 4 4 4 SP D3 SD: 6.9824 c. M: 11.9641 c. 51: 4 5 4 3 5 4 5 4 4 4 4 5 SP D3 SD: 6.4896 c. M:-11.7474 c. 52: 4 5 4 4 5 4 4 4 5 4 4 5 SP D3 SD: 6.3210 c. M:-9.7763 c. 53: 4 5 4 4 5 4 5 4 5 4 4 5 SP M ME SD: 3.1371 c. M:-10.5484 c. 63: 5 6 4 5 6 5 6 5 5 5 5 6 SP D3 SD: 4.7551 c. M: 8.4207 c. 65: 5 6 5 5 6 5 6 5 5 6 5 6 SP M ME SD: 2.7956 c. M: 8.9582 c. 67: 5 6 5 6 6 5 6 5 6 6 5 6 SP M ME SD: 4.7623 c. M:-7.7161 c. 70: 5 7 5 5 7 5 7 5 6 6 5 7 SP D3 S SD: 4.7359 c. M: 9.1709 c. 75: 6 7 5 6 7 6 7 6 6 6 6 7 SP D3 SD: 3.9087 c. M:-7.8200 c. 77: 6 7 6 6 7 6 7 6 6 7 6 7 SP M ME SD: 2.7968 c. M: 6.8003 c. 82: 6 8 6 6 8 6 8 6 7 7 6 8 SP D3 S SD: 2.9726 c. M: 5.8259 c. 89: 7 8 7 7 8 7 8 7 7 8 7 8 SP M ME SD: 3.0956 c. M: 5.2244 c. 94: 7 9 7 7 9 7 9 7 8 8 7 9 SP D3 S SD: 1.6853 c. M: 3.3350 c. 106: 8 10 8 8 10 8 10 8 9 9 8 10 SP D3 S SD: 0.7865 c. M: 1.5445 c. 118: 9 11 9 9 11 9 11 9 10 10 9 11 SP D3 S SD: 0.5976 c. M:-1.3004 c. 224: 17 21 17 17 21 17 21 17 19 19 17 21 SP D3 S SD: 0.5605 c. M: 0.9380 c. 323: 24 31 24 25 30 25 30 24 28 27 25 30 SP SD: 0.5066 c. M: 1.0609 c. 335: 25 32 25 26 31 26 31 25 29 28 26 31 SP SD: 0.4748 c. M:-0.8211 c. 417: 31 40 31 32 39 32 39 31 36 35 32 39 SP SD: 0.5268 c. M: 1.0164 c. 429: 32 41 32 33 40 33 40 32 37 36 33 40 SP SD: 0.3342 c. M: 0.7145 c. 441: 33 42 33 34 41 34 41 33 38 37 34 41 SP SD: 0.2152 c. M: 0.4291 c. 547: 41 52 41 42 51 42 51 41 47 46 42 51 SP SD: 0.1488 c. M: 0.2798 c. 559: 42 53 42 43 52 43 52 42 48 47 43 52 SP SD: 0.0616 c. M:-0.1162 c. 1224: 92 116 92 94 114 94 114 92 105 103 94 114 SP SD: 0.0216 c. M: 0.0392 c. 1783: 134 169 134 137 166 137 166 134 153 150 137 166 SP SD: 0.0189 c. M: 0.0399 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: 7.6080 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: 10.7654 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 8.4728 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 7.9743 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.7210 cents 11 x 12:13:14:15 x x 17:18:20:21:22 SD: 11.7217 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 6.7801 cents 13:14:15 x 16:17:18:20:21:22:23:24:26 S SD: 9.2479 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 7.9587 cents 15:16:17:18:19:20:21:23:24:25:27:28:30 SD: 5.4145 cents 16:17:18:19:20:21:23:24:25:27:28:30:32 SD: 5.0421 cents 21:22:24:25:26:28:30:32:33:35:37:39:42 SD: 4.8468 cents 22:23:25:26:28:29:31:33:35:37:39:41:44 SD: 3.8860 cents 24:25:27:28:30:32:34:36:38:40:43:45:48 SD: 3.3307 cents 31:33:35:37:39:41:44:47:49:52:55:58:62 SD: 3.1827 cents 32:34:36:38:40:43:45:48:51:54:57:60:64 SD: 2.2882 cents 39:41:44:46:49:52:55:59:62:65:69:73:78 SD: 2.2493 cents 40:42:45:47:50:53:56:60:63:67:71:75:80 SD: 1.8232 cents 54:57:61:64:68:72:76:81:85:91:96:101:108 SD: 1.5847 cents 55:58:62:65:69:73:77:83:87:92:98:103:110 SD: 1.5653 cents 56:59:63:66:70:75:79:84:88:94:100:105:112 SD: 1.5349 cents 64:67:72:76:80:85:90:96:101:107:114:120:128 SD: 1.2416 cents 71:75:80:84:89:95:100:107:112:119:126:133:142 SD: 1.1622 cents 72:76:81:85:90:96:101:108:114:121:128:135:144 SD: 0.8338 cents 96:101:108:114:120:128:135:144:152:161:171:180:192 SD: 0.5247 cents 136:143:153:161:170:181:191:204:215:228:242:255:272 SD: 0.4840 cents 152:160:171:180:190:203:214:228:240:255:270:285:304 SD: 0.3788 cents 168:177:189:199:210:224:236:252:265:282:299:315:336 SD: 0.3762 cents 184:194:207:218:230:245:259:276:291:309:327:345:368 SD: 0.3588 cents 216:228:243:256:270:288:304:324:341:362:384:405:432 SD: 0.3501 cents 224:236:252:265:280:299:315:336:354:376:398:420:448 SD: 0.3439 cents 248:261:279:294:310:331:349:372:392:416:441:465:496 SD: 0.2421 cents 264:278:297:313:330:352:371:396:417:443:469:495:528 SD: 0.1946 cents 352:371:396:417:440:469:495:528:556:590:626:660:704 SD: 0.1784 cents 432:455:486:512:540:576:608:648:683:724:768:810:864 SD: 0.1722 cents 448:472:504:531:560:597:630:672:708:751:796:840:896 SD: 0.1305 cents 544:573:612:645:680:725:765:816:860:912:967:1020:1088 SD: 0.1207 cents 576:607:648:683:720:768:810:864:910:966:1024:1080:1152 SD: 0.0903 cents 600:632:675:711:750:800:844:900:948:1006:1067:1125:1200 SD: 0.0805 cents