17:
1363:
30:
234:
224:
210:
312:, however, inflate the kleisma up to an entire semitone instead of tempering it out, as six minor thirds are equal to 18 semitones, while a perfect twelfth is 19 semitones. The same is true for the difference between five minor thirds (15 semitones) and one major tenth (16 semitones).
322:
Larry Hanson independently discovered this interval which also manifested in a unique mapping using a generalized keyboard capable of accommodating all the above temperaments as well as just intonation constant structures (periodicity blocks) with these numbers of scale degrees.
106:
1084:
1385:
347:
427:
876:
856:
16:
982:
988:
771:
1390:
1366:
1104:
1043:
420:
1164:
1159:
970:
380:
1346:
1099:
1006:
1000:
586:
316:
299:
295:
291:
287:
1129:
994:
976:
69:
1242:
679:
413:
123:
1184:
1124:
685:
661:
649:
630:
444:
343:
305:
1254:
1079:
1024:
1018:
1012:
916:
816:
789:
696:
673:
655:
600:
594:
436:
122:(of size 5/2, formed by a 2/1 octave plus a 5/4 major third) or as the difference between a
65:
29:
1222:
946:
734:
624:
618:
606:
549:
512:
73:
279:
1202:
1196:
1190:
962:
489:
127:
49:
233:
223:
209:
1379:
1326:
1175:
1094:
1089:
1035:
896:
836:
722:
710:
575:
538:
495:
95:
61:
1341:
1298:
1293:
1154:
800:
557:
520:
309:
99:
45:
98:). It is equal to a frequency ratio of 15625/15552 = 2 3 5, or approximately 8.1
1303:
1134:
569:
563:
532:
526:
119:
115:
77:
1336:
1230:
728:
641:
1236:
740:
1268:
716:
464:
1274:
1262:
667:
612:
282:
after the Greek for "closure", who noted that it was tempered out to a
82:
114:). It can be also defined as the difference between five justly tuned
1331:
1308:
1216:
1210:
501:
483:
283:
91:
364:
1/1: The
Quarterly Journal of the Just Intonation Network, Volume 8
704:
393:
15:
409:
379:, Band 6, Nr. 1, Breitkopf und Härtel, Leipzig 1890, pp. 1-90 (
405:
80:(each with a frequency ratio of 6/5) and one justly tuned
1317:
1284:
1173:
1143:
1113:
1070:
1061:
1034:
961:
944:
786:
769:
758:
695:
640:
585:
548:
511:
476:
458:
443:
90:(with a frequency ratio of 3/1, formed by a 2/1
20:Kleisma as minor thirds versus one twelfth on F
421:
8:
375:Studien im Gebiete der reinen Stimmung, in:
315:The interval was described but not used by
1362:
1067:
958:
783:
766:
473:
455:
428:
414:
406:
377:Vierteljahrsschrift für Musikwissenschaft
394:Development of a 53-Tone Keyboard Layout
132:
340:The Mathematical Theory of Tone Systems
331:
358:
356:
60:, is a minute and barely perceptible
7:
72:. It is the difference between six
14:
1361:
362:Just Intonation Network (1993).
232:
222:
208:
28:
1:
290:. It is also tempered out in
1386:5-limit tuning and intervals
983:septimal chromatic semitone
1407:
989:septimal diatonic semitone
777:(Numbers in brackets refer
278:The interval was named by
1359:
779:to fractional semitones.)
772:24-tone equal temperament
1367:List of pitch intervals
1105:Subminor and supermajor
1044:minor diatonic semitone
954:refer to pitch ratios.)
392:Hanson, Larry (1989). "
1165:Undecimal quarter tone
41:
1160:Septimal quarter tone
971:septimal quarter tone
338:Haluska, Jan (2003).
118:and one justly tuned
68:important to musical
19:
1347:Incomposite interval
1100:Pythagorean interval
952:(Numbers in brackets
460:(Numbers in brackets
310:24 equal temperament
300:72 equal temperament
288:53 equal temperament
1130:Pythagorean apotome
977:septimal third tone
1243:Septimal semicomma
124:chromatic semitone
42:
1373:
1372:
1355:
1354:
1185:Pythagorean comma
1125:Pythagorean limma
1057:
1056:
1053:
1052:
1019:supermajor fourth
995:supermajor second
940:
939:
754:
753:
750:
749:
462:are the number of
398:Xenharmonikon XII
348:978-0-8247-4714-5
276:
275:
1398:
1365:
1364:
1255:Septimal kleisma
1068:
1025:subminor seventh
1007:supermajor third
959:
947:Just intonations
932:
931:
927:
924:
912:
911:
907:
904:
892:
891:
887:
884:
872:
871:
867:
864:
852:
851:
847:
844:
832:
831:
827:
824:
812:
811:
807:
784:
767:
474:
456:
430:
423:
416:
407:
401:
390:
384:
373:
367:
360:
351:
336:
237:
236:
227:
226:
213:
212:
203:
202:
133:
113:
112:
111:
109:
58:semicomma majeur
39:
38:
33:
32:
25:
24:
1406:
1405:
1401:
1400:
1399:
1397:
1396:
1395:
1376:
1375:
1374:
1369:
1351:
1313:
1280:
1223:Septimal diesis
1169:
1139:
1109:
1063:
1049:
1030:
953:
950:
936:
929:
925:
922:
920:
909:
905:
902:
900:
889:
885:
882:
880:
869:
865:
862:
860:
849:
845:
842:
840:
829:
825:
822:
820:
809:
805:
804:
794:
793:
792:
788:
778:
775:
762:
760:
746:
691:
636:
581:
544:
507:
468:
463:
461:
451:
449:
446:
439:
434:
404:
391:
387:
374:
370:
361:
354:
337:
333:
329:
231:
221:
207:
200:
199:
194:
107:
105:
104:
103:
88:perfect twelfth
36:
35:
27:
22:
21:
12:
11:
5:
1404:
1402:
1394:
1393:
1391:Commas (music)
1388:
1378:
1377:
1371:
1370:
1360:
1357:
1356:
1353:
1352:
1350:
1349:
1344:
1339:
1334:
1329:
1323:
1321:
1315:
1314:
1312:
1311:
1306:
1301:
1296:
1290:
1288:
1282:
1281:
1279:
1278:
1272:
1266:
1259:
1258:
1252:
1246:
1240:
1234:
1227:
1226:
1220:
1217:Greater diesis
1214:
1207:
1206:
1203:Septimal comma
1200:
1197:Holdrian comma
1194:
1191:Syntonic comma
1188:
1181:
1179:
1171:
1170:
1168:
1167:
1162:
1157:
1151:
1149:
1141:
1140:
1138:
1137:
1132:
1127:
1121:
1119:
1111:
1110:
1108:
1107:
1102:
1097:
1092:
1087:
1082:
1076:
1074:
1065:
1059:
1058:
1055:
1054:
1051:
1050:
1048:
1047:
1040:
1038:
1032:
1031:
1029:
1028:
1022:
1016:
1013:subminor fifth
1010:
1004:
1001:subminor third
998:
992:
986:
980:
974:
967:
965:
956:
942:
941:
938:
937:
935:
934:
914:
894:
874:
854:
834:
814:
797:
795:
787:
781:
764:
756:
755:
752:
751:
748:
747:
745:
744:
738:
732:
726:
720:
714:
708:
701:
699:
693:
692:
690:
689:
683:
677:
671:
665:
659:
653:
646:
644:
638:
637:
635:
634:
628:
622:
616:
610:
604:
598:
591:
589:
583:
582:
580:
579:
573:
567:
561:
554:
552:
546:
545:
543:
542:
536:
530:
524:
517:
515:
509:
508:
506:
505:
499:
493:
487:
480:
478:
471:
453:
441:
440:
435:
433:
432:
425:
418:
410:
403:
402:
385:
368:
352:
330:
328:
325:
274:
273:
270:
267:
264:
261:
258:
255:
252:
249:
245:
244:
242:
240:
238:
228:
218:
215:
204:
196:
190:
189:
186:
183:
180:
177:
174:
171:
168:
165:
161:
160:
157:
154:
151:
148:
145:
142:
139:
136:
128:greater diesis
126:(25/24) and a
56:(κλείσμα), or
13:
10:
9:
6:
4:
3:
2:
1403:
1392:
1389:
1387:
1384:
1383:
1381:
1368:
1358:
1348:
1345:
1343:
1340:
1338:
1335:
1333:
1330:
1328:
1325:
1324:
1322:
1320:
1316:
1310:
1307:
1305:
1302:
1300:
1297:
1295:
1292:
1291:
1289:
1287:
1283:
1276:
1273:
1270:
1267:
1264:
1261:
1260:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1228:
1224:
1221:
1218:
1215:
1212:
1211:Lesser diesis
1209:
1208:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1182:
1180:
1178:
1177:
1172:
1166:
1163:
1161:
1158:
1156:
1153:
1152:
1150:
1148:
1147:
1146:Quarter tones
1142:
1136:
1133:
1131:
1128:
1126:
1123:
1122:
1120:
1118:
1117:
1112:
1106:
1103:
1101:
1098:
1096:
1095:Pseudo-octave
1093:
1091:
1088:
1086:
1083:
1081:
1078:
1077:
1075:
1073:
1069:
1066:
1060:
1045:
1042:
1041:
1039:
1037:
1033:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
968:
966:
964:
960:
957:
955:
949:
948:
943:
918:
915:
898:
895:
878:
875:
858:
855:
838:
835:
818:
815:
802:
799:
798:
796:
791:
785:
782:
780:
774:
773:
768:
765:
757:
742:
739:
736:
733:
730:
727:
724:
721:
718:
715:
712:
709:
706:
703:
702:
700:
698:
694:
687:
684:
681:
678:
675:
672:
669:
666:
663:
660:
657:
654:
651:
648:
647:
645:
643:
639:
632:
629:
626:
623:
620:
617:
614:
611:
608:
605:
602:
599:
596:
593:
592:
590:
588:
584:
577:
574:
571:
568:
565:
562:
559:
556:
555:
553:
551:
547:
540:
537:
534:
531:
528:
525:
522:
519:
518:
516:
514:
510:
503:
500:
497:
494:
491:
488:
485:
482:
481:
479:
475:
472:
470:
466:
457:
454:
448:
442:
438:
431:
426:
424:
419:
417:
412:
411:
408:
399:
395:
389:
386:
382:
378:
372:
369:
365:
359:
357:
353:
349:
345:
341:
335:
332:
326:
324:
320:
318:
313:
311:
307:
303:
301:
297:
293:
289:
285:
281:
271:
268:
265:
262:
259:
256:
253:
250:
247:
246:
243:
241:
239:
235:
229:
225:
219:
216:
211:
205:
197:
192:
191:
187:
184:
181:
178:
175:
172:
170:(6 : 5)
169:
166:
163:
162:
158:
155:
152:
149:
146:
143:
140:
137:
135:
134:
131:
129:
125:
121:
117:
110:
101:
97:
96:perfect fifth
93:
89:
85:
84:
79:
75:
71:
67:
63:
59:
55:
51:
47:
31:
18:
1318:
1285:
1271:(0.72 cents)
1265:(1.95 cents)
1248:
1245:(13.8 cents)
1239:(10.1 cents)
1233:(19.5 cents)
1225:(35.7 cents)
1219:(62.6 cents)
1213:(41.1 cents)
1205:(27.3 cents)
1199:(22.6 cents)
1193:(21.5 cents)
1187:(23.5 cents)
1174:
1155:Quarter tone
1145:
1144:
1115:
1114:
1071:
1036:Higher-limit
951:
945:
857:major fourth
801:quarter tone
776:
770:
459:
397:
388:
376:
371:
363:
342:, p.xxviii.
339:
334:
321:
314:
304:
280:Shohé Tanaka
277:
116:minor thirds
87:
81:
78:minor thirds
70:temperaments
57:
53:
46:music theory
43:
1304:Millioctave
1286:Measurement
1277:(0.4 cents)
1257:(7.7 cents)
1251:(8.1 cents)
1135:Major limma
877:minor fifth
173:3 : 2
167:6 : 5
141:6 just m3s
130:(648/625).
120:major tenth
94:plus a 3/2
1380:Categories
1337:Semiditone
1231:Diaschisma
1046:(17-limit)
737:(22 or 23)
735:fourteenth
731:(20 or 21)
729:thirteenth
725:(18 or 19)
719:(17 or 18)
713:(15 or 16)
707:(13 or 14)
642:Diminished
469:interval.)
450:(post-Bach
381:Goole-Scan
327:References
260:700 / 600
1299:Centitone
1237:Semicomma
1116:Semitones
1080:Microtone
1064:intervals
741:fifteenth
587:Augmented
465:semitones
437:Intervals
319:in 1726.
1269:Breedsma
717:eleventh
697:Compound
452:Western)
447:semitone
201:♭
144:Just P5
138:Just m3
66:interval
37:♯
23:♯
1275:Ragisma
1263:Schisma
1249:Kleisma
1085:5-limit
991:(15:14)
985:(21:20)
979:(28:27)
973:(36:35)
963:7-limit
928:⁄
917:seventh
908:⁄
888:⁄
868:⁄
848:⁄
828:⁄
808:⁄
790:Neutral
763:systems
723:twelfth
680:seventh
631:seventh
576:seventh
539:seventh
477:Perfect
445:Twelve-
366:, p.19.
269:701.89
266:705.88
263:694.74
257:701.96
254:693.84
251:315.64
83:tritave
54:kleisma
1332:Ditone
1319:Others
1309:Savart
1176:Commas
1072:Groups
1021:(10:7)
817:second
761:tuning
686:octave
662:fourth
650:second
613:fourth
601:second
595:unison
558:second
521:second
502:octave
490:fourth
484:unison
467:in the
346:
317:Rameau
298:, and
284:unison
248:Cents
193:Letter
176:2 / 2
164:Ratio
159:72TET
156:53TET
153:34TET
150:19TET
147:12TET
92:octave
76:tuned
74:justly
52:, the
50:tuning
1342:Secor
1090:Comma
1062:Other
1027:(7:4)
1015:(7:5)
1009:(9:7)
1003:(7:6)
997:(8:7)
897:sixth
837:third
759:Other
711:tenth
705:ninth
674:sixth
668:fifth
656:third
625:sixth
619:fifth
607:third
570:sixth
564:third
550:Minor
533:sixth
527:third
513:Major
496:fifth
230:G / A
220:G / A
195:name
100:cents
64:type
62:comma
34:vs. C
1327:Wolf
1294:Cent
743:(24)
688:(11)
633:(12)
627:(10)
578:(10)
541:(11)
504:(12)
344:ISBN
308:and
272:700
108:Play
48:and
682:(9)
676:(7)
670:(6)
664:(4)
658:(2)
652:(0)
621:(8)
615:(6)
609:(5)
603:(3)
597:(1)
572:(8)
566:(3)
560:(1)
535:(9)
529:(4)
523:(2)
498:(7)
492:(5)
486:(0)
396:",
286:by
86:or
44:In
26:: D
1382::
921:10
355:^
306:12
302:.
296:34
294:,
292:19
217:G
214:+
188:2
185:2
182:2
179:2
933:)
930:2
926:1
923:+
919:(
913:)
910:2
906:1
903:+
901:8
899:(
893:)
890:2
886:1
883:+
881:6
879:(
873:)
870:2
866:1
863:+
861:5
859:(
853:)
850:2
846:1
843:+
841:3
839:(
833:)
830:2
826:1
823:+
821:1
819:(
813:)
810:2
806:1
803:(
429:e
422:t
415:v
400:.
383:)
350:.
206:A
198:E
102:(
40:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
