300:
107:
65:
20:
1398:
250:, in order to avoid confusing them with the more familiar, mathematically simple integer harmonics; both are often relevant in the same sentence. Partials measured in the sounds produced by real musical instruments almost always have a slightly higher pitch than the corresponding idealized harmonic, with the discrepancy being less important for high-pitched instruments (above
38:
278:
must match, makes it necessary to widen every interval very slightly. Generally, it's more than sufficient to sharpen only whole octaves slightly, rather than separately modifying all intervals that reach individual pitches in the upper octaves
83:
296:
are never tuned 2:1, but instead are stretched or compressed in a consistent manner throughout the range of each individual gamelan, due to the physical characteristics of their instruments. Another example is the tritave
301:
36:
37:
81:
82:
29:
344:(hence making a single piano better able to be perceived over the volume of an entire orchestra) and giving them longer sustain than similar, smaller instruments.
141:, but is perceived or treated as equivalent to this ratio, and whose pitches are considered equivalent to each other as with octave equivalency.
193:, stretched octaves are commonly encountered in instruments where string thickness and high string tension causes some strings to approach their
1119:
428:
197:, which makes the string respond to stretching and bending with a pull to restore its original shape and position a little
462:
74:
621:
316:
Octave stretching is less apparent on large pianos which have longer strings and hence less curvature for a given
433:
911:
891:
1017:
186:
when played with lower notes whose overtones are themselves naturally stretched by an equivalent amount.
1420:
1023:
731:
210:
584:
547:
806:
368:
240:. In detailed discussions of pitch and tuning the actual overtones in the sounded note are called
1401:
1139:
1078:
455:
310:
1199:
676:
493:
106:
1194:
1005:
317:
229:
1381:
1134:
1041:
1035:
221:
8:
1164:
1029:
1011:
1277:
714:
448:
1333:
1219:
1159:
720:
696:
684:
665:
479:
1289:
1114:
1059:
1053:
1047:
951:
851:
824:
708:
690:
635:
629:
471:
353:
325:
283:
150:
127:
336:. Another reason is that long strings under high tension can store more acoustic
1376:
1257:
981:
769:
659:
653:
641:
241:
206:
270:
The practical consequence of the discrepancy between the sharpened pitches in a
1237:
1231:
1225:
997:
524:
275:
228:
expected for its overtone series, which would all be integer multiples of the
1414:
1361:
1210:
1124:
1070:
931:
871:
757:
745:
610:
573:
530:
411:"The Paradoxes of Octave Identities", p.213. Author(s): Jenő Keuler. Source:
329:
264:
237:
202:
194:
1328:
1189:
835:
592:
555:
391:
271:
258:
190:
168:
1338:
1169:
604:
598:
567:
561:
395:
363:
321:
320:; that is one reason why orchestras go to the expense of using very long
402:, Vol. 4, No. 4, (Winter, 1980), pp. 15–22. Published by: The MIT Press.
1371:
1265:
763:
415:, T. 40, Fasc. 1/3, (1999), pp. 211–224. Published by: Akadémiai Kiadó.
198:
98:
53:
1271:
775:
358:
225:
179:
131:
1303:
751:
499:
341:
214:
176:
1309:
1297:
1283:
702:
647:
293:
1366:
1343:
1251:
1245:
536:
518:
337:
333:
233:
138:
739:
134:
440:
290:
236:
of the note that the string actually produces has slightly
137:
is not 2:1 (2.3:1 or 1.9:1, for example), that of the
413:
263:whose high-level overtones fall above the range of
340:than can short strings, making larger instruments
1412:
390:"Interview with Max Mathews", p.21. Author(s):
205:causes small differences between the string's
456:
1397:
463:
449:
201:to how far it was bent or stretched. That
386:
384:
209:frequencies and the mathematically ideal
105:
1413:
821:
381:
444:
324:rather than shorter, less expensive
144:
13:
63:
18:
14:
1432:
422:
369:piano acoustics § Railsback curve
1396:
97:Problems playing this file? See
79:
52:Problems playing this file? See
34:
224:is that rather than the simple
405:
1:
470:
374:
220:The effect of strings' small
7:
1018:septimal chromatic semitone
347:
10:
1437:
1024:septimal diatonic semitone
812:(Numbers in brackets refer
211:simple harmonic oscillator
171:pitch difference), sounds
148:
1394:
1352:
1319:
1208:
1178:
1148:
1105:
1096:
1069:
996:
979:
814:to fractional semitones.)
807:24-tone equal temperament
804:
793:
730:
675:
620:
583:
546:
511:
478:
1402:List of pitch intervals
1140:Subminor and supermajor
1079:minor diatonic semitone
989:refer to pitch ratios.)
175:when played with ideal
1200:Undecimal quarter tone
400:Computer Music Journal
111:
68:
23:
1195:Septimal quarter tone
1006:septimal quarter tone
274:overtone series that
230:fundamental frequency
110:Pseudo-octave (2.1:1)
109:
67:
22:
1382:Incomposite interval
1135:Pythagorean interval
987:(Numbers in brackets
495:(Numbers in brackets
322:concert grand pianos
238:inharmonic overtones
213:'s integer multiple
1165:Pythagorean apotome
1012:septimal third tone
311:Bohlen–Pierce scale
1278:Septimal semicomma
222:inelastic response
124:paradoxical octave
112:
69:
24:
1421:Intervals (music)
1408:
1407:
1390:
1389:
1220:Pythagorean comma
1160:Pythagorean limma
1092:
1091:
1088:
1087:
1054:supermajor fourth
1030:supermajor second
975:
974:
789:
788:
785:
784:
497:are the number of
303:play on clarinets
199:out of proportion
84:
39:
1428:
1400:
1399:
1290:Septimal kleisma
1103:
1102:
1060:subminor seventh
1042:supermajor third
994:
993:
982:Just intonations
967:
966:
962:
959:
947:
946:
942:
939:
927:
926:
922:
919:
907:
906:
902:
899:
887:
886:
882:
879:
867:
866:
862:
859:
847:
846:
842:
819:
818:
802:
801:
509:
508:
491:
490:
465:
458:
451:
442:
441:
416:
409:
403:
388:
354:electronic tuner
308:
307:
306:
304:
284:stretched tuning
262:
257:
256:
166:
162:
157:stretched octave
151:stretched tuning
145:Stretched octave
86:
85:
66:
41:
40:
21:
1436:
1435:
1431:
1430:
1429:
1427:
1426:
1425:
1411:
1410:
1409:
1404:
1386:
1348:
1315:
1258:Septimal diesis
1204:
1174:
1144:
1098:
1084:
1065:
988:
985:
971:
964:
960:
957:
955:
944:
940:
937:
935:
924:
920:
917:
915:
904:
900:
897:
895:
884:
880:
877:
875:
864:
860:
857:
855:
844:
840:
839:
829:
828:
827:
823:
813:
810:
797:
795:
781:
726:
671:
616:
579:
542:
503:
498:
496:
486:
484:
481:
474:
469:
434:BillBremmer.com
425:
420:
419:
410:
406:
389:
382:
377:
350:
302:
299:
298:
289:The octaves of
254:
252:
251:
164:
161:2.01 : 1 ,
160:
153:
147:
126:in music is an
104:
103:
95:
93:
92:
91:
90:
87:
80:
77:
70:
64:
59:
58:
50:
48:
47:
46:
45:
42:
35:
32:
25:
19:
12:
11:
5:
1434:
1424:
1423:
1406:
1405:
1395:
1392:
1391:
1388:
1387:
1385:
1384:
1379:
1374:
1369:
1364:
1358:
1356:
1350:
1349:
1347:
1346:
1341:
1336:
1331:
1325:
1323:
1317:
1316:
1314:
1313:
1307:
1301:
1294:
1293:
1287:
1281:
1275:
1269:
1262:
1261:
1255:
1252:Greater diesis
1249:
1242:
1241:
1238:Septimal comma
1235:
1232:Holdrian comma
1229:
1226:Syntonic comma
1223:
1216:
1214:
1206:
1205:
1203:
1202:
1197:
1192:
1186:
1184:
1176:
1175:
1173:
1172:
1167:
1162:
1156:
1154:
1146:
1145:
1143:
1142:
1137:
1132:
1127:
1122:
1117:
1111:
1109:
1100:
1094:
1093:
1090:
1089:
1086:
1085:
1083:
1082:
1075:
1073:
1067:
1066:
1064:
1063:
1057:
1051:
1048:subminor fifth
1045:
1039:
1036:subminor third
1033:
1027:
1021:
1015:
1009:
1002:
1000:
991:
977:
976:
973:
972:
970:
969:
949:
929:
909:
889:
869:
849:
832:
830:
822:
816:
799:
791:
790:
787:
786:
783:
782:
780:
779:
773:
767:
761:
755:
749:
743:
736:
734:
728:
727:
725:
724:
718:
712:
706:
700:
694:
688:
681:
679:
673:
672:
670:
669:
663:
657:
651:
645:
639:
633:
626:
624:
618:
617:
615:
614:
608:
602:
596:
589:
587:
581:
580:
578:
577:
571:
565:
559:
552:
550:
544:
543:
541:
540:
534:
528:
522:
515:
513:
506:
488:
476:
475:
468:
467:
460:
453:
445:
439:
438:
429:"Octave Types"
424:
423:External links
421:
418:
417:
404:
379:
378:
376:
373:
372:
371:
366:
361:
356:
349:
346:
159:, for example
149:Main article:
146:
143:
94:
89:Perfect Octave
88:
78:
75:Perfect Octave
73:
72:
71:
62:
61:
60:
49:
43:
33:
28:
27:
26:
17:
16:
15:
9:
6:
4:
3:
2:
1433:
1422:
1419:
1418:
1416:
1403:
1393:
1383:
1380:
1378:
1375:
1373:
1370:
1368:
1365:
1363:
1360:
1359:
1357:
1355:
1351:
1345:
1342:
1340:
1337:
1335:
1332:
1330:
1327:
1326:
1324:
1322:
1318:
1311:
1308:
1305:
1302:
1299:
1296:
1295:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1263:
1259:
1256:
1253:
1250:
1247:
1246:Lesser diesis
1244:
1243:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1217:
1215:
1213:
1212:
1207:
1201:
1198:
1196:
1193:
1191:
1188:
1187:
1185:
1183:
1182:
1181:Quarter tones
1177:
1171:
1168:
1166:
1163:
1161:
1158:
1157:
1155:
1153:
1152:
1147:
1141:
1138:
1136:
1133:
1131:
1130:Pseudo-octave
1128:
1126:
1123:
1121:
1118:
1116:
1113:
1112:
1110:
1108:
1104:
1101:
1095:
1080:
1077:
1076:
1074:
1072:
1068:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1003:
1001:
999:
995:
992:
990:
984:
983:
978:
953:
950:
933:
930:
913:
910:
893:
890:
873:
870:
853:
850:
837:
834:
833:
831:
826:
820:
817:
815:
809:
808:
803:
800:
792:
777:
774:
771:
768:
765:
762:
759:
756:
753:
750:
747:
744:
741:
738:
737:
735:
733:
729:
722:
719:
716:
713:
710:
707:
704:
701:
698:
695:
692:
689:
686:
683:
682:
680:
678:
674:
667:
664:
661:
658:
655:
652:
649:
646:
643:
640:
637:
634:
631:
628:
627:
625:
623:
619:
612:
609:
606:
603:
600:
597:
594:
591:
590:
588:
586:
582:
575:
572:
569:
566:
563:
560:
557:
554:
553:
551:
549:
545:
538:
535:
532:
529:
526:
523:
520:
517:
516:
514:
510:
507:
505:
501:
492:
489:
483:
477:
473:
466:
461:
459:
454:
452:
447:
446:
443:
436:
435:
430:
427:
426:
414:
408:
401:
397:
393:
387:
385:
380:
370:
367:
365:
362:
360:
357:
355:
352:
351:
345:
343:
339:
335:
334:spinet pianos
331:
327:
323:
319:
314:
312:
305:
295:
292:
287:
285:
282:
277:
273:
268:
266:
265:human hearing
260:
249:
248:
244:
243:partial tones
239:
235:
231:
227:
223:
218:
216:
212:
208:
207:real overtone
204:
203:non-linearity
200:
196:
195:elastic limit
192:
187:
185:
181:
178:
174:
170:
167:(an 8.6
158:
152:
142:
140:
136:
133:
129:
125:
121:
117:
116:pseudo-octave
108:
102:
100:
76:
57:
55:
44:Pseudo Octave
31:
30:Pseudo Octave
1353:
1320:
1306:(0.72 cents)
1300:(1.95 cents)
1280:(13.8 cents)
1274:(10.1 cents)
1268:(19.5 cents)
1260:(35.7 cents)
1254:(62.6 cents)
1248:(41.1 cents)
1240:(27.3 cents)
1234:(22.6 cents)
1228:(21.5 cents)
1222:(23.5 cents)
1209:
1190:Quarter tone
1180:
1179:
1150:
1149:
1129:
1106:
1071:Higher-limit
986:
980:
892:major fourth
836:quarter tone
811:
805:
494:
432:
412:
407:
399:
318:displacement
315:
288:
280:
276:treble notes
269:
246:
242:
219:
191:piano tuning
188:
183:
172:
163:rather than
156:
154:
123:
120:pseudooctave
119:
115:
113:
96:
51:
1339:Millioctave
1321:Measurement
1312:(0.4 cents)
1292:(7.7 cents)
1286:(8.1 cents)
1170:Major limma
912:minor fifth
396:Max Mathews
364:octave band
272:bass note's
173:out of tune
165:2 : 1
1372:Semiditone
1266:Diaschisma
1081:(17-limit)
772:(22 or 23)
770:fourteenth
766:(20 or 21)
764:thirteenth
760:(18 or 19)
754:(17 or 18)
748:(15 or 16)
742:(13 or 14)
677:Diminished
504:interval.)
485:(post-Bach
398:. Source:
375:References
326:baby grand
99:media help
54:media help
1334:Centitone
1272:Semicomma
1151:Semitones
1115:Microtone
1099:intervals
776:fifteenth
622:Augmented
500:semitones
472:Intervals
359:mel scale
226:harmonics
215:harmonics
180:overtones
132:frequency
1415:Category
1304:Breedsma
752:eleventh
732:Compound
487:Western)
482:semitone
392:C. Roads
348:See also
294:gamelans
291:Balinese
247:partials
177:harmonic
128:interval
1310:Ragisma
1298:Schisma
1284:Kleisma
1120:5-limit
1026:(15:14)
1020:(21:20)
1014:(28:27)
1008:(36:35)
998:7-limit
963:⁄
952:seventh
943:⁄
923:⁄
903:⁄
883:⁄
863:⁄
843:⁄
825:Neutral
798:systems
758:twelfth
715:seventh
666:seventh
611:seventh
574:seventh
512:Perfect
480:Twelve-
330:upright
313:(3:1).
309:of the
184:in tune
1367:Ditone
1354:Others
1344:Savart
1211:Commas
1107:Groups
1056:(10:7)
852:second
796:tuning
721:octave
697:fourth
685:second
648:fourth
636:second
630:unison
593:second
556:second
537:octave
525:fourth
519:unison
502:in the
342:louder
338:energy
234:timbre
232:, the
182:, but
139:octave
130:whose
1377:Secor
1125:Comma
1097:Other
1062:(7:4)
1050:(7:5)
1044:(9:7)
1038:(7:6)
1032:(8:7)
932:sixth
872:third
794:Other
746:tenth
740:ninth
709:sixth
703:fifth
691:third
660:sixth
654:fifth
642:third
605:sixth
599:third
585:Minor
568:sixth
562:third
548:Major
531:fifth
332:, or
135:ratio
122:, or
1362:Wolf
1329:Cent
778:(24)
723:(11)
668:(12)
662:(10)
613:(10)
576:(11)
539:(12)
394:and
169:cent
155:The
717:(9)
711:(7)
705:(6)
699:(4)
693:(2)
687:(0)
656:(8)
650:(6)
644:(5)
638:(3)
632:(1)
607:(8)
601:(3)
595:(1)
570:(9)
564:(4)
558:(2)
533:(7)
527:(5)
521:(0)
286:).
281:see
255:000
245:or
189:In
1417::
956:10
431:,
383:^
328:,
267:.
259:Hz
217:.
118:,
114:A
968:)
965:2
961:1
958:+
954:(
948:)
945:2
941:1
938:+
936:8
934:(
928:)
925:2
921:1
918:+
916:6
914:(
908:)
905:2
901:1
898:+
896:5
894:(
888:)
885:2
881:1
878:+
876:3
874:(
868:)
865:2
861:1
858:+
856:1
854:(
848:)
845:2
841:1
838:(
464:e
457:t
450:v
437:.
279:(
261:)
253:5
101:.
56:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.