296:
422:
486:
412:
was divided differently by these two theorists, but in both cases the two intervals were not equal to one another. Archytas, who was the first theorist to give ratios for all of the genera, chose 28:27 and 36:35, and
Didymus, some four centuries later, gave 32:31 and 31:30.
517:
119:). The positions of the inner notes vary from one genus to another, for which reason they are called "movable notes". In its basic theoretical form, the largest interval of a tetrachord is at the top, and the smallest at the bottom. The existence of a
553:, book 2), whenever tetrachords are combined to form a scale filling an octave, "Two consecutive pycna may not occur in ascent or descent. A ditone may precede or follow in ascent or descent. A tone may follow only in descent".
452:—translated as "incomposite" (or "noncomposite") "trihemitone" (Bower, Hagel, Levin, and Barker prefer a descriptive translation, "an individed interval of three semitones"; Strunk uses "trisemitone"), the modern term being "
460:
of some type of whole tone to be divided into two semitones. There is a larger number of variations in the tuning of the chromatic than in the enharmonic. Up to the beginning of the 4th century BC the chromatic
91:). The Greek word πυκνόν is an adjective meaning "close", "compact", "close-packed", or "crowded". In Ancient Greek music theory, this term is used to describe a pair of intervals within a
1125:
Mathiesen Thomas J. 2001. "Greece, §I: Ancient", The New Grove
Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers.
477:
were in ratios of 28:27 and 15:14 for the soft chromatic and 22:21 and 12:11 for the intense. The larger remaining interval was 6:5 in the soft chromatic and 7:6 in the intense.
231:
A "composite interval" is one made up of two or more smaller intervals; an "incomposite interval" has no smaller components In these terms, if the composite interval between the
784:, ninth edition, revised and augmented throughout by Sir Henry Stuart Jones and Roderick McKenzie. Oxford: Clarendon Press; New York City: Oxford University Press.
469:
into ascending semitone intervals of 256:243 and 2187:2048. Ptolemy defined two different tunings of the chromatic genus: the "soft" chromatic with a smaller
167:. A second tetrachord is added above, after a disjunctive tone, and the corresponding names (together with the interval ratios of the standing tones) are:
67:
made by compounding two such tetrachords, and the rules governing the ways in which such compounds may be made centre on the relationships of the two
1087:
Strunk, Oliver. 1998. Source
Readings in Music History. Revised Edition by Leo Treitler. New York City, London: W. W. Norton and Company. pp. 36-7.
536:
must be smaller than or equal to the upper one. Didymus in the chromatic genus and
Archytas in the enharmonic broke this rule, however, and in the
139:. For this reason, the enharmonic and chromatic genera are sometimes called the "pyknic genera", in order to distinguish them from the diatonic.
1134:. Cambridge Classical Studies 2. Cambridge: The University Press. Reprinted, Chicago, Argonaut Inc., 1967; Amsterdam: Adolf M. Hakkert, 1968.
111:
bounded by the interval of a perfect fourth, the outer notes of which remain fixed in all genera and therefore are called "standing notes" (
1148:
540:(2. 13) Ptolemy criticized this feature in Didymus, holding that it is unmelodic and out of agreement with the evidence of our ears.
1035:
789:
726:
669:
108:
659:
982:
974:
350:
was exactly (that is by exact mathematic calculation) divided into its two component intervals is not known. The tuning of
135:
consists of two whole tones and one semitone, no single interval is larger than the other two combined, and so there is no
466:
123:
therefore depends on the uppermost interval being larger than half of a perfect fourth, which occurs only in the
1153:
95:, the sum of which is less than the remainder of the tetrachord. Although in modern usage, a tetrachord may be
1158:
780:
1059:
1012:
397:
52:
362:
in the ratios of 40:39 and 39:38. Although
Aristoxenus also implies that the two intervals of the
753:
699:
646:
638:
609:
580:
545:
319:
295:
421:
1031:
978:
970:
869:
817:
785:
722:
718:
665:
343:
745:
691:
630:
601:
572:
128:
490:
124:
1077:. Vol. II: Harmonic and Acoustic Theory. Cambridge University Press. pp. 261, 267.
821:
445:
437:
429:
303:
132:
112:
84:
64:
40:
32:
20:
1142:
703:
650:
100:
60:
485:
107:, in ancient Greek music theory a tetrachord consists of a four-note segment of the
351:
335:
473:
and the "intense" chromatic with a larger one. The unequal semitones dividing the
712:
283:(a whole tone), the lowest three notes of the diatonic tetrachord are designated
1051:
679:
529:
453:
426:
401:
355:
315:
300:
104:
563:
Barbera, André (1977). "Arithmetic and
Geometric Divisions of the Tetrachord".
695:
92:
48:
714:
Apollo's Lyre: Greek Music and Music Theory in
Antiquity and the Middle Ages
366:
in the enharmonic genus may be equal, the anonymous author of the
Euclidean
621:
Barker, Andrew (1981). "Methods and Aims in the
Euclidean Sectio Canonis".
51:
in which a composite of two smaller intervals is less than the remaining (
393:
327:
310:
In the enharmonic genus, the large incomposite interval was originally a
147:
The notes of the central tetrachord of the system in ascending order are
385:
275:(a minor third) is larger than the remaining incomposite interval from
969:. Oxford: Clarendon Press; New York: Oxford University Press. p. 170.
757:
642:
613:
584:
465:
spanned a major whole tone with a 9:8 ratio, and this was divided by
311:
358:, uses a major third of 19:15 with the two unequal intervals of the
263:), the three notes in that composite interval are together called a
83:
was an important criterion in the classification of melodic genera (
749:
634:
605:
576:
47:) in the music theory of Antiquity is a structural property of any
484:
420:
294:
493:
octave species on E in the chromatic genus: conjunct tetrachords
299:
Two pyknic enharmonic tetrachords, together comprising the Greek
528:
A further refinement of tetrachordal construction, according to
425:
Two pyknic chromatic tetrachords, together comprising the Greek
1011:. Translated, with Introduction and Notes by Calvin M. Bower.
658:
Chalmers, John (1990). Larry
Polansky; Carter Scholz (eds.).
267:. In the diatonic genus, because the composite interval from
513:
are separated by the larger interval between steps 3 and 4
880:
878:
436:
In the chromatic genus, the largest interval was called a
1009:
Fundamentals of Music. Anicius Manlius Severinus Boethius
922:
920:
907:
905:
1056:
The Manual of Harmonics of Nicomachus the Pythagorean
404:
major third with the number ratio of 5:4, making the
247:) is smaller than the incomposite interval from the
1030:. Cambridge University Press. pp. 105, 266–7.
736:Solomon, Jon (1984). "Towards a History of Tonoi".
330:. The Pythagorean ditone is equivalent to two 9:8
338:), together an interval of 81:64, thus leaving a
1058:. Translation and commentary by Flora R. Levin.
961:
959:
778:Liddell, Henry George, and Robert Scott. 1996.
103:, or indeed any (unordered) collection of four
1015:and London: Yale University Press. p. 43.
63:(also called "genus of a tetrachord") and the
8:
1028:Ancient Greek Music. A New Technical History
682:(2007). "Ἀπειρία in Aristoxenian Theory".
346:(minor Pythagorean semitone), but how the
1130:Winnington-Ingram, Reginald Pepys. 1936.
1108:
938:
884:
865:
813:
592:Barbera, André (1984). "Octave Species".
39:close, close-packed, crowded, condensed;
994:
926:
911:
861:
400:, replaced the ditone with the smaller,
1096:
896:
849:
837:
771:
950:
801:
7:
1062:MI: Phanes Press. pp. 125, 174.
532:, is that the lower interval of the
27:), sometimes also transliterated as
392:(2. 14) that two other theorists,
109:Greater and Lesser Perfect Systems
14:
721:: University of Nebraska Press.
664:. Lebanon NH: Frog Peak Music.
505:, and interval of disjunction
55:) interval. The makeup of the
1:
711:Mathiesen, Thomas J. (1999).
408:correspondingly larger. This
326:with a total width of just a
820:, 344, 350, et passim; from
1132:Mode in Ancient Greek Music
661:Divisions of the Tetrachord
623:Journal of Hellenic Studies
501:, with note of conjunction
370:(P18) is unequivocal: "The
1175:
1149:Ancient Greek music theory
543:According to Aristoxenus'
825:
738:The Journal of Musicology
696:10.25162/hermes-2007-0038
594:The Journal of Musicology
450:triemitonium incompositum
441:
116:
88:
36:
24:
143:Theoretical applications
1073:Barker, Andrew (1989).
1026:Hagel, Stephan (2009).
781:A Greek-English Lexicon
744:(3 - Summer): 242–251.
600:(3 - Summer): 229–241.
571:(2 - Autumn): 294–323.
565:Journal of Music Theory
382:into equal intervals".
306:in the enharmonic genus
99:four-note segment of a
59:serves to identify the
1075:Greek Musical Writings
1007:Bower, Calvin (1989).
525:
449:
442:τριημιτόνιόν ἀσύνθετον
433:
432:in the chromatic genus
307:
287:: "not close-packed".
216:must remain above the
212:Although movable, the
44:
816:, pp. 301, 312,
488:
424:
298:
89:γένη τῶν μελῳδουμένων
967:Ancient Greek Music
551:Elements of Harmony
1099:, pp. 229–30.
965:West, M. L. 1992.
826:κινούμενοι φθόγγοι
546:Elementa harmonica
526:
434:
378:do not divide the
320:Pythagorean tuning
308:
985:(electronic bk.).
354:, as reported by
129:enharmonic genera
1166:
1135:
1126:
1112:
1106:
1100:
1094:
1088:
1085:
1079:
1078:
1070:
1064:
1063:
1048:
1042:
1041:
1023:
1017:
1016:
1004:
998:
992:
986:
963:
954:
948:
942:
936:
930:
924:
915:
909:
900:
894:
888:
882:
873:
859:
853:
847:
841:
835:
829:
827:
811:
805:
799:
793:
776:
761:
732:
707:
675:
654:
617:
588:
524:
523:
522:
520:
443:
208:(3:2) (standing)
178:(2:1) (standing)
118:
90:
38:
26:
1174:
1173:
1169:
1168:
1167:
1165:
1164:
1163:
1154:Music of Greece
1139:
1138:
1129:
1124:
1121:
1119:Further reading
1116:
1115:
1107:
1103:
1095:
1091:
1086:
1082:
1072:
1071:
1067:
1052:Levin, Flora R.
1050:
1049:
1045:
1038:
1025:
1024:
1020:
1006:
1005:
1001:
997:, pp. 7–8.
993:
989:
964:
957:
949:
945:
937:
933:
925:
918:
910:
903:
895:
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883:
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856:
848:
844:
836:
832:
812:
808:
800:
796:
777:
773:
735:
729:
710:
680:Levin, Flora R.
678:
672:
657:
620:
591:
562:
559:
518:
516:
515:
514:
483:
481:Scale structure
419:
388:reports in his
293:
145:
117:ἑστῶτες φθόγγοι
77:
12:
11:
5:
1172:
1170:
1162:
1161:
1159:Musical scales
1156:
1151:
1141:
1140:
1137:
1136:
1127:
1120:
1117:
1114:
1113:
1111:, p. 331.
1109:Mathiesen 1999
1101:
1089:
1080:
1065:
1043:
1036:
1018:
999:
987:
955:
943:
941:, p. 333.
939:Mathiesen 1999
931:
916:
901:
889:
887:, p. 312.
885:Mathiesen 1999
874:
866:Mathiesen 1999
854:
852:, p. 246.
842:
840:, p. 229.
830:
814:Mathiesen 1999
806:
804:, p. 413.
794:
770:
769:
763:
762:
750:10.2307/763814
733:
727:
708:
690:(4): 406–428.
676:
670:
655:
635:10.2307/629840
618:
606:10.2307/763813
589:
577:10.2307/843492
558:
555:
482:
479:
430:octave species
418:
415:
368:Sectio Canonis
304:octave species
292:
289:
210:
209:
199:
189:
179:
144:
141:
133:diatonic genus
131:. Because the
76:
73:
65:octave species
13:
10:
9:
6:
4:
3:
2:
1171:
1160:
1157:
1155:
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1150:
1147:
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1039:
1037:9780521517645
1033:
1029:
1022:
1019:
1014:
1010:
1003:
1000:
996:
995:Chalmers 1990
991:
988:
984:
980:
976:
972:
968:
962:
960:
956:
952:
947:
944:
940:
935:
932:
928:
927:Chalmers 1990
923:
921:
917:
913:
912:Chalmers 1990
908:
906:
902:
898:
893:
890:
886:
881:
879:
875:
871:
867:
864:, p. 4;
863:
862:Chalmers 1990
858:
855:
851:
846:
843:
839:
834:
831:
823:
819:
815:
810:
807:
803:
798:
795:
791:
790:0-19-864226-1
787:
783:
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759:
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728:9780803230798
724:
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671:0-945996-04-7
667:
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387:
383:
381:
377:
373:
369:
365:
361:
357:
353:
349:
345:
342:of 256:243—a
341:
337:
336:major seconds
333:
329:
325:
322:), leaving a
321:
317:
313:
305:
302:
297:
290:
288:
286:
282:
278:
274:
270:
266:
262:
258:
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207:
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166:
162:
158:
154:
150:
142:
140:
138:
134:
130:
126:
122:
114:
110:
106:
105:pitch classes
102:
98:
94:
86:
82:
74:
72:
70:
66:
62:
61:melodic genus
58:
54:
50:
46:
42:
34:
30:
22:
18:
1131:
1104:
1097:Barbera 1984
1092:
1083:
1074:
1068:
1060:Grand Rapids
1055:
1046:
1027:
1021:
1008:
1002:
990:
966:
953:, p. 6.
946:
934:
929:, p. 8.
914:, p. 9.
899:, p. 321n11.
897:Barbera 1977
892:
857:
850:Solomon 1984
845:
838:Barbera 1984
833:
809:
797:
779:
774:
765:
764:
741:
737:
713:
687:
683:
660:
626:
622:
597:
593:
568:
564:
550:
544:
542:
537:
533:
527:
510:
506:
502:
498:
494:
474:
470:
462:
457:
456:"—leaving a
435:
409:
405:
389:
384:
379:
375:
371:
367:
363:
359:
352:Eratosthenes
347:
339:
331:
323:
309:
284:
280:
276:
272:
268:
264:
260:
256:
252:
248:
244:
240:
236:
232:
230:
225:
221:
217:
213:
211:
205:
201:
195:
191:
185:
181:
175:
171:
164:
160:
156:
152:
148:
146:
136:
120:
96:
80:
78:
68:
56:
28:
16:
15:
951:Barker 1981
530:Aristoxenus
454:minor third
356:Aristoxenus
316:major third
53:incomposite
1143:Categories
983:0585229929
975:0198149751
802:Levin 2007
557:References
509:; the two
491:Mixolydian
467:Gaudentius
372:parhypatai
291:Enharmonic
224:above the
220:, and the
93:tetrachord
75:Definition
71:involved.
49:tetrachord
1013:New Haven
766:Footnotes
704:151484983
651:162356270
538:Harmonics
417:Chromatic
390:Harmonics
218:parhypate
198:(movable)
192:parhypate
188:(movable)
161:hypermese
153:parhypate
125:chromatic
1054:(1994).
977:(pbk.);
629:: 1–16.
394:Archytas
328:semitone
277:lichanos
273:lichanos
257:paranete
249:lichanos
245:paranete
241:paramese
237:lichanos
235:and the
222:paranete
214:lichanos
206:paramese
204:(1:1) –
186:paranete
182:lichanos
174:(4:3) –
157:lichanos
719:Lincoln
398:Didymus
386:Ptolemy
332:epogdoa
285:apyknon
251:to the
163:), and
45:spissus
1034:
981:
973:
870:p. 245
788:
758:763814
756:
725:
702:
684:Hermes
668:
649:
643:629840
641:
614:763813
612:
585:843492
583:
534:pyknon
489:Greek
471:pyknon
463:pyknon
458:pyknon
427:Dorian
410:pyknon
406:pyknon
380:pyknon
376:tritai
364:pyknon
360:pyknon
348:pyknon
340:pyknon
324:pyknon
312:ditone
301:Dorian
269:hypate
265:pyknon
233:hypate
202:hypate
149:hypate
137:pyknon
121:pyknon
81:pyknon
57:pyknon
37:πυκνός
31:(from
29:pycnon
25:πυκνόν
19:(from
17:Pyknon
822:Greek
754:JSTOR
700:S2CID
647:S2CID
639:JSTOR
610:JSTOR
581:JSTOR
511:pykna
475:pykna
446:Latin
438:Greek
344:limma
334:, or
314:(the
226:trite
196:trite
113:Greek
101:scale
85:Greek
69:pykna
41:Latin
33:Greek
21:Greek
1032:ISBN
979:ISBN
971:ISBN
786:ISBN
723:ISBN
666:ISBN
519:Play
497:and
402:just
396:and
374:and
281:mese
261:nete
255:(or
253:mese
243:and
239:(or
176:nete
172:mese
165:mese
159:(or
127:and
79:The
818:322
746:doi
692:doi
688:135
631:doi
627:101
602:doi
573:doi
318:of
279:to
271:to
259:to
97:any
1145::
958:^
919:^
904:^
877:^
868:,
824::
752:.
740:.
717:.
698:.
686:.
645:.
637:.
625:.
608:.
596:.
579:.
569:21
567:.
448::
444:,
440::
228:.
194:–
184:–
155:,
151:,
115::
87::
43::
35::
23::
1040:.
872:.
828:.
792:.
760:.
748::
742:3
731:.
706:.
694::
674:.
653:.
633::
616:.
604::
598:3
587:.
575::
549:(
507:d
503:c
499:b
495:a
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
