444:
418:
398:
123:
1155:
277:
493:("making the small numbers smaller," versus making, "the larger numbers ... smaller"). For many years it was accepted that there were only five instances in which the two algorithms differ. However, in 2017, music theorist Ian Ring discovered that there is a sixth set class where Forte and Rahn's algorithms arrive at different prime forms. Ian Ring also established a much simpler algorithm for computing the prime form of a set, which produces the same results as the more complicated algorithm previously published by John Rahn.
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may be considered either the normal form of the set or the normal form of its inversion, whichever is more tightly packed. Forte (1973) and Rahn (1980) both list the prime forms of a set as the most left-packed possible version of the set. Forte packs from the left and Rahn packs from the right
472:
ordering of the pitches in a set. Tomlin defines the "most compact" ordering as the one where, "the largest of the intervals between any two consecutive pitches is between the first and last pitch listed". For example, the set (0,2) (a
445:
224:) used to structure a work. These authors speak of "twelve tone sets", "time-point sets", "derived sets", etc. (See below.) This is a different usage of the term "set" from that described above (and referred to in the term "
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38:
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Each of the four trichords (3-note sets) thus displays a relationship which can be made obvious by any of the four serial row operations, and thus creates certain
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894:
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268:
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204:
where the distance in time units between attack points, or time-points, is the distance in semitones between pitch classes.
280:
987:
784:
693:
685:
677:
644:
605:
575:
1035:
138:. Nevertheless, it is often musically important to consider sets that are equipped with an order relation (called
126:
Set 3-1 has three possible rotations/inversions, the normal form of which is the smallest pie or most compact form
688:(Prentice Hall International). Reprinted 1987 (New York: Schirmer Books; London: Collier Macmillan, 1980), p.27.
220:) use the term "set" where others would use "row" or "series", namely to denote an ordered collection (such as a
1215:
1199:
377:. These invariances in serial music are analogous to the use of common-tones and common-chords in tonal music.
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887:
96:, but theorists have extended its use to other types of musical entities, so that one may speak of sets of
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1134:
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17:
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37:
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153:(occasionally "triads", though this is easily confused with the traditional meaning of the word
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1097:
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726:
142:); in such contexts, bare sets are often referred to as "unordered", for the sake of emphasis.
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221:
122:
46:
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1210:
1194:
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635:
See any of his writings on the twelve-tone system, virtually all of which are reprinted in
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252:
8:
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is one which is generated or derived from consistent operations on a subset, for example
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386:
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31:
672:(New York: Longman; London and Toronto: Prentice Hall International, 1980), pp.27–28.
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601:
571:
547:
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The fundamental concept of a non-serial set is that it is an unordered collection of
244:
85:
804:
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997:
566:
Wittlich, Gary (1975). "Sets and
Ordering Procedures in Twentieth-Century Music",
130:
A set by itself does not necessarily possess any additional structure, such as an
1092:
992:
201:
112:
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1112:
1087:
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347:
E F) being the retrograde of the first, transposed up (or down) six semitones:
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154:
570:, p.475. Wittlich, Gary (ed.). Englewood Cliffs, New Jersey: Prentice-Hall.
131:
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Rather than the "original" (untransposed, uninverted) form of the set, the
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213:
146:
81:
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the term is traditionally applied most often to collections of pitches or
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327:) being the retrograde-inverse of the first, transposed up one semitone:
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93:
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945:
197:
162:
158:
173:(heptads or, sometimes, mixing Latin and Greek roots, "septachords"),
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962:
777:
Abstract
Musical Intervals: Group Theory for Composition and Analysis
665:
411:
272:, Op.24, in which the last three subsets are derived from the first:
166:
391:
933:
150:
101:
598:
823:
Analyzing Atonal Music: Pitch-Class Set Theory and Its
Contexts
359:
A) being the inverse of the first, transposed up one semitone:
97:
30:"Set class" redirects here. For the concept in set theory, see
294:
This can be represented numerically as the integers 0 to 11:
639:, S. Peles et al., eds. Princeton University Press, 2003.
239:) is a particular arrangement of such an ordered set: the
485:
of a major second) is not, its normal form being (10,0).
282:
Audio playback is not supported in your browser. You can
902:
88:
and general parlance, is a collection of objects. In
1228:
546:, p.165. New York: Cambridge University Press.
435:Inverted minor seventh on C (major second on B
350:3 11 0 retrograde + 6 6 6 ------ 9 5 6
1036:
888:
799:
797:
787:. Algorithms given in Morris, Robert (1991).
751:. Archived from the original on Dec 23, 2017.
745:"Two Algorithms for Computing the Prime Form"
366:mod 12 0 1 9 inverse, interval-string =
27:Collection of objects studied in music theory
710:"All About Set Theory: What is Normal Form?"
477:) is in normal form while the set (0,10) (a
157:). Sets of higher cardinalities are called
727:"All About Set Theory: What is Prime Form?"
650:
620:
334:mod 12 3 7 6 inverse, interval-string =
41:Six-element set of rhythmic values used in
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895:
881:
794:
562:
560:
430:
410:
390:
121:
111:Prime form of five pitch class set from
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36:
805:"A study of musical scales by Ian Ring"
557:
544:The Cambridge Introduction to Serialism
14:
1229:
1050:
850:. Calculates normal form, prime form,
742:
637:The Collected Essays of Milton Babbitt
362:0 11 3 prime form, interval-vector =
1024:
876:
736:
704:
702:
370:mod 12 + 1 1 1 ------- 1 2 10
330:3 11 0 retrograde, interval-string =
309:0 11 3 prime-form, interval-string =
789:Class Notes for Atonal Music Theory
338:mod 12 + 1 1 1 ------ = 4 8 7
24:
815:
699:
568:Aspects of Twentieth-Century Music
25:
1248:
837:
216:, however, some authors (notably
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858:for a given set and vice versa.
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719:
600:, p.27. Yale University Press.
1216:Structure implies multiplicity
1200:Generic and specific intervals
659:
629:
611:
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581:
536:
13:
1:
529:
380:
255:(backwards and upside down).
279:
145:Two-element sets are called
7:
507:
10:
1253:
1179:Cardinality equals variety
821:Schuijer, Michiel (2008).
500:
496:
384:
353:And the fourth subset (C C
297:0 11 3 4 8 7 9 5 6 1 2 10
29:
1162:
1151:
1058:
919:
791:, p.103. Frog Peak Music.
761:: CS1 maint: unfit URL (
542:Whittall, Arnold (2008).
207:
117:In memoriam Dylan Thomas
1063:All-interval tetrachord
844:"Set Theory Calculator"
656:Wittlich (1975), p.474.
626:Wittlich (1975), p.476.
617:E.g., Rahn (1980), 140.
596:Morris, Robert (1987).
587:Whittall (2008), p.127.
284:download the audio file
1068:All-trichord hexachord
454:
428:
408:
127:
119:
49:
1186:(Deep scale property)
856:interval class vector
743:Nelson, Paul (2004).
434:
414:
394:
368:⟨+1 −4⟩
364:⟨−1 +4⟩
336:⟨+4 −1⟩
332:⟨−4 +1⟩
311:⟨−1 +4⟩
300:The first subset (B B
231:For these authors, a
149:, three-element sets
125:
110:
40:
1211:Rothenberg propriety
1195:Generated collection
1118:Pitch-interval class
315:The second subset (E
189:, and, finally, the
43:Variazioni canoniche
1202:(Myhill's property)
1135:Similarity relation
775:Tsao, Ming (2007).
670:Basic Atonal Theory
524:Similarity relation
503:List of set classes
415:Minor seventh on C
341:The third subset (G
1237:Musical set theory
1052:Musical set theory
455:
429:
409:
395:Major second on C
387:Set theory (music)
253:retrograde inverse
243:(original order),
128:
120:
50:
32:Class (set theory)
1224:
1223:
1018:
1017:
988:"Ode-to-Napoleon"
863:PC Set Calculator
831:978-1-58046-270-9
749:ComposerTools.com
552:978-0-521-68200-8
288:
251:(backwards), and
212:In the theory of
16:(Redirected from
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1206:Maximal evenness
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468:of a set is the
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90:musical contexts
78:pitch collection
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1093:Interval vector
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816:Further reading
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247:(upside down),
222:twelve-tone row
210:
113:Igor Stravinsky
104:, for example.
62:pitch-class set
35:
28:
23:
22:
15:
12:
11:
5:
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1190:Diatonic scale
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1140:Transformation
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1113:Pitch interval
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1098:Multiplication
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1088:Interval class
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904:Pitch segments
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838:External links
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793:
779:, p.99, n.32.
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519:Pitch interval
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501:Main article:
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385:Main article:
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218:Milton Babbitt
209:
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198:time-point set
165:(or pentads),
161:(or tetrads),
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9:
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848:JayTomlin.com
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785:9781430308355
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731:JayTomlin.com
728:
725:Tomlin, Jay.
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715:
714:JayTomlin.com
711:
708:Tomlin, Jay.
705:
703:
695:
694:0-02-873160-3
691:
687:
686:0-02-873160-3
683:
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678:0-582-28117-2
675:
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645:0-691-08966-3
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606:0-300-03684-1
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479:minor seventh
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459:pitch classes
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169:(or hexads),
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94:pitch-classes
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48:
44:
39:
33:
19:
1122:
1078:Forte number
968:All-trichord
951:All-interval
907:
903:
866:
852:Forte number
847:
822:
788:
776:
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748:
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669:
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631:
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583:
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514:Forte number
489:
487:
475:major second
470:most compact
469:
465:
463:
456:
372:
352:
340:
314:
299:
293:
267:
257:
236:
232:
230:
214:serial music
211:
202:duration set
195:
190:
187:undecachords
186:
182:
178:
174:
170:
144:
139:
129:
116:
82:music theory
77:
73:
69:
65:
61:
57:
53:
51:
42:
1184:Common tone
1108:Pitch class
1103:Permutation
908:cardinality
680:(Longman);
466:normal form
375:invariances
260:derived set
191:dodecachord
171:heptachords
163:pentachords
159:tetrachords
136:permutation
86:mathematics
18:Musical set
1166:set theory
1145:Z-relation
1073:Complement
1003:Schoenberg
958:Pentachord
946:Tetrachord
530:References
490:prime form
381:Non-serial
306:D) being:
249:retrograde
241:prime form
226:set theory
185:(decads),
183:decachords
181:(nonads),
179:nonachords
177:(octads),
175:octachords
167:hexachords
47:Luigi Nono
1010:Aggregate
993:Petrushka
973:Chromatic
963:Hexachord
666:John Rahn
483:inversion
151:trichords
98:durations
74:set genus
66:set class
58:pitch set
1231:Category
1174:Bisector
1164:Diatonic
1083:Identity
978:Diatonic
939:Viennese
934:Trichord
757:cite web
508:See also
438:♭
356:♯
344:♯
324:♯
318:♭
303:♭
269:Concerto
237:row form
233:set form
140:segments
132:ordering
84:, as in
70:set form
497:Vectors
245:inverse
102:timbres
998:Sacher
983:Mystic
867:MtA.Ca
854:, and
829:
783:
692:
684:
676:
643:
604:
574:
554:(pbk).
550:
481:, the
264:Webern
208:Serial
924:Monad
200:is a
155:triad
147:dyads
80:) in
1128:List
929:Dyad
912:list
827:ISBN
781:ISBN
763:link
690:ISBN
682:ISBN
674:ISBN
641:ISBN
602:ISBN
572:ISBN
548:ISBN
464:The
447:Play
421:Play
401:Play
235:(or
228:").
1123:Set
906:by
865:",
321:G F
266:'s
134:or
115:'s
100:or
54:set
45:by
1233::
846:,
825:.
796:^
759:}}
755:{{
747:.
729:,
712:,
701:^
668:,
559:^
461:.
441:)
258:A
196:A
193:.
76:,
72:,
68:,
64:,
60:,
52:A
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914:)
910:(
896:e
889:t
882:v
869:.
861:"
833:.
807:.
765:)
733:.
716:.
696:.
647:.
608:.
578:.
453:.
427:.
407:.
286:.
56:(
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
