274:
120:
490:("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
469:
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
442:
221:) 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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35:
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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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where the distance in time units between attack points, or time-points, is the distance in semitones between pitch classes.
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34:
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1175:
135:. Nevertheless, it is often musically important to consider sets that are equipped with an order relation (called
123:
Set 3-1 has three possible rotations/inversions, the normal form of which is the smallest pie or most compact form
119:
685:(Prentice Hall International). Reprinted 1987 (New York: Schirmer Books; London: Collier Macmillan, 1980), p.27.
217:) use the term "set" where others would use "row" or "series", namely to denote an ordered collection (such as a
374:. These invariances in serial music are analogous to the use of common-tones and common-chords in tonal music.
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93:, but theorists have extended its use to other types of musical entities, so that one may speak of sets of
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150:(occasionally "triads", though this is easily confused with the traditional meaning of the word
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139:); in such contexts, bare sets are often referred to as "unordered", for the sake of emphasis.
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See any of his writings on the twelve-tone system, virtually all of which are reprinted in
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is one which is generated or derived from consistent operations on a subset, for example
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214:
669:(New York: Longman; London and Toronto: Prentice Hall International, 1980), pp.27–28.
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The fundamental concept of a non-serial set is that it is an unordered collection of
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260:
210:
143:
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563:
Wittlich, Gary (1975). "Sets and
Ordering Procedures in Twentieth-Century Music",
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127:
A set by itself does not necessarily possess any additional structure, such as an
17:
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256:
90:
344:
E F) being the retrograde of the first, transposed up (or down) six semitones:
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155:
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662:
567:, p.475. Wittlich, Gary (ed.). Englewood Cliffs, New Jersey: Prentice-Hall.
163:
128:
485:
Rather than the "original" (untransposed, uninverted) form of the set, the
89:
the term is traditionally applied most often to collections of pitches or
1006:
930:
324:) being the retrograde-inverse of the first, transposed up one semitone:
147:
98:
170:(heptads or, sometimes, mixing Latin and Greek roots, "septachords"),
1017:
774:
Abstract
Musical Intervals: Group Theory for Composition and Analysis
269:, Op.24, in which the last three subsets are derived from the first:
94:
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407:
387:
118:
103:
33:
595:
1021:
873:
820:
Analyzing Atonal Music: Pitch-Class Set Theory and Its
Contexts
356:
A) being the inverse of the first, transposed up one semitone:
27:"Set class" redirects here. For the concept in set theory, see
291:
This can be represented numerically as the integers 0 to 11:
636:, S. Peles et al., eds. Princeton University Press, 2003.
236:) is a particular arrangement of such an ordered set: the
482:
of a major second) is not, its normal form being (10,0).
279:
Audio playback is not supported in your browser. You can
85:
and general parlance, is a collection of objects. In
1159:
543:, p.165. New York: Cambridge University Press.
432:Inverted minor seventh on C (major second on B
347:3 11 0 retrograde + 6 6 6 ------ 9 5 6
1033:
885:
796:
794:
784:. Algorithms given in Morris, Robert (1991).
748:. Archived from the original on Dec 23, 2017.
742:"Two Algorithms for Computing the Prime Form"
363:mod 12 0 1 9 inverse, interval-string =
8:
707:"All About Set Theory: What is Normal Form?"
474:) is in normal form while the set (0,10) (a
154:). Sets of higher cardinalities are called
724:"All About Set Theory: What is Prime Form?"
331:mod 12 3 7 6 inverse, interval-string =
38:Six-element set of rhythmic values used in
1040:
1026:
1018:
892:
878:
870:
559:
557:
108:Prime form of five pitch class set from
802:"A study of musical scales by Ian Ring"
541:The Cambridge Introduction to Serialism
532:
847:. Calculates normal form, prime form,
751:
634:The Collected Essays of Milton Babbitt
359:0 11 3 prime form, interval-vector =
701:
699:
367:mod 12 + 1 1 1 ------- 1 2 10
327:3 11 0 retrograde, interval-string =
306:0 11 3 prime-form, interval-string =
7:
786:Class Notes for Atonal Music Theory
335:mod 12 + 1 1 1 ------ = 4 8 7
565:Aspects of Twentieth-Century Music
25:
213:, however, some authors (notably
1150:
855:for a given set and vice versa.
597:, p.27. Yale University Press.
1213:Structure implies multiplicity
1197:Generic and specific intervals
1:
252:(backwards and upside down).
276:
142:Two-element sets are called
1250:
1176:Cardinality equals variety
818:Schuijer, Michiel (2008).
497:
381:
350:And the fourth subset (C C
294:0 11 3 4 8 7 9 5 6 1 2 10
26:
1148:
1055:
916:
788:, p.103. Frog Peak Music.
758:: CS1 maint: unfit URL (
539:Whittall, Arnold (2008).
114:In memoriam Dylan Thomas
1060:All-interval tetrachord
841:"Set Theory Calculator"
653:Wittlich (1975), p.474.
623:Wittlich (1975), p.476.
614:E.g., Rahn (1980), 140.
593:Morris, Robert (1987).
584:Whittall (2008), p.127.
281:download the audio file
1065:All-trichord hexachord
451:
425:
405:
124:
116:
46:
1183:(Deep scale property)
853:interval class vector
740:Nelson, Paul (2004).
431:
411:
391:
365:⟨+1 −4⟩
361:⟨−1 +4⟩
333:⟨+4 −1⟩
329:⟨−4 +1⟩
308:⟨−1 +4⟩
297:The first subset (B B
228:For these authors, a
146:, three-element sets
122:
107:
37:
1208:Rothenberg propriety
1192:Generated collection
1115:Pitch-interval class
312:The second subset (E
186:, and, finally, the
40:Variazioni canoniche
1199:(Myhill's property)
1132:Similarity relation
772:Tsao, Ming (2007).
667:Basic Atonal Theory
521:Similarity relation
500:List of set classes
412:Minor seventh on C
338:The third subset (G
1234:Musical set theory
1049:Musical set theory
452:
426:
406:
392:Major second on C
384:Set theory (music)
250:retrograde inverse
240:(original order),
125:
117:
47:
29:Class (set theory)
18:Prime form (music)
1221:
1220:
1015:
1014:
985:"Ode-to-Napoleon"
860:PC Set Calculator
828:978-1-58046-270-9
746:ComposerTools.com
549:978-0-521-68200-8
285:
248:(backwards), and
209:In the theory of
16:(Redirected from
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1203:Maximal evenness
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465:of a set is the
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75:pitch collection
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1090:Interval vector
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813:Further reading
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244:(upside down),
219:twelve-tone row
207:
110:Igor Stravinsky
101:, for example.
59:pitch-class set
32:
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22:
15:
12:
11:
5:
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1187:Diatonic scale
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1137:Transformation
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1110:Pitch interval
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1095:Multiplication
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1085:Interval class
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901:Pitch segments
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836:
835:External links
833:
832:
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776:, p.99, n.32.
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516:Pitch interval
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498:Main article:
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382:Main article:
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293:
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215:Milton Babbitt
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195:time-point set
162:(or pentads),
158:(or tetrads),
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845:JayTomlin.com
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728:JayTomlin.com
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722:Tomlin, Jay.
719:
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712:
711:JayTomlin.com
708:
705:Tomlin, Jay.
702:
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691:0-02-873160-3
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683:0-02-873160-3
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675:0-582-28117-2
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642:0-691-08966-3
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476:minor seventh
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456:pitch classes
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166:(or hexads),
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91:pitch-classes
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68:
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52:
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41:
36:
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19:
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1075:Forte number
965:All-trichord
948:All-interval
904:
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863:
849:Forte number
844:
819:
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666:
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633:
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619:
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594:
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511:Forte number
486:
484:
472:major second
467:most compact
466:
462:
460:
453:
369:
349:
337:
311:
296:
290:
264:
254:
233:
229:
227:
211:serial music
208:
199:duration set
192:
187:
184:undecachords
183:
179:
175:
171:
167:
141:
136:
126:
113:
79:music theory
74:
70:
66:
62:
58:
54:
50:
48:
39:
1181:Common tone
1105:Pitch class
1100:Permutation
905:cardinality
677:(Longman);
463:normal form
372:invariances
257:derived set
188:dodecachord
168:heptachords
160:pentachords
156:tetrachords
133:permutation
83:mathematics
1163:set theory
1142:Z-relation
1070:Complement
1000:Schoenberg
955:Pentachord
943:Tetrachord
527:References
487:prime form
378:Non-serial
303:D) being:
246:retrograde
238:prime form
223:set theory
182:(decads),
180:decachords
178:(nonads),
176:nonachords
174:(octads),
172:octachords
164:hexachords
44:Luigi Nono
1007:Aggregate
990:Petrushka
970:Chromatic
960:Hexachord
663:John Rahn
480:inversion
148:trichords
95:durations
71:set genus
63:set class
55:pitch set
1228:Category
1171:Bisector
1161:Diatonic
1080:Identity
975:Diatonic
936:Viennese
931:Trichord
754:cite web
505:See also
435:♭
353:♯
341:♯
321:♯
315:♭
300:♭
266:Concerto
234:row form
230:set form
137:segments
129:ordering
81:, as in
67:set form
494:Vectors
242:inverse
99:timbres
995:Sacher
980:Mystic
864:MtA.Ca
851:, and
826:
780:
689:
681:
673:
640:
601:
571:
551:(pbk).
547:
478:, the
261:Webern
205:Serial
921:Monad
197:is a
152:triad
144:dyads
77:) in
1125:List
926:Dyad
909:list
824:ISBN
778:ISBN
760:link
687:ISBN
679:ISBN
671:ISBN
638:ISBN
599:ISBN
569:ISBN
545:ISBN
461:The
444:Play
418:Play
398:Play
232:(or
225:").
1120:Set
903:by
862:",
318:G F
263:'s
131:or
112:'s
97:or
51:set
42:by
1230::
843:,
822:.
793:^
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752:{{
744:.
726:,
709:,
698:^
665:,
556:^
458:.
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255:A
193:A
190:.
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69:,
65:,
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911:)
907:(
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886:t
879:v
866:.
858:"
830:.
804:.
762:)
730:.
713:.
693:.
644:.
605:.
575:.
450:.
424:.
404:.
283:.
53:(
31:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
