View source for Set (music)

{{Short description|Collection of objects studied in music theory}}
{{Redirect|Set class|the concept in set theory|Class (set theory)}}
[[Image:Nono - Variazioni canoniche, rhythmic values row.png|thumb|right|350px|Six-element set of rhythmic values used in ''Variazioni canoniche'' by [[Luigi Nono (composer)|Luigi Nono]]<ref>Whittall, Arnold (2008). ''The Cambridge Introduction to Serialism'', p.165. New York: Cambridge University Press. {{ISBN|978-0-521-68200-8}} (pbk).</ref>]]

A '''set''' ('''pitch set''', '''pitch-class set''', '''set class''', '''set form''', '''set genus''', '''pitch collection''') in [[music theory]], as in [[Set (mathematics)|mathematics]] and general parlance, is a collection of objects. In [[Set theory (music)|musical contexts]] the term is traditionally applied most often to collections of pitches or [[Pitch class|pitch-classes]], but theorists have extended its use to other types of musical entities, so that one may speak of sets of [[rhythm|durations]] or [[timbres]], for example.<ref name="Wittlich">Wittlich, Gary (1975). "Sets and Ordering Procedures in Twentieth-Century Music", ''Aspects of Twentieth-Century Music'', p.475. Wittlich, Gary (ed.). Englewood Cliffs, New Jersey: Prentice-Hall. {{ISBN|0-13-049346-5}}.</ref>

[[Image:Stravinsky - In memoriam Dylan Thomas five-tone row.png|thumb|right|350px|Prime form of five pitch class set from [[Igor Stravinsky]]'s ''In memoriam Dylan Thomas''<ref>Whittall (2008), p.127.</ref>]]
[[File:Set theory 3-1 in the chromatic circle fix.svg|thumb|Set 3-1 has three possible rotations/inversions, the normal form of which is the smallest pie or most compact form]]

A set by itself does not necessarily possess any additional structure, such as an [[wikt:ordering|ordering]] or [[permutation (music)|permutation]]. Nevertheless, it is often musically important to consider sets that are equipped with an order relation (called ''segments''); in such contexts, bare sets are often referred to as "unordered", for the sake of emphasis.<ref>Morris, Robert (1987). ''Composition With Pitch-Classes: A Theory of Compositional Design'', p.27. Yale University Press. {{ISBN|0-300-03684-1}}.</ref>

{{anchor|7|8|9|10|11|12}}Two-element sets are called [[dyad (music)|dyad]]s, three-element sets [[trichord]]s (occasionally "triads", though this is easily confused with the traditional meaning of the word [[Triad (music)|triad]]).  Sets of higher cardinalities are called [[tetrachord]]s (or tetrads), [[pentachord]]s (or pentads), [[hexachord]]s (or hexads), '''heptachords''' (heptads or, sometimes, mixing Latin and Greek roots, "septachords"),<ref>E.g., Rahn (1980), 140.</ref> '''octachords''' (octads), '''nonachords''' (nonads), '''decachords''' (decads), '''undecachords''', and, finally, the '''dodecachord'''.

A [[time-point|time-point set]] is a [[duration series|duration set]] where the distance in time units between attack points, or time-points, is the distance in semitones between pitch classes.<ref name="Wittlich 476">Wittlich (1975), p.476.</ref>

==Serial==
In the theory of [[serial music]], however, some authors{{weasel inline|date=September 2012}} (notably [[Milton Babbitt]]<ref>See any of his writings on the twelve-tone system, virtually all of which are reprinted in ''The Collected Essays of Milton Babbitt'', S. Peles et al., eds. Princeton University Press, 2003. {{ISBN|0-691-08966-3}}.</ref>{{page needed|date=October 2018}}{{Quote needed|date=February 2017|Specific claim made without any support, the claim cites everything Babbitt ever wrote.}}) use the term "set" where others would use "row" or "series", namely to denote an ordered collection (such as a [[Twelve-tone technique|twelve-tone row]]) used to structure a work. These authors{{weasel inline|date=September 2012}} 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 "[[Set theory (music)|set theory]]").
  
For these authors,{{weasel inline|date=September 2012}} a ''set form'' (or ''row form'') is a particular arrangement of such an ordered set: the [[Prime form (music)|prime form]] (original order), [[Melodic inversion|inverse]] (upside down), [[Retrograde (music)|retrograde]] (backwards), and [[Retrograde inversion|retrograde inverse]] (backwards and upside down).<ref name="Wittlich"/>

A [[derived row|derived set]] is one which is generated or derived from consistent operations on a subset, for example [[Anton Webern|Webern]]'s ''[[Concerto for Nine Instruments (Webern)|Concerto]]'', Op.24, in which the last three subsets are derived from the first:<ref name="Wittlich 474">Wittlich (1975), p.474.</ref>

: <score vorbis="1" lang="lilypond">
{
\override Score.TimeSignature
#'stencil = ##f
\override Score.SpacingSpanner.strict-note-spacing = ##t
  \set Score.proportionalNotationDuration = #(ly:make-moment 1/1)
    \relative c'' {
        \time 3/1
        \set Score.tempoHideNote = ##t \tempo 1 = 60
        b1 bes d  
        es, g fis  
        aes e f  
        c' cis a
    }
}
</score>

This can be represented numerically as the integers 0 to 11:
 0 11 3 4 8 7 9 5 6 1 2 10

The first subset (B B{{music|b}} D) being:
 0 11 3 prime-form, interval-string = {{angbr|−1 +4}}
The second subset (E{{music|b}} G F{{music|#}}) being the retrograde-inverse of the first, transposed up one semitone:
   3 11 0 retrograde, interval-string = {{angbr|−4 +1}} mod 12
   
   3  7 6 inverse, interval-string = {{angbr|+4 −1}} mod 12
 + 1  1 1
   ------
 = 4  8 7 
The third subset (G{{music|#}} E F) being the retrograde of the first, transposed up (or down) six semitones:
   3 11 0 retrograde
 + 6  6 6
   ------
   9  5 6 
And the fourth subset (C C{{music|#}} A) being the inverse of the first, transposed up one semitone:
   0 11  3 prime form, interval-vector = {{angbr|−1 +4}} mod 12 
 
   0  1  9 inverse, interval-string = {{angbr|+1 −4}} mod 12
 + 1  1  1
   -------
   1  2 10

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 [[invariance (music)|invariances]]. These invariances in serial music are analogous to the use of common-tones and common-chords in tonal music.{{citation needed|date=September 2012}}

==Non-serial== <!--[[Normal form (music)]] and [[Prime form (music)]] redirect directly here.-->
{{Main|Set theory (music)}}

[[Image:Major second on C.svg|thumb|right|Major second on C {{audio|Major second on C.mid|Play}}.]]
[[Image:Minor seventh on C.png|thumb|left|Minor seventh on C {{audio|Minor seventh on C.mid|Play}}.]]
[[Image:Inverted minor seventh on C.png|thumb|right|Inverted minor seventh on C (major second on B{{music|b}}) {{audio|Inverted minor seventh on C.mid|Play}}.]]

The fundamental concept of a non-serial set is that it is an unordered collection of [[pitch class]]es.<ref>[[John Rahn]], ''Basic Atonal Theory'' (New York: Longman; London and Toronto: Prentice Hall International, 1980), pp.27–28. {{ISBN|0-582-28117-2}} (Longman); {{ISBN|0-02-873160-3}} (Prentice Hall International). Reprinted 1987 (New York: Schirmer Books; London: Collier Macmillan, 1980), p.27. {{ISBN|0-02-873160-3}}.</ref>

The '''normal form''' of a set is the '''most compact''' ordering of the pitches in a set.<ref name="NF">Tomlin, Jay. [http://www.jaytomlin.com/music/settheory/help.html#normalform "All About Set Theory: What is Normal Form?"], ''JayTomlin.com''.</ref> 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".<ref name="NF"/> For example, the set (0,2) (a [[major second]]) is in normal form while the set (0,10) (a [[minor seventh]], the [[Inversion (interval)|inversion]] of a major second) is not, its normal form being (10,0).

Rather than the "original" (untransposed, uninverted) form of the set, the '''prime form''' may be considered either the normal form of the set or the normal form of its inversion, whichever is more tightly packed.<ref>Tomlin, Jay. [http://www.jaytomlin.com/music/settheory/help.html#primeform "All About Set Theory: What is Prime Form?"], ''JayTomlin.com''.</ref> 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 ("making the small numbers smaller," versus making, "the larger numbers ... smaller"<ref name="Tools">{{Cite web |last=Nelson |first=Paul |date=2004 |title=Two Algorithms for Computing the Prime Form |url=http://composertools.com/Tools/PCSets/desc_alg.html |url-status=unfit |archive-url=https://web.archive.org/web/20171223210600/http://composertools.com/Tools/PCSets/desc_alg.html |archive-date=Dec 23, 2017 |website=ComposerTools.com}}</ref>). For many years it was accepted that there were only five instances in which the two algorithms differ.<ref>Tsao, Ming (2007). ''Abstract Musical Intervals: Group Theory for Composition and Analysis'', p.99, n.32. {{ISBN|9781430308355}}. Algorithms given in Morris, Robert (1991). ''Class Notes for Atonal Music Theory'', p.103. Frog Peak Music.</ref> 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.<ref name="RingScales">{{Cite web|url=https://ianring.com/musictheory/scales/#primeform|title = A study of musical scales by Ian Ring}}</ref> Ian Ring also established a much simpler algorithm for computing the prime form of a set,<ref name="RingScales" /> which produces the same results as the more complicated algorithm previously published by John Rahn.

==Vectors==
{{Main|List of set classes}}

== See also ==
* [[Forte number]]
* [[Pitch interval]]
* [[Set list]]
* [[Similarity relation]]

== References ==
{{reflist}}

==Further reading==
*Schuijer, Michiel (2008). ''Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts''. {{ISBN|978-1-58046-270-9}}.

==External links==
*[http://www.jaytomlin.com/music/settheory/default.htm "Set Theory Calculator"], ''JayTomlin.com''. Calculates normal form, prime form, [[Forte number]], and [[interval class vector]] for a given set and vice versa.
*"[https://www.mta.ca/pc-set/calculator/pc_calculate.html PC Set Calculator]", ''MtA.Ca''.

{{Pitch segments}}
{{Set theory (music)}}

[[Category:Musical set theory]]