View source for Multiplication (music)

[[File:Bartok - Third Quartet tetrachord multiplication.png|thumb|Example from [[Béla Bartók]]'s [[String Quartet No. 3 (Bartók)|Third Quartet]]:<ref>{{harvnb|Antokoletz|1993|loc=260}}, cited in {{harvnb|Schuijer|2008|loc=77–78}}</ref> multiplication of a chromatic [[tetrachord]] [[File:Bartok - Third Quartet tetrachord multiplication top.mid]] to a [[quartal chord|fifths]] chord [[File:Bartok - Third Quartet tetrachord multiplication bottom.mid]] C{{music|#}}=0: 0·''7''='''0''', 1·''7''='''7''', 2·''7''='''2''', 3·''7''='''9''' (mod 12).]]
[[File:Bartok - Music for Strings, Percussion and Celesta interval expansion.png|thumb|350px|right|Bartók—''[[Music for Strings, Percussion and Celesta]]'' interval expansion example, mov.&nbsp;I, mm.&nbsp;1–5 and mov.&nbsp;IV, mm.&nbsp;204–209{{sfn|Schuijer|2008|loc=79}}[[File:Bartok - Music for Strings, Percussion and Celesta interval expansion.mid]]]]

The mathematical operations of '''multiplication''' have several applications to [[music]]. Other than its application to the frequency ratios of [[Interval (music)|intervals]] (for example, [[Just intonation]], and the [[twelfth root of two]] in [[equal temperament]]), it has been used in other ways for [[twelve-tone technique]], and [[set theory (music)|musical set theory]]. Additionally [[ring modulation]] is an electrical audio process involving multiplication that has been used for musical effect.

A multiplicative operation is a [[Map (mathematics)|mapping]] in which the [[Argument of a function|argument]] is multiplied.{{sfn|Rahn|1980|loc=53}} Multiplication originated intuitively in '''interval expansion''', including [[tone row]] order number [[Rotation (mathematics)|rotation]], for example in the music of [[Béla Bartók]] and [[Alban Berg]].{{sfn|Schuijer|2008|loc=77–78}} Pitch number rotation, ''Fünferreihe'' or "five-series" and ''Siebenerreihe'' or "seven-series", was first described by [[Ernst Krenek]] in ''Über neue Musik''.{{sfn|Krenek|1937}}{{sfn|Schuijer|2008|loc=77–78}} Princeton-based theorists, including [[James K. Randall]],{{sfn|Randall|1962}} [[Godfrey Winham]],{{sfn|Winham|1970}} and Hubert S. Howe{{sfn|Howe|1965}} "were the first to discuss and adopt them, not only with regards {{sic}} to twelve-tone series".{{sfn|Schuijer|2008|loc=81}}

== Pitch-class multiplication modulo 12 ==

When dealing with [[pitch class|pitch-class]] sets, multiplication [[modular arithmetic|modulo]] 12 is a common operation. Dealing with all [[twelve tone technique|twelve tones]], or a [[tone row]], there are only a few numbers which one may multiply a row by and still end up with a set of twelve distinct tones. Taking the prime or unaltered form as P<sub>0</sub>, multiplication is indicated by ''M<sub>x</sub>'', ''x'' being the multiplicator:
: ''M<sub>x</sub>''(''y'') ≡ ''xy'' mod 12

The following table lists all possible multiplications of a chromatic twelve-tone row:

{| class="wikitable" style="text-align:center" align="center"
 |-
 ! M
 ! colspan="12" | M × (0,1,2,3,4,5,6,7,8,9,10,11) mod 12
 |-
 ! width="20" | 0
 | width="20" | 0
 | width="20" | 0
 | width="20" | 0
 | width="20" | 0
 | width="20" | 0
 | width="20" | 0
 | width="20" | 0
 | width="20" | 0
 | width="20" | 0
 | width="20" | 0
 | width="20" | 0
 | width="20" | 0
 |- style="background:#ffdddd"
 ! 1
 | 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11
 |-
 ! 2
 | 0 || 2 || 4 || 6 || 8 || 10 || 0 || 2 || 4 || 6 || 8 || 10
 |-
 ! 3
 | 0 || 3 || 6 || 9 || 0 || 3 || 6 || 9 || 0 || 3 || 6 || 9
 |-
 ! 4
 | 0 || 4 || 8 || 0 || 4 || 8 || 0 || 4 || 8 || 0 || 4 || 8
 |- style="background:#ffdddd"
 ! 5
 | 0 || 5 || 10 || 3 || 8 || 1 || 6 || 11 || 4 || 9 || 2 || 7
 |-
 ! 6
 | 0 || 6 || 0 || 6 || 0 || 6 || 0 || 6 || 0 || 6 || 0 || 6
 |- style="background:#ffdddd"
 ! 7
 | 0 || 7 || 2 || 9 || 4 || 11 || 6 || 1 || 8 || 3 || 10 || 5
 |-
 ! 8
 | 0 || 8 || 4 || 0 || 8 || 4 || 0 || 8 || 4 || 0 || 8 || 4
 |-
 ! 9
 | 0 || 9 || 6 || 3 || 0 || 9 || 6 || 3 || 0 || 9 || 6 || 3
 |-
 ! 10
 | 0 || 10 || 8 || 6 || 4 || 2 || 0 || 10 || 8 || 6 || 4 || 2
 |- style="background:#ffdddd"
 ! 11
 | 0 || 11 || 10 || 9 || 8 || 7 || 6 || 5 || 4 || 3 || 2 || 1
 |}

Note that only M<sub>1</sub>, M<sub>5</sub>, M<sub>7</sub>, and M<sub>11</sub> give a [[Bijection|one-to-one]] mapping (a complete set of 12 unique tones). This is because each of these numbers is [[relatively prime]] to 12. Also interesting is that the [[chromatic]] scale is mapped to the [[circle of fourths]] with M<sub>5</sub>, or fifths with M<sub>7</sub>, and more generally under M<sub>7</sub> all even numbers stay the same while odd numbers are transposed by a [[tritone]]. This kind of multiplication is frequently combined with a [[Transposition (music)|transposition]] operation. It was first described in print by [[Herbert Eimert]], under the terms "Quartverwandlung" (fourth transformation) and "Quintverwandlung" (fifth transformation),{{sfn|Eimert|1950|loc=29–33}} and has been used by the composers [[Milton Babbitt]],{{sfn|Morris|1997|loc=238 & 242–43}}{{sfn|Winham|1970|loc=65–66}} [[Robert Morris (composer)|Robert Morris]],{{sfn|Morris|1997|loc=238–239, 243}} and [[Charles Wuorinen]].{{sfn|Hibbard|1969|loc=157–158}} This operation also accounts for certain harmonic transformations in jazz.{{sfn|Morris|1982|loc=153–154}}

Thus multiplication by the two meaningful operations (5 & 7) may be designated with '''''M'''''<sub>5</sub>(''a'') and '''''M'''''<sub>7</sub>(''a'') or '''''M''''' and '''''IM'''''.{{sfn|Schuijer|2008|loc=77–78}}

*M<sub>1</sub> = Identity
*M<sub>5</sub> = Cycle of fourths transform
*M<sub>7</sub> = Cycle of fifths transform
*M<sub>11</sub> = Inversion
*M<sub>11</sub>M<sub>5</sub> = M<sub>7</sub>
*M<sub>7</sub>M<sub>5</sub> = M<sub>11</sub>
*M<sub>5</sub>M<sub>5</sub> = M<sub>1</sub>
*M<sub>7</sub>M<sub>11</sub>M<sub>5</sub> = M<sub>1</sub>
*...

== Pitch multiplication ==

[[Pierre Boulez]]{{sfn|Boulez|1971|loc=39-40, 79-80}}{{Dubious|date=July 2021|reason=On pp. 79–80 Boulez speaks of "isomorphic families", "a double network of privileged series", "dividing" a series and "the whole is multiplied by each of its parts in turn", thereby creating "multiple isomorphic relationships", objects "multiplied by themselves", etc., but where is this expression "pitch multiplication"?}} described an operation he called '''pitch multiplication''', which is somewhat akin {{Clarify|date=March 2013}} to the [[Cartesian product]] of pitch-class sets. Given two sets, the result of pitch multiplication will be the set of sums ([[modular arithmetic|modulo]] 12) of all possible pairings of elements between the original two sets. Its definition:

:<math>X \times Y = \{ (x+y)\bmod 12 | x\in X, y\in Y\}</math>

For example, if multiplying a C-major chord <math>\{ 0,4,7 \}</math> with a dyad containing '''C''','''D''' <math>\{ 0,2 \}</math>, the result is:

:<math>\{ 0,4,7 \} \times \{ 0,2 \} = \{ 0,2,4,6,7,9 \}</math>

In this example, a set of three pitches multiplied with a set of two pitches gives a new set of 3 &times; 2 pitches. Given the limited space of modulo 12 arithmetic, when using this procedure very often duplicate tones are produced, which are generally omitted. This technique was used most famously in Boulez's 1955 ''[[Le Marteau sans maître]]'', as well as in his [[Piano sonatas (Boulez)|Third Piano Sonata]], ''[[Structures (Boulez)|Structures II]]'', "Don" and "Tombeau" from ''[[Pli selon pli]]'', ''[[Éclat]]'' (and ''[[Éclat/Multiples]]''), ''[[Figures—Doubles—Prismes]]'', ''[[Domaines (Boulez)|Domaines]]'', and ''[[Cummings ist der Dichter]]'', as well as the withdrawn choral work, ''Oubli signal lapidé'' (1952).{{sfn|Koblyakov|1990|loc=32}}{{sfn|Heinemann|1993}}{{sfn|Heinemann|1998}} This operation, like arithmetic multiplication and transpositional combination of set classes, is [[commutative]].{{sfn|Heinemann|1993|loc=24}}

[[Howard Hanson]] called this operation of [[commutative]] mathematical [[convolution]] "superposition"{{sfn|Hanson|1960|loc=44, 167}} or "@-projection" and used the "/" notation interchangeably. Thus "p@m" or "p/m" means "perfect fifth at major third", e.g.: { C E G B }. He specifically noted that two triad forms could be so multiplied, or a triad multiplied by itself, to produce a resultant scale. The latter "squaring" of a triad produces a particular scale highly saturated in instances of the source triad.{{sfn|Hanson|1960|loc=167}} Thus "pmn", Hanson's name for common the major triad, when squared, is "PMN", e.g.: { C D E G G{{music|sharp}} B }.

[[Nicolas Slonimsky]] used this operation, non-generalized, to form 1300 scales by multiplying the [[Symmetry|symmetric]] [[tritone]]s, [[augmented chord]]s, [[diminished seventh chord]]s, and [[wholetone scale]]s by the sum of 3 factors which he called interpolation, infrapolation, and ultrapolation.{{sfn|Slonimsky|1947|loc=v}} The combination of interpolation, infrapolation, and ultrapolation, forming obliquely infra-interpolation, infra-ultrapolation, and infra-inter-ultrapolation, [[Addition|additively]] sums to what is effectively a second sonority. This second sonority, multiplied by the first, gives his formula for generating scales and their [[harmonization]]s.

[[Joseph Schillinger]] used the idea, undeveloped, to categorize common 19th- and early 20th-century harmonic styles as product of horizontal harmonic root-motion and vertical harmonic structure.{{sfn|Schillinger|1941|loc=147}} Some of the composers' styles which he cites appear in the following multiplication table.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none; text-align:center;"
|-
! colspan="1" |
! colspan="4" | Chord type
|-
! Root Scale !! [[Major chord]] !! [[Augmented chord]]!! [[Minor chord]] !! [[Diminished seventh chord]]
|-
! [[Diminished seventh chord]]
| [[Octatonic scale]]<br />[[Richard Wagner]] || [[Chromatic scale]] || [[Octatonic scale]] ||
|-
! [[Augmented chord]]
| [[Augmented scale]]<br />[[Franz Liszt]] || [[Claude Debussy]]<br />[[Maurice Ravel]] || [[Augmented scale]]<br />[[Nikolai Rimsky-Korsakov]] ||
|-
! [[Whole tone scale]]
| [[Chromatic scale]]<br />[[Claude Debussy]]<br />[[Maurice Ravel]] || [[Whole tone scale]]<br />[[Claude Debussy]]<br />[[Maurice Ravel]] || [[Chromatic scale]]<br />[[Claude Debussy]]<br />[[Maurice Ravel]] ||
|-
! [[Chromatic scale]]
| [[Chromatic scale]]<br />[[Richard Wagner]] || [[Chromatic scale]] || [[Chromatic scale]] || [[Chromatic scale]]
|-
! [[Quartal chord]]
| [[Major scale]] || || [[Aeolian scale|Natural]] [[minor scale]] ||
|-
! [[Major chord]]
| [[Hexatonic scale|6-note analog of]] [[harmonic major scale]] || [[Augmented scale]] || || [[Octatonic scale]]
|-
! [[Minor chord]]
| || [[Augmented scale]] || [[Hexatonic scale|6-note analog of]] [[harmonic major scale]] || [[Octatonic scale]]
|-
! [[Diatonic scale]]
| Undecatonic scale || [[Chromatic scale]] || Undecatonic scale || [[Chromatic scale]]
|}

The [[Approximation theory|approximation]] of the 12 pitches of Western music by [[Modular arithmetic|modulus-12 math]], forming the [[Necklace (combinatorics)|Circle of Halfsteps]], means that musical intervals can also be thought of as [[angle]]s in a [[polar coordinate system]], stacking of identical intervals as functions of [[Simple harmonic motion|harmonic motion]], and [[transposition (music)|transposition]] as [[Axis–angle representation|rotation around an axis]]. Thus, in the multiplication example above from Hanson, "p@m" or "p/m" ("perfect 5th at major 3rd", e.g.: { C E G B }) also means "perfect fifth, superimposed upon perfect fifth rotated 1/3 of the circumference of the Circle of Halfsteps". A conversion table of intervals to angular measure (taken as negative numbers for clockwise rotation) follows:

{| class="wikitable" style="text-align:center" align="center"
|-
! colspan="1" rowspan="2"| Interval
! colspan="3" | Circle of halfsteps
! colspan="3" | Circle of fifths
|-
! Halfsteps !! Radians !! Degrees !! Fifths !! Radians !! Degrees
|-
! [[Unison]]
| 0 || 0 || 0 || 0 || 0 || 0
|-
! [[Minor second]]
| 1 || {{sfrac|{{pi}}|6}} || 30 || 7 || {{sfrac|7{{pi}}|6}} || 210
|-
! [[Major second]]
| 2 || {{sfrac|{{pi}}|3}} || 60 || 2 || {{sfrac|{{pi}}|3}} || 60
|-
! [[Minor third]]
| 3 || {{sfrac|{{pi}}|2}} || 90 || 9 || {{sfrac|3{{pi}}|2}} || 270
|-
! [[Major third]]
| 4 || {{sfrac|2{{pi}}|3}} || 120 || 4 || {{sfrac|2{{pi}}|3}} || 120
|-
! [[Perfect fourth]]
| 5 || {{sfrac|5{{pi}}|6}} || 150 || 11 || {{sfrac|11{{pi}}|6}} || 330
|-
! [[Diminished fifth]] or [[Augmented fourth]]
| 6 || {{pi}} || 180 || 6 || {{pi}} || 180
|-
! [[Perfect fifth]]
| 7 || {{sfrac|7{{pi}}|6}} || 210 || 1 || {{sfrac|{{pi}}|6}} || 30
|-
! [[Minor sixth]]
| 8 || {{sfrac|4{{pi}}|3}} || 240 || 8 || {{sfrac|4{{pi}}|3}} || 240
|-
! [[Major sixth]]
| 9 || {{sfrac|3{{pi}}|2}} || 270 || 3 || {{sfrac|{{pi}}|2}} || 90
|-
! [[Minor seventh]]
| 10 || {{sfrac|5{{pi}}|3}} || 300 || 10 || {{sfrac|5{{pi}}|3}} || 300
|-
! [[Major seventh]]
| 11 || {{sfrac|11{{pi}}|6}} || 330 || 5 || {{sfrac|5{{pi}}|6}} || 150
|-
! [[Octave]]
| 12 || 2{{pi}} || 360 || 12 || 2{{pi}} || 360
|}

This angular interpretation of intervals is helpful to visualize a very practical example of multiplication in music: [[Euler–Fokker genus|Euler-Fokker genera]] used in describing the [[Just intonation]] [[Musical tuning|tuning]] of keyboard instruments.{{sfn|Fokker|1987}} Each genus represents an harmonic function such as "3 perfect fifths stacked" or other sonority such as { C G D F{{music|sharp}} }, which, when multiplied by the correct angle(s) of copy, approximately [[Tessellation|fills]] the [[12TET]] [[Circumference|circumferential]] space of the [[Circle of fifths]]. It would be possible, though not musically pretty, to tune an [[Augmented chord|augmented triad]] of two perfect non-beating [[major third]]s, then (multiplying) tune two tempered [[Perfect fifth|fifths]] above and 1 below each note of the augmented chord; this is Euler-Fokker genus [555]. A different result is obtained by starting with the "3 perfect fifths stacked", and from these non-beating notes tuning a tempered [[major third]] above and below; this is Euler-Fokker genus [333].

==Time multiplication==
[[Joseph Schillinger]] described an operation of "[[Polynomial multiplication|polynomial time multiplication]]" (''polynomial'' refers to any rhythm consisting of more than one duration) corresponding roughly to that of [[#Pitch multiplication|Pitch multiplication]] above.{{sfn|Schillinger|1941|loc=70–? {{Page needed|date=September 2014|reason='70ff' is not adequate, since it might mean 70–72, 70–9361, or any terminal number larger than 71.}}}} A theme, reduced to a consistent series of integers representing the quarter, 8th-, or 16th-note duration of each of the notes of the theme, could be [[Polynomial multiplication|multiplied]] by itself or the series of another theme to produce a coherent and related variation. Especially, a theme's series could be squared or cubed or taken to higher powers to produce a saturation of related material.

==Affine transformation==
{{See also|Affine transformation}}
[[File:Multiplication as mirror operation.png|thumb|upright=1.4|[[Chromatic scale]] into circle of fourths and/or fifths through multiplication as mirror operation,<ref>{{harvnb|Eimert|1950|loc={{Page needed|date=September 2014}}}} as reproduced with minor alterations in {{harvnb|Schuijer|2008|loc=80}}</ref> [[File:Multiplication as mirror operation.mid]] or chromatic scale, [[File:Chromatic scale ascending on C.mid]] circle of fourths, [[File:Circle of fifths desc within octave.mid]] or circle of fifths. [[File:Circle of fifths ascending within octave.mid]]]]

[[Herbert Eimert]] described what he called the "eight modes" of the twelve-tone series, all mirror forms of one another. The [[Melodic inversion|inverse]] is obtained through a horizontal mirror, the [[retrograde (music)|retrograde]] through a vertical mirror, the [[retrograde inversion|retrograde-inverse]] through both a horizontal and a vertical mirror, and the "cycle-of-fourths-transform" or ''Quartverwandlung'' and "cycle-of-fifths-transform" or ''Quintverwandlung'' obtained through a slanting mirror.{{sfn|Eimert|1950|loc=28–29}} With the retrogrades of these transforms and the prime, there are eight [[permutation (music)|permutations]].

{{blockquote|Furthermore, one can sort of move the mirror at an angle, that is the 'angle' of a fourth or fifth, so that the chromatic row is reflected in both cycles. ... In this way, one obtains the cycle-of-fourths transform and the cycle-of-fifths transform of the row.<ref>{{harvnb|Eimert|1950|loc=29}}, translated in {{harvnb|Schuijer|2008|loc=81}}</ref>}}

[[Joseph Schillinger]] embraced not only contrapuntal [[Melodic inversion|inverse]], [[retrograde (music)|retrograde]], and [[retrograde inversion|retrograde-inverse]]—operations of [[matrix multiplication]] in [[Euclidean space|Euclidean vector space]]—but also their rhythmic counterparts as well. Thus he could describe a variation of theme using the same pitches in same order, but employing its original time values in [[Retrograde motion|retrograde]] order. He saw the scope of this [[Map (mathematics)|multiplicatory universe]] beyond simple [[Reflection (mathematics)|reflection]], to include [[Translation (geometry)|transposition]] and [[Rotation (mathematics)|rotation]] (possibly with [[Projection (linear algebra)|projection]] back to source), as well as [[Dilation (affine geometry)|dilation]] which had formerly been limited in use to the time dimension (via [[Augmentation (music)#Augmentation in composition|augmentation]] and [[Diminution#Diminution in composition|diminution]]).{{sfn|Schillinger|1941|loc=187ff {{Page needed|date=February 2014|reason=Use of 'ff' is discouraged because it is vague; please supply exact terminal page number instead.}}}} Thus he could describe another variation of theme, or even of a basic scale, by multiplying the halfstep counts between each successive pair of notes by some factor, possibly [[Normalized vector|normalizing]] to the octave via [[Modulo operation|Modulo]]-12 operation.{{sfn|Schillinger|1941|loc=115ff, 208ff {{Page needed|date=February 2014|reason=Use of 'ff' is discouraged because it is vague; please supply exact terminal page number instead.}}}}

==Z-relation==
Some [[Interval vector|Z-related]] chords are connected by ''M'' or ''IM'' (multiplication by 5 or multiplication by 7), due to identical entries for 1 and 5 on the [[interval vector|APIC vector]].{{sfn|Schuijer|2008|loc=98n18}}

== References ==
{{reflist|15em}}
'''Sources'''
{{div col|colwidth=45em}}
* {{wikicite|ref={{harvid|Antokoletz|1993}}|reference=Antokoletz, Elliott. 1993. "Middle Period String Quartets". In ''The Bartok Companion'', edited by [[Malcolm Gillies]], 257–277. London: Faber and Faber. {{ISBN|0-571-15330-5}} (cased); {{ISBN|0-571-15331-3}} (pbk).}}
* {{wikicite|ref={{harvid|Boulez|1971}}|reference=[[Pierre Boulez|Boulez, Pierre]]. 1971. ''Boulez on Music Today''. Translated by [[Susan Bradshaw]] and [[Richard Rodney Bennett]]. Cambridge, Massachusetts: Harvard University Press. {{ISBN|0-674-08006-8}}.}}
* {{wikicite|ref={{harvid|Eimert|1950}}|reference=[[Herbert Eimert|Eimert, Herbert]]. 1950. ''Lehrbuch der Zwölftontechnik''. Wiesbaden: Breitkopf & Härtel.}}
* {{wikicite|ref={{harvid|Fokker|1987}}|reference=[[Adriaan Fokker|Fokker, Adriaan Daniël]]. 1987. ''Selected Musical Compositions''. Utrecht: The Diapason Press. {{ISBN|90-70907-11-9}}.}}
* {{wikicite|ref={{harvid|Hanson|1960}}|reference=[[Howard Hanson|Hanson, Howard]]. 1960. ''Harmonic Materials of Modern Music''. New York: Appleton-Century-Crofts.}}
* {{wikicite|ref={{harvid|Heinemann|1993}}|reference=Heinemann, Stephen. 1993. ''Pitch-Class Set Multiplication in Boulez's Le Marteau sans maître''. D.M.A. diss., University of Washington.}}
* {{wikicite|ref={{harvid|Heinemann|1998}}|reference=Heinemann, Stephen. 1998. "Pitch-Class Set Multiplication in Theory and Practice." ''[[Music Theory Spectrum]]'' 20, no. 1 (Spring): 72–96.}}
* {{wikicite|ref={{harvid|Hibbard|1969}}|reference=Hibbard, William. 1969. "Charles Wuorinen: ''The Politics of Harmony''". ''[[Perspectives of New Music]]'' 7, no. 2 (Spring-Summer): 155–166.}}
* {{wikicite|ref={{harvid|Howe|1965}}|reference=Howe, Hubert S. 1965. "Some Combinational Properties of Pitch Structures." ''Perspectives of New Music'' 4, no. 1 (Fall-Winter): 45–61.}}
* {{wikicite|ref={{harvid|Koblyakov|1990}}|reference=Koblyakov, Lev. 1990. ''Pierre Boulez: A World of Harmony''. Chur: Harwood Academic Publishers. {{ISBN|3-7186-0422-1}}.}}
* {{wikicite|ref={{harvid|Krenek|1937}}|reference=[[Ernst Krenek|Krenek, Ernst]]. 1937. ''Über neue Musik: Sechs Vorlesungen zur Einführung in die theoretischen Grundlagen''. Vienna: Ringbuchhandlung.}}
* {{wikicite|ref={{harvid|Morris|1982}}|reference=Morris, Robert D. 1982. Review: "[[John Rahn]], ''Basic Atonal Theory''. New York: Longman, 1980". ''[[Music Theory Spectrum]]'' 4:138–154.}}
* {{wikicite|ref={{harvid|Morris|1997}}|reference=Morris, Robert D. 1997. "Some Remarks on ''Odds and Ends''". ''[[Perspectives of New Music]]'' 35, no. 2 (Summer): 237–256.}}
* {{wikicite|ref={{harvid|Rahn|1980}}|reference=[[John Rahn|Rahn, John]]. 1980. ''Basic Atonal Theory''. Longman Music Series. New York and London: Longman. Reprinted, New York: Schirmer Books; London: Collier Macmillan, 1987.}}
* {{wikicite|ref={{harvid|Randall|1962}}|reference=[[James K. Randall|Randall, James K.]] 1962. "Pitch-Time Correlation". Unpublished. Cited in {{harvnb|Schuijer|2008|loc=82}}.}}
* {{wikicite|ref={{harvid|Schillinger|1941}}|reference=[[Joseph Schillinger|Schillinger, Joseph]]. 1941. ''The Schillinger System of Musical Composition''. New York: Carl Fischer. {{ISBN|0306775220}}.}}
* {{wikicite|ref={{harvid|Schuijer|2008}}|reference=Schuijer, Michiel. 2008. ''Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts''. Eastman Studies in Music 60. Rochester, New York: University of Rochester Press. {{ISBN|978-1-58046-270-9}}.}}
* {{wikicite|ref={{harvid|Slonimsky|1947}}|reference=[[Nicolas Slonimsky|Slonimsky, Nicolas]]. 1947. ''Thesaurus of Scales and Melodic Patterns''. New York: Charles Scribner Sons. {{ISBN|002-6118505}}.}}
* {{wikicite|ref={{harvid|Winham|1970}}|reference=[[Godfrey Winham|Winham, Godfrey]]. 1970. "Composition with Arrays". ''[[Perspectives of New Music]]'' 9, no. 1 (Fall-Winter): 43–67.}}
{{div col end}}

== Further reading ==
{{div col|colwidth=45em}}
* Losada, Catherine C. 2014. "Complex Multiplication, Structure, and Process: Harmony and Form in Boulez’s Structures II". ''[[Music Theory Spectrum]]'' 36, no. 1 (Spring): 86–120.
* Morris, Robert D. 1977. "On the Generation of Multiple-Order-Function Twelve-Tone Rows". ''[[Journal of Music Theory]]'' 21, no. 2 (Autumn): 238–262.
* Morris, Robert D. 1982–83. "[[Combinatoriality]] without the [[tone row#total chromatic|Aggregate]]". ''[[Perspectives of New Music]]'' 21, nos. 1 & 2 (Autumn-Winter/Spring-Summer): 432–486.
* Morris, Robert D. 1990. "Pitch-Class Complementation and Its Generalizations". ''[[Journal of Music Theory]]'' 34, no. 2 (Autumn): 175–245.
* Starr, Daniel V. 1978. "Sets, Invariance, and Partitions." ''Journal of Music Theory'' 22, no. 1:1–42.
{{div col end}}

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{{Twelve-tone technique}}
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