Editing Tetrachord (section)

==Variations==

===Romantic era===
[[File:Locrian tetrachord.png|thumb|upright=1.3|[[Descending tetrachord]] in the modern [[locrian mode|B Locrian]] (also known as the upper minor tetrachord): <sub>{{music|scale|8}}</sub>–{{music|b}}<sub>{{music|scale|7}}</sub>–{{music|b}}<sub>{{music|scale|6}}</sub>–{{music|b}}<sub>{{music|scale|5}}</sub> (b–a–g–f). This tetrachord spans a tritone instead of a perfect fourth.[[File:Locrian tetrachord.mid]]]]

[[File:Phrygian half cadence in C.png|thumb|upright=1.3|The [[Phrygian mode|Phrygian]] [[chord progression|progression]] creates a descending tetrachord<ref>[http://classicalmusicblog.com/2007/08/phrygian-progression.html "Phrygian Progression"], ''Classical Music Blog''. {{Webarchive|url=https://web.archive.org/web/20111006081406/http://classicalmusicblog.com/2007/08/phrygian-progression.html |date=2011-10-06 }}</ref>{{Unreliable source?|reason=just a blog, describes itself as a "work in progress" (i.e., it is unstable), makes several dubious claims, such as that there were musicologists in ancient Greece|date=November 2015}} [[bassline]]: <sub>{{music|scale|8}}</sub>–{{music|b}}<sub>{{music|scale|7}}</sub>–{{music|b}}<sub>{{music|scale|6}}</sub>–<sub>{{music|scale|5}}</sub>. [[Phrygian half cadence]]: i–v6–iv6–V in C minor (bassline: c–b{{music|b}}–a{{music|b}}–g)[[File:Phrygian half cadence in C.mid]]]]

Tetrachords based upon [[equal temperament]] tuning were used to explain common [[heptatonic scale]]s. Given the following vocabulary of tetrachords (the digits give the number of semitones in consecutive intervals of the tetrachord, adding to five):
{| class="wikitable" style="margin-left: 4em;"
|-
! Tetrachord !! Halfstep String
|-
| Major || 2 2 1
|-
| Minor || 2 1 2
|-
| Harmonic || 1 3 1
|-
| Upper Minor || 1 2 2
|}
the following scales could be derived by joining two tetrachords with a [[whole step]] (2) between:{{sfn|Dupré|1962|loc=2:35}}<ref>[[Joseph Schillinger]], ''The Schillinger System of Musical Composition'', 2 vols. (New York: Carl Fischer, 1941), 1:112–114. {{ISBN|978-0306775215}}.</ref>
{| class="wikitable" style="margin-left: 4em;"
|-
! Component tetrachords !! Halfstep string !! Resulting scale
|-
| Major + major || 2 2 1 : 2 : 2 2 1 || Diatonic major
|-
| Minor + upper minor || 2 1 2 : 2 : 1 2 2 || Natural minor
|-
| Major + harmonic || 2 2 1 : 2 : 1 3 1 || Harmonic major
|-
| Minor + harmonic || 2 1 2 : 2 : 1 3 1 || Harmonic minor
|-
| Harmonic + harmonic || 1 3 1 : 2 : 1 3 1 ||[[Double harmonic scale]]<ref>Joshua Craig Podolsky, ''Advanced Lead Guitar Concepts'' (Pacific, Missouri: Mel Bay, 2010): 111. {{ISBN|978-0-7866-8236-2}}.</ref><ref>{{cite web|url=http://www.docs.solfege.org/3.21/C/scales/dha.html|title=Double harmonic scale and its modes|website=docs.solfege.org|access-date=2015-04-12|url-status=dead|archive-url=https://web.archive.org/web/20150618200916/http://www.docs.solfege.org/3.21/C/scales/dha.html|archive-date=2015-06-18}}</ref> or Gypsy major<ref>[[Jonathan Bellman]], ''The "Style hongrois" in the Music of Western Europe'' (Boston: Northeastern University Press, 1993): 120. {{ISBN|1-55553-169-5}}.</ref>
|-
| Major + upper minor || 2 2 1 : 2 : 1 2 2 || Melodic major
|-
| Minor + major || 2 1 2 : 2 : 2 2 1 || Melodic minor
|-
| Upper minor + harmonic || 1 2 2 : 2 : 1 3 1 || Neapolitan minor
|}
All these scales are formed by two complete disjunct tetrachords: contrarily to Greek and Medieval theory, the tetrachords change here from scale to scale (i.e., the C major tetrachord would be C–D–E–F, the D major one D–E–F{{music|sharp}}–G, the C minor one C–D–E{{music|flat}}–F, etc.). The 19th-century theorists of ancient Greek music believed that this had also been the case in Antiquity, and imagined that there had existed Dorian, Phrygian or Lydian tetrachords. This misconception was denounced in Otto Gombosi's thesis (1939).<ref>Otto Johannes Gombosi, ''Tonarten und Stimmungen der Antiken Musik'', Kopenhagen, Ejnar Munksgaard, 1939.</ref>

===20th-century analysis===
Theorists of the later 20th century often use the term "tetrachord" to describe any four-note set when analysing music of a variety of styles and historical periods.<ref>See the following:
* {{cite journal
 |first=Benedict |last=Taylor
 |date=Spring 2010
 |title=Modal four-note pitch collections in the music of [[Antonín Dvořák|Dvořák]]'s American period
 |journal=[[Music Theory Spectrum]]
 |volume=32 |issue=1 |pages=44–59
}}
* {{cite journal
 |first1=Steven |last1=Block  |author1-link=Steven Block
 |first2=Jack   |last2=Douthett
 |date=Spring 1994
 |title=[[Vector product]]s and intervallic weighting
 |journal=[[Journal of Music Theory]]
 |volume=38 |issue=1 |pages=21–41
}}
* {{cite journal
 |first=Ian |last=Quinn
 |date=Summer 2001
 |title=Listening to similarity relations
 |journal=[[Perspectives of New Music]]
 |volume=39 |issue=2 |pages=108–158
}}
* {{cite journal
 |first=Joseph N. |last=Straus
 |date=Spring 1999
 |title=[[Igor Stravinsky|Stravinsky]]'s ''Construction of twelve verticals'': An aspect of harmony in the serial music
 |journal=[[Music Theory Spectrum]]
 |volume=21 |issue=1 |pages=43–73
}}
* {{cite journal
 |first=Tuire |last=Kuusi
 |date=September 2007
 |title=Subset-class relation, common pitches, and common interval structure guiding estimations of similarity
 |journal=[[Music Perception]]
 |volume=25 |issue=1 |pages=1–11
}}
* {{cite journal
 |first=J.B. |last=Mailman |author-link=Joshua Banks Mailman
 |date=July–October 2009
 |title=An imagined drama of competitive opposition in [[Elliott Carter|Carter]]'s ''Scrivo in Vento'', with notes on narrative, symmetry, quantitative flux, and [[Heraclitus]]
 |journal=[[Music Analysis (journal)|Music Analysis]]
 |volume=28 |issue=2-3 |pages=373–422
}}
* {{cite journal
 |first1=J. |last1=Harbison |author1-link=John Harbison
 |first2=E. |last2=Cory     |author2-link=Eleanor Cory
 |date=Spring–Summer 1973
 |title=[[Martin Boykan]]: ''String Quartet'' (1967): Two views
 |journal=[[Perspectives of New Music]] |volume=11 |issue=2 |pages=204–209 }}
* {{cite journal
 |first=M. |last=Babbitt |author-link=Milton Babbitt
 |date=Spring–Summer 1973
 |title=[[Edgard Varèse]]: A few observations of his music
 |journal=[[Perspectives of New Music]]
 |volume=4 |issue=2 |pages=14–22
}}
* {{cite journal
 |first=Annie K. |last=Yih
 |date=July 2000
 |title=Analysing [[Claude Debussy|Debussy]]: Tonality, motivic sets and the referential pitch-class specific collection
 |journal=[[Music Analysis (journal)|Music Analysis]]
 |volume=19 |issue=2 |pages=203–229
}}
* {{cite journal
 |first=J.K. |last=Randall |author-link=James K. Randall
 |date=Autumn–Winter 1963
 |title=[[Godfrey Winham]]'s ''Composition for Orchestra'' &hairsp;
 |journal=[[Perspectives of New Music]]
 |volume=2 |issue=1 |pages=102–113
}}</ref> 
The expression "chromatic tetrachord" may be used in two different senses: to describe the special case consisting of a four-note segment of the chromatic scale,<ref>
{{cite journal
 |first=Brent |last=Auerbach
 |date=Fall 2008
 |title=Tiered polyphony and its determinative role in the piano music of [[Johannes Brahms]]
 |journal=[[Journal of Music Theory]]
 |volume=52 |issue=2 |pages=273–320
}}
</ref>
or, in a more historically oriented context, to refer to the six chromatic notes used to fill the interval of a perfect fourth, usually found in descending bass lines.<ref>
{{cite journal
 |first=R. |last=Gauldin |author-link=Robert Gauldin
 |year=1991
 |title=[[Beethoven]]'s interrupted tetrachord and the [[Symphony No. 7 (Beethoven)|''Seventh Symphony'']] &hairsp;
 |journal=[[Intégral (journal)|Intégral]]
 |volume=5 |pages=77–100
}}
</ref>
It may also be used to describes sets of fewer than four notes, when used in scale-like fashion to span the interval of a perfect fourth.<ref>
{{cite journal
 |first=Nors S. |last=Josephson
 |year=2004
 |title=On some apparent sketches for [[Jean Sibelius|Sibelius's]] &hairsp; [[Symphony No. 8 (Sibelius)|''Eighth Symphony'']] &hairsp;
 |journal=[[Archiv für Musikwissenschaft]]
 |volume=61 |issue=1 |pages=54–67
}}
</ref>

===Atonal usage===
[[Allen Forte]] occasionally uses the term ''tetrachord'' to mean what he elsewhere calls a ''[[Tetrad (music)|tetrad]]'' or simply a "4-element set" – a set of any four pitches or ''pitch classes''.<ref>[[Allen Forte]] (1973). ''The Structure of Atonal Music'', pp. 1, 18, 68, 70, 73, 87, 88, 21, 119, 123, 124, 125, 138, 143, 171, 174, and 223. New Haven and London: Yale University Press. {{ISBN|0-300-01610-7}} (cloth) {{ISBN|0-300-02120-8}} (pbk). Allen Forte (1985). "Pitch-Class Set Analysis Today". ''[[Music Analysis (journal)|Music Analysis]]'' 4, nos. 1 & 2 (March–July: Special Issue: [[King's College London]] ''Music Analysis'' Conference 1984): 29–58, citations on 48–51, 53.</ref> In [[Twelve-tone technique|twelve-tone theory]], the term may have the special sense of any consecutive four notes of a twelve-tone row.<ref>Reynold Simpson, "New Sketches, Old Fragments, and Schoenberg's Third String Quartet, Op. 30",  ''Theory and Practice'' 17, In Celebration of Arnold Schoenberg (1) (1992): 85–101.</ref>