all-interval tetrachord
File:All-interval tetrachords.png
An all-interval tetrachord is a tetrachord, a collection of four pitch classes, containing all six interval classes.Whittall, Arnold. 2008. The Cambridge Introduction to Serialism, p.271. Cambridge Introductions to Music. New York: Cambridge University Press. {{ISBN|978-0-521-86341-4}} (hardback) {{ISBN|978-0-521-68200-8}} (pbk). There are only two possible all-interval tetrachords (to within inversion), when expressed in prime form. In set theory notation, these are [0,1,4,6] (4-Z15)Schuijer, Michiel (2008). Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts, p.109. {{ISBN|978-1-58046-270-9}}. and [0,1,3,7] (4-Z29).Forte, Allen (1998), The Atonal Music of Anton Webern, p.17. {{ISBN|0-300-07352-6}}. Their inversions are [0,2,5,6] (4-Z15b) and [0,4,6,7] (4-Z29b).{{cite web |url=http://solomonsmusic.net/intstring.htm |url-status=dead |archive-url=https://web.archive.org/web/20060204234206/http://solomonsmusic.net/intstring.htm |archive-date=2006-02-04 |title=Interval String Table}} The interval vector for all all-interval tetrachords is [1,1,1,1,1,1].
Table of interval classes as relating to all-interval tetrachords
File:All-interval tetrachord dyads.png
File:All-interval tetrachord 4-Z15 diagram.svg
In the examples below, the tetrachords [0,1,4,6] and [0,1,3,7] are built on E.
| class="wikitable"
|+ Interval class table for [0,1,4,6] | ||
| ic | notes of [0,1,4,6] built on E | diatonic counterparts |
|---|---|---|
| 1 | E to F | minor 2nd and major 7th |
| 2 | A{{music|b}} to B{{music|b}} | major 2nd and minor 7th |
| 3 | F to A{{music|b}} | minor 3rd and major 6th |
| 4 | E to G{{music|#}} | major 3rd and minor 6th |
| 5 | F to B{{music|b}} | perfect 4th and perfect 5th |
| 6 | E to B{{music|b}} | augmented 4th and diminished 5th |
| class="wikitable"
|+ Interval class table for [0,1,3,7] | ||
| ic | notes of [0,1,3,7] built on E | diatonic counterparts |
|---|---|---|
| 1 | E to F | minor 2nd and major 7th |
| 2 | F to G | major 2nd and minor 7th |
| 3 | E to G | minor 3rd and major 6th |
| 4 | G to B | major 3rd and minor 6th |
| 5 | E to B | perfect 4th and perfect 5th |
| 6 | F to B | augmented 4th and diminished 5th |
Use in modern music
The unique qualities of the all-interval tetrachord have made it very popular in 20th-century music. Composers including Frank Bridge, Elliott Carter (First String Quartet) and George Perle used it extensively.{{Citation needed|date=February 2012}}
See also
References
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External links
- [https://archive.today/20121209154714/http://uwacadweb.uwyo.edu/smaw/AIT.ppt The All-Interval Tetrachord, A Musical Application of Almost Difference Sets] all-interval tetrachord tutorial
- [https://web.archive.org/web/20090220110735/http://www.wpunj.edu/coac/music/link/sonus/sonuspaper.html The Composition of Elliott Carter's Night Fantasies] the use of all-interval tetrachords in Elliott Carter
- [https://web.archive.org/web/20071115223321/http://solomonsmusic.net/intstring.htm Interval String Table]
- [https://web.archive.org/web/20010308210605/http://www.jaytomlin.com/music/settheory/help.html Musical Set Theory]
- [http://mto.societymusictheory.org/issues/mto.06.12.4/mto.06.12.4.childs.html Structural and Transformational Properties of All-Interval Tetrachords] a comprehensive analysis of all-interval tetrachords
{{Pitch segments}}
{{Set theory (music)}}
Category:Post-tonal music theory
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