All-interval tetrachord

An all-interval tetrachord is a tetrachord, a collection of four pitch classes, containing all six interval classes.[1] 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)[2] and [0,1,3,7] (4-Z29).[3] Their inversions are [0,2,5,6] (4-Z15b) and [0,4,6,7] (4-Z29b).[4] 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

In the examples below, the tetrachords [0,1,4,6] and [0,1,3,7] are built on E.

Interval class table for [0,1,4,6]
icnotes of [0,1,4,6] built on Ediatonic counterparts
1E to Fminor 2nd and major 7th
2A to Bmajor 2nd and minor 7th
3F to Aminor 3rd and major 6th
4E to Gmajor 3rd and minor 6th
5F to Bperfect 4th and perfect 5th
6E to Baugmented 4th and diminished 5th
Interval class table for [0,1,3,7]
icnotes of [0,1,3,7] built on Ediatonic counterparts
1E to Fminor 2nd and major 7th
2F to Gmajor 2nd and minor 7th
3E to Gminor 3rd and major 6th
4G to Bmajor 3rd and minor 6th
5E to Bperfect 4th and perfect 5th
6F to Baugmented 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 Elliott Carter (First String Quartet) and George Perle used it extensively.

See also

Sources

  1. 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).
  2. Schuijer, Michiel (2008). Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts, p.109. ISBN 978-1-58046-270-9.
  3. Forte, Allen (1998), The Atonal Music of Anton Webern, p.17. ISBN 0-300-07352-6.
  4. http://solomonsmusic.net/intstring.htm
  5. Schiff, David (1998). The Music of Elliott Carter, p.34. ISBN 0-8014-3612-5.


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