All-interval tetrachord
![](http://upload.wikimedia.org/wikipedia/commons/thumb/4/44/All-interval_tetrachords.png/220px-All-interval_tetrachords.png)
All-interval tetrachords (
![About this sound](http://upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Loudspeaker.svg/11px-Loudspeaker.svg.png)
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[]
![](http://upload.wikimedia.org/wikipedia/commons/thumb/7/76/All-interval_tetrachord_4-Z15_diagram.svg/60px-All-interval_tetrachord_4-Z15_diagram.svg.png)
4-Z15
In the examples below, the tetrachords [0,1,4,6] and [0,1,3,7] are built on E.
ic | notes of [0,1,4,6] built on E | diatonic counterparts |
---|---|---|
1 | E to F | minor 2nd and major 7th |
2 | A♭ to B♭ | major 2nd and minor 7th |
3 | F to A♭ | minor 3rd and major 6th |
4 | E to G♯ | major 3rd and minor 6th |
5 | F to B♭ | perfect 4th and perfect 5th |
6 | E to B♭ | augmented 4th and diminished 5th |
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 Elliott Carter (First String Quartet) and George Perle used it extensively.[citation needed]
See also[]
Sources[]
- ^ 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).
- ^ Schuijer, Michiel (2008). Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts, p.109. ISBN 978-1-58046-270-9.
- ^ Forte, Allen (1998), The Atonal Music of Anton Webern, p.17. ISBN 0-300-07352-6.
- ^ http://solomonsmusic.net/intstring.htm
- ^ Schiff, David (1998). The Music of Elliott Carter, p.34. ISBN 0-8014-3612-5.
External links[]
- The All-Interval Tetrachord, A Musical Application of Almost Difference Sets all-interval tetrachord tutorial
- The Composition of Elliott Carter's Night Fantasies the use of all-interval tetrachords in Elliott Carter
- Interval String Table
- Musical Set Theory
- Structural and Transformational Properties of All-Interval Tetrachords a comprehensive analysis of all-interval tetrachords
Categories:
- Chords
- Post-tonal music theory
- Hemitonic scales
- Musical set theory
- Music theory stubs