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].
All-interval tetrachords (ⓘ).
Table of interval classes as relating to all-interval tetrachords
All-interval tetrachord dyads[5] (ⓘ).
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.
References
- 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.
- "Interval String Table". Archived from the original on 2006-02-04.
- 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
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