All-interval tetrachord
An all-interval tetrachord is a tetrachord, a collection of four pitch classes, from which all six interval classes can be derived[1] by means of various inversions. There are only two possible all-interval tetrachords. In set theory notation, these are [0,1,4,6] (4-Z15[2]) and [0,1,3,7] (4-Z29[3]). The interval vector for both 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]
| 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 |
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 such as Elliott Carter and George Perle used it extensively, as did Webern and Schoenberg. (Although the term all-interval tetrachord and indeed set theory itself was not in use during most of their life times.)
See also
Source
- ^ 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.
- ^ Schiff, David (1998). The Music of Elliott Carter, p.34. ISBN 0801436125.
External links