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. (hardback) (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. . and [0,1,3,7] (4-Z29). The following are examples of each all-interval tetrachord.
Their inversions are [0,2,5,6] (4-Z15b) and [0,4,6,7] (4-Z29b), respectively. 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 including Frank Bridge, Elliott Carter (First String Quartet) and George Perle used it extensively.
See also All-interval twelve-tone row All-trichord hexachord Perfect ruler Serialism Trichord
References
External links 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