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 (Play ).

Table of interval classes as relating to all-interval tetrachords

 

All-interval tetrachord dyads^[5] ( Play ).

 

4-Z15

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 Elliott Carter (First String Quartet) and George Perle used it extensively.^{[citation needed]}

See also

• All-interval twelve-tone row
• All-trichord hexachord
• Perfect ruler
• Serialism
• Trichord

References

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. . Archived from the original on 2006-02-04.
5. 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 use of all-interval tetrachords in Elliott Carter
• Interval String Table
• Structural and Transformational Properties of All-Interval Tetrachords a comprehensive analysis of all-interval tetrachords