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 Frank Bridge, 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