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].
In the examples below, the tetrachords [0,1,4,6] and [0,1,3,7] are built on E.
| 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 |
| 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 |
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.