All-interval tetrachord
From Wikipedia, the free encyclopedia
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). The following are examples of each all-interval tetrachord.
![{
\override Score.TimeSignature
#'stencil = ##f
\relative c' {
\time 4/4
\set Score.tempoHideNote = ##t \tempo 1 = 20
\set Score.proportionalNotationDuration = #(ly:make-moment 1/2)
<c des e ges>1^\markup { [0,1,4,6] } <c des es g?>^\markup { [0,1,3,7] }
}
}](http://upload.wikimedia.org/score/2/1/214mcq7bg1nr6wpynl1n8gx7x0q4tp3/214mcq7b.png)
Their inversions are [0,2,5,6] (4-Z15b) and [0,4,6,7] (4-Z29b), respectively.[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.
| ic | notes of [0,1,4,6] built on E | diatonic counterparts |
|---|---|---|
| 1 | E to F | minor 2nd and major 7th |
| 2 | Aâ to Bâ | major 2nd and minor 7th |
| 3 | F to Aâ | minor 3rd and major 6th |
| 4 | E to G⯠| major 3rd and minor 6th |
| 5 | F to Bâ | perfect 4th and perfect 5th |
| 6 | E to Bâ | augmented 4th and diminished 5th |
| 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]
