All-interval tetrachord
Encyclopedia
An all-interval tetrachord is a tetrachord
, a collection of four pitch classes
, from which all six interval classes
can be derived by means of various inversions
. There are only two possible all-interval tetrachords. In set theory notation, these are [0,1,4,6] (4-Z15) and [0,1,3,7] (4-Z29). The interval vector
for both all-interval tetrachords is [1,1,1,1,1,1].
.
and George Perle
used it extensively, as did Webern and Schoenberg
. (Although the term all-interval tetrachord and indeed set theory itself was not in use during most of their life times.)
Tetrachord
Traditionally, a tetrachord is a series of three intervals filling in the interval of a perfect fourth, a 4:3 frequency proportion. In modern usage a tetrachord is any four-note segment of a scale or tone row. The term tetrachord derives from ancient Greek music theory...
, a collection of four pitch classes
Pitch class
In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...
, from which all six interval classes
Interval class
In musical set theory, an interval class , also known as unordered pitch-class interval, interval distance, undirected interval, or interval mod...
can be derived by means of various inversions
Inversion (music)
In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and inverted voices...
. There are only two possible all-interval tetrachords. In set theory notation, these are [0,1,4,6] (4-Z15) and [0,1,3,7] (4-Z29). The interval vector
Interval vector
In musical set theory, an interval vector is an array that expresses the intervallic content of a pitch-class set...
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 EE (musical note)
E or mi is the third note of the solfège.When calculated in equal temperament with a reference of A above middle C as 440 Hz, the frequency of Middle E is approximately 329.628 Hz. See pitch for a discussion of historical variations in frequency.-Designation by octave:...
.
| ic Interval class In musical set theory, an interval class , also known as unordered pitch-class interval, interval distance, undirected interval, or interval mod... | 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 |
| ic Interval class In musical set theory, an interval class , also known as unordered pitch-class interval, interval distance, undirected interval, or interval mod... | 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 such as Elliott CarterElliott Carter
Elliott Cook Carter, Jr. is a two-time Pulitzer Prize-winning American composer born and living in New York City. He studied with Nadia Boulanger in Paris in the 1930s, and then returned to the United States. After a neoclassical phase, he went on to write atonal, rhythmically complex music...
and George Perle
George Perle
George Perle was a composer and music theorist. He was born in Bayonne, New Jersey. Perle was an alumnus of DePaul University...
used it extensively, as did Webern and Schoenberg
Schoenberg
Schoenberg is the surname of several persons:* Arnold Schoenberg , Austrian-American composer* Claude-Michel Schoenberg , French record producer, actor, singer, popular songwriter, and musical theatre composer...
. (Although the term all-interval tetrachord and indeed set theory itself was not in use during most of their life times.)
External links
- The All-Interval Tetrachord, A Musical Application of Almost Difference Sets all-interval tetrachord tutorial
- The Composition of Elliott Carter's Night Fantasies the use of all-interval tetrachords in Elliott CarterElliott CarterElliott Cook Carter, Jr. is a two-time Pulitzer Prize-winning American composer born and living in New York City. He studied with Nadia Boulanger in Paris in the 1930s, and then returned to the United States. After a neoclassical phase, he went on to write atonal, rhythmically complex music...
- Interval String Table
- Musical Set Theory
- Structural and Transformational Properties of All-Interval Tetrachords a comprehensive analysis of all-interval tetrachords