music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspect ...

, an all-interval twelve-tone row, series, or chord, is a

twelve-tone The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and (in British usage) twelve-note composition—is a method of musical composition first devised by Austrian composer Josef Matthias Hauer, who published his "law o ...

tone row In music, a tone row or note row (german: Reihe or '), also series or set, is a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in musical set theory of the chromatic scale, though both larger and smaller sets ar ...

arranged so that it contains one instance of each interval within the octave, 1 through 11 (an ordering of every interval, 0 through 11, that contains each (ordered) pitch-interval class, 0 through 11). A "twelve-note ''spatial set'' made up of the eleven intervals etween consecutive pitches"Schiff, David (1998). ''The Music of Elliott Carter'', second edition (Ithaca: Cornell University Press), pp. 34–36. . Labels added to image. There are 1,928 distinct all-interval twelve-tone rows. These sets may be ordered in time or in register. "Distinct" in this context means in transpositionally and rotationally normal form (yielding 3856 such series), and disregarding inversionally related forms. These 1,928 tone rows have been independently rediscovered several times, their first computation probably was by Andre Riotte in 1961, see. Since the sum of numbers 1 through 11 equals 66, an all-interval row must contain a

tritone In music theory, the tritone is defined as a musical interval composed of three adjacent whole tones (six semitones). For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can be decomposed into the three a ...

between its first and last notes, as well as in their middle.

Examples

Mother chord

The first known all-interval row, F, E, C, A, G, D, A, D, E, G, B, C, was named the ''Mutterakkord'' (mother chord) by Fritz Heinrich Klein, who created it in 1921 for his chamber-orchestra composition ''Die Maschine''. 0 e 7 4 2 9 3 8 t 1 5 6 The intervals between consecutive pairs of notes are the following (t = 10, e = 11): e 8 9 t 7 6 5 2 3 4 1 Klein used the Mother chord in his ''Die Maschine'', Op. 1, and derived it from the ''Pyramid chord'' 'Pyramidakkord'' 0 0 e 9 6 2 9 3 8 0 3 5 6 difference e t 9 8 7 6 5 4 3 2 1 by transposing the underlined notes (0369) down two semitones. The Pyramid chord consists of every interval stacked, low to high, from 12 to 1 and while it contains all intervals, it does not contain all pitch classes and is thus not a tone row. Klein chose the name ''Mutterakkord'' in order to avoid a longer term such as ''all-interval twelve-tone row'' and because it is a chord which unites all other chords by containing them within itself. The Mother chord row was also used by

Alban Berg Alban Maria Johannes Berg ( , ; 9 February 1885 – 24 December 1935) was an Austrian composer of the Second Viennese School. His compositional style combined Romantic lyricism with the twelve-tone technique. Although he left a relatively sma ...

in his '' Lyric Suite'' (1926) and in his second setting of

Theodor Storm Hans Theodor Woldsen Storm (; 14 September 18174 July 1888), commonly known as Theodor Storm, was a German writer. He is considered to be one of the most important figures of German realism. Life Storm was born in the small town of Husum, on th ...

's poem '' Schliesse mir die Augen beide''. In contrast, the

chromatic scale The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the ...

only contains the interval 1 between each consecutive note: 0 1 2 3 4 5 6 7 8 9 t e 1 1 1 1 1 1 1 1 1 1 1 and is thus not an all-interval row.

Grandmother chord

The ''Grandmother chord'' is an eleven-interval, twelve-note, invertible chord with all of the properties of the Mother chord. Additionally, the intervals are so arranged that they alternate odd and even intervals (counted by semitones) and that the odd intervals successively decrease by one whole-tone while the even intervals successively increase by one whole-tone. It was invented by

Nicolas Slonimsky Nicolas Slonimsky ( – December 25, 1995), born Nikolai Leonidovich Slonimskiy (russian: Никола́й Леони́дович Сло́нимский), was a Russian-born American conductor, author, pianist, composer and lexicographer. B ...

on February 13, 1938. 0 e 1 t 2 9 3 8 4 7 5 6 \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / odd: e , 9 , 7 , 5 , 3 , 1 even: 2 4 6 8 t

Link chords

'Link' chords are all-interval twelve-tone sets containing one or more uninterrupted instances of the

all-trichord hexachord In music, the all-trichord hexachord is a unique hexachord that contains all twelve trichords, or from which all twelve possible trichords may be derived. The prime form of this set class is and its Forte number is 6-Z17. Its complement is 6-Z43 ...

(). Found by John F. Link, they have been used by

Elliott Carter Elliott Cook Carter Jr. (December 11, 1908 – November 5, 2012) was an American modernist composer. One of the most respected composers of the second half of the 20th century, he combined elements of European modernism and American "ultra- ...

in pieces such as ''Symphonia''. 0 1 4 8 7 2 e 9 3 5 t 6 1 3 4 e 7 9 t 6 2 5 8 0 4 e 5 2 1 3 8 9 7 t 6 4 7 6 9 e 2 5 1 t 3 8 There are four 'Link' chords which are RI-

invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...