All-interval twelve-tone row

All-interval row from Alban Berg's Lyric Suite  Play .
Elliott Carter often bases his all-interval sets on the list generated by Bauer-Mendelberg and Ferentz and uses them as a "tonic" sonority[1]  Play .
All-interval series from Luigi Nono's Il canto sospeso Play .[2] (Equivalent to Nicolas Slonimsky's "Grandmother Chord".)[3]

In music, an all-interval twelve-tone row, series, or chord, is a twelve-tone tone row arranged so that it contains one instance of each interval within the octave, 1 through 11. A "twelve-note spatial set made up of the eleven intervals [between consecutive pitches]."[1] There are 1,928 distinct all-interval twelve-tone rows.[4] 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.[5]

The sum of numbers 1 through 11 = 66 and thus the chord contains a tritone between its outer notes[6] and as its sixth (middle) interval, and between the two notes directly outside of those, etc.

Examples

Mother chord

Mother chord[7]  Play 

For example, the first all-interval row, the Mother chord, was devised by Fritz Heinrich Klein: F, E, C, A, G, D, A♭, D♭, E♭, G♭, B♭, C♭.[8]

0 e 7 4 2 9 3 8 t 1 5 6

with the intervals between consecutive pairs of notes being (t = 10, e = 11):

 e 8 9 t 7 6 5 2 3 4 1

This row was also used by Alban Berg in his Lyric Suite (1926).

Chromatic scale  Play .

In contrast, the chromatic scale 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

Grandmother chord[9]  Play 

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.[10] It was invented by Nicolas Slonimsky on February 13, 1938.[11]

    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' chord used once in Carter's "End of Chapter".[12]  Play 

'Link' chords are all-interval twelve-tone sets whose first and second halves consist of an all-triad hexachord ({012478}). Found by John F. Link, they have been used by Elliott Carter in pieces such as Symphonia,[13][14]

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.[15]

0 t 3 e 2 1 7 8 5 9 4 6
 t 5 8 3 e 6 1 9 4 7 2
0 t 9 5 8 1 7 2 e 3 4 6
 t e 8 3 5 6 7 9 4 1 2

See also

Sources

  1. ↑ 1.0 1.1 Schiff, David (1998). The Music of Elliott Carter, second edition (Ithaca: Cornell University Press), pp. 34–36. ISBN 0-8014-3612-5. Labels added to image.
  2. ↑ Leeuw, Ton de (2005). Music of the Twentieth Century: A Study of Its Elements and Structure , translated from the Dutch by Stephen Taylor (Amsterdam: Amsterdam University Press), p. 177. ISBN 90-5356-765-8. Translation of Muziek van de twintigste eeuw: een onderzoek naar haar elementen en structuur. Utrecht: Oosthoek, 1964. Third impression, Utrecht: Bohn, Scheltema & Holkema, 1977. ISBN 90-313-0244-9.
  3. ↑ Slonimsky, Nicolas (1975). Thesaurus of Scales and Melodic Patterns, p. 185. ISBN 0-8256-1449-X.
  4. ↑ Carter, Elliott (2002). Harmony Book, p.15. Nicholas Hopkins and John F. Link, eds. ISBN 9780825845949.
  5. ↑ Robert Morris and Daniel Starr (1974). "The Structure of All-Interval Series", Journal of Music Theory 18/2: pp. 364-89, citation on p. 366.
  6. ↑ Slonimsky (1975), p.iv.
  7. ↑ Schuijer, Michiel (2008). Analyzing Atonal Music: Pitch-class Set Theory and Its Contexts, p.116. University Rochester Press. ISBN 9781580462709.
  8. ↑ Whittall, Arnold (2008). The Cambridge Introduction to Serialism, p. 271 and 68–69. ISBN 978-0-521-68200-8.
  9. ↑ Slonimsky (1975), p.243.
  10. ↑ Slonimsky (1975), p.iii.
  11. ↑ Slonimsky (1975), p.vii.
  12. ↑ Boland, Marguerite and Link, John (2012). Elliott Carter Studies, p.281. Cambridge University. ISBN 9780521113625.
  13. ↑ Schiff (1998), p.41.
  14. ↑ Boland and Link (2012), p.67.
  15. ↑ Boland and Link (2012), p.208.

Further reading

External links