# All-interval twelve-tone row

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 (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 [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] These 1,928 tone rows have been independently rediscovered several times, their first computation probably was by Andre Riotte in 1961.[6]

Since the sum of numbers 1 through 11 equals 66, an all-interval row must contain a tritone between its first and last notes,[7] as well as in their middle.

## Examples

### Mother chord

Pyramid chord

Mother chord[8]

Grandmother chord[9]

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.[10][11]

```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.[12]

The Mother chord row was also used by Alban Berg in his Lyric Suite (1926) and in his second setting of Theodor Storm's poem Schliesse mir die Augen beide.

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

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.[13] It was invented by Nicolas Slonimsky on February 13, 1938.[14]

```    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
```

A similar row, although with all the notes within the same octave by alternating ascending and descending intervals (where the size of all the intervals decrease by one), was used by Kaikhosru Shapurji Sorabji as the first subject in the sextuple fugue of his Organ Symphony No. 3.[15]

'Link' chords are all-interval twelve-tone sets containing one or more uninterrupted instances of the all-trichord hexachord ({012478}). Found by John F. Link, they have been used by Elliott Carter in pieces such as Symphonia.[17][18]

```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.[19]

```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
```

## References

1. ^ a b 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. ^ a b Slonimsky, Nicolas (1975). Thesaurus of Scales and Melodic Patterns Archived 2017-01-09 at the Wayback Machine, 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. 364–389, citation on p. 366.
6. ^ André Riotte (1963) Génération des cycles équilibrés, Rapport interne n°353 Euratom. Ispra.
7. ^ Slonimsky (1975), p. iv.
8. ^ Schuijer, Michiel (2008). Analyzing Atonal Music: Pitch-class Set Theory and Its Contexts, p. 116. University Rochester Press. ISBN 9781580462709.
9. ^ Slonimsky (1975), p. 243.
10. ^ Whittall, Arnold (2008). The Cambridge Introduction to Serialism, p. 271 and 68–69. ISBN 978-0-521-68200-8.
11. ^ Arved Ashby, "Klein, Fritz Heinrich", The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell (London: Macmillan, 2001).
12. ^ Klein, p.283. "Die Grenze der Halbtonwelt" ["The Boundary of the Semitone World"], Die Musik 17/4 (1924), pp. 281–286.
13. ^ Slonimsky (1975), p. iii.
14. ^ Slonimsky (1975), p.vii.
15. ^ Sorabji, Kaikhosru (2014). Bowyer, Kevin (ed.). Organ Symphony No. 3. Sorabji Archive. p. 231.
16. ^ Boland, Marguerite and Link, John (2012). Elliott Carter Studies, p. 281. Cambridge University. ISBN 9780521113625.
17. ^ Schiff (1998), p. 41.
18. ^ Boland and Link (2012), p. 67.
19. ^ Boland and Link (2012), p. 208.
• Bauer-Mendelberg, Stefan, and Melvin Ferentz (1965). "On Eleven-Interval Twelve-Tone Rows", Perspectives of New Music 3/2: 93–103.
• Cohen, David (1972–73). "A Re-examination of All-Interval Rows", Proceedings of the American Society of University Composers 7/8: 73–74.