All-interval series

The all-interval series is a special form of the twelve-tone row , which forms the compositional basis of twelve-tone music. While a regular twelve-tone series is required to contain all twelve pitches of the chromatic scale once, in the case of the all-interval series this rule is also extended to the intervals . The eleven possible different intervals of the octave space are arranged in the all-interval series in such a way that they constitute a regular twelve-tone series in relation to an initial tone.

In terms of the totality of composing with twelve tones that are only related to one another , according to Ernst Krenek, the all-interval series in dodecaphony has a higher degree of wholeness, since the totality of the tone series finds its completion in the totality of the intervals.

Among the possible 12! = 479,001,600 twelve-tone rows (with a fixed starting tone) only 3,856 rows have the property that qualifies them as an all-interval series.

History of the all-interval series

The first all-interval series was discovered by the composer Fritz Heinrich Klein in 1921. For 15 years it was considered the only possible series formation of this kind. Alban Berg used it for the second “Stormlied” in 1925 and for the “Lyrische Suite” in 1926.

Ernst Krenek published a second all-interval series in 1937 (see note example above) and also discovered the first regularities of this series form.

The number of all-interval series was first calculated at the suggestion of the Austrian composer Hanns Jelinek by the information theorist Heinz Zemanek with the help of a self-made electronic computer called “Mai-Fan'l”.

Herbert Eimert presented an orderly catalog of the twelve-tone interval series for the first time in his work "Basics of musical series technology".

The all-interval series opened the way from the original twelve-tone music to serial music , in which an attempt was made to order all parameters of the tone (i.e. pitch, tone duration, volume and sometimes also the timbre) in series. This meant that other series of intervals other than twelve-tone intervals became up-to-date, as tone durations and volume levels are not necessarily based on the number 12. Paul Irmen proved in 1974 that the regularity of the all-interval series and its transformation possibilities can be applied to all even-numbered element series.

Series transformations

There are four unambiguous transformations (conversions) by means of which one all-interval series can be converted into another:

cancer

Row backwards.

reversal

Conventional inversion of the intervals: The intervals are replaced by their complementary interval to the octave. (Reflection on the major seventh , interval 11).

Fourth transformation, fifth transformation

The intervals are replaced by their 5-fold modulo 12 (reflection on the pure fourth , interval 5). When Quint transformation corresponding seventh with Interval The bottom transformation results in the reversal of the fourth transformation and is therefore not independent.

Tritone change

The row is cut apart at the tritone (interval 6) and the two parts are mixed up again. This again results in an all-interval series, since the beginning and end notes of an all-interval series are always a tritone apart. (The sum of all eleven intervals: 66 modulo 12 is namely 6, the tritone.) The tritone transformation is a specific rotation of the series.

A combination of these transformations is commutative ; H. the order in which the transformations are applied is irrelevant. In addition, every transformation to itself is reciprocal: applying it twice again results in the initial series.

With the help of these transformations, the number of all-interval series can be reduced to 267 so-called basic series.

Symmetries and the 267 basic all-interval series

From 267 all-interval series, which are called base series, all 3,856 can be derived through the transformations cancer, inversion, fourth transformation, tritone transformation and the possible combinations thereof.

The series changes combine the series into groups with a close degree of relationship. Since there are four regular metamorphoses (i.e. the result is again an all-interval series), in principle 2 ⁴  = 16 series can be derived from each other (basic sequence, cancer, inversion, cancer inversion, fourth transformation and its cancer, inversion and cancer inversion, as well as the tritone transformation of all of the above ).

However, due to the possible symmetries, this number is sometimes reduced to eight.

A series is called symmetric if it is equal to its cancer, counter-symmetric if it is equal to the inverse of its tritone transformation (i.e. the second half of the series is the inversion of the first), line- symmetric if at least the tritone is in the middle of the series ( this is actually not a real symmetry, since the two halves of the row otherwise have no relation to each other; the tritone is inevitably in the middle even with the symmetrical and oppositely symmetrical rows). All others are asymmetrical . Equal quarters can be straight or asymmetrical; they are like the Cancer inversion of the fourth transformation of their tritone transformation. This is a more distant, but all the more interesting, relationship.

There are also axially symmetric all-interval series. If you “tilt” them, the result is the same row.

The symmetry properties occur in the basic series:

• asymmetrical 211 times,
• symmetrical 22 times,
• route symmetrically 19 times,
• counter-symmetrical 15 times,
• quarterly 15 times, of which 12 times asymmetrical, 3 times symmetrical.

From this it follows for the totality of the all-interval series:

• (211 - 12) × 16 = 3184 rows are asymmetrical,
• 12 × 8 = 96 asymmetrical and equal quarters,
• 22 × 8 = 176 symmetrical,
• 15 × 8 = 120 counter-symmetrical,
• (19 - 3) × 16 = 256 symmetrical lines,
• 3 × 8 = 24 symmetrical lines and equal quarters.

Only just under 17.5% of all all-interval series show any symmetries. If one disregards the less essential "route symmetry", only 10.8% remain.

Web links

• Manuel Gervink : The structuring of the tonal space. Attempts to systematize twelve-tone rows in the 1920s to 1970s.
• Overview of the definition and elementary properties of all-interval series

literature

• Herbert Eimert: Basics of the musical serial technique. Universal Edition, Vienna 1964.
• Herbert Eimert: Textbook of the twelve-tone technique. Breitkopf and Härtel, Wiesbaden 1966.

Individual evidence

1. ↑ Hanns Jelinek: The cancer-like all-interval series. In: Archives for Musicology . XVIII / 2. Vienna 1961
2. ↑ Herbert Eimert: Fundamentals of the musical serial technique. Vienna 1964
3. ^ Paul Irmen: On the mathematical calculation of all interval series , Cologne 1974