Integral Serialism II

Integral serialist composers:

Karlheinz Stockhausen
Pierre Boulez
Luigi Nono
Luciano Berio
Luigi Dallapiccolla

and others. These were part of the same generation who first adopted this method of composing. The festival and summer courses at Darmstadt have continued since the beginning of the 50s and were for a long time the focal point of contemporary and avant-garde music. Die Reihe was a publication dedicated to the ideas of the Darmstadt group.

The Domain technique of Boulez. Used in Le marteau sans maitre and other subsequent pieces, the domain technique involves a tone-row:

Eb - F - D - C# - Bb - B - A - C - G# - E - G - F#

which is divided into unequal segments of 2, 4, 2, 1, and 3 sounds:

[Eb, F]; [D, C#, Bb, B]; [A, C]; [G#]; [E, G, F#] The numbers representing the groupings are permutated:

			2 4 2 1 3       I
			4 2 1 3 2       II
			2 1 3 2 4       III
			1 3 2 4 3       IV
			3 2 4 3 1       V

Each permutation generates a "domain" by multiplying these chords and intervals with each other. The multiplication may be understood as transposing one chord on each of the notes of another. If we notate, for example, the chords of domain I with

a [Eb, F]
b [D, C#, Bb, B]
c [A, C]
d [G#]
e [E, G, F#]

and multiply them, we obtain a matrix of the form:

		aa  ab  ac  ad  ae
		ba  bb  bc  bd  be
		ca  cb  cc  cd  ce
		da  db  dc  dd  de
		ea  eb  ec  ed  ee

where aa consists of the interval Eb - F transposed on Eb and F, i.e. the chord: Eb, F, Db; ab will consist of the same minor seventh interval (Eb - F) built or transposed on each sound of the second chord (D, C#, Bb, B), i.e. the chord: D, C, C#, B, Bb, Ab, (B), A; etc.

Boulez observes that elements ab and ba contain exatly the same sounds and so do ca=ac, de=ed, etc. He calls these total isomorphisms. Partial isomorphisms are created by all the sounds in the same row or in the same column since they all have one element in common. The diagonal: aa, bb, cc, dd, ee is made of unique elements. The composer organizes then the music as a play between these similar, partially similar, and unique groupings.