I invented this form while working on Krähenalles. The basic idea is that the proportions of the whole work should also be present in each of the sections of the work.

The following description of the form is taken from work no. 80 for solo trombone and tape: WHALE

The sections of WHALE were labelled W, H, A, L, and E, with durations taken from the Fibonacci series: 13, 8, 3, 5, 3.

In **Abelian Form** the sections W, H, A, L, E will have subsections whose proportions are also 13, 8, 3, 5, 3. We can designate the subsections of W as Ww, Wh, Wa, Wl and We, and similarly for the others. The whole process can be shown in the following table:

This table looks like a mathematical Abelian Group—hence the name.

I wanted the piece to be about ten minutes long, ie. 600 seconds. When one works out the time in seconds for these proportions one gets (rounding off to whole numbers):

**Reflection about the Leading Diagonal**

The only fact of mathematical group theory which I now remember (from over 60 years ago!) is that Abelian Groups are symmetric about their leading diagonals. This symmetry led me to adopt the name for my form. It is also the most fascinating point about the form. It says that these symmetrically placed subsections are the same length and suggests that the contents should either be the same or closely related. This is the essential feature of any form: to determine where repetition or variation of material takes place. In my use of this form I have often experimented with this “repetition” or “variation”. When one considers the Abelian Form of WHALE (for example) the related subsections are: Wh + Hw, Wa + Aw, Wl + Lw, etc. The use of letters in this way suggested that the contents should be reflections of each other (i.e. retrograde). This is (with some exceptions) the case in WHALE (see score of

WHALE).

Also interesting is what material should be used in the subsections on the *leading diagonal*. In the song cycle on texts by Robert Walser, Scheint denn die Sonne heut’ nicht? the nine songs are arranged in a 3x3 Abelian Form:

Naturally, in the case of songs, the texts have to be chosen in advance to be suitable (and roughly the same length).

This problem of songs in retrograde was also a salient feature of Vlaminck's Tie

**Further Aspect**

I had set out to find a form in which the sections were also divided in the same proportions as those of the whole piece.

What I had also produced, however, was a new way of organising musical material. In contrast to traditional forms which finish with a recapitulation of the opening material, Abelian Form produces a sort of musical journey from A to Z with stations on the way which are heard twice (often in variation) — a musical trip with constant reminders of the past. With careful selection of the material one can create a gradual transition from the beginning to an end which, in view of the route taken, is not only new, but has a logic which makes arriving here completely convincing.

The more I have used this form, the more important has this second aspect become, indeed sometimes I have abandoned the concept of the proportions completely and used it solely as a way of organising my material (After Babel) for example.

**Interactive Demonstration ***(not available at the moment)*

The following interactive diagrams have been devised to show how Abelian form works.

The reader can enter his/her own proportions and the diagrams automatically show

- the corresponding matrix and
- a graphic representation, where the bars of the graph have the lengths of the entered proportions.

(You can also adjust the size of the pictures with the slider knob)

**To help you start:**
Enter the numbers for the proportions of my piece WHALE: 13 8 3 5 3
(the numbers can be separated with spaces or commas or dots)

You can also enter a word, e.g. WHALE or a phrase: e.g. Scheint denn die Sonne heut’ nicht

**Works using Abelian Form:**