of numbers

tion, the octave shift gives a melody its expression: the Adagio from Anton Bruckner’s 9th Symphony starts with the jump B- c2, a leap of a minor ninth, played by the first violins on the G-string; the remarkable colour that comes from this leap into a high position on a low string is decisive for how this melodic line works. In the pitch class this is reduced to a minor 2nd as basic movement, but this reduction no longer has any characteristic by which we can tell that the beginning of this melody has such a strong expression.

There is a second pitch-class theory, which takes the reduction of intervals even further, assuming that there are only intervals between the prime (= 0) and the tritone (=6) and that all other intervals are a reversal of these basic intervals. In this pitch-class theory it is assumed, analogous to the idea that, for example, C and c4 sound ‘the same’, that the intervals 2nd and 7th, prime and octave, 3rd and 6th, and 4th and 5th also sound ‘the same’. - The pitches are enharmonically equated. In atonal music this makes sense, but in (enriched) tonal music a difference between c and d may very well be audible. - Each pitch is assigned a number, starting at 0. These numbers are called the ‘pitch-class representatives’. I have always found that rather awkward, since the twelve-tone technique is based on 1-12, and ultimately there are 12 pitches in both systems. The principle of 0 for the prime in the pitch-class set implies that this is not so much about the pitches themselves but rather about the interval relation between the pitches. And if that is the case, why then leave out all the octaves? - For convenience’s sake, in this text we will assume that 0 = c, and thus: c = 1 and so on, up to b = 11. - This makes it relatively easy to express motives in numbers, for example g-g -d-f = 7-8-2-5.

12.3 OVERVIEW OF THE WAYS TO MANIPULATE THIS SERIES OF NUMBERS Next, I will give a brief overview of the ways to manipulate this series of numbers, which also reveals the risk of this method: the sound result of the act of composing is pushed into the background by the number manipulation and the system tempts the composer to resort to manipulations and decisions that are like the frivolous movements of a virtuoso football player who has lost sight of the other team’s goal. The system can easily take the initiative from the composer as it produces, quickly and easily, a great number of pitches that are at least strongly structur-

ally connected, to the satisfaction of the modern music scene, which in the U.S.A. is centred at the universities.187

First, we establish the ‘prime form’. This is the simplest form of the set. It means that the numbers are arranged from low to high. As prime form, our g-g -d-f=7-8-2-5 becomes 2-5-7-8. The composer can now look at the specific properties of his prime form in a list: ‘related prime form sets’ and ‘software tools’ (for the aficionados: www.ComposerTool. com).

Another source is Allen Forte’s book The Structure of Atonal Music (1973). Forte (1926-2014) is the author of the pitch-class set theory (in his 1964 essay ‘A Theory of Set-Complexes for Music’). He was a musicologist and music theorist and without jumping to conclusions about these noble professions, it may explain the over-theoretical nature of the pitch-class set theory. Yet another source can be found in the works of Elliott Carter, who produced a number listing of pitch-class sets188 , which he called his ‘chords’ and used as basic material for his compositions.

A second example for finding a prime form: g -c-e-f =8-0-4-6: - Change into 0-4-6-8 - All positions of the set are: 0-4-6-8 4-6-8-(0+12=)12 6-8-12-(4+12=)16 8-12-16-(6+12=)18 - Which sets have the smallest distance from the first and the fourth number? Answer: the first and the second (distance = 8). - When two distances are the same, we look at the distance between the first and second number. The second set wins (distance = 2). - We write this as 0-2-4-8 All these prime forms are simple, there are only 208. Transposition: add a fixed number to all numbers of the set. Transpose an octave up/down: add 12 to a number or subtract 12. Invention: subtract the numbers from 12. - Interval vector This is a list of all possible intervals within the pitch-class set. If the set consists of only 2 tones, there is only 1 interval, 3 tones means 3 intervals, 4 tones 6 intervals, 5 tones 10 intervals, and so on. Reminder: the intervals 2nd and 7th, prime and octave, 3rd and 6th, and 4th and 5th sound ‘the same’ and belong to the same vector.

187An almost routine question in those circles is: ‘Where did you get your notes?’ 188To Carter this was a rather mechanical activity. He used to say that he calculated these chords during long and boring flights.