From Scale to Index Set
Siegfried Koepf, 1997/1999 (English by Clarence Barlow)
Aristoxenus says: "Nay, rather there is no error so fatal and so preposterous as to base the natural laws of harmony on any instrument. The essence and order of harmony depend not upon any of the properties of instruments. It is not because the clarinet has finger-holes and bores, and the like, nor is it because it submits to certain operations of the hands and of the other parts naturally adapted to raise and lower the pitch, that the Fourth, and the Fifth, and the Octave are concords, or that each of the other intervals possesses its proper magnitude."(1)
A tone system is traditionally taken to mean a complex of terms, rules and operations relating to tones, which in this context are presupposed or defined as constants arguable in the manner of numerical mysticism, psychoacoustics but mainly physics.
The more the argumentation focuses on the "constants" aspect, the more Helmholtzian physics comes into play, with its ability to make statements about materials and their (physical) attributes. What physics hereby achieves is exactly the deliverance of physical equivalents and thus a description strategy for musical phenomena at first for tones and consequentially for intervals etc. But it is precisely this constant-ness of the observed objects that is contested by recent physics.
Although tone system theories tend in general to stress their differentness to other tone system theories (for example in respect to the innovative or the critical), there is one aspect they all have in common: the interest in a theory as generally acceptable and as universally justifiable as possible.
Euclid says: "If there were only quiet and no motion, there would be silence. If however there were silence and nothing were to move, one would also hear nothing. If therefore one wishes to hear something, impact and motion would have to take place beforehand.
Now since every tone results from an impact, an impact without preceding motion is impossible, a motion, the individual movements whereof are closer or less close together - and the tones succeeding the closer movements will be of higher pitch and the others by contrast of lower pitch. The higher tones must be so, because they consist of denser, more numerous movements; the lower tones must be so because they consist of rarer, less numerous movements.
If a tone is too high, it will be relaxed, i.e. by reducing movement it achieves the correct pitch. If it is too low, it will be more strongly strained, i.e. the tone achieves the correct pitch by increasing movement.
For this reason one must say that tones are assembled from parts, since they achieve the correct measure by adding and taking away. Everything that is assembled from parts, behaves mutually like whole numbers; therefore tones must also necessarily behave like whole numbers."(2)
Whereas Aristoxenus deduces the universal status of his theory by means of the nature concept, i.e. "the natural laws of harmony", Euclid postulates (in Pythagorean tradition) the general validity of his model with the help of comprehensive universal terms: "Everything that is assembled from parts, behaves mutually like whole numbers".
These and similar theories (irrespective of historic placement) are obviously typically concerned with totally grasping the subject and controlling it maximally and as definitively as possible.
The legitimacy, or even the usefulness of such a tendentially totalistic procedure seems doubtful to me, to say the least.
Totalistic tone systems are naturally best suited to totalistic music productions, situations and discourses.
These considerations can (should) be confronted by the following:
One can imagine a complex of terms, rules and operations, at first referring to arbitrary objects, which can (but don't have to) be subsequently defined as "tones".
This condition would be accommodated by certain declaration strategies, long established in any case in the art of the 20th century (e.g. Duchamp, Cage, Tudor).
Things can either be invented or be selected from the set of all available things. A selection of this nature can involve elements already prestructured within a historical or tone-systematic sense or which are semantically neutral. These distinctions can (but don't have to) be made.
A selection always involves reduction through the exclusion of the unselected. In focussing on justifying the selection, which appears legitimate due to the probable obscurity of the external, one is tempted to shift the unselected from the area of the obscure to that of the irrelevant. The tendency is to overlook the obscure or on occasion to reject it.
The sense in first permitting any objects therefore involves their repeated possibly (or necessarily) considered exclusion.
One can (but doesn't have to) agree on a convention that selectable things are associable with audible events.
Tone systems always first have potential character, because making selections is one thing and exhausting their combinatorial possibilities is another.
"Combinatorics is the art or the science of exhausting the possible through inclusive disjunctions."(3)
If the strategy of argumentation employs a formal language, one can demand of such a complex of terms, rules and operations that it consists of formalizable sentences, statements etc.
Formalized representational methods enable the machine-based further treatment of selected materials and therewith the application of combinatorial operations to larger quantities of data.
As an example, selected objects can be summarized and indexed into (ordered) sets. One can then continue to work with the corresponding index sets as substitutes.
In principle a purely intuitive procedure is just as legitimate and able to constitute something like a tone system. The intellectual act of description renders the matter objective, i.e. conscious. By the systematic act of reasoning it becomes a theory.
Intuitive methods skip the step of theory making, which does not preclude the application of the former to the latter. From the viewpoint of tone system theory they are latent and emphasize their potential nature.
The times are (long) gone, in which one could (want to) agree on one tone system and in which composers (and artists) limited themselves to dabbling with its predefined material.
Thus this is not about a comprehensive ultimate tone system. It is central (also to systematic or organizing methods) not to occupy the material but to release it again and to expect to re-encounter it in exactly the same form.
(1) Macran, Henry Stewart: The Harmonics of Aristoxenus, Hildesheim 1974.
(2) Euclid: SECTIO CANONIS, in: van der Waerden, B.L.: Die Pythagoreer, Zurich 1979, S. 382.
(3) Deleuze, Gilles: Erschöpft, in: Beckett, Samuel: Quadrat, Frankfurt a.M. 1996.
Originally published in German: Von der Tonleiter zur Indexmenge, in: Feedback Papers 42, 1999, further in: Komposition und Musikwissenschaft im Dialog, editors Christoph von Blumröder and Imke Misch, Signale aus Köln, Vol. 3, Saarbrücken 1999