Organizzazione delle altezze nello spazio temperato, Treviso, Diastema 1998, 384 pp. Seconda edizione, Collana Urania 2018.

PRESENTATION: Treviso, Ca' dei Carraresi, 20 November 1998, on the occasion of the Festival "Finestre sul Novecento".

Reviews:
NonSoloPiano , 8, luglio-agosto 1999 (Roberto Piana)
Il Saggiatore Musicale, VI, 1-2, 1999, pp.339-41 (Nicolas Meeùs)
L'informazione bibliografica, Il Mulino, 4, ottobre-dicembre 1999, pp. 543-555 (Giovanni Morelli)
Musiktheorie, XV, 1, 2000, pp.93-94 (Detlef Gojowy)
Musikforschung, 4, 2000, pp.493-4 (Detlef Gojowy)
Analisi, XII, 34, 1/2001, pp.28-31 (Moreno Andreatta)
Musurgia, VIII, 3-4, 2001 (Moreno Andreatta)

A deep knowledge of the numerical laws ruling certain phenomena of aggregation of sounds is indispensable for a more correct comprehension of the phenomena themselves; however it is necessary to point out that the use of numerical proceedings in the working out of a musical composition can only lead to vain sorts of  formulas, if they are not accompanied by an adequate insight rather than a wanting-in-judgement acceptance. A quantitative approach is easily schematizable in numerical terms: harmony, melody, rhythm, form are translatable in figures. A quantitative analysis may be perfect from every point of view, yet  unable to explain nor demonstrate anything; it runs the risk of contradicting the essence of music itself if it is not followed by a qualitative elaboration; for this reason,  the information contained in this book should be considered from its merely quantitative point of view, so as to leave  the most receptive readers the task of working out  the possible qualitative implications. The ability of solving “well  defined”  problems is mechanical, and can be taught; the ability of solving “badly defined” problems cannot be taught, but in order to become explicit, it needs mechanical skill as an unavoidable condition. The human mind is not a computer, and if this may be a disadvantage from the point of view of mechanics and data cataloguing, it is yet irreplaceable as regards the working out and the qualitative transformation of data themselves; therefore, it is necessary to turn the human mind to this direction, by exalting and making the most of what it already owns and is not mechanical or “badly defined”. This does not mean that it is necessary to give up defining the quantitative aspects of a problem in the best way, but only that it is convenient to use them as a basis and an assumption for a following qualitative elaboration: Art , infact, is man’s tendency to set about problems and to solve them in different forms, so as to have again the opportunity of setting about newer and newer ones, by a continuous redefinition of what “must” be and what “can” be on the basis on what is “expected”.

This work  intends first of all  to organize  the tempered sound space on the basis of the several possibilities to arrange pitches inside it, and then to investigate their main combinatorial properties. The processes ruling the arrangement of the several combinations are expressed through simple mathematical formulas, complete with elementary examples, which are of easy application. The first part of the book summarizes and develops a few notions of general theory, based in particular on research carried out by some of the main exponents of the American analytical school: from Hanson to Martino, from Lewin to Forte, until Howe, Starr and Rahn. In the same way, wide space has been devoted  to the theories of European scholars and composers as Busoni, Costère, Hauer, Simbriger, which , although very relevant, often precede chronologically the ones worked out in America and show a less systematic nature, an easy empiricism or even a tendency towards too abstract or esoteric research which in conclusion results in being  aimed at itself. If as a foundation of this study there is a rich  and manifold series of stimulations, there is also a great number of completely new conceptions and elaborations; the store of original examples is also very rich. The classification of all sets in a series of tables summarizing and synthetizing the proposals of thirteen among the most authoritative theoretic researchers from different geographical and cultural origins (Babbit, Busoni, Costère, Forte, Hauer, Martino, Mazzola, Perle, Pinos, Rahn, Solomon, Simbriger, Starr) is in this view. The wide theoretic part is followed by a short chapter which exemplifies some possible practical applications, based on the concepts developed in the first part, while the historical section widens some typical aspects of the several theories on chord classification. In the historical section, a wide space is also reserved to the composers of the Russian and Ukrainian schools of the early 1900s, who developed composing techniques of serial and dodecaphonic sort which are completely independent of the West. In the great variety of the theoretically possible sets, the ones which are symmetrical through transposition have been widely used in  the work of 20th century composers. In this view, the subsequent wide section, complete with several examples, deals with a deep analysis of these particular forms of aggregation. The text is not necessarily to be read by following the succession of the topics; themes which are hinted to in the theoretical part are often widened in the historical part and viceversa, so that the reader is invited to choose the itinerary he/she prefers.

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