Luigi Verdi Organizzazione delle altezze nello spazio temperato Ensemble '900, Treviso 1998, pp.382.

Review by Moreno Andreatta, in Analisi, XII, 34, 1/2001, pp.28-31

It is tempting to start reviewing Luigi Verdi’s brilliant study with the same praising tones used by American theorist John Rahn about a text by David lewin (1987), “important book (…) partly thanks to the topics dealt with, partly thanks to the way they are dealt with” (Rahn, 1987; p.305). But maybe superior is the wish to accept the author’s challenge in the preface, warning the reader of the necessity to consider “the information contained in this book (…) from their merely quantitative point of view” and inviting him/her to look somewhere else for the “possible qualitative implications” (p.5). Yet, as we will try to show by outlining the main topics dealt with and pointing particularly out some subtler theoretical matters, it is sufficient to adopt one different point of view to highlight the qualitative feature of many of the problems dealt with in the text. The subject matter of Verdi’s study is the problem of the organization of pitches in the temperate  sound space. The choice of the temperate space as a particular  sound system reference, just to mention a concept which is  often recurring in musical theory treatises (1), is a necessary step for an accurate definition of elementary musical objects (pitches and intervals), initial generalizations (classes of pitches and classes of intervals) and relations (transpositions, inversions, permutations, reductions, equivalences, transposition symmetries) which through the mathematical notion of set permit to understand the main theoretical elaborations of the so called “American analytical school” (primary form of a set,  interval function and  vector…). The initial chapter deals with the systematic presentation of these concepts (“General Theory”) and includes both research work by several United States theoricians (from Babbitt to Rahn, including Lewin, Hanson, Martino, Perle, Forte, Howe…) and contributions from Central and Eastern Europe scholars and composers (Busoni, Hauer, Hindemith, Krenek, Messiaen…). The list of relevant theoretical propositions ideally continues in the fourth chapter (“Historical outline”)  pointing out again the links between the American analytical school and the “continental” tradition, which in general analyzes the theoretical problems of music in a less systematic way, often showing some partiality for its most esoteric aspects. This is , for instance, the case of Hauer’s trope theory (p.174), of which the neo-Pythagoric component is found in Simbriger’s theoretic system (p.177), or in Costère’s échelonnement theory (p.179); anyway, the list of theoretic propositions based on numerological preference of a neo-Pythagoric sort cannot necessarily be exhaustive (2). The main theoretic constructions examined in the first chapter lead the reader progressively to the central definition of “interval vector”, a term adopted by Hanson in the ‘50s (p.76) and beside which it is appropriate to place, as suggested by Verdi, conceptual tools representing the forerunning ideals, such as fuction and interval contents (p.73).   After a few simple considerations as to the interval contents (common note theorem, p.74; exachord theorem, p.82), the author lists the main properties of the interval vector (pp.78-81), successfully demonstrating the fact that it is actually an extremely practical and effective means to analyze the internal construction of a combination” (p.78). The second chapter (“An outline of combinatorial technique” ) completes the framework of theoretic references which are necessary to cope with the central problem of the classification of chord structures. Thanks to the inclusion relation, it is possible to draw one first piece of information about the amount of  m cardinality subsets (i.e. having m elements) contained in a  n cardinality set (where n has the value of 12 in the usual temperate space). On an amount of 212=4096 subsets of the chromatic whole (including the borderline case of the void set and of the whole set), the chord amount of m sounds corresponds to the amount of subsets containing 12-m elements. For instance, for m=5 the result is that the number of possible pentachords amounts to the number of eptachords. This general consideration allows to limit the problem of chord classification to the study of cardinalities included between i and 6 (or, similarly, between 6 and 11) without loss of generality, as said in mathematics (3). Such formal symmetry property is obtained by introducing some equality relations which determine a progressive reduction of the number of “chords” which are structurally different in the chromatic scale. One first drastic reduction of the amount of subsets which are theoretically possible is obtained by introducing the concept of  equivalence by transposition (p.38). Two A and B subsets in the tempered sound space are equivalent by transposition (and represent, as a matter of fact, the same chord) if one is the image of the other by one transposition, or if a Tn transposition of n semitones, so that B is equal to Tn (A), exists (4). It is necessary to note that A is always  transposed of itself (reflexivity). Besides, if B is transposed of A by Tn, necessarily A will be transposed of B by Tm, where m=12-n (symmetry property). One third property (transitivity) is obtained by considering three A,B,C chords where B is transpossed of A and C is transposed of B. Hence it is possible to deduct that C is transposed of A. The musical notion of transposition, therefore, defines as a matter of fact one equality relation in a mathematical sense, a concept which the author should probably have tried to emphasize more (p.38). In this way it is possible to obtain the 352 equality classes (chord structures which are equivalents by transposition) which have been subject to several theoretical elaborations, often independent, as it is possible to realize from the rich collection of terminological proposals which are collected and commented by the author (p.46). American  composer and theoretic Milton Babbitt has undoubtedly been among the first to apply the mathematical relation of congruence to music (Babbitt, 1960), relation which is a necessary basis to cope with the problem of classification of structures which are equivalent by transposition and/or by inversion. In the case of equivalence by transposition, the internal structure of a chord, intended as a subset of the group of “whole numbers by pitch class” (p.16), is preserved, and this is the reason why it is possible to identify every chord combination with one unique interval structure, a conclusion which East European less known theorists as Polish M. Zalewski and Rumenian A. Vieru (5) had independently drawn. The reader can enjoy himself/herself by following some of the best known criteria of classification of chord sets which are discussed in the first four chapters and which are dealt with again in the very useful summary tables of the fifth chapter (“Classification of sets”). It is interesting to point out that the same problem can be faced through a complementary approach to the combinatorial numerical one described by the author. The operation of “module 12 addition” (p.16) defines some important properties on the “whole numbers by pitch class” which the reader can easily verify. First of all, it makes the set of whole number by pitch class an “algebrically closed” set, that is to say it contains the sum (module 12) of two of any of its elements (6).  There is only one element (the 0)  which if summed to another whole number by pitch class leaves it unchanged (neutral element). In correspondence with every whole number by pitch class, besides, is a unique whole number which, if summed to the preceeding one gives back the neutral element (inverted element) Finally, given any three elements it is equivalent to sum the third one to the sum of the first two or the first one to the sum of the second and the third (associative property). These four properties define a group algebrical structure over the set of the whole numbers by pitch class. More in general, every division of the musical octave in a n number of equal parts (that is to say every tempered sound space) brings forth a group structure over the set of the corresponding whole numbers by pitch class. It is worth while noticing that the group structure, which does not find space in the original theory by Forte, is then fundamental in the generalization of the Set Theory by D. Lewin (1987). Anyway, it is already present in the early theoretical works by M. Babbitt (1960) as well as in the several theoretical/composing, often contemporary reflections by I. Xenakis (1965). G. Mazzola’s theory itself , which Verdi follows with attention, considering it “one of the most interesting synthesis attempt, thanks to the mathematical rigour with which research is carried out” (p.204), uses the module algebrical structure as space-setting for the 88 chord classes of the classical temperate system (7). Going back to the main mathematical/musical properties considered by Verdi, it is worth while noticing that  some of these are naturally suited to an algebrical survey. Among these, there are surely the limited transposability (p.41), the transpositional partition (p.119), the symmetry by inversion and by complementarity (p.64) and p.81) and the reversibility (p.66 and p.83).  The whole of the sixth chapter (“The symmetrical scales in 20th century music) is devoted to a detailed presentation of the concept of transpositional symmetry” from as theoretical, analytical and composing point of view. The careful historical reconstruction by Verdi points out the centrality of the problem of octave division in equal parts in several music theorists and 20th century composers (Alaleona, Slonimskij, Lendvai, Zalevski, Schillinger, Messiaen, Haba, Wischnegradsky, Frazzi, Dallapiccola, Strawinsky, Skrjabin, Bartòk…). As observed in the first chapter, the transpositional symmetry conveys the fact that “some combinations have the property of presenting themselves as equal to the original one at a transposition level which is inferior to 12” (p.41). They are the so called “limited transposition modes” by Messiaen’s, one of the first theorists and composers to try and generalize their study and composing use. We agree with Verdi on the fact that many of the composer’s theoretic propositions are lacking and not rigorous, but it is necessary  to combine the author’s criticism with a series of additional specifications. First of all the term “limited transposition mode” does not seem us to be inappropriate, as on the contrary Verdi suggests (p.41), since it expresses the property of a m  well defined musical object (mode) to solve the equation M = Tn (M) for a n  limited value (or different from the trivial case represented by 0 or by whole 12 multiples).  We remind the reader that one of the possible geometrical representations of the set of whole numbers by pitch class is, as Verdi well points out, that of “ a (regular) dodecagon inscribed within a circle, the twelve vertexes of which correspond to the different pitch classes” (p.22). The transpositions are geometrically equivalent to rotations, while the inversions correspond to all geometrical figures which, if revolved by a certain angle (different from 360° and its multiples), are superimposed  over themselves. They are not necessarily “regular and inscribed in a dodecagon” poligons (p.44), as, moreover,the figure of example 1.27 on the same page clearly shows. It is true that Messiaen “lists only 7 limited transposition modes” (p.41) but maintaining, as Verdi does, that they are,as a matter of fact, 16 means making the concept of “mode” itself problematical. In other words, the bichord corresponding to the tritone interval, for example, could not be included, conceptually, in Messiaen’s definition of mode. We can just smile at the confidence with which the composer states, as Verdi informs us, that it is “mathematically impossible to find some others (limited transposition modes) at least in our 12 semitone temperate system” (p.269),  especially as some of the transposition modes which are not taken into consideration by Messiaen (for example the ones corresponding to sets 86 and 87 in Mazzola’s classification) own some interesting algebrical properties (Mazzola, 1990; p.142) and have not yet been explored completely from the point of view of composition. Besides, one exhaustive classification of all limited transposition chord structures had been effected, at the beginning of 1980’s, by Rumenian mathematician D.T. Vuza through an algebrical approach which, as a positive aspect, does not limit study to the case of octave division of 12 equal parts. Starting from the so called “modal theory” by composer A. Vieru (1980), the transpositional symmetry property is generalized, very naturally, to the so called microtonal systems, of which it is possible to point out the structural porperties in a more systematic way than what is proposed by such theorists as A. Haba and I. Wischnegradsky (p.271). It should be specified that not necessarily “the possibilities of forming limited transposition modes increase (…) in microtonal systems” (p.271). As a counter example, it is sufficient to consider all microinterval systems corresponding to octave divisions in a prime number of parts, some of which are still interesting for theorists and composers (one may think of enneadecaphonic system, that is to say  the one which divides the octave into 19  equal parts, or of the microtonal system corresponding to the octave division into 31 equal parts). It is not difficult to realize that in these cases no set of pitch class having transpositional symmetry exists. It is also possible to note that many are the still open theorical problems linked to this apparently simple concept, as it is possible to realize when trying to transpose the concept of transpositional symmetry onto the domain of musical rhythm. This problem offers some elements of reflection to integrate within the third chapter (“some composition applications”) (8) and brings us nearer the contents of the seventh and last chapter (“Reversibility of musical time”), in particular as regards the rythmic retrogradation technique (p. 340). Verdi points out that it is an extremely delicate problem for an art as music, which “lives in time and (in which) time specular symmetry belongs to a dimension which the human mind can hardly perceive” (p.343). In this view, a reference to Messiaen’s  so called non-retrogradable rythms, which characterize themselves as length palindrome successions, in the sense of  remaining unchanged after an inversion of the time axis, is inevitable. Again in this case it is not possible to accept, dogmatically, Messiaen’s theoretical assertions, especially as regards the existence of a perfect analogy between non retrogradable rythms and limited transposition modes (Messiaen, 1944) (9). In order to be able to read, rythmically, structural properties which are similar to the ones considered in the vertical organization of sounds, it is necessary to build an appropriate algebrical model of musical rhythm. This has been done in successive stages by D.T. Vuza (Vuza, 1988; 1991) who has come to the point of a rigorous formalization of the concept of “rythmic canon”, a starting point for a systematical study of the structural properties of this musical form. The many-centuried interest of composers in the techniques which are at the basis of the musical canon is undoubtedly linked to the “occult and esoteric meanings of Phytagorical origin” (p.324) which such technique could recall, just because based on numerical procedures. And if it is true that the rediscovery – and the use at the beginning of 20th century – of canon techniques often surrounded by mysticism and esoterism is due to the Second School of Wien, what is sure is that the rythmic retrogradation technique could be fully integrated as a composing proceeding thanks to Messiaen. Infact we owe to him the introduction and systematic use of canon forms where the imitation of different parts involves rythmic values only, independently of other parametres as melody and harmony. Within this new theoretic setting, and thanks to the rigorous formalization given by Vuza, such as property as limited transposability is all but trivial, especially if it is studied in the relation with other algebrical properties as “transpositional partition” , which is in many stages discussed by Verdi (p.119; p.137; p.206). It is, also, at the basis of one particular m  rythmic  canon class, which are able to completely  fill the sound space by regular pulsation, without any intersection between different voices nor empty spaces. Such canon structures take the name of “complementary regural canons of maximal category” (Vuza, 1991) and represent an appropriate example of musical reading of some geometrical properties Verdi refers to several times. It is a matter of plane regular division, a mathematical problem which finds one of the most famous artistic applications in Dutch graphic designer M.C.Escher’s work (p.37). The family of rythmic canons formalized by Vuza, besides, is the formal model to which many canons used by Messiaen ideally tend. It is worth while mentioning, as an example, the triple canon upon non- retrogradable rythms which is the seventh part of Harawi (Adieu) and where the maximum complementarity  between the three parts is closely linked to the idea of a “well arranged disorder” (Messiaen, 1992; p.46). It is clear, now, that a precise and rigorous formalization of a musical phenomenon may well lead to the comprehension of a series of extremely complex composing problems, like those, as generally known, linked to the construction of musical canons. It is well worth while ending this survey on the main topics dealt with by Verdi by expressing an extremely critical  position towards those pseudoscientific theories of music which have often been considered as the only contact point between music and mathematics. Many of these theories are expressly quoted by Verdi and it would be  difficult not to include among the latter also what, according to its author, should have been “the first scientific system able to go beyond the threshold  of a musical creation sanctuary” (Schillinger, 1941; p.1063). Without considering Josip Schillinger’s composing theories completely useless, as groundless from a physical/mathematical point of view (Backus, 1960), it seems to be possible to admit that this work is unable to give rigorous grounds to theoretical musical problems, despite the fact that many composers (from Gershwin to Cowell) have been able to get composing ideas from it. What is well different is Verdi’s precious work which develops, in a clear and convincing way, a field of topics where theoretic reflections, analytical needs and original ideas for possible composing applications are harmoniously balanced and cohexist.  The present reading does not intend to correct practically anything but some little imprecision in what the author discusses. Maybe it suggests possible parallel readings of one same issue and therefore confirms the crucial importance of the topics dealt with in the book and the necessity that new points of meeting may foster the dialogue which has been so successfully opened.

Moreno Andreatta


1.    See, for example, L. Azzaroni’s recent theoretical contribution (1997), reviewed by Miguel A. Roig-Francoli in issue no. 30 of the journal.

2.    Readers who are interested in integrating the framework of  (pseudo) scientific theories on music may find a useful reference point in L. Fichet’s ample study (1996)

3.    As an example of use of this property, see the two different classifications porposed by Martino and Mazzola, both included in the very useful summary table (pp.216-233)

4.    For the sake of convenience, the term “trasnsposed”, as a matter of fact not very appropriate due to its rooted and completely different use in Mathematics, will be used.

5.    For a deep analysis of Polish theorist M. Zalewski’s work and several points of contact with Allen’s Forte’s Set theory, see Moscariello (1995; 1996; 1997)

6.    the property, which is a direct consequence of the definition of addition (module 12), is also expressed by stating that the operation is a law of interior composition for the set considered

7.    Not to be mistaken with the concept of module previously discussed as regards the equivalence relation

8.    See in particular the presentation of the composing technique based on the transposition of an interval structure upon its pitches, found for example in N. Roslavec (p.155), from our point of view a particular case of  more general theoretical/composing proceedings, as  Boulez’s “chord multiplication” (1963; p.89) or Vieru’s “modal class composition” (Vuza, 1988;p.282)

9.    The statement, considered again the second tome of the monumental Traité de Rythme, de Couleur et d’Ornithologie (Messiaen, 1992), lends itself to several criticisms if analyzed, for example, through the tools of mathematical theory in music (;azzola, 1990; p.99)

10.    The example, quoted by Azzaroni (1997; p.205), is discussed in a recent study on the relations between rythmic canons and mathematical theory of music (Andreatta, 1999)  


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