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The following overview of the book will provide a good point of departure.Chapter 1 is purely mathematical; it presents terminology and notation thatwill be needed later, along with a fewimportant theorems. I am not happy tobegin a book about music with a mathematical essay. On the other hand, I dofeel that it is helpful for the reader to have this material collated and isolatedfrom the rest of the book. Chapter 1 can be used for quick reference where itstands, and the material obtrudes only minimally into musical discussionslater on. Readers w ho find themselves put off or fatigued in the middle of thischapter are urged to move on into the rest of the book; they can return tochapter 1 later, when later applications of the material make the referenceback seem natural or desirable.

Chapter 2 takes as its point of departure the general situation portrayedschematically by figure 0.1.

FIGURE 0 .1The figure shows two points s and t in a symbolic musical space. Thearrow marked i symbolizesa characteristic directed measurement, distance, ormotion from s to t. W e intuit such situations in many musical spaces, and weare used to calling i "the interval from s to t" when the symbolic points are xxix

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Introductionpitches or pitch classes. Chapter 2 begins by run nin g through twelve exam plesof musical spaces fo r which we have the intuition of figure 0.1. Six involvepitches or pitch classes in melodic or harm on ic relations; six invo lve aspects ofmeasured rhythm. The general intu ition at hand is then made formal by amathematical model which I call a Generalized Interval System, GIS forshort. A few basic fo rm al properties of the m odel are explored. Then thetwelve examples are reviewed to see how each (w ith one exception) instancesthe generalized structu re.Chapter 3 concerns itself with further formal properties of the GISm odel. In that m odel, the points of the space m ay b e labeled by the ir intervalsfrom one referen tial point; this has advantages and disadvantages. New GISstructures may be constructed from old in various ways. A passage fromWebern is examined in connection with a combined pitch-and-rhythm GISconstructed in one such way. Generalized analogs of transposition an d inver-sion operations are explored. So are "interval-preserving operations"; thesecoincide w ith transposition s in som e G IS m odels but not in others, specificallynot in GISs that are "non-commutative."The bulk of chapter 4 explores one non-commutat ive GIS o f musicalinterest. The elements of the system are fo rm al tim e-spans. E xtended dis-cussion of a passage from Carter's F irst Q uartet dem on strates the pertinenceof this GIS to exploring m usic in which there are functional measured rela-tions am ong tim e spans, but no one overriding tim e span that acts as a unit tomeasure all others. After that, chapter 4 presents tw o exam ples of tim bralGISs, and ends with a methodological note on the relations of m usic theory,perception, and the intuitions of a listener. Som e m otivic w ork by Chopin isconsidered in this connection.Chapter 5 begins a study of generalized set theory, that is, the interrela-tionships among finite sets of objects in m usical spaces. The first constructionstudied is the Interval Fun ction between sets X and Y; this function assignsto each interval i in a GIS the n u m b e r of ways i can be spanned between amember of X and a member of Y. Then the Embedding Number of X in Yis studied; this is the num ber o f distinct form s of X that are subsets of Y. Tostudy that number, we have to establish what we mean by a "form" of the setX , a notion that involves stipulating a Canonical Group of operations. Bothth e Interval F unction and the E m bedding N um ber generalize Forte's IntervalVector. Passages from Webern, Chopin, and Brahms illustrate applicationsof the constructs.Chapter 6 continues the study of set theory, generalizing the work ofchapter 5 even farther. The basic construction is now the Injection F unction:Given a space S, finite subsets X and Y of S, and a transformation f mapping Sinto itself, IN J(X , Y ) (f) counts how m any mem bers of X are mapped by f intomembers of Y. This number is m eaningful even w hen S does not have a GISstructure, and even when the transformation f js no t so well behaved as arexx

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Introduction

transpositions, inversions, and the like. Passages from Schoenberg and fromBabbitt are studied by way of illustration.Instead of starting with a GIS and deriving certain characteristic trans-

formations therefrom, it is possible to start with a family of characteristictransformations on a musical space and derive a GIS structure therefrom.That is, instead of regarding the i-arrow on figure 0.1 as a measurement ofextension between points s and t observed passively "ou t there" in a Cartesianre s ex tensa, one can regard the situation actively, like a singer, player, orcomposer, thinking: "I am at s; what characteristic transformation do Iperform in order to arrive at t?" Chapter 7 explores this conceptual inter-relation betw een interval-as-extension and transposition-as-characteristic-motion-through-space. After developing the m athem atics that shows a logicalequivalence between G IS structures and certain structures of transform ationson spaces, the work proceeds by example. Passages from Schoenberg, Wag-ner, Brahms, and Beethoven indicate how suggestive it can be to considernetworks of "intervals" an d networks of "transpositions" (modulations, andso forth) as various aspects of the same basic phenomenon.The m orphology of such networks can be carried over to that of networksinvolving other sorts of transform ation s. Chapter 8 studies netw ork s invo lv-ing transformations of Klangs in the sense of R iem ann, n etworks involvingserial transform ations of various sorts, an d networks involving inversionaltransformations. The Beethoven exam ple from chapter 7 is reconsidered, an dthere are further examples from Wagner, Webern, and Bach.Chapter 9 develops the form alities of transform ation netw orks in arigorous way . The structure of a network allows us to assign a formal "input"function to som e things and a formal "output" function to oth er thing s; thesefunctions seem of considerable m usical interest in some cases. The networkshave intrinsic rhythm ic properties which can also be studied formally. Net-work structure can accommodate hierarchic levels in a quasi-Schenkeriansetting, as an exam ple shows.Chapter 10 applies the n etw ork concept in a variety of w ays to passagesfrom M ozart, Bartok, Prokofieff, an d Debussy.Note on Musical TerminologyAll references to specific pitches in this book will be made according to thenotation suggested by the Acoustical Society of America: The pitch class issymbolized by an upper-case letter and its specific octave placement by anumber following the letter. A n octave num ber refers to pitches from a givenC thro ug h the B a m ajor seven th above it. Cello C is C2, viola C is C3, m iddleC is C4, and so on. An y B# gets the sam e octave n um ber as the B jus t below it;thus B#3 is enharmonically C4. Likewise, an y Cb gets the same octave num beras the C ju st above it; thu s Q?4 is enharm onically B 3. x x x i