Generalized Musical Intervals and Transformations

By David Lewin

David Lewin's Generalized Musical durations and modifications is well-known because the seminal paintings paving the way in which for present stories in mathematical and systematic techniques to song research. Lewin, one of many twentieth century's so much sought after figures in song idea, pushes the bounds of the learn of pitch-structure past its belief as a static procedure for classifying and inter-relating chords and units. recognized via so much tune theorists as "GMIT", the booklet is via some distance the main major contribution to the sphere of systematic song thought within the final half-century, producing the framework for the "transformational conception" flow. showing virtually two decades after GMIT's preliminary ebook, this Oxford college Press variation contains a formerly unpublished preface via David Lewin, in addition to a foreword via Edward Gollin contextualizing the work's value for the present box of tune theory.

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6. 2, along side the formulation of Theorem three. five. 2, tells us that any interval-reversing transformation needs to be a few inversion, in particular a few Vtcf. It continues to be to teach that any I* reverses durations. surroundings i = LABEL(v) and j = LABEL(u), we will write fifty eight intai(s), W)) = LABELS))-1 LABEL(I^(s)) = (iLABEL(t)~1j)-1(iLABEL(s)-1j) = j~1LABEL(t)i-1iLABEL(s)-1j = LABEL(s)~1LABEL(t), (3. 1. 2) (3. five. 2) Generalized period platforms (2) three. 6. four due to the fact that IVLS is commutative! And that = int(t,s) through three. 1. 2. q. e. d. three. 6. four THEOREM (optional): If IVLS is non-commutative, then there exists no interval-reversing transformation on S. facts: We think that Y is an interval-reversing transformation and arrive at a contradiction. For all s and all t, LABEL(Y(t))~1LABEL(Y(s)) = LABEL(s)-1 LABEL(t), as within the facts for Lemma three. 6. 2. Having the formulation of that lemma at our disposal now, we will alternative for the LABELs of Y(s) and Y(t) within the above equation, utilizing the exact period i of the lemma. We get the hot equation (i(LABEL(t)r1)-1 • iLABEL(s)-1 = LABEL(s)~1LABEL(t). Or: (LABELCOr^iLABELts)-1 = LABEL(s)~1LABEL(t). Or: LABEL(t)LABELCs)-1 = LABEL(s)~1LABEL(t). yet that equation, conserving for all s and t, says that IVLS is a commutative staff. And that contradicts the idea of the theory, q. e. d. fifty nine 4 Generalized period structures (3): A Non-Commutative GIS; a few Timbral GIS types through the dialogue of transpositions, interval-preserving operations, and inversions, the reader can have been questioned by means of the care with which noncommutative GIS constructions have been separated from commutative. finally, we now have thus far no longer encountered any specimen GIS that is non-commutative. Why should still we be troubled in any respect in regards to the non-commutative case? Why no longer store ourselves a few hassle, and simply stipulate within the definition of a GIS that the gang IVLS could be commutative? The paintings of the current bankruptcy will reply to those matters by means of exploring a musically major noncommutative GIS. i've got already awarded a few of the paintings somewhere else, however it could have a distinct impression within the current context. 1 four. 1. 1 DEFINITION: by means of a time span, we are going to comprehend an ordered pair (a, x), the place a is any genuine quantity and x is any optimistic genuine quantity. The pair of numbers is known to version our feel of place and extension a few musical occasion that "begins at time a" and "extends x devices of time" thereafter. The relations of all time spans may be denoted TMSPS. we have now already encountered one rhythmic GIS whose items have been sure time spans; that used to be instance three. three. 2. There, we limited the values for the numbers a to integers, and we constrained the values for the numbers x to definite proportions. utilizing an identical direct-product development, shall we additionally build a GIS for time spans within which the quantity a may suppose any rational worth, and the quantity x any confident rational worth. utilizing the direct- 60 1. David Lewin, "On Formal durations among Time-Spans," tune conception vol. 1, no. four (Summer 1984), 414-23.

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