A Theory of Voice-leading Sets for Post-tonal Music.
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1 Justin Lundberg SMT A Theory of Voice-leading Sets for Post-tonal Music justin.lundberg@necmusic.edu Voice-leading Set (vlset): an ordered series of transpositions or inversions that maps one pitchclass set onto another. Voice-leading Set-class (vlclass): a class of vlsets created through transpositional, inversional, and/or rotational equivalence. Figure 1: transformational voice leading B A B A Tt or I8 Figure 2: Vlset <442> spanning trichords {65e} and {t91} T2 e 1 T4 5 9 T4 6 t <442> Figure 3: Vlset <420>i spanning trichords {65e} and {t91} I0 e 1 I2 5 9 I4 6 t <420>i
2 Justin Lundberg SMT Figure 4: Drei Klavierstuecke Op. 11 no. 1, by Arnold Schoenberg, mm. 1-5 e 9 e 1 e e t t <tt8> <442> <tt0> <446> a b c d a c b d Figure 5: Drei Klavierstuecke Op. 11 no. 1, by Arnold Schoenberg, mm e e t e <113> <99e> <442> <335> g h i j e g f i
3 Justin Lundberg SMT Figure 6a: vlset <05> toroidal voice-leading space. 6b: <05> Moebius Strip.
4 Justin Lundberg SMT Table 1: dyadic voice-leading spaces. Vlset Sum Number of Pcsets Tn-class cycles Vlset Sum Number of Pcsets <01> <0e> e <2e> <1t> e 48 3 <3t> <29> e <49> <38> e <58> <47> e 48 3 <67> <56> e <05> <07> <14> <8e> <23> <9t> <6e> <16> <7t> <25> <89> <34> Tn-class cycles <02> <0t> t 36 2 <11> <ee> t <3e> <19> t 36 4 <4t> <28> t 12 6 <59> <37> t 36 4 <68> <46> t 36 2 <77> <55> t <03> <09> <12> <te> <4e> <18> <5t> <27> <69> <36> <78> <45> <04> <08> <13> <9e> <22> <tt> <5e> <17> <6t> <26> <79> <35> <88> <44> <00> <06> <1e> <15> <2t> <24> <39> <33> <48> <7e> <57> <8t> <66> <99>
5 Justin Lundberg SMT Figure 7: permutational reduction of trichordal orbifold.
6 Justin Lundberg SMT Table 2: trichordal voice-leading spaces. Vlset Inversional Pair Sum Number of Pcsets Tn-class spaces <001>, <445>, <889> <00e>, <443>, <887> 1/e <02e>, <346>, <78t> <01t>, <245>, <689> 1/e <03t>, <247>, <68e> <029>, <146>, <58t> 1/e <049>, <148>, <058> <038>, <047>, <48e> 1/e <067>, <4te>, <238> <056>, <49t>, <128> 1/e <11e>, <355>, <799> <1ee>, <335>, <779> 1/e <12t>, <256>, <69t> <2te>, <236>, <67t> 1/e <139>, <157>, <59e> <39e>, <137>, <57e> 1/e <166>, <5tt>, <229> <66e>, <3tt>, <227> 1/e <337>, <77e>, <3ee> <599>, <119>, <155> 1/e <005>, <449>, <188> <007>, <44e>, <388> 5/ <014>, <458>, <089> <08e>, <034>, <478> 5/ <023>, <467>, <8te> <09t>, <124>, <568> 5/ <06e>, <34t>, <278> <016>, <45t>, <289> 5/ <07t>, <24e>, <368> <025>, <469>, <18t> 5/ <113>, <557>, <99e> <9ee>, <133>, <577> 5/ <122>, <566>, <9tt> <tte>, <223>, <667> 5/ <15e>, <359>, <179> <17e>, <35e>, <379> 5/ <16t>, <25t>, <269> <26e>, <36t>, <27t> 5/ <33e>, <377>, <7ee> <199>, <115>, <559> 5/ <002>, <446>, <88t> <00t>, <244>, <688> 2/t <011>, <455>, <899> <0ee>, <334>, <778> 2/t <03e>, <347>, <78e> <019>, <145>, <589> 2/t <04t>, <248>, <068> <028>, <046>, <48t> 2/t <059>, <149>, <158> <037>, <47e>, <38e> 2/t <077>, <4ee>, <338> <055>, <499>, <118> 2/t <12e>, <356>, <79t> <1te>, <235>, <679> 2/t <13t>, <257>, <69e> <29e>, <136>, <57t> 2/t <167>, <5te>, <239> <56e>, <39t>, <127> 2/t <22t>, <266>, <6tt> <2tt>, <226>, <66t> 2/t 54 5 <004>, <448>, <088> <008>, <044>, <488> 4/ <013>, <457>, <89e> <09e>, <134>, <578> 4/ <022>, <466>, <8tt> <0tt>, <224>, <668> 4/ <05e>, <349>, <178> <017>, <45e>, <389> 4/ <06t>, <24t>, <268> <026>, <46t>, <28t> 4/ <079>, <14e>, <358> <035>, <479>, <18e> 4/ <112>, <556>, <99t> <tee>, <233>, <677> 4/ <15t>, <259>, <169> <27e>, <36e>, <37t> 4/ <23e>, <367>, <7te> <19t>, <125>, <569> 4/ <277>, <6ee>, <33t> <55t>, <299>, <116> 4/ <000>, <444>, <888> 0 1, 3 <01e>, <345>, <789> <02t>, <246>, <68t> <039>, <147>, <58e> 0 16, 48 4
7 Justin Lundberg SMT <048> 0 9 <057>, <49e>, <138> <066>, <4tt>, <228> 0 8, <11t>, <255>, <699> <2ee>, <336>, <77t> 0 16, 48 4 <129>, <156>, <59t> <237>, <67e>, <3te> <003>, <447>, <88e> <009>, <144>, <588> 3/9 64, <012>, <456>, <89t> <0te>, <234>, <678> 3/ <04e>, <348>, <078> <018>, <045>, <489> 3/ <05t>, <249>, <168> <027>, <46e>, <38t> 3/ <069>, <14t>, <258> <036>, <47t>, <28e> 3/9 64, <111>, <555>, <999> <eee>, <333>, <777> 3/9 4, 12 <13e>, <357>, <79e> <19e>, <135>, <579> 3/ <159> <37e> 3/9 36 <177>, <5ee>, <339> <55e>, <399>, <117> 3/9 16, <22e>, <366>, <7tt> <1tt>, <225>, <669> 3/9 64, <23t>, <267>, <6te> <29t>, <126>, <56t> 3/ <006>, <44t>, <288> 6 8, <015>, <459>, <189> <07e>, <34e>, <378> <024>, <468>, <08t> <033>, <477>, <8ee> <099>, <114>, <558> 6 32, 96 4 <123>, <567>, <9te> <16e>, <35t>, <279> <17t>, <25e>, <369> 6 32, 96 4 <222>, <666>, <ttt> 6 2, 6 <26t> 6 18 Figure 8: Vlclass [047] ordered tn-class space.
8 Justin Lundberg SMT Figure 9: Examples of complete ordered set-class spaces. a) vlclass [056] b) vlclass [025]
9 Justin Lundberg SMT c) vlclass [014] d) vlclass [013]
10 Justin Lundberg SMT Figure 10: Vlset <t16> in A; Webern Op. 5 n. 3, mm Figure 11: Vlclass [038] in B, mm
11 Justin Lundberg SMT Figure 12: Variations of A in mm Figure 13: A and B on the [047] ordered tn-class space.
12 Justin Lundberg SMT Selected Bibliography Callender, Clifton. Voice-Leading Parsimony in the Music of Alexander Scriabin. Journal of Music Theory 42, n. 2 (1998): , Ian Quinn, and Dmitri Tymoczko. Generalized Voice-leading Spaces. Science 320, (2008): Cohn, Richard. A Tetrahedral Graph of Tetrachordal Voice-Leading Space. Music Theory Online 9, n. 4 (2003). Available at Forte, Allen. Pitch-class Set Genera and the Origin of Modern Harmonic Species. Journal of Music Theory 32, n. 2 (1988): Gollin, Edward. Some Aspects of Three-Dimensional Tonnetze. Journal of Music Theory 42, n. 2 (1998): Multi-Aggregate Cycles and Multi-Aggregate Serial Techniques in the Music of Bela Bartok. Music Theory Spectrum 30, n. 1 (2008): Hall, Rachel. Geometrical Music Theory. Science 320 (2008): Haimo, Ethan. Atonality, Analysis, and the Intentional Fallacy. Music Theory Spectrum 18, n. 2 (1996): Headlam, David. Introduction. Theory and Practice 33, (2008): Hook, Julian. Uniform Triadic Transformations. Journal of Music Theory 46, nos. 1&2 (2002): Lewin, David. Notes on the Opening of the F# Minor Fugue from WTCI. Journal of Music Theory 26, n. 1 (1982): Some Ideas about Voice-Leading between PCSets. Journal of Music Theory 42, n. 1 (1998): Transformational Techniques in Atonal and Other Music Theories. Perspectives of New Music 21, nos. 1-2 ( ): Morris, Robert. Composition with Pitch-Classes. New Haven: Yale University Press, Voice-Leading Spaces. Music Theory Spectrum 20, n. 2 (1998): O Connell, Walter. Tone Spaces. Die Reihe 8 (English ed. 1968): Perle, George. Serial Composition and Atonality. 6 th ed. Berkeley: University of California Press, Twelve-Tone Tonality. 2 nd ed. Berkeley: University of California press, Quinn, Ian. Listening to Similarity Relations. Perspectives of New Music 39, n. 2 (2001): Roeder, John. A Theory of Voice Leading for Atonal Music. Ph.D. Diss. Yale University, A Geometric Representation of Pitch-Class Series. Perspectives of New Music 25, n. 1/2 (1987): Voice Leading as Transformation. In Musical Transformation and Musical Intuition: Essays in Honor of David Lewin. Edited by Raphael Atlas and Michael Cherlin. Roxbury, MA: Ovenbird Press, 1994: Straus, Joseph. Uniformity, Balance, and Smoothness in Atonal Voice Leading. Music Theory Spectrum 25, n. 2 (2003): Voice Leading in Atonal Music. Music Theory in Concept and Practice. Rochester, NY: University of Rochester Press, 1997: Voice Leading in Set-Class Space. Journal of Music Theory 49, n. 1 (2005): Tymoczko, Dmitri. The Geometry of Musical Chords. Science 313, n (2006): A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. New York: Oxford University Press, Wilson, Andrew. (2011) Voice Leading as Set. Unpublished manuscript.
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