A Theory of Voice-leading Sets for Post-tonal Music.

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Justin Lundberg SMT 2014 1 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

Justin Lundberg SMT 2014 2 Figure 4: Drei Klavierstuecke Op. 11 no. 1, by Arnold Schoenberg, mm. 1-5 e 9 e 1 e e 9 1 8 5 5 9 8 5 5 9 7 4 6 t 7 6 4 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. 6-13 8 5 0 8 6 8 8 0 4 e e 2 2 4 t e 6 7 3 9 0 6 9 3 <113> <99e> <442> <335> g h i j e g f i

Justin Lundberg SMT 2014 3 Figure 6a: vlset <05> toroidal voice-leading space. 6b: <05> Moebius Strip.

Justin Lundberg SMT 2014 4 Table 1: dyadic voice-leading spaces. Vlset Sum Number of Pcsets Tn-class cycles Vlset Sum Number of Pcsets <01> 1 144 1 <0e> e 144 1 <2e> 1 48 3 <1t> e 48 3 <3t> 1 144 1 <29> e 144 1 <49> 1 144 1 <38> e 144 1 <58> 1 48 3 <47> e 48 3 <67> 1 144 1 <56> e 144 1 <05> 5 144 1 <07> 7 144 1 <14> 5 48 3 <8e> 7 48 3 <23> 5 144 1 <9t> 7 144 1 <6e> 5 144 1 <16> 7 144 1 <7t> 5 48 3 <25> 7 48 3 <89> 5 144 1 <34> 7 144 1 Tn-class cycles <02> 2 36 2 <0t> t 36 2 <11> 2 12 12 <ee> t 12 12 <3e> 2 36 4 <19> t 36 4 <4t> 2 12 6 <28> t 12 6 <59> 2 36 4 <37> t 36 4 <68> 2 36 2 <46> t 36 2 <77> 2 12 12 <55> t 12 12 <03> 3 16 3 <09> 9 16 3 <12> 3 48 1 <te> 9 48 1 <4e> 3 48 1 <18> 9 48 1 <5t> 3 48 1 <27> 9 48 1 <69> 3 16 3 <36> 9 16 3 <78> 3 48 1 <45> 9 48 1 <04> 4 9 4 <08> 8 9 4 <13> 4 36 2 <9e> 8 36 2 <22> 4 6 12 <tt> 8 6 12 <5e> 4 12 6 <17> 8 12 6 <6t> 4 18 4 <26> 8 18 4 <79> 4 36 2 <35> 8 36 2 <88> 4 3 12 <44> 8 3 12 <00> 0 1 12 <06> 6 4 6 <1e> 0 12 2 <15> 6 12 4 <2t> 0 6 4 <24> 6 12 2 <39> 0 4 6 <33> 6 4 12 <48> 0 3 4 <7e> 6 12 4 <57> 0 12 2 <8t> 6 12 2 <66> 0 2 12 <99> 6 4 12

Justin Lundberg SMT 2014 5 Figure 7: permutational reduction of trichordal orbifold.

Justin Lundberg SMT 2014 6 Table 2: trichordal voice-leading spaces. Vlset Inversional Pair Sum Number of Pcsets Tn-class spaces <001>, <445>, <889> <00e>, <443>, <887> 1/e 1728 1 <02e>, <346>, <78t> <01t>, <245>, <689> 1/e 1728 1 <03t>, <247>, <68e> <029>, <146>, <58t> 1/e 1728 1 <049>, <148>, <058> <038>, <047>, <48e> 1/e 1728 1 <067>, <4te>, <238> <056>, <49t>, <128> 1/e 1728 1 <11e>, <355>, <799> <1ee>, <335>, <779> 1/e 432 2 <12t>, <256>, <69t> <2te>, <236>, <67t> 1/e 1728 1 <139>, <157>, <59e> <39e>, <137>, <57e> 1/e 432 2 <166>, <5tt>, <229> <66e>, <3tt>, <227> 1/e 1728 1 <337>, <77e>, <3ee> <599>, <119>, <155> 1/e 108 5 <005>, <449>, <188> <007>, <44e>, <388> 5/7 1728 1 <014>, <458>, <089> <08e>, <034>, <478> 5/7 1728 1 <023>, <467>, <8te> <09t>, <124>, <568> 5/7 1728 1 <06e>, <34t>, <278> <016>, <45t>, <289> 5/7 1728 1 <07t>, <24e>, <368> <025>, <469>, <18t> 5/7 1728 1 <113>, <557>, <99e> <9ee>, <133>, <577> 5/7 432 2 <122>, <566>, <9tt> <tte>, <223>, <667> 5/7 1728 1 <15e>, <359>, <179> <17e>, <35e>, <379> 5/7 432 2 <16t>, <25t>, <269> <26e>, <36t>, <27t> 5/7 1728 1 <33e>, <377>, <7ee> <199>, <115>, <559> 5/7 108 5 <002>, <446>, <88t> <00t>, <244>, <688> 2/t 216 2 <011>, <455>, <899> <0ee>, <334>, <778> 2/t 864 1 <03e>, <347>, <78e> <019>, <145>, <589> 2/t 864 1 <04t>, <248>, <068> <028>, <046>, <48t> 2/t 216 2 <059>, <149>, <158> <037>, <47e>, <38e> 2/t 864 1 <077>, <4ee>, <338> <055>, <499>, <118> 2/t 864 1 <12e>, <356>, <79t> <1te>, <235>, <679> 2/t 864 1 <13t>, <257>, <69e> <29e>, <136>, <57t> 2/t 864 1 <167>, <5te>, <239> <56e>, <39t>, <127> 2/t 864 1 <22t>, <266>, <6tt> <2tt>, <226>, <66t> 2/t 54 5 <004>, <448>, <088> <008>, <044>, <488> 4/8 27 5 <013>, <457>, <89e> <09e>, <134>, <578> 4/8 432 1 <022>, <466>, <8tt> <0tt>, <224>, <668> 4/8 108 2 <05e>, <349>, <178> <017>, <45e>, <389> 4/8 432 1 <06t>, <24t>, <268> <026>, <46t>, <28t> 4/8 108 2 <079>, <14e>, <358> <035>, <479>, <18e> 4/8 432 1 <112>, <556>, <99t> <tee>, <233>, <677> 4/8 432 1 <15t>, <259>, <169> <27e>, <36e>, <37t> 4/8 432 1 <23e>, <367>, <7te> <19t>, <125>, <569> 4/8 432 1 <277>, <6ee>, <33t> <55t>, <299>, <116> 4/8 432 1 <000>, <444>, <888> 0 1, 3 <01e>, <345>, <789> 0 144 3 <02t>, <246>, <68t> 0 36 6 <039>, <147>, <58e> 0 16, 48 4

Justin Lundberg SMT 2014 7 <048> 0 9 <057>, <49e>, <138> 0 144 3 <066>, <4tt>, <228> 0 8, 12 10 <11t>, <255>, <699> <2ee>, <336>, <77t> 0 16, 48 4 <129>, <156>, <59t> <237>, <67e>, <3te> 0 144 3 <003>, <447>, <88e> <009>, <144>, <588> 3/9 64, 192 4 <012>, <456>, <89t> <0te>, <234>, <678> 3/9 576 3 <04e>, <348>, <078> <018>, <045>, <489> 3/9 576 3 <05t>, <249>, <168> <027>, <46e>, <38t> 3/9 576 3 <069>, <14t>, <258> <036>, <47t>, <28e> 3/9 64, 192 4 <111>, <555>, <999> <eee>, <333>, <777> 3/9 4, 12 <13e>, <357>, <79e> <19e>, <135>, <579> 3/9 144 6 <159> <37e> 3/9 36 <177>, <5ee>, <339> <55e>, <399>, <117> 3/9 16, 48 10 <22e>, <366>, <7tt> <1tt>, <225>, <669> 3/9 64, 192 4 <23t>, <267>, <6te> <29t>, <126>, <56t> 3/9 576 3 <006>, <44t>, <288> 6 8, 24 10 <015>, <459>, <189> <07e>, <34e>, <378> 6 288 3 <024>, <468>, <08t> 6 72 6 <033>, <477>, <8ee> <099>, <114>, <558> 6 32, 96 4 <123>, <567>, <9te> 6 288 3 <16e>, <35t>, <279> 6 288 3 <17t>, <25e>, <369> 6 32, 96 4 <222>, <666>, <ttt> 6 2, 6 <26t> 6 18 Figure 8: Vlclass [047] ordered tn-class space.

Justin Lundberg SMT 2014 8 Figure 9: Examples of complete ordered set-class spaces. a) vlclass [056] b) vlclass [025]

Justin Lundberg SMT 2014 9 c) vlclass [014] d) vlclass [013]

Justin Lundberg SMT 2014 10 Figure 10: Vlset <t16> in A; Webern Op. 5 n. 3, mm. 1-4. Figure 11: Vlclass [038] in B, mm. 9-10.

Justin Lundberg SMT 2014 11 Figure 12: Variations of A in mm. 10-14 Figure 13: A and B on the [047] ordered tn-class space.

Justin Lundberg SMT 2014 12 Selected Bibliography Callender, Clifton. Voice-Leading Parsimony in the Music of Alexander Scriabin. Journal of Music Theory 42, n. 2 (1998): 219-233., Ian Quinn, and Dmitri Tymoczko. Generalized Voice-leading Spaces. Science 320, (2008): 346-348. Cohn, Richard. A Tetrahedral Graph of Tetrachordal Voice-Leading Space. Music Theory Online 9, n. 4 (2003). Available at http://societymusictheory.org/mto/. Forte, Allen. Pitch-class Set Genera and the Origin of Modern Harmonic Species. Journal of Music Theory 32, n. 2 (1988): 187-270. Gollin, Edward. Some Aspects of Three-Dimensional Tonnetze. Journal of Music Theory 42, n. 2 (1998): 195-206.. Multi-Aggregate Cycles and Multi-Aggregate Serial Techniques in the Music of Bela Bartok. Music Theory Spectrum 30, n. 1 (2008): 139-151. Hall, Rachel. Geometrical Music Theory. Science 320 (2008): 328-329. Haimo, Ethan. Atonality, Analysis, and the Intentional Fallacy. Music Theory Spectrum 18, n. 2 (1996): 167-199. Headlam, David. Introduction. Theory and Practice 33, (2008): 1-47. Hook, Julian. Uniform Triadic Transformations. Journal of Music Theory 46, nos. 1&2 (2002): 57-126. Lewin, David. Notes on the Opening of the F# Minor Fugue from WTCI. Journal of Music Theory 26, n. 1 (1982): 23-60.. Some Ideas about Voice-Leading between PCSets. Journal of Music Theory 42, n. 1 (1998): 15-72.. Transformational Techniques in Atonal and Other Music Theories. Perspectives of New Music 21, nos. 1-2 (1982-83): 312-71. Morris, Robert. Composition with Pitch-Classes. New Haven: Yale University Press, 1987.. Voice-Leading Spaces. Music Theory Spectrum 20, n. 2 (1998): 175-208. O Connell, Walter. Tone Spaces. Die Reihe 8 (English ed. 1968): 35-67. Perle, George. Serial Composition and Atonality. 6 th ed. Berkeley: University of California Press, 2001.. Twelve-Tone Tonality. 2 nd ed. Berkeley: University of California press, 1996. Quinn, Ian. Listening to Similarity Relations. Perspectives of New Music 39, n. 2 (2001): 108-158. Roeder, John. A Theory of Voice Leading for Atonal Music. Ph.D. Diss. Yale University, 1984.. A Geometric Representation of Pitch-Class Series. Perspectives of New Music 25, n. 1/2 (1987): 362-409.. 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: 41-58. Straus, Joseph. Uniformity, Balance, and Smoothness in Atonal Voice Leading. Music Theory Spectrum 25, n. 2 (2003): 305-352.. Voice Leading in Atonal Music. Music Theory in Concept and Practice. Rochester, NY: University of Rochester Press, 1997: 237-74.. Voice Leading in Set-Class Space. Journal of Music Theory 49, n. 1 (2005): 45-108. Tymoczko, Dmitri. The Geometry of Musical Chords. Science 313, n. 5783 (2006): 72-74.. A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. New York: Oxford University Press, 2011. Wilson, Andrew. (2011) Voice Leading as Set. Unpublished manuscript.

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