1 of 9 Volume 2, Number 2, Jue 214 Copyright 214 Society for Music Theory Total Voice Leadig Joseph N. Straus NOTE: The examples for the (text-oly) PDF versio of this item are available olie at: http://www.mtosmt.org/issues/mto.14.2.2/mto.14.2.2.straus.php KEYWORDS: Voice leadig, IFUNC, iterval multisets, traspositioal combiatio ABSTRACT: The total voice leadig betwee sets X ad Y is defied as the complete collectio of ordered pc itervals formed betwee the pcs i X ad the pcs i Y. These iterval multisets (imultisets) describe the soic sigature of the motio from X to Y, ad they always have the property of traspositioal combiatio. I some cases, the same total voice leadig may be produced by differet pairs of X ad Y (Coh 1988 refers to this as multiple paretage ). I such cases, the total voice leadig may remai the same eve though the geeratig chords differ. Received December 213 [1] People usually thik about voice leadig as i Example 1, with each ote i Chord X movig to a destiatio i Chord Y. (1) I propose to thik of it istead as i Example 2, takig ito accout all the possible destiatios i Chord Y of each ote i Chord X. A bit more formally, I defie the total voice leadig betwee sets X ad Y as the complete collectio of ordered pc itervals formed betwee the pcs i X ad the pcs i Y. (2) Istead of cocerig ourselves with the behavior of idividual toes, we will focus o the overall voice-leadig gesture, the soud evelope, the soic eviromet created by the motio from X to Y. [2] Amog the itervals formed betwee two sets i ay particular cotext, some will be more marked for attetio tha others, but all are at least potetially i play. The less marked oes create a imbus withi which the more marked oes are heard. Take together, they create a total soic package i which all itervals play a role. It is the same as i hearig the soic idetity of a sigle chord oe may be more aware of some itervals tha others, but it is the total package of itervals that characterizes the chord. Ideed, the iterval cotet of Chord X is equivalet to the total voice leadig from Chord X to itself. I that sese, the iterval cotet of a harmoy is a special case of total voice leadig X Y, where X=Y. Total voice leadigs are thus roughly equivalet to the iterval cotet of a sigle chord, geeralized ad horizotalized, with some of the same coceptual ad perceptual advatages ad challeges. [3] It is easy to trump up a cotext i which oe is ivited ad almost compelled to atted to all of the voice-leadig itervals simultaeously. I Example 3, for istace, oe simultaeously hears the itervals formed withi each istrumetal part (flute, violi ad trumpet), withi each register (high, middle, ad low), ad amog the similarly articulated otes (sustaied whole otes, tremolo half otes, ad staccato eighth otes). But eve i a much simpler, more covetioal progressio like the oe i Example 4, most of the itervals that comprise the total voice leadig are ear the forefrot of musical cosciousess, or ca easily be summoed there. [4] If x is the cardiality of X ad y is the cardiality of Y, the umber of itervals i the total voice leadig betwee X ad Y will be xy. I Example 2, for istace, there are ie arrows coectig the three pcs i X with the three pcs i Y. Similarly, the total voice leadig betwee two tetrachords would be 4 x 4 = 16. If the perceptual task is uderstood to be the direct perceptio of all sixtee itervals simultaeously, this might seem a dautig propositio. But, at least i priciple, the task is o differet from recogizig the idetity of a particular hexachord, ad it s ot all that differet i quatity either. A
2 of 9 hexachord cotais fiftee itervals, ad recogizig it would seem to deped more o the perceptio of the total package of itervals tha o the discrete discermet of each oe idividually. The same is true of the total voice leadig betwee two tetrachords, or betwee two sets of whatever size. I ay case, total voice leadig, like total iterval cotet, is probably best thought of as a uderlyig potetial i the relatioship betwee two sets (which may be the same set), with local cotextual factors ievitably emphasizig some itervals more tha others, ad probably doig so i differet ways. IFUNC ad imultisets [5] My cocept of total voice leadig represets a slight reorietatio of David Lewi s IFUNC(X, Y)(i), which tells us i how may differet ways the iterval i ca be spaed betwee (members of) X ad (members of) Y (Lewi 1987, 88). (3) As i rages over the twelve possible ordered pc itervals ( through 11), we ca gather the values ito a twelve-place IFUNC-vector (see Example 5). The first etry i the IFUNC-vector idicates the umber of times the ordered pc-iterval is foud betwee X ad Y; the secod etry idicates the umber of times the ordered pc-iterval 1 is foud betwee X ad Y; ad so o. [6] The cotets of the IFUNC-vector ca be arraged as a set (or multiset) of itervals, which I call a imultiset. A imultiset is the collectio of ordered pc itervals spaed betwee X ad Y, that is, the total voice leadig betwee X ad Y. Just as a pcset or pcmultiset cotais pcs, a imultiset cotais ordered pc itervals. These imultisets ca be treated ad uderstood somewhat i the maer of pcsets or pcmultisets. (4) [7] Like pcsets ad pcmultisets, imultisets ca be placed i a usable ormal form, followig a stadard algorithm. Whe there is more tha oe occurrece of oe or more itervals i a imultiset, I follow the procedure described i Robiso (29): elimiate the doubligs; put the simple set ito ormal form i the usual way; reistate the doubligs. Equivalece relatios (imultiset ad imultiset-class) [8] Order permutatio ad imultiset iversio. If the order of X ad Y is reversed (that is, if we are cocered with the voice leadig Y X istead of X Y), each iterval i the imultiset will be replaced with its complemet mod 12. I other words, reversig the order of the geeratig pcsets has the effect of ivertig the imultiset formed betwee them. (5) Two imultisets related by iversio will be said to be I-equivalet members of a shared imultiset-class. Imultiset prime forms will be the member of the imultiset class with the lowest umerical value (see Example 6). (6) The idea that the total voice leadig from X to Y ad the total voice leadig from Y to X could be uderstood as equivalet, to withi iversio, does ot seem to me to pose ay particular coceptual or perceptual problem. Each iterval i oe total voice leadig is simply replaced by its complemet mod 12 (i.e., its iversio) i the other. What wet up ow goes dow, ad vice versa. [9] Traspositio. Whe Set Y is trasposed by T with respect to Set X, the imultiset is also trasposed at T. (7) Two imultisets related by traspositio, will be said to be T-equivalet members of a shared imultiset-class (see Example 7). I will desigate the traspositio of a imultiset as <T >, to be read as hyper- T. <T > is a traspositio of traspositios or, to put it slightly differetly, a iterval betwee itervals. (8) The possibility of <T > arises whe we are comparig two imultisets (i.e., collectios of itervals). If each iterval i the secod imultiset is semitoes larger tha a correspodig iterval i the first multiset, the the two imultisets are related at <T >. [1] I Example 8, each iterval i the secod imultiset is 3 semitoes larger tha the correspodig iterval i the first imultiset (iterval 4 becomes iterval 7, iterval 1 becomes iterval 1, ad so o). The two imultisets are thus related at <T3>. We could also talk i the same way about hyper-hyper-t <<T>> that would be the iterval betwee the itervals betwee itervals. [11] Compared to iversio, the traspositio of a imultiset poses serious coceptual ad perceptual issues. It requires oe to atted to the shape of the total voice leadig rather tha to the qualities of the itervals. For example, if the total voice leadig from X to Y has three 5s ad zero 6s, the total voice leadig from X to T (Y) will have three 6s ad zero 7s: 1 everythig has shifted oe place over. The itervallic profile (the itervals amog the itervals) will stay the same, but the itervals themselves will chage. With referece to Example 7, oe might liste for <T > by attedig to the repeated 3 itervals i the first voice leadig: two 6 s (G D ad A E) ad two 8 s (G E ad B G), differig i size by 2. I the secod voice leadig, we fid two 9 s (A G ad G F) ad two 11 s (G G ad B A ), agai differig i size by 2. The actual iterval sizes have chaged, but a distictive feature of the total voice leadig repeated itervals that differ i size by 2 remais recogizably the same. The optimal ear-traiig strategy would thus ivolve attedig to the distictive features of the imultiset that mark its cotour or shape. (9) [12] Iversio. The iversio of pcset Y with respect to X has o cosistet effect o IFUNC(X, Y), or o the total voice leadig (imultiset) betwee X ad Y. (1) However, ivertig both X ad Y about the same axis does have a predictable effect, ad it is the same effect as reversig the order of X ad Y, amely ivertig the imultiset at <I > (see Example 9). [13] I will desigate the iversio of a imultiset as <I >, to be read as hyper-i. <I > is a way of thikig about the axis of iversio aroud which two itervals might be heard to balace. Iterval 3 ad iterval 7, for example, could be thought
3 of 9 of as lyig equidistat o either side of iterval 5, which lies halfway betwee them. I that sese, iterval 3 ad iterval 7 are related at <I 1 >, ad their axis of iversio is 1/2 = 5. I comparig two imultisets, if each iterval i oe imultiset is as far below some midpoit as each correspodig iterval i the other imultiset is above it, the the two imultisets are related at <I > (see Example 1). We could talk i the same way about hyper-hyper I <<I>> that would idicate the poit of iversioal balace betwee two poits of iversioal balace. (11) [14] Imultiset-class ad prime form. If two imultisets are related by either traspositio or iversio, they may be cosidered members of the same imultiset-class. All the members of a imultiset-class, like all the members of a pcset-class, have the same total iterval cotet. However, whe speakig of imultiset-class, we are ow talkig about the ivariace of the itervals betwee itervals. A imultiset-class cotais total voice leadigs that are related by traspositio or iversio ad that thus share a particular shape or arragemet or dispersio of itervals. As oted above, imultiset-classes ca be idetified with a represetative (prime) form. Iversioal Symmetry (IS) [15] X ad Y are pc idetical. Whe X ad Y are pc idetical (i.e., related at T ), the imultiset that results from the total voice leadig betwee them will correspod to the traditioal twelve-place iterval vector for scx (ad scy) (what I referred to above as a IFUNC-vector): the umber of s i the vector will correspod to the cardiality of the set, ad each o-zero iterval will be balaced by its complemet mod 12. The voice-leadig imultiset is thus iversioally symmetrical (see Example 11). I a iversioally symmetrical imultiset, the itervals balace above ad below a cetral pair (iversioal axis) of tritoe-related itervals. Whe pcset X ad pcset Y are idetical, the itervals i the imultiset are distributed equally aroud itervals ad 6 (for every iterval, there is aother iterval -). The imultiset is thus I-symmetrical at <I >. [16] X ad Y are related by traspositio. If X ad Y are members of the same T -type (i.e., related by T ), the imultiset that results from the voice leadig betwee them will be iversioally symmetrical at <I > (see Example 12). Whe the 2 pcsets are related at T, the imultiset is self-iversioal at <I >, as just oted. Whe the pcsets are related at T (as i the 2 left side of Example 12), the imultiset is self-iversioal at <I > = <I >. I such a imultiset, the itervals are distributed 2 2 4 equally aroud itervals 2 ad 8 (for every iterval, there is aother iterval 4-). [17] If we reverse the order of X ad Y, the imultisets of X Y ad Y X will be related by iversio at <I >, as oted above. But because X ad Y i this case are related by traspositio, ad their imultiset is thus self-iversioal, the imultisets of X Y ad Y X will also be related by traspositio. [18] X ad Y are related by iversio. If X ad Y are related by iversio, however, the voice-leadig imultiset is geerally ot iversioally symmetrical (see Example 13). [19] X ad Y are both iversioally symmetrical. If both X ad Y are iversioally symmetrical pcsets, the the imultiset that results from the total voice leadig betwee them will also be iversioally symmetrical (see Example 14). I that sese, there is somethig i commo betwee two sets related by traspositio ad two sets that are pc-symmetrical: both produce total voice leadigs that are iversioally symmetrical. (12) Traspositioal Combiatio (TC) [2] All imultisets produced as the total voice leadig betwee two pcsets will have the TC-property. Accordig to Coh, Ay pitch- or pc-set has the TC-property if it may be disuited ito two or more traspositioally related subsets (1988, 23). A set (or set class) with the TC-property may be uderstood as geerated by the traspositio of oe of its operads to the traspositioal levels of the other: TC is a biary operatio which takes as its operads two set-classes, ad adds the value of each elemet i the prime form of the first operad to that of each elemet i the prime form of the secod operad. The result is a larger set which bears the TC-property (27 28). (13) [21] I the case of the imultisets uder discussio here, the operads are the geeratig pcsets, X ad Y. But because of the differece betwee TC (with X trasposed to the level of the pcs i Y) ad total voice leadig (where the otes i X move to the otes i Y), the imultiset will be equivalet to I(X)*Y rather tha X*Y. Noetheless, because I(X) has the same iterval cotet as X, it is still true to say that the total voice leadig from X to Y bears the imprit of the geeratig sets, ad that X ad Y are always embedded i the total voice leadig they joitly geerate. [22] Whe X ad Y are trichords, for example, the ie-elemet imultiset will embed three I(X)-type trichords (related by traspositio by the itervals i Y) ad three Y-type trichords (related by traspositio by the itervals i I(X)) (see Example 15). The TC property ca be easier to visualize if the imultiset is writte as a matrix. (14) The colums are all the T -type of Y ad the rows are all the T-type of I(X). For the first pair of pcsets i Example 15 ([G, A, B] [D, E, G]), the colums of the matrix cotai forms of (25) related at T, T, ad T while the rows of the matrix cotai forms of (13) related at T, T, ad T. (15) 1 3 I will use this matrix otatio occasioally, whe the TC property of a imultiset is at 2 5 issue. [23] The I(X)-type trichords are the subsets of the total voice leadig that have arrows from all three of the pcs of X to the
4 of 9 same sigle pc of Y (these are foud i the rows of the array). The Y-type trichords are the subsets of the total voice leadig that have arrows from oe sigle pc of X to all three of the pcs of Y (these are foud i the colums of the array). Example 16 demostrates with referece to the first pair of pcsets i Example 15. [24] If both X ad Y are iversioally symmetrical, as i the third pair of chords i Example 15 (24 ad 27), I(X)*Y ad X*Y are equivalet. Example 17 examies the effect of TC o the voice-leadig imultiset betwee two pairs of iversioally symmetrical pcsets. If X is a member of sc(3) ad Y is a member of sc(4), the total voice leadig betwee them will be the imultiset-class i(347), which is equivalet to (3)*(4). If X is a member of sc(5) ad Y is a member of sc(24), the total voice leadig betwee them will be the imultiset-class i(24579), which is equivalet to (5)*(24). [25] Multiple paretage. Just as pcsets ad pcsc may have the TC-property i more tha oe way, imultisets ca be produced by differet pairs of geeratig sets. (16) Iterestigly, the multiple paretage of imultiset i(2468t) plays a cetral role i Lewi s (Lewi 1987) descriptio of IFUNC. Example 18 offers a recofiguratio of Lewi s Figure 5.1, with some additioal iformatio added. [26] As Lewi shows, there are three differet ways to produce the same total voice leadig. He does t explore this aspect of the situatio, but I thik the clear implicatio is that the larger gesture here is substatially ad audibly the same i all three cases, despite the differeces i the geeratig sets. This is the place where the preset cocept of total voice leadig ca make a particularly valuable cotributio: showig that disparate scs may be coected by effectively the same voice leadig. X ad Y are both dyads (2x2) [27] Whe X ad Y are both dyads, their total voice leadig ca be represeted by a four-elemet imultiset. Because dyads are iversioally symmetrical, the voice-leadig imultiset betwee ay pair of dyads will always be I-symmetrical. All of the imultisets have the TC property X*Y, where X ad Y are the geeratig dyads. The chart i Example 19 idetifies the imultiset-classes resultig from the total voice leadig betwee ay two two-elemet pcset-classes. (17) [28] Amog the four-elemet imultisets produced by the voice leadig from a dyad to aother dyad (2 x 2), two have multiple paretage (see Example 2). Example 21 provides a simple schematic realizatio of these two multiply pareted total voice leadigs. Musical examples with dyads [29] We look ow at three short passages to get a sese of how total voice leadig (icludig imultisets, hyper-t, ad multiple paretage) might play out i a musical cotext. I Example 22, the opeig measures of Dimiished Fifth, from Bartók s Mikrokosmos, the multiple paretage of i(167) is at issue. Five simultaeities are idetified i Example 22a (the fourth of these is a theoretical recostructio A ad A belog together, but the A has moved o by the time A arrives). Because there is a alteratio of (6) ad (1), the total voice leadig i every case is i(167), represeted by either i[56e] or i[167]. [3] I Example 22b, the harmoic tritoes are heard i relatio to the melodic perfect fourths. Oce agai, the total voice leadig is i(167) represeted by i[56e] ad i[167]. To put it i a slightly differet way, the melodic fourths bear the same relatio to the harmoic tritoes that the harmoic semitoes do. The hyper-traspositios of the imultisets reflect the pricipal traspositios (i.e., itervals) amog the pcs, amely 1 ad 5. That is, the voice-leadig itervals (mostly 1s ad 5s) are recursively recalled i the shift of the total voice leadig (hyper-t) by 1 or 5. I this case, <T 1 > is equivalet to <T -5 > (or <T 7 >). The choice of label is iteded to emphasize the recursive relatioship betwee the itervals ad the itervalsbetwee-the-itervals. [31] There are obviously sigificat perceptual issues i hearig total voice leadig, icludig multiple paretage ad especially recursio. Let us begi to address these issues by focusig o the first progressio i Example 22a (E -A to E -D) i compariso with the first progressio i Example 22b (E -A to A-D). I both cases, the total voice leadig is i[56e], but produced i differet ways. I the first progressio, the move from A to D (5) is extremely promiet, filled i with passig toes, ad it is equally easy to hear the E as sustaied (). Hearig E as goig to D (11) is also ot much of a challege: the lowest ote is draw upward to the highest ote. The coectio betwee A ad E (6) is also perfectly audible, although i this case oe hears more a associatio of two toes tha ay sese of directed motio from oe to the other: it is more a case of hearig with tha of goig to. Thus all the itervals i this particular total voice leadig are fully preset to ay willig listeer, although the itesity of the presece, ad the ature of the associatio, ecessarily varies. [32] I the first progressio of Example 22b, it is easy to hear E goig to A (6) ad A goig to D as i Example 22a (5). E goes to D, the lowest ote draw up to the highest, as i Example 22a (11). Hearig the A as retaied () is perhaps a bit more of a stretch, but its recurrece o the secod beat of the followig measure reiforces ay attempt to do so. As i the first progressio of Example 22a, the four itervals that comprise the total voice leadig are all fully accessible to ay willig listeer, but the itesity of their presece varies: some are experieced as simple, directed motios; others are experieced more as associatios that provide a cotext for these simpler motios. A bit of extra effort is required to brig all four itervals to cosciousess, but the effort is rewarded by the ability to hear oe total voice leadig, [56e] i this case,
5 of 9 produced by differet dyads. Such multiple paretage becomes a distict ad uifyig feature of the passage. [33] Stravisky s Epitaphium begis with a duet for flute ad clariet (Example 23). The harmoic dyads are all members of ic1, ic2, ad ic3, ad I wish to make three observatios about their successio from the poit of view of their total voice leadig. First, the progressio from ➂ to ➃ (C -E to C -C) ad the progressio from ➅ to ➆ (E -F to E -D) have the same total voice leadig, show i the example as <T >. Secod, the relatioship that ➄ ➅ bears to ➃ ➄ is the same as the relatioship that ➇ ➈ bears to ➄ ➅, amely <T >. I both cases, the itervallic profile of the total voice leadig has 7 shifted over to the same degree: each iterval i the secod imultiset is 7 semitoes larger tha i the first imultiset. Third, the relatioship that ➁ ➂ bears to ➀ ➁ is the same as the relatioship that ➇ ➈ bears to ➅ ➇, amely <I >. This is true 3 eve though the sizes of the dyads are differet i the two cases. Hyper-relatios thus permit compariso of progressios amog pc sets of differet types. [34] The perceptual challeges ivolved with hearig recursive relatioships are worth explorig i a bit more detail. We focus o a specific istace, amely the <T > produced by the compariso betwee the progressios of ➃ ➄ ad ➄ ➅. I 7 the first of those progressios (C -C to B-C), the total voice leadig is i[tee]. C is retaied as a commo toe () while C moves to B (1). (C is followed by C i attack order (11) ad C is the followed by B (11). We ca thik of the total voice leadig as ivolvig two itervals of the same size (11), produced by attack order, flaked by oe slightly smaller iterval (1) ad oe slightly larger iterval (). I the followig progressio (B-C to E -F ), the total voice leadig has the same cotour: two itervals of the same size (6), produced by the motio withi each istrumetal part (C-F ad B-E ), flaked by oe slightly smaller iterval (5 ad 7, both produced by attack order). [35] To hear both of these total voice leadigs, ad to hear that they share the same distributio of itervals, strikes me as a perfectly attaiable goal of focused attetio. Recogizig that this distributio has bee shifted up seve semitoes <T > is 7 obviously a bit harder to produce as a immediate feature of aural experiece. Eve more problematic is the otio that the 7 i <T >, which measures the iterval betwee the itervals, is i some sese the same as the 7 i, for example, T that 7 7 associates B with F. I thik it is possible to make this coectio, both coceptually ad experietially (I base this assertio o quiet, sustaied itrospectio), ad that effort is worth it, because the recursive relatioship adds a erichig ad uifyig dimesio to my hearig of the passage. But decidig whether the rewards are worth the effort takes us ito the realm of persoal prefereces, which may vary a great deal. [36] I the opeig of Varèse s Desity 21.5 (Example 24), the first phrase (measures 1 2) approaches its cocludig tritoe (C -G) from a perfect fourth (C -F ), via the total voice leadig i[167]. I the secod phrase (measures 3 5), the same tritoe is approached from a mior secod (F -G), via the same total voice leadig, i[167]. To put it i a slightly differet way, approachig or leavig a tritoe from a perfect fourth or semitoe with which it shares a toe will produce the same total voice leadig. I the music that follows, the same total voice leadig bids together all of the melodic dyads. [37] As oted earlier, the extet to which the four itervals of these total voice leadigs are directly experieced aurally ecessarily varies. I the first progressio (C -F to C -G), F -G (1) ad C -C () are the registral lies, while F -C (7) ad C -G (6) are produced by the attack order. Oe might imagie that 1 ad are somethig aki to a more traditioal idea of voice leadig while 6 ad 7 provide a cotextual peumbra for them. The itervals are ot all equally vivid or fuctioally the same, but they all participate i creatig the color of the progressio. I the secod progressio (F -G to C -G), the situatio is slightly differet: ow ad 7 comprise the registral lies (G-G ad F -C ) while 1 ad 6 result from attack order (G-C ad F -G). But that the same total voice leadig is ivolved i both cases, amely i[167], ad that it results from two differet types of dyad pairs, amely (5)*(6) ad (1)*(6), strike me as a experietially immediate ad erichig way of hearig the passage. [38] Whe X is a dyad ad Y is a trichord (or vice versa), their total voice leadig ca be represeted as a six-elemet imultiset (see Example 25). Amog these six-elemet imultisets, seve have multiple paretage, ad the umber of differet geeratig pairs rages from two to four (see Example 26). [39] Example 27 cotais a chord progressio that exploits the multiple paretage of i(13679), which results from the combiatio of a tritoe (6) with oe of four possible trichords: (13), (14), (25), (37). (18) Despite the variety of the chords, each adjacet pair of chords expresses the same total voice leadig. There are also iterestig patters of recurrece amog the hyper-t ad hyper-i relatios. Musical examples with dyads ad trichords [4] The total voice leadig explored abstractly i Example 27, ivolvig the multiple paretage of i(13679), is realized throughout the opeig measures of Crawford s Diaphoic Suite No. 1, secod movemet (see Example 28). The melodic lie is frequetly puctuated with tritoes, ad these are typically followed by or overlapped with (13), (14), or (37) to produce a cosistet total voice leadig amid the harmoic diversity. If we look more closely at the first phrase of the melody (delimited by the double bar at the ed of the fourth measure), we ca hear it as two related gestures i which a sigle tritoe (E -A or A-(E ) is followed by two differet trichords (C-C -E ad (E -G -F). Withi these two pairs of chords, the secod
6 of 9 total voice leadig is related to the first at <T -1 >. I other words, all of the itervals i the first total voice leadig have shruk by semitoe. That larger gesture is reflected at the melodic surface by the descedig semitoe, G -F, which cocludes the phrase. Somethig similar happes i the varied repeat of the opeig music, begiig i measure 8. There, the two voice leadig gestures are related at <T +3 >, which is reflected i the ascedig mior thirds that coclude each of the costituet trichords. [41] Somethig similar happes i the first two phrases of the first movemet of the same work (see Example 29). I the first phrase, combiatios of (12) ad (3) produce a total voice leadig represeted by i(12345). I the secod phrase, combiatios of a differet pair of scs, (1) ad (24), produce the same total voice leadig. As before, the aalysis offers a way of hearig a sigle class of total voice leadig, i(12345), as remaiig the same eve as the harmoies that produce it vary. The itervals ivolved are ot all experieced with the same degree of immediacy some are registral lies, some are produced by successive attacks, ad some are more associatios tha directed motios, providig a cotext or atmosphere for the others. But take together, these itervals comprise a sigle voice-leadig gesture that uifies ad eriches a hearig of the passage. X ad Y are both trichords (3x3) [42] Whe X ad Y are both trichords, their total voice leadig ca be represeted as a ie-elemet imultiset (see Example 3). Five of these imultisets have multiple paretage (see Example 31). Musical examples with trichords [43] I three passages of music by Schoeberg, from the Book of the Hagig Gardes ad Pierrot Luaire, the augmeted triad plays a promiet role withi a prevailig trichordal texture (see Examples 32, 33, ad 34). Approachig ad leavig a augmeted triad from these particular trichords ivolves the same total voice leadig. It might be possible to make a recursive argumet about these passages that the hyper-t ivolved reflect aspects of the itervallic surface of the music but they would seem a bit forced to me. I would prefer to coclude with the observatio that the augmeted triad i these passages fuctios as somethig like a uiversal solvet it ca be approached from differet places ad left i differet directios, but the total voice leadig remais the same, impartig a sese of gestural uiformity amid the harmoic diversity. Joseph N. Straus Graduate Ceter, City Uiversity of New York 365 Fifth Aveue New York, NY 116 jstraus@gc.cuy.edu Works Cited Buchler, Michael. 27. Recosiderig Klumpehouwer Networks. Music Theory Olie 13, o. 2. Calleder, Clifto, Ia Qui, ad Dmitri Tymoczko. 28. Geeralized Voice Leadig Spaces. Sciece 32: 346 48. Coh, Richard. 1986. Traspositioal Combiatio i Twetieth-Cetury Music. PhD diss., Eastma School of Music.. 1988. Iversioal Symmetry ad Traspositioal Combiatio i Bartók. Music Theory Spectrum 1: 19 42.. 1991. Properties ad Geerability of Traspositioally Ivariat Sets. Joural of Music Theory 35, o. 1 2: 1 32. Klumpehouwer, Hery. 1991. A Geeralized Model of Voice-Leadig for Atoal Music. PhD diss., Harvard Uiversity.. 27. Recosiderig Klumpehouwer Networks: a Respose. Music Theory Olie 13, o. 3. Lewi, David. 1959. Re: Itervallic Relatios Betwee Two Collectios of Notes. Joural of Music Theory 4, o. 1: 298 31.. 1977a. Forte s Iterval Vector, My Iterval Fuctio, ad Regeer s Commo-Note Fuctio. Joural of Music Theory 21, o. 2: 194 237.. 1977b. A Label-Free Developmet for 12-Pitch-Class Systems. Joural of Music Theory 21, o. 1: 29 48.. 1987. Geeralized Musical Itervals ad Trasformatios. New Have: Yale Uiversity Press.. 199. Klumpehouwer Networks ad Some Isographies That Ivolve Them. Music Theory Spectrum 12, o. 1:
7 of 9 83 12.. 1998. Some Ideas About Voice-Leadig Betwee Pcsets. Joural of Music Theory 42, o. 1: 15 72. Ludberg, Justi. 212. A Theory of Voice-leadig Sets for Post-toal Music. PhD diss., Eastma School of Music. Morris, Robert. 1998. Voice-Leadig Spaces. Music Theory Spectrum 2, o. 2 (1998): 175 28. Robiso, Thomas. 29. Pitch-Class Multisets. PhD diss., Graduate Ceter of the City Uiversity of New York. Roeder, Joh. 1985. A Theory of Voice Leadig for Atoal Music. PhD diss., Yale Uiversity.. 1994. Voice Leadig as Trasformatio. I Musical Trasformatio ad Musical Ituitios: Essays i Hoor of David Lewi, ed. Raphael Atlas ad Michael Cherli, 41 58. Roxbury, MA: Ovebird Press. Straus, Joseph. 1997. Voice Leadig i Atoal Music. I Music Theory i Cocept ad Practice, ed. James Baker, David Beach, ad Joatha Berard, 237 74. Rochester: Uiversity of Rochester Press, 1997).. 23. Uiformity, Balace, ad Smoothess i Atoal Voice Leadig. Music Theory Spectrum 25, o. 2: 35 52.. 25. Voice Leadig i Set-Class Space. Joural of Music Theory 49, o. 1: 45 18. Tymoczko, Dmitri. 211. A Geometry of Music: Harmoy ad Couterpoit i the Exteded Commo Practice. New York: Oxford Uiversity Press. Wilso, Adrew. 212. Voice Leadig as Set. Paper preseted at the aual meetig of the Music Theory Society of the Mid-Atlatic. Foototes 1. There has bee a upsurge of scholarly iterest i atoal voice leadig i recet years, icludig Roeder 1985; Klumpehouwer 1991; Roeder 1994; Straus 1997; Lewi 1998; Morris 1998; Straus 23; Straus 25; Calleder, Qui, ad Tymoczko, 28; ad Tymoczko 211. All of these sources, i oe way or aother, take the view of voice leadig eshried i Example 1. I am grateful to Julia Hook, Philip Lambert, Drew Nobile, Richard Coh, ad two aoymous reviewers for this joural for valuable commets o earlier drafts of this article. Some of the iitial impetus for this study, ad all of the extesive computatioal work associated with it are attributable to Will Orzo. 2. I take the phrase ad the cocept of total voice leadig from a brief discussio i Morris 1998: Give two pcsets A ad B, the total voice-leadig from A to B icludes ay ad all moves from ay pcs of A to ay pcs of B that is, all the ways oe ca associate the pcs of A with those of B i as may voices as ecessary or desired (178). Morris is primarily iterested i restrictig the total voice leadig by parsig it ito the more usual sorts of voices (as i Example 1). The preset study, i cotrast, focuses o the total package of itervals formed betwee pcsets A ad B, without restrictios ad without traditioal parsig. 3. Lewi 1959 itroduces the cocept of iterval fuctio ad Lewi 1977a explores it i cosiderable detail. I a particularly revealig passage, ad oe that directly egages the topic of the curret study, Lewi calls attetio to the cotraputal ature of his coceptio he is cocered about the itervals formed betwee two harmoies rather tha those formed withi a sigle harmoy: Forte s iterval vector is a rigorously harmoic cocept. My iterval fuctio, from which I derive my iterval cotet is as rigorously cotraputal i coceptio. Give pcsets X ad Y, ad a iterval i, INTF(X, Y, i) is the umber of distict ways i which the iterval i ca be spaed from a member of X to a member of Y; that is, the umber of distict pairs of pcs (x, y) such that x is a member of X, y is a member of Y, ad the iterval from x to y is i. The cocept is a useful model for the followig situatio. Imagie yourself improvisig at the keyboard, usig oly otes of X i your left had ad oly otes of Y i your right had. How may differet ways ca a iterval of i occur i the resultig couterpoit? I this coectio, oe eeds all the itervals from through 11, sice X ad Y may have commo otes, the umber of possible cotraputal 7 s may be differet from the umber of cotraputal 5 s, etc. (1977a, 21 2 ). Similarly, Lewi 1977a refers to the potetial for IFUNC to reflect the total potetial couterpoit betwee two harmoies: The iterval fuctio is a useful tool for coceptualizig such musical pheomea as the total potetial couterpoit betwee a hexachord X ad its complemet X (23). I the same spirit, Lewi 1977b speaks of the way i which his iterval fuctio ca reflect the overall soud of the couterpoit betwee two pitch-class sets (4). 4. Ludberg 212 ad Wilso 212 idepedetly develop the cocept of what they both call a voice-leadig set or vlset. This is a set of itervals formed betwee two pcsets ad, like pcsets, a vlset may be uderstood as part of a larger class of
8 of 9 sets related by traspositio or iversio. For both Ludberg ad Wilso, voice leadig is uderstood i the traditioal maer of my Example 1, ot i terms of the total voice leadigs discussed here. 5. Compare Lewi s (1987) Theorem 5.1.4, which asserts: IFUNC(Y, X)(i) = IFUNC(X, Y)(i -1 ). I other words, however may occurreces of some iterval i there are i the first IFUNC-vector, there will be exactly that umber of occurreces of the complemet of i i the secod IFUNC. I this sese, IFUNC(Y, X) ad IFUNC(X, Y) are iversios of each other. 6. This differs from the procedure i Robiso 29: elimiate doubligs; put the set i prime form followig the stadard algorithm; replace the doubligs. 7. Compare Lewi s (1987) Theorem 5.1.6A ad 5.1.6B which assert: IFUNC(T (X), Y)(i) = IFUNC(X, Y)(i) ad IFUNC(X, T (Y))(i) = IFUNC(X, Y)(i -1 ). I other words, whe X is trasposed by semitoes with respect to Y (or vice versa), the IFUNC-vector will rotate clockwise or couterclockwise by places. 8. My cocept of hyper-t ad hyper-i (to be discussed below) are obviously derived from ad related to the use of these terms ad cocepts i relatio to Klumpehouwer etworks see Klumpehouwer 1991 ad Lewi 199. I both cases, we are talkig about the traspositio of traspositios (itervals) or the iversio of iversioal axes (sums). For more recet assessmets of hyper-t ad hyper-i i the cotext of Klumpehouwer etworks, see Buchler 27 ad the umerous resposes i Music Theory Olie 13/3 (27), especially Klumpehouwer 27. 9. I the related cotext of the traspositio of what they call voice-leadig sets, both Wilso (212) ad Ludberg (212) cosider the perceptual challeges of the traspositio of itervals (what I am callig hyper-t). 1. As Lewi (1987) observes, There is ot much to be said i geeral about the effect of applyig a iversio operatio to X, Y, or both, so far as IFUNC is cocered (11). The last part of Lewi s assertio ( or both ) appears to be icorrect. 11. As with hyper-t, the perceptual issues associated with hyper-i are discussed i the K-et literature cited above. 12. What I am callig the iversioal symmetry of the imultiset (represetig the total voice leadig betwee two sets), Lewi (1977a) thiks of as eablig a kid of ivertible couterpoit, i.e., a couterpoit whose itervals are ot chaged by reversig the order of the two sets: We ca coceptualize the couterpoit as beig ivertible, i this cotext, if a exchage of pcsets betwee left ad right had does ot affect the total distributio of itervallic possibilities. Formally, we ca call the iterval fuctio from X to Y ivertible if, for every iterval i, INTF(Y, X, i) = INTF(X, Y, i) (22). Lewi goes o to say, It would be iterestig to fid what geeral relatios betwee X ad Y would be ecessary ad sufficiet for their iterval fuctio to be ivertible. Such geeral relatios have as yet ot bee foud. I believe that the precedig sectio of the preset study does idetify these geeral relatios. I my terms, the imultisetclass that represets the total voice leadig betwee pcsetx ad pcsety will remai the same whe X ad Y are reordered if X ad Y are idetical, or if X ad Y are related by traspositio, or if X ad Y are both iversioally symmetrical. 13. See also Coh 1986, which icludes a list of the 137 set classes that bear the TC-property, alog with the operads that geerate them. 14. Coh 1988 uses matrices for precisely this purpose. Morris 1998 describes the use of t-matrices i describig total voice leadig: The t-matrix costructed from two pcsets A ad B lists all the itervals from A to B.... Sice it cotais all of the possible itervals from A to B, it registers the total voice-leadig from A to B (23). Morris is primarily iterested i subsets of the cotets of these matrices. 15. The matrix otatio also makes it easy to grasp the traditioal (oe-to-oe) voice leadigs embedded withi the total voice leadig. If oe extracts from oe of these matrices a group of itervals such that o two occupy the same row or colum, the result will be oe of the ways of mappig the three pcs of X oto the three pcs of Y. I the first of the matrices i Example 15, there are six ways of extractig three umbers such that o two are i the same row or the same colum: 3-6-e; 3-8-9; 4-5-e; 4-8-9; 6-6-8; ad 6-5-9. These are the six traditioal, oe-to-oe ways of voice leadig from Set X to Set Y (Morris 1998).
9 of 9 16. Multiple paretage is a coiage of Coh 1988: Most of the 137 TC-set-classes are the product of several or eve may differet operatios (28). 17. Coh 1988 Example 9, presets the same iformatio i staff otatio, icludig the multiple paretage of (167) ad (268), but excludig the multisets with pc duplicatio. 18. Coh 1991 explais why the traspositioal combiatio of these four trichord-types uder tritoe traspositio produces the imultiset 13679. See Coh s Table 3 (p. 11) ad surroudig discussio. Copyright Statemet Copyright 214 by the Society for Music Theory. All rights reserved. [1] Copyrights for idividual items published i Music Theory Olie (MTO) are held by their authors. Items appearig i MTO may be saved ad stored i electroic or paper form, ad may be shared amog idividuals for purposes of scholarly research or discussio, but may ot be republished i ay form, electroic or prit, without prior, writte permissio from the author(s), ad advace otificatio of the editors of MTO. [2] Ay redistributed form of items published i MTO must iclude the followig iformatio i a form appropriate to the medium i which the items are to appear: This item appeared i Music Theory Olie i [VOLUME #, ISSUE #] o [DAY/MONTH/YEAR]. It was authored by [FULL NAME, EMAIL ADDRESS], with whose writte permissio it is reprited here. [3] Libraries may archive issues of MTO i electroic or paper form for public access so log as each issue is stored i its etirety, ad o access fee is charged. Exceptios to these requiremets must be approved i writig by the editors of MTO, who will act i accordace with the decisios of the Society for Music Theory. This documet ad all portios thereof are protected by U.S. ad iteratioal copyright laws. Material cotaied herei may be copied ad/or distributed for research purposes oly. Prepared by Carmel Raz, Editorial Assistat