Stochastic analysis of Stavinsky s vaied ostinati Daniel Bown Depatment of Music, Univesity of Califonia at Santa Cuz, USA dalaow@ucsc.edu Poceedings of the Xenakis Intenational Symposium Southank Cente, London, 1-3 Apil 2011 - www.gold.ac.uk/ccmc/xenakis-intenational-symposium An analogy is dawn etween the vaied ostinato, a common musical device in Stavinsky s music, and Xenakis stochastically geneated soundmasses. The analogy is constucted as follows: Stavinsky s vaied ostinati ae made up of cellula melodies; these cells can e epesented as Makov chains, as can the sequence of cells themselves; the Makov chain fo an sequence of cells has the popety of egodicity; this egodicity allows two stochastic vaiales, the mean and vaiance of cell density in a sequence, to uniquely detemine the chain. The iegulaity of Stavinsky s melodic epetition, in the past consideed lagely inscutale, is pehaps accessile to musical analysis in tems of these stochastic vaiales. An example of compute-geneated output that uses these vaiales to simulate Stavinsky s style is given. Iegulaly vaied epetition is one of the most salient featues of Stavinsky's music, and also pehaps the most esistant to analysis. A vaied ostinato is a device that Stavinsky used fequently; it featues one o seveal small musical figues that ae epeated iegulaly at vaying time intevals, with vaying duations, and in vaious odes. Boulez cellula analysis of ite of Sping intoduced a vivid teminology fo desciing the musical events eing epeated iegulaly; this notion of cells was futhe developed in susequent analyses of Stavinsky y Messiaen and Jonathan Kame. These analyses stop shot, though, of analyzing the natue of the iegulaity itself. In his analysis of Symphonies of Wind Instuments, fo example, Kame confesses that "thee is little moe I can do than show seveal sequences of duations whee one might odinaily find some egulaity and then state that egulaity is not thee" (221). Desciptions of cells which use tems like of iegulaity, andomness, pemutations, etc. depict them as stochastic ojects, though the authos don't actually lael them with this wod. A cell, like a motive, is a small configuation of pitches and duations; the diffeence etween a cell and a motive is that the ode of pitches and duations in the cell is not fixed. Howeve, thee ae usually cetain odes which ae pivileged; as Kame puts it, "afte we have head a nume of sequences of the same cells, we undestand which odeings ae 'pemissile' and which ones ae not" (223). His tem "sequence" efes to a section of music that contains a nume of epetitions, also iegulaly placed, of a cell o set of cells; a Stavinsky ostinato is thus a sequence. Kame notes that "unpedictaility within caefully defined ounday conditions exists on two adjacent hieachic levels: etween and within cells" (224). This suggests that, in ode to distinguish the two types of unpedictaility, thee ae two diffeent andom pocesses occuing in Stavinsky s ostinati, one govening the elements that make up the cells, and one govening the odeing of the cells themselves in a sequence. Futhemoe, the notion of pivileged odes suggests that the andomness in cells and sequences is constained y some quantity o quantities. This intepetation invites a method of analysis in which stochastic vaiales ae deived that quantitatively measue some aspects of cells' highe-level collective ehavio. The teminology and method fo deiving such vaiales comes fom Xenakis fomulation of stochastically poduced musical events. In Xenakis wok, these events wee soundmasses o clouds. In this way, instances of cells in a cell sequence loosely esemle gains of sound in a Xenakian soundmass: each consists of small elements (cells o gains) that occu in a andom manne; the oveall sound quality of these elements is what gives the ostinato o cloud its chaacte. In oth cases, thee ae oth individual and collective popeties that shape the sound of the whole. In a cloud, the sonic chaacte is a timal esult of the gains lending togethe to fom a complex spectum of patials. In a vaied ostinato, it s the highly syncopated hythm and placement of cells that gives it a distinguishing quality. Even though clouds and sequences ae topologically diffeent, it s not unintuitive to ecognize some common qualities etween the two; fo instance, 1
the notion of cell density the faction of time that instances of the cell occu in a sequence is simila to ganula density in a cloud, the nume of gains of some type that occu ove a faction of time. ""$%&' "$%&'()'*'+,-./' +0,,&'()'*'&01.0%+0' Figue 1 In this pape I will make this analogy into a moe igoous mapping y poviding a method of quantitatively deiving the mean and vaiance of a cell s density in a sequence. Futhemoe, I ll show that these two vaiales uniquely define a Makov chain fo a cell sequence, and theefoe ae apt, distinguishing desciptos of the sequence. The eason I've chosen a Makov chain model is ecause of the diffeence etween cells and gains, which is analogous to the diffeence etween thei natual countepats. Gains ae small: they geneally last a faction of a second, and lend with the othe gains to the point of eing indistinguishale. Cells ae igge; they endue pehaps seveal seconds, and contain discete musical elements that maintain thei identities ove many epetitions. These elements ae geneally melodic, in the sense that some odeing pinciple is imposed upon the events. Xenakis contolled oth the individual sonic aspects of gains and thei collective textue with stochastic vaiales. egading the collective aspect, he contolled the density of gains with algoithms like the ST algoithm, placing gains andomly on a gid in accodance with a given density and distiution. The intenal melodic stuctue of cells, and the fact that they geneally don't ovelap in Stavinsky's ostinati, ut ae contiguous, peclude the use of simila methods to poduce Stavinskian ostinati with desied cell densities. But a Makov model peseves cells' odeing aspects, while still yielding stochastic infomation aout the oveall musical sound. The collective aspect will e the aspect I conside, not the individual. But to do this, the smallest individual elements must fist e modeled in ode to get infomation fom them that will then e used to detemine the collective sonic qualities. Sucells, Cells, and Sequences Kame defines thee types of musical elements in Stavinsky's ostinati: sucells, cells, and sequences. This ode is hieachic: sucells compise cells, and cells compise pocesses. As mentioned ealie, diffeent types of andomness occu among these thee layes. At times the distinction etween these categoies may seem luy that is, it can e had to detemine if a paticula element is in one categoy o anothe. Fo my puposes, I diffe slightly fom Kame s (and Boulez s, and Messiaen s) categoization of elements. The following definitions of the thee categoies seve as the famewok fo the analyses to follow. 2
Sucell: A sucell is a set of one o moe musical events whose ode, if thee ae moe than one event, is invaiant. The duation of individual events can vay, howeve (in the following analyses, an event is simply a single note o chod). Cell: A cell is a set of sucells. As a musical oject, it is an astact class; that is, an instance of a cell could feasily consist of any contiguous set of instances of the cell s sucells, including epetitions. In a given instance of a cell, thee is no single fixed ode in which its sucells occu, o a fixed nume of sucells that need occu. Howeve, the ode and nume of sucells in an instance of a cell also follow a poaility distiution, which detemines the pivileged sucell odes Kame emaked on. Cells ae distinguished y thei sucells. Stavinsky usually aids in the distinction though ochestation: sucells within a cell ae simila in tems of egiste and instumentation, and ae diffeent in oth espects fom sucells in othe cells. I add one constaint to the definition of a cell: an instance of a cell always has a eginning sucell and an ending sucell; that is, evey instance of a cell has a finite length. Sequence: A sequence is a set of cells. Its elationship to its cells is analogous to a cell's elationship to its sucells; an instance of a sequence is a linea odeing of cell instances. Cell instances vay in thei stuctue thoughout a sequence, as descied in the peceding paagaph. Stavinsky's vaied ostinati ae sequences. The stuctual diffeence etween a sequence and a cell is that the sequence does not have a "stop" state. It is a pocess which, once egun, will continue indefinitely. The motivation fo defining a sequence like this is intuitive: ostinati in ite ae long sections containing many iegula epetitions of a small nume of cells, and thus don't convey a sense of eginning o ending. When a sequence ends, it sounds as if it was cut off, o inteupted y a new sequence. Conside the sting ostinato that occus etween eheasal numes 75 and 78 in "Dance of the Eath" in The ite of Sping: & 4 3 ΠΠ& ΠFigue 2. Dance of the Eath, The ite of Sping, ostinato in stings To detemine what constitute sucells in an ostinato, I choose the lagest ode-invaiant units. Sometimes this is staightfowad; othe times the segmentation is suject to intepetation. Segmentation is an impotant notion which, unfotunately, goes eyond the scope of this pape. I have picked cells which ae easy to segment, ut not tivial; hopefully thei usefulness in the model constucted justifies my intepetation. The "Dance of the Eath" sting ostinato can e segmented into five invaiant figues of fou sixteenth-notes each, each occuing on a eat, which constitute the entie ostinato. They ae the sucells: a 0 a 1 a 2 a 3 a 4 & Figue 3. Sucells in Dance of the Eath sting ostinato 3
Between eheasal nos. 75 and 76, cell instances egin with sucell a 0 and end with sucell a 4. Between these eginning and ending sucells, the ode of sucells is vaiale. The netwok diagam elow depicts all the vaious odes of figues in the phase. I include a "stop" node, which epesents the end of a cell instance. The numes on each edge epesent the poaility of that tansition occuing, which ae calculated diectly fom the scoe. By defining a cell as a netwok, a cell instance can e intepeted as a andom walk (i.e., a Makov chain) ove its sucells. )"* " " ( ( )"* ( C )(( )+, $%&' )+( - ( D )-. Figue 4. Netwok diagam fo Dance of the Eath cell Makov Analysis Fo a cell with n sucells, define its tansition matix as an (n+1) (n+1) matix, whee enty s ij gives the tansition poaility fom sucell i to sucell j fo i,j n, and the enty s i(n+1) gives the poaility that sucell i will go to the "stop" state i.e., that sucell i will e the last sucell in an instance. To calculate a tansition matix fo a cell ove a set of instances, sum the nume of times each sucell i goes to each sucell j o the "stop" state, use that sum fo the i,jth enty in the tansition matix, and then nomalize each ow of the matix so that its enties sum to 1. The (n+1)th ow consists of all zeoes except fo the last value: s (n+1)(n+1) = 1. This designates the stop state as an asoing state: once a Makov chain has eached the stop state, it stays thee. It can e shown that evey cell modeled this way is an asoing Makov chain; that is, any instance will each the stop state eventually. Because of this, we can calculate the expected length of an instance of a cell fom its matix. Let t i e the expected length of time to go fom sucell i to the stop state. The vecto t = [t 0 t 1... t n ] is calculated y solving the equation whee t = (I - Q) -1 (Eq. 1) I is the n n identity matix, Q is the matix of tansition poailities etween all sucells excluding the stop state is the vecto of expected sucell lengths j fo each sucell j. The expected length of the cell is the dot poduct of t and the vecto a of stat-state poailities, i.e. the poaility that a chain will egin with sucell i: 4
E length(n) = t a (Eq. 2) Fom the netwok diagam given ealie, we have the following matix: A = a0 a1 a2 a3 a4 stop a0 0 1 0 0 0 0 a1 0.06.78.06 0.11 a2 0 0 0 0.29.71 a3 0 1 0 0 0 0 a4 0 0 0 0 0 1 stop 0 0 0 0 0 1 Solving fo equations 1 and 2 aove gives t = [3.31 2.31 1.29 3.31 1.0] and t a = [3.31 2.31 1.29 3.31 1] [1 0 0 0 0] = 3.31 The chain will geneate cells with a mean length of 3.31 quate-notes. This is in ageement with the mean length of the cell's instances in the scoe. This models the distiution of the sucells in a cell; the distiution of the cells in a sequence uses anothe Makov matix. In this paticula ostinato, the peiod etween cell instances always occus in multiples of one quate-note (filled in with a single epeating sixteenth-note y the hons). This, too, can e epesented as a cell with a single sucell and an expected duation of 1; the "esting state. Counting the tansitions etween the sting cell and the esting state yields the following: P = cell est cell.4.6 est.59.41 Thee is one impotant diffeence etween the sequence matix and the cell matix. Unlike a cell, a cell sequence does not have a clea eginning o ending cell. It is a pocess which, once egun, will continue indefinitely. Stavinsky's ostinati ae long sections containing many iegula epetitions of a small nume of cells, and thus don't convey a sense of eginning o ending o movement. When a sequence ends, it usually sounds as if it was cut off aitaily, o was inteupted y a new sequence. The sequence's matix thus has no stop state: it is nonasoing. The sequence matix also has two moe impotant popeties. It is ecuent, meaning that when a chain leaves a cell, it will, with poaility 1, etun to that cell at some point. It is apeiodic, meaning the nume of steps it takes to etun to the cell it just left is not always the same, no will it always e a multiple of some nume geate than one (that is, thee ae no "hidden" cycles). A Makov chain that is oth ecuent and apeiodic is called an egodic chain. An egodic Makov chain is significant ecause of its steady-state poailities. The steady-state poailities of a Makov chain ae the aveage amounts of time the chain will spend in each of its states "ove the long un." An egodic chain will convege to its steady-state poaility distiution egadless of the initial value of the chain. This means that the poaility distiution, o any stochastic vaiales that define it, also completely define the chain itself the odeing of events ecomes 5
insignificant; only thei distiution is impotant. This is diffeent fom the Makov chains that model cells, in which the ode of sucells is impotant to some extent. The diffeence Kame mentions etween the unpedictaility on two adjacent hieachic levels can now e epesented in moe pecise tems: tansition matices fo cells ae non-egodic, while those fo cell sequences ae egodic. The egodicity of a cell sequence matix will allow the futhe deivation of stochastic vaiales that define the sequence. To find the vecto w = [w 1 w 2 w n ] consisting of the steady-state poaility fo each cell in a cell sequence matix P, solve the equation The mean density of the ith cell is then given y wp = w (Eq. 3) wiei (wjej) (Eq. 4) whee Ej epesents the expected duation of cell j, and the sum in the denominato is taken ove all the cells in the sequence. Fom these equations, the mean densities of the sting cell and the est cell in the Dance of the Eath ostinato ae detemined to e.76 and.24, espectively. The vaiance of the densities can also e calculated; it will necessaily e the same fo each cell, since cells in a sequence ae contiguous. The tansition matix fo a two-cell sequence can e epesented in tems of two vaiales, as follows: 1 p q p 1 q (Eq. 5) The vaiance σ of the densities is then given y σ = 2w1(w 2 + q) 2 + w1+ w1 (p q)(w2 w1) (Eq. 6) Fo the "Dance of the Eath" ostinato, the vaiance of the cell density is equal to.17. Convesely, given a desied mean µ and vaiance σ fo the density of one cell, unique values fo p and q can e deived: w1 = µe1 µe1 + (1 µ)e 2 (Eq. 7) w2 = (1 µ)e 2 µe1 + (1 µ)e 2 (Eq. 8) whee α = 2w1w 2. σ + w1w2 p = αw 2 (Eq. 9) q = αw 1 (Eq. 10) The tansition matix fo a two-cell sequence is thus completely detemined y these two stochastic vaiales, the mean and vaiance of the density of one of the cells. The two vaiales that goven sequences ae analogues to the stochastic vaiales Xenakis used to contol the oveall sonic textue of his soundmasses. In othe wods, given two cells with defined tansition matices ove thei sucells, a unique tansition matix fo the two-cell sequence can e calculated fom any given mean and vaiance of one of the cells density. This uniqueness suggests that 6
these two vaiales can e used as desciptive quantities in musical analysis of music composed using cellula melodies. Application: compute-geneated music in the style of Stavinsky It is easonale to conside whethe, afte all this fomal teatment, the quantities defined in this pape actually convey a significant musical quality. To test this, I have implemented the analytical techniques pesented in this pape in a compute pogam (witten in Python). The sucell pitch and hythm data and tansition matices fo two cells ae loaded into the pogam. The cell density mean and vaiance of one of the cells can then e set aitaily; the pogam then calculates the matix given in equation 5 and uses this matix to output a sequence with the given values of these vaiales. I pesent a sample of the pogam s output elow. The following except (figue 5) fom the Sacificial Dance in The ite of Sping (lasting fom eheasal nume 192 to 196) consists of two cells, (laeled ( ascending and a descending ). The sucells and tansition matices associated with each cell ae listed afte the except (figue 6). The mean cell density of the ascending cell is.24, and the vaiance of the density is.09. Figue 7 shows an example of the output of the pogam when the mean and vaiance of the ascending cell s density ae set to these two values. While diffeent fom the actual scoe, the quality of iegulaity in the Stavinsky except and the compute-geneated output is vey simila. "$%&'(&)*+$%,,+ '%$%&'(&)*+$%,,+ & n & n & Figue 5. ite of Sping, Sacificial Dance N 192-195 7
Ascending cell sucells and tansition matix: & a 0 a 1 a 2 a 3 a 4 a 5 & " a 0 a 1 a 2 a 3 a 4 a 5 stop a0 0.33.67 0 0 0 0 a1 0 0 1 0 0 0 0 a2 0 0 0 1 0 0 0 a3 0 0 0 0 1 0 0 a4 0 0 0 0 0 1 0 a5 0 0 0 0 0 0 1 stop 0 0 0 0 0 0 0 Descending cell sucells and tansition matix: & d 0 d 1 d 2 d 3 d 4 d 5 d 6 n d 0 d 1 d 2 d 3 d 4 d 5 d 6 stop d0 0.12.25.5.13 0 0 0 d1 0 0 1 0 0 0 0 0 d2 0 0 0 1 0 0 0 0 d3.14 0 0 0.86 0 0 0 d4 0 0 0 0 0 1 0 0 d5 0 0 0 0 0 0 1 0 d6 0 0 0 0 0 0 0 1 stop 0 0 0 0 0 0 0 1 Figue 6. Sacificial Dance cells and sucell tansition matices, N 192-195 8
& 6 & n 12 & 17 22 & & 27 & n n 16 2 16 2 n n n Figue 7. Compute-geneated output using cells fom Sacificial Dance with mean cell density =.24, cell density vaiance =.09 9
While the musical examples pesented in this pape wee all taken fom (o simulate) Stavinsky s woks, the analytical techniques ae solely ased on the use of stochastic vaiales, as invented y Xenakis. It has een my hope to show that this notion of stochastic paametes can meaningfully apply to a lage class of musical elements than the timal and textual aspects of taditional stochastic music, and susequently to othe composes styles in which cellula techniques ae employed. Consideing Stavinsky s vaied ostinati in tems of oveall stochastic paametes opens up new avenues fo oth analysis and composition. In tems of analysis, it povides pecise quantities that can e deived fom the music, and which descie aspects of Stavinsky s iegula epetition. The two quantities pesented in this pape ae cell density mean and vaiance; futhe eseach could deive moe such vaiales. Fo composes, it offes a means of applying lage-scale stochastic contol to a musical paamete melody that is not typically associated with stochastic music. efeences Asenault, Linda M. 2002. "Iannis Xenakis's Achoipsis: The Matix Game." Compute Music Jounal 26/1: 58-72. Boulez, Piee. 1968. Notes of an Appenticeship. Ed. Paule Thévenin. Tans. Heet Weinstock. New Yok: A. A. Knopf. Feldman, ichad M., and Ciiaco Valdez-Floes. 1996. Applied Poaility and Stochastic Pocesses. Boston: PWS. Kelle, Damián and Bian Feneyhough. 2004. "Analysis y Modeling: Xenakis's ST/10-1 080262." Jounal of New Music eseach 33/2: 161-171. Kame, Jonathan D. 1988. The Time of Music: new meanings, new tempoalities, new listening stategies. New Yok: Schime Books. Messiaen, Olivie. 1995. Taité de ythme, de couleu, et d'onithologie, Tome II. Pais: Alphonse Leduc. Xenakis, Iannis. 1971. Fomalized Music: thought and mathematics in composition. Bloomington, Indiana Univesity Pess. 10