John Cage's Number Pieces as Stochastic Processes: a Large-Scale Analysis
JJohn Cage’s Number Pieces as StochasticProcesses: a Large-Scale Analysis
Alexandre Popoffal.popoff@free.frFranceNovember 25, 2013
Starting from 1987 to 1992, the composer John Cage began writing a series ofscores named the ”Number Pieces”. The Number Pieces are easily identifiablethrough their titles, which refer to the number of performers involved, and therank of the piece among those with the same number of performers. For example
Four is the first Number Piece written for four performers, whereas
Four is thethird one. From 1987 till Cage’s death, forty-seven such pieces were written.In all the Number Pieces except One and Two , John Cage used a partic-ular time-structure for determining the temporal location of sounds which wasnamed ”time-bracket”. These time-brackets already appeared in earlier workssuch as
Thirty Pieces for Five Orchestras and
Music for... . However, in theNumber Pieces, Cage simplified the contents of the time-brackets, most of themcontaining only a single tone or sound, especially in his late works.A time-bracket is basically made of three parts : a fragment of one or manystaves, lying under two time intervals, one on the left and one on the right. Atypical time-bracket can be seen on Figure 1. The time intervals consist of tworeal-time values separated by a two-way arrow. The staves contain one or moresound events without any duration indications. The time-bracket is performedas follow : the performer decides to start playing the written sounds anywheninside the first time interval on the left, and chooses to end them anywheninside the second one. These parameters are thus left free to the performer,provided he respects the time-bracket structure. In the example of Figure 1,the performer can start playing the note F whenever between 0 and 45 seconds,and can choose to end it whenever between 30 and 75 seconds (assuming ofcourse that the note has started before). Note that there exists an overlap,which we will call the internal overlap , between the starting time interval andthe ending time interval, which is 15 seconds long in this specific case.In a few cases, a time-bracket may also be fixed. In that case, time inter-vals are replaced with single indications of time, for example 2’15” and 2’45”,meaning that sounds should always begin and end at the indicated times.1 a r X i v : . [ phy s i c s . s o c - ph ] N ov igure 1: A typical time-bracket from Cage’s Number Piece Five
Successive time-brackets occurs in a Number Piece score with possible over-lap between each other, which we will call external overlaps , meaning that theending time interval of one time-bracket may overlap the beginning interval ofthe next one.In the case of Number Pieces written for multiple performers, the super-position of different voices each playing time-brackets according to their choicecreates a polyphonic landscape in constant evolution. Previous authors ([1],[2], [3]) have shown how the Number Pieces were written as a consequence ofCage’s new insights about harmony. In particular, Haskins ([3]) has commentedin a detailed dissertation on Cage’s views about harmony throughout his career.Cage had been critical of traditional Western harmony, and even of twelve-tonemethods, as they were based on rules which prevented the appearance of cer-tain combination of pitches. A majority of his work arose as expressions of a”liberated” harmony where any sound or chord, or transition between them,could happen, mainly through chance operations. It is notable that, towardsthe end of his career, Cage seems to have adopted a conception of harmonywhich simply consists in the principle of sounds sounding together at the sametime. To quote: ”harmony means that there are several sounds...being noticedat the same time, hmm ? It’s quite impossible not to have harmony, hmm ?” ([4]).In his dissertation, Haskins points out ([3], p. 196) that the analysis of theNumber Pieces is complex ”...because the brackets offer a flexibility that createsmany possibilities” . He later adresses the same problem when analyzing
Five : ”Coping with the myriad possibilities of pitch combinations - partially orderedsubsets - within each time-bracket of Five remains an important issue” ([3], p.207), and cites the work of Weisser on Four . In [1] and [2], Weisser enumeratedthe possible pitch-class sets in
Four , classifying them in ”certain triads/seventhchords” , ”possible triads/seventh chords” , ”thwarted triads/seventh chords” and ”triadic segments” . By doing so, Weisser is able to identify some of the possiblepitch-class sets which can occur during a time-bracket. However, his analysispresents some drawbacks. The first one is that Weisser concentrates on triadsand seventh chords and neglects the other possible pitch-class sets. As will beseen below, a performance of Four opens the possibility of hearing 49 differentpitch-class sets (including silence, single sounds and dyads). The second draw-back is that Weisser’s analysis poorly takes into account the inherently randomtemporal structure of the time-bracket, in which sounds from different parts2ay begin and end at different times. For example, he classifies ([2], p. 202,Example 10a) a seventh chord in the last time-bracket of
Four , section C, as ”virtually certain” chord. However, if one of the player stops playing before theothers have entered, as is possible given the rules of time-brackets interpreta-tion, then this chord will not be heard. Haskins faced the same difficulties and,in the case of
Five , turned to the analysis of one particular performance takenfrom a recording.In previous works ([5], [6]), we have advocated a statistical approach to theanalysis of the time-bracket structure, focusing on a single time-bracket con-taining a single pitch. This approach allows to deal with the entire possibilitiesoffered in terms of starting and ending times (and thus durations and tempo-ral location in the time-bracket). The purpose of this paper is therefore toextend this approach to the analysis of an entire Number Piece, by consider-ing all time-brackets and all parts. The determination of an entire part allowsthe determination of its sonic content over time. Having this information foreach part allows the determination of the chords occuring during a performance.We use here the methods of (musical) set theory in which chords are identifiedby their corresponding pitch-class set. By averaging over a large number ofrealizations (which is achieved through a computer program running the deter-mination of the parts repeatedly) we can access the probability distributions ofeach pitch-class set over time, thus turning the Number Pieces into stochasticprocesses. By doing so, we solve the problem posed by Haskins and Weisser ofcoping with all the possibilities offered by the Number Pieces. We have cho-sen to focus on two Number Pieces, namely Four and
Five , as these are shortpieces with a reduced number of players which therefore allows for a convenientcomputer implementation. This will also allow us to compare the results about
Four with Weisser’s analysis. Section 2 of this paper describes the methodologyused for the analysis, while section 3 and 4 present the analysis of
Five and
Four respectively. A part in a Number Piece is the set of all time-brackets and their pitch contentassociated with a player’s score. Given the score of a Number Piece, i.e thedescription of all time-brackets and their pitch content, we call realization of aNumber Piece the knowledge of starting and ending times for all pitches, aftertheir selection from the time-brackets. .In order to study a Number Piece as a whole from a statistical point ofview, a computer program wa written in order to generate a large number N (typically 10 − ) realizations of the Number Piece, and derive probabilitiesfor the possible pitch-class sets. The programs for the analysis of Five and
Four are written in ANSI C, mainly for speed issues. We give here the generaloverview of the program, while the specifics pertaining to particular choices in3he algorithm will be given in the following subsections.For each realization, the programs successively and independently generatethe parts corresponding to each player. Whether parts in actual performancesof the Number Pieces are indeed independent, or should be chosen so, is worthquestioning. In commenting the specific example of
Seven , Weisser ([1]) un-derlines the fact that performers should work cooperatively to fullfill Cage’sinstructions. Nevertheless, we have chosen to select each part independently asit is practical (such a choice is the easiest to implement algorithmically), and forlack of a proper model of human behavior, which would be difficult to describein such a case.Once parts have been selected, the pitch-class content is known for each time t in each part. By using Starr’s algorithm (see below), we can thus derive thecorresponding pitch-class set at each time t .By averaging over N = 10 − realizations, it is possible to derive theprobabilities P r ( P CS t = i ) of obtaining the pitch-class set i at time t . In otherwords, we obtain a collection of random variables P CS t indexed over time, withvalues in the possible pitch-class sets. In the framework of a statistical anal-ysis, we thus see that there is a stochastic process naturally associated withthe Number Piece being studied. In the rest of this paper, the probabilities P r ( P CS t = i ) will be presented under the form of a heat map with respect totime and possible pitch-class sets. Moreover, since we have access for each real-ization to the entire pitch-class set content over time, we can derive conditionalprobabilities of the form P r ( P CS t + τ = j | P CS t = i ), τ >
0. These probabilitiesare calculated using
P r ( P CS t + τ = j | P CS t = i ) = P r ( P CS t + τ = j ∩ P CS t = i ) P r ( P CS t = i )assuming that P r ( P CS t + τ = j | P CS t = i ) = 0 if P r ( P CS t = i ) for con-tinuity. These conditional probabilities are useful to determine the possibleevolutions of pitch-class sets during a realization, as they express the probabil-ity of having pitch-class set j at τ time in the future, knowing that we havepitch-class set i in the present. We have described in [5] the analysis of a single time-bracket containing a uniquepitch, using a temporal selection procedure based on uniform distributions overthe starting and ending time intervals. In [6] we have studied the effect of usingdifferent distributions on the characteristics of the sounds thus produced. Wewill use in this analysis a similar strategy, as described below.In all cases, we have discretized time, using a smallest unit of 0 . .
01 seconds. The devices used for timemeasurement thus already discretize time, and it is very unlikely that normal4erformers are able to choose starting and ending time with a 0 .
01s precision.Hence a resolution of 0 . , , t = 720, then the starting time of thepitch of the second time-bracket cannot be selected inside [600 , , P r ( T = t ) anddepending on a parameter t P rec . This parameter is either • The previously chosen starting time in the same time-bracket, if the timeinterval is an ending time interval, or • the chosen ending time in the previous time-bracket, if the time intervalis a starting time interval.Thus, if the considered time interval is [ t s , t e ] and t P rec < t s , the randomselection will be performed on the interval [ t s , t e ], otherwise we replace it with[ t P rec , t e ].For a time interval [ t s , t e ], we assume here a gaussian probability distributionof the form P r ( T = t ) = A.e − ( t − c ) σ , c = t s + t e , σ = t e − t s −3 t P r( T = t ) Figure 2: The gaussian distribution used for selecting starting and ending timesin their corresponding intervals. The distribution is presented here on a unittime interval with resolution 0.001 seconds. It is scaled accordingly dependingon the time interval considered.1. The outer limits t s and t e of the sonic content are selected in an identicalway as they are for a single-pitch time-bracket. Additional time marksare then selected successively to determine the location of each pitch.2. The presence of a pause indication necessitates the determination of twotime marks. For example, to determine the temporal location of pitchesF t is selectedin the interval [ t s , t e ], following the distribution of Figure 2, then a secondtime mark t is selected in the remaining interval [ t , t e ]. A third selectionin the time interval [ t , t e ] marks the end of G P CS t may be affected on short time-scales, the largescale behavior remains essentially the same. One may also object to the un-equal treatment of the internal time marks as compared to the outer temporal6 ttttt Figure 3: The selection procedure for a complex time-bracket (top). The outerlimits of the content are selected identically to a single pitch time-bracket. Ad-ditional time marks are then selected successively inside the obtained limits inorder to determine the temporal location of each individual pitch (representedat the bottom of the figure by color bars).7 xtended Forte number Pitch-class set0-1 ∅ , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
7] Extended Forte number Pitch-class set5-1 [0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Table 1: List of pitch-class sets up to pentachords with their correspondingextended Forte number 8imits. We have used this procedure to ensure that the determined time markswould respect the time-bracket structure. In any case, we are aware that otherprocedures exist, which could be applied for analysis. The remaining questionof whether they accurately reflect human behavior is difficult, and has beenadressed in [6], noting in particular that human behavior may be too complexto model easily.
Given the knowledge of pitch-classes in each part at each time t , we determinethe corresponding pitch-class set using Daniel Starr’s algorithm, which has beendescribed in [7] and [8]. The pitch-class set are given using Forte’s notation ([9]),which we extend to take into account silence (notated by 0-1), single sounds (1-1)and dyads (2-1 to 2-6). These Forte numbers and their corresponding pitch-classsets can be found in Table 1. Note that the notation of pitch-class sets onlydiffers from Forte’s original notation for pitch-class set 5-20, which is [0 , , , , We have emphasized the calculation of probabilities
P r ( P CS t = i ) in section2.1, as the probability distributions of P CS t for each t give a large-scale descrip-tion of the Number Pieces. However it should be noted that this description isreductive as it does not consider the possible dynamics between pitch-class sets.The calculation of conditional probabilities P r ( P CS t + τ = j | P CS t = i ) allowsa partial description of such dynamics. Yet, considering only these conditionalprobabilities would amount to assume that the stochastic process is a Markovone, which, as will be made clear below, is false.To account for the possible relations between pitch-class sets over time, wecan study the possible paths taken during the performance of a Number Piece.We define a path as the set of successive and different pitch-classes sets whichcan occur during a realization of a Number Piece. For example { } is a valid path, while { } is not as pitch-class set 2-3 is repeated twice. While we can study paths overthe whole time of a Number Piece, we will mainly focus on paths on a singletime-bracket.The computer program written for the analysis can be used for the determi-nation of paths, and by averaging over all realizations we can determine theirstatistic. 9ime-Bracket Starting time interval Ending time interval1 [0 , , , , , , , , Five
Five
The Number Piece
Five was written by Cage during 1988 and is dedicated toWilfried Brennecke and the Wittener Tage [10]. This piece is written for fivevoices or instruments or mixture of voices and instruments. Each part containsfive time-brackets, the third one being fixed. The time-brackets are identical forall players. The temporal structure of
Five is given on Table 2.The pitch class structure of each time-bracket is also given in Table 3. Eachcolored circle in the usual circle of semitones represent one player’s pitch contentfor the considered time-bracket, according to the caption given on top. When aplayer’s time-bracket contains multiple pitches, their order is given by the asso-ciated numbers, and the pauses and slurs indications are given by the diagramson the right.
The plot of the probabilities
P r ( P CS t = i ) calculated over the 87 possible pitch-class sets (in ordinate) at each time t (in abscissa) is presented in a heatmapplot on Figure 4. Notice that the colorbar, which indicates the correspondingprobability values, is given in pseudo-logarithmic scale: if p is a calculatedprobability and N is the total number of realizations, the color corresponds tothe value 1 − log ( p + 1 /N ) log (1 /N ) .As a first observation, one can note that the five different time-brackets areclearly identifiable on this plot. They are generally separated by either silence orsingle sounds, as one can see that the probability of obtaining the correspondingpitch-class sets is superior to 0.5 in the external overlaps between the time-brackets. The external overlaps are also characterized by the fact that theycontain very rare events. For example, the pitch-class set 4-10 can occur in theexternal overlap between time-brackets 1 and 2 with a probability of roughly10 − . Such rare events correspond to extreme results in the random selectionprocedure, such as pitches changing at the very end of a time-bracket and soon. 10ime-Bracket Pitch-class content12345Table 3: Pitch-class structure of Five , represented on the usual circle of semi-tones. When a time-bracket in a part contains multiple pitches (ordered bythe represented numbers), the diagram on the right indicates whether they areseparated by pauses (’) or slurs (-). 11igure 4: Heatmap of the probabilities
P r ( P CS t = i ) over the 87 possiblepitch-class sets (in ordinate) at each time t (in abscissa) in Five . The colorbarindicates the corresponding probabilities in pseudo-logarithmic scale (see text).12he fixed time-bracket is also clearly visible in this plot, and is characterizedby the absence of silence, single sounds or dyads. Indeed, since all players aresupposed to start at the same time, there cannot be any such event inside thetime-bracket.It can also be noted that the time-brackets are very different with respect totheir possible content. The first time-bracket is characterized by the appearanceof only seven possible triads, and just three possible tetrachords. Indeed, it iseasy to check from Table 3 that no pentachord can occur even if the playersare all playing at the same time. Time-bracket 2, a contrario , is characterizedby the appearance of all possible triads except 3-12, many tetrachords withsimilar probability values, and seven possible pentachords. As said before, time-bracket 3 is characterized by only one possible triad, four tetrachords and fourpentachords. The material of time-bracket 4 is reduced, with only four possibletriads and two tetrachords. as can be verified on Table 3. Finally time-bracket5 is of moderate complexity between time-bracket 1 and 2.Another feature shown in this heatmap is that, while many n -chords arepossible in each time bracket, they do not all occur with the same probabilities,nor at the same time. For example, pitch-class sets 2-3 and 3-7 are prevalentthroughout time-bracket 1. Pitch-class set 2-5 may occur at the beginning ofthis time-bracket with a high probability, while pitch-class sets 3-2 and 3-10 aremore probable towards its end. In time-bracket 2, the distributions of P CS t seem more uniformly spread over the possible pitch-class sets. Yet we can seethat pitch-class sets 4-z15, 5-6 and 5-11 have higher probabilities of occurence.A the same time, some pitch-class sets may appear at the beginning or end butare less probable in the middle of the time-bracket, such as 3-3. The peculiartime-structure of time-bracket 3 imposes to begin with pitch-class set 4-3 andto end with pitch-class set 5-z18. Time-bracket 4 is dominated by the highprobabilities of obtaining pitch-class sets 3-3 and 4-3. Finally, time-bracket 5shows higher probabilities of obtaining pitch-class sets 5-2, 4-11, 4-4 or 3-4 inthe middle of its structure.In order to get a better insight about the probabilities of occurence of thedifferent pitch-class sets, we will now focus on a specific analysis of the first time-bracket in Five . Table 4 lists the possible pitch-class sets of cardinality superiorto 2 which can occur in this time-bracket. Pitch-class set 3-7 (respectively 3-2)may occur in two different ways: they will be designated by 3-7 α , 3-7 β (resp.3-2 α , 3-2 β ) as indicated in the Table.Most of these chords contain the pitch-class G in contrario havevery low probabilities of occurence.Considering the probabilities of the pitch-class sets as represented in Figure4 amounts to studying a static description of the pitch-class set content of time-13able 4: The possible triads and tetrachords which can occur in the first time-bracket of Five . Two possibilities exist in the case of 3-7 and 3-2, which arenotated with the indicated symbols α and β .14racket 1. However, the low probability associated with pitch-class set 3-5 canalso be explained from a dynamic point of view by studying the stability of thistriad. Indeed, we can see that 3-5 is composed of the second pitch of player2, the unique pitch of player 3 and the third pitch of player 5. Players 1 and4 are silent which means either that they have not started their time-bracketyet, or that they have already finished playing it. The fact that time-bracketscoming from separate parts are treated independently in our model implies thatthe mean of the difference between the starting times (or ending times) of twodifferent players is null. In other terms, the time-brackets start and finish onaverage at the same time. We thus have two possible evolutions depending onplayers’ 1 and 4 behavior: • Players 1 and 4 have not started their time-bracket. Their entry is there-fore imminent since player 2 is already playing its second pitch. The pitch-class set 3-5 is thus unstable and will evolve quickly towards pitch-classset 4-13. • Players 1 and 4 have finished their time-bracket. All the remaining playersare therefore expected to finish soon and the pitch-class set 3-5 is alsounstable. It can evolve towards either 2-1, 2-5 or 2-6.In both cases, we see that the pitch-class set has a short life time, whichcontributes to its low probability of occurence in time-bracket 1.The evolution of chords can be given a more quantitative treatment by study-ing the conditional probabilities
P r ( P CS t + τ = j | P CS t = i ). As stated inSection 2, these probabilities P r ( P CS t + τ = j | P CS t = i ) (with τ >
0) ex-press the probability of transitioning from pitch-class set i to pitch-class j at τ times in the future. Since this is not a Markov process, these probabili-ties depend on the time t . Consider for example the evolution of pitch-classset 3-7, which is prevalent throughout the time-bracket. Figure 5 presentsthe graphs of P r ( P CS t +10 = j | P CS t = 3-7) over the time interval [100,600]for the pitch-class sets 3-7, 2-5, 4-27, 2-7, 2-2, 4-13 and 1-1. The probability P r ( P CS t +10 = 3-7 | P CS t = 3-7) is the highest, meaning that there is a highchance of finding the same chord at one second in the future. However this prob-ability decreases over time as the chord become more and more unstable. Thebottom part of Figure 5 shows the probability of transitioning to other chords.The second highest probability corresponds to the transition from 3-7 to 2-3.Indeed, if we are listening to pitch-class set 3-7 α , there is a high probability thatplayer 5 would transition from his first pitch to its second, ultimately endingwith pitch-class set 3-2. Since there is a pause between these two pitches, wewould hear pitch-class set 2-3 played on pitch-classes F and G P r ( P CS t +10 = j | P CS t = 3-7) for differentpitch-class sets j over the interval [100,600], calculated over 10 realizations.Only the pitch-class sets with highest probabilities have been represented: 3-7(black), 2-3 (green), 4-27 (pink), 2-5 (cyan), 2-2 (yellow), 4-13 (blue), 1-1 (red).The bottom figure is identical to the top one, with P r ( P CS t + τ = 3-7 | P CS t =3-7) removed for clarity. 16 ath Probability0-1, 1-1, 2-3, 3-7, 2-3, 3-2, 3-7, 2-3, 3-10, 2-3, 1-1, 0-1 0.0147200-1, 1-1, 2-5, 3-7, 2-3, 3-2, 3-7, 2-3, 3-10, 2-3, 1-1, 0-1 0.0101500-1, 1-1, 2-3, 3-7, 2-3, 3-2, 3-7, 2-3, 1-1, 2-3, 1-1, 0-1 0.0070700-1, 1-1, 2-3, 3-7, 2-3, 3-2, 3-7, 2-3, 3-10, 2-3, 1-1, 0-1 0.0061400-1, 1-1, 2-3, 3-7, 2-3, 3-2, 3-7, 2-3, 1-1, 0-1, 1-1, 0-1 0.0045500-1, 1-1, 2-3, 3-7, 2-3, 3-2, 3-7, 2-3, 3-10, 2-3, 1-1 0.0045200-1, 1-1, 2-5, 3-7, 2-3, 3-2, 3-7, 2-3, 1-1, 2-3, 1-1, 0-1 0.0044800-1, 1-1, 2-5, 1-1, 2-3, 3-2, 3-7, 2-3, 3-10, 2-3, 1-1, 0-1 0.0042900-1, 1-1, 2-5, 3-7, 2-3, 3-2, 3-7, 2-3, 3-10, 2-3, 1-1, 0-1 0.0041200-1, 1-1, 2-3, 3-7, 2-3, 3-2, 3-7, 4-13, 3-10, 2-3, 1-1, 0-1 0.0037600-1, 1-1, 2-3, 3-7, 4-27, 3-7, 2-3, 3-2, 3-7, 2-3, 1-1, 0-1 0.0036900-1, 1-1, 2-3, 3-7, 2-3, 3-2, 3-7, 2-3, 3-10, 2-6, 1-1, 0-1 0.0035700-1, 1-1, 2-3, 3-7, 2-3, 3-2, 3-7, 2-3, 1-1, 2-6, 1-1, 0-1 0.0034100-1, 1-1, 2-3, 3-7, 2-3, 3-10, 4-12, 4-13, 3-10, 2-3, 1-1, 0-1 0.0034100-1, 1-1, 2-5, 3-7, 2-3, 3-2, 3-7, 4-13, 3-10, 2-3, 1-1, 0-1 0.0032700-1, 1-1, 2-2, 3-7, 2-3, 3-2, 3-7, 2-3, 3-10, 2-3, 1-1, 0-1 0.0032000-1, 1-1, 2-5, 3-7, 2-3, 3-2, 3-7, 2-3, 3-10, 2-3, 1-1 0.0030700-1, 1-1, 2-3, 3-7, 4-27, 3-10, 4-12, 4-13, 3-10, 2-3, 1-1, 0-1 0.0029400-1, 1-1, 2-3, 3-7, 2-3, 3-2, 3-7, 2-3, 1-1, 2-3, 1-1, 0-1 0.0028600-1, 1-1, 2-5, 3-7, 2-3, 3-2, 3-7, 2-3, 1-1, 0-1, 1-1, 0-1 0.0028300-1, 1-1, 2-3, 3-7, 2-3, 3-2, 4-12, 4-13, 3-10, 2-3, 1-1, 0-1 0.0025900-1, 1-1, 2-3, 3-7, 4-27, 3-10, 2-3, 3-2, 3-7, 2-3, 1-1, 0-1 0.0025900-1, 1-1, 2-5, 3-7, 2-3, 3-2, 3-7, 2-3, 3-10, 2-6, 1-1, 0-1 0.0025800-1, 1-1, 2-5, 3-7, 2-3, 3-10, 4-12, 4-13, 3-10, 2-3, 1-1, 0-1 0.0024700-1, 1-1, 2-3, 3-7, 2-3, 3-2, 3-7, 2-3, 1-1, 2-3, 1-1 0.0023300-1, 1-1, 2-3, 3-7, 2-3, 3-10, 2-3, 3-2, 3-7, 2-3, 1-1, 0-1 0.0023100-1, 1-1, 2-5, 1-1, 2-1, 2-2, 1-1, 2-3, 3-10, 2-3, 1-1, 0-1 0.0022000-1, 1-1, 2-5, 3-7, 2-3, 3-2, 3-7, 2-3, 1-1, 2-6, 1-1, 0-1 0.0021800-1, 1-1, 2-5, 3-7, 2-3, 3-2, 4-12, 4-13, 3-10, 2-3, 1-1, 0-1 0.0021600-1, 1-1, 2-5, 3-7, 2-3, 3-10, 2-3, 3-2, 3-7, 2-3, 1-1, 0-1 0.0021400-1, 1-1, 2-3, 3-7, 2-3, 3-2, 3-7, 2-3, 1-1, 0-1 0.0021200-1, 1-1, 2-3, 3-7, 2-3, 3-2, 3-7, 2-3, 1-1, 0-1, 1-1, 0-1 0.0020300-1, 1-1, 2-5, 3-7, 4-27, 3-7, 2-3, 3-2, 3-7, 2-3, 1-1, 0-1 0.0019400-1, 1-1, 2-5, 1-1, 2-3, 3-2, 3-7, 2-3, 1-1, 2-3, 1-1, 0-1 0.0018600-1, 1-1, 2-3, 3-7, 4-27, 3-7, 2-5, 1-1, 2-1, 2-2, 1-1, 0-1 0.0018500-1, 1-1, 2-3, 3-7, 2-3, 1-1, 2-1, 2-2, 1-1, 2-3, 1-1, 0-1 0.0018300-1, 1-1, 2-3, 3-7, 2-3, 3-2, 3-7, 2-2, 1-1, 2-3, 1-1, 0-1 0.0018200-1, 1-1, 2-3, 3-7, 2-5, 1-1, 2-1, 2-2, 1-1, 2-3, 1-1, 0-1 0.0018200-1, 1-1, 2-5, 3-7, 2-3, 3-2, 3-7, 2-3, 1-1, 2-3, 1-1, 0-1 0.0018100-1, 1-1, 2-3, 3-7, 2-3, 3-2, 3-7, 2-3, 3-10, 2-6, 1-1, 0-1 0.0018100-1, 1-1, 2-5, 1-1, 2-3, 3-2, 3-7, 2-3, 3-10, 2-3, 1-1, 0-1 0.0018000-1, 1-1, 2-5, 1-1, 2-1, 3-2, 3-7, 2-3, 3-10, 2-3, 1-1, 0-1 0.0017300-1, 1-1, 2-3, 3-7, 2-3, 3-2, 3-7, 4-13, 3-10, 2-3, 1-1, 0-1 0.0016700-1, 1-1, 2-5, 3-7, 2-3, 3-2, 3-7, 2-2, 1-1, 2-3, 1-1, 0-1 0.0016400-1, 1-1, 2-3, 3-7, 4-27, 3-7, 2-3, 3-2, 3-7, 2-3, 1-1, 0-1 0.0016200-1, 1-1, 2-3, 3-7, 4-27, 3-10, 2-3, 1-1, 2-1, 2-2, 1-1, 0-1 0.0016100-1, 1-1, 2-2, 3-7, 2-3, 3-2, 3-7, 2-3, 1-1, 2-3, 1-1, 0-1 0.0015900-1, 1-1, 2-5, 3-7, 4-27, 3-10, 2-3, 3-2, 3-7, 2-3, 1-1, 0-1 0.0015900-1, 1-1, 2-3, 3-7, 4-27, 3-7, 2-3, 1-1, 2-1, 2-2, 1-1, 0-1 0.0015500-1, 1-1, 2-5, 3-7, 2-3, 1-1, 2-1, 2-2, 1-1, 2-3, 1-1, 0-1 0.001550 Table 5: List of the fifty paths with highest probabilities (right) in time-bracket1. 17ounter pitch-class set 3-7 β at this time, these transitions correspond to playerswho have finished playing their time-bracket.Incidentally, the above discussion makes it clear that the stochastic processat hand should not be assimilated to a zero- or first-order Markov process. Inthe case of pitch-class set 3-7, we have seen that its evolution is determined byall the past events and depends on the time considered. To account for the entirepast of a pitch-class set, we can study the possible paths inside a time-bracket.Table 5 lists the fifty paths with highest probabilities occuring in time-bracket1, calculated over 10 realizations (27225 different paths were found in total).It can be seen that some of the paths are merely variants of others, and thatthey confirm the prevalence of pitch-class sets 3-7, 3-2, 3-10 and 4-27 inside thistime-bracket.With the knowledge of the possible paths and the transition probabilities P r ( P CS t + τ = j | P CS t = i ), we can propose a simplified model for pitch-classset evolution in time-bracket 1. This model, which is shown on Figure 6, con-siders only the most probable transitions between pitch-class sets. We wouldlike to highlight the fact that, as represented, the model suggests there is a re-versible transition between pitch-class sets 4-27 and 3-7. As we have seen above,this assertion is false and while the evolution from 3-7 to 4-27 is reversible, theevolution from 4-27 to 3-7 is not. Though the model is not perfect, it reproducesfairly well the paths in Table 5. This network of possible transitions also high-lights the recurring role of pitch-class sets 3-7 and 2-3, and to a lesser extent ofpitch-class sets 3-2 and 3-10. It also highlights the peculiar role of pitch-classset 4-27: while most of the pitch-classes are accessible through any path in thenetwork, 4-27 is only accessible from 3-7 α towards the beginning of the time-bracket. If 3-7 α transitions instead to 2-3, 3-10 or 3-2 α , there is no possibility ofhearing 4-27 in the time-bracket anymore. The first time-bracket therefore of-fers two alternate paths, contributing to the rich dynamics of possibilities duringperformance.The same analysis can be carried out for the other time-brackets, thoughit is likely that the complexity of time-bracket 2, for example, would make itdifficult to identify the most probable paths. Four
Four
The Number Piece
Four was composed by Cage during the same year as
Five .This Number Piece is written for a string quartet, and is dedicated to theArditti Quartet [11]. The piece consists in three sections A, B and C of fiveminutes each. Each section contains a part for each member, and each partcontains ten time-brackets, with the exception of one part in section A in whichtwo time-brackets have been merged into one. Among the ten time-brackets,one of them is always fixed. The parts in each section are not associated withan instrument in particular, as they can be played by any of the players. A18igure 6: Model for pitch-class set evolution in the first time-bracket of
Five .This model represents only the most probable transitions between pitch-classsets. Lines of reduced thickness indicate less probable transitions. Single soundsand silence have been omitted at the end of the model. Pitch-class set 4-27 mayevolve to 3-7, in which case the only evolution possible is 2-3.performance of
Four should last either 10, 20 or 30 minutes. If the performancelasts 10 minutes, all players play section B, exchange their parts, then play Bagain. If a performance duration of 20 minutes has been chosen, players willplay sections A and C without pause, exchange their parts, then play sectionsA and C again. In the last case, players should play sections A, then B and C.
Using the same methodology as for
Five we study the distribution of possiblepitch-class sets in each section, the sections being treated independently fromone another. Since no individual time-bracket contain chords in any of thesections, the cardinality of the possible pitch-class sets is therefore limited to 4,which encompasses 49 different pitch-class sets from silence to tetrachords.The heatmaps of the probabilities
P r ( P CS t = i ) calculated over the 49possible pitch-class sets (in ordinate) at each time t (in abscissa) is presentedfor each section on Figures 7, 8 and 9. As before, the colorbar is given in pseudo-logarithmic scale. Similarly to Five , we can readily identify the location of eachtime-bracket in these heatmaps, as their sonic content is well separated fromeach other.Weisser has emphasized about the level consonance heard in
Four , focusingon major/minor triads (pitch-class set 3-11) and seventh chords, among whichthe dominant seventh (4-27, which also corresponds to the half-diminished sev-enth), the major seventh (4-20), and the minor seventh (4-26). The plots pre-sented here allow to quantify this level by looking at the corresponding proba-bilities of occurence.Section A is characterized by the scarcity of major/minor triads, as pitch-class set 3-11 is absent from six time-brackets, barely present in one, rare intwo, and prevalent in the last time-bracket. Incidentally, this last time-bracket19lso exhibits moderate probabilities for pitch-class sets 4-20 and 4-26, thoughthere is a higher probability of obtaining the all-interval tetrachord 4-z15. Thetetrachords 4-20 and 4-26 are absent from all other time-brackets, and dominantsevenths occur only rarely in time-brackets 5 and 9.The situation is identical for section B, in which major/minor triads areonly prevalent in the fixed time-bracket, more rarely heard in time-bracket 9,and barely present in the remaining four time-brackets where they occur. It isinteresting to note that the possible minor triad which can occur in the fixedtime-bracket as been categorized as part of a ”thwarted triad/seventh chord” byWeisser (see Example 10.c in [2]), missing the possibility that this minor triadcould exist on its own if player 4 has started playing its second pitch before anyother player. Pitch-class set 4-26 is absent from the entire section, while 4-27has only two low-probability occurences, and 4-20 only one.Section C stands again the other sections, given the greater possibilitiesof hearing pitch-class set 3-11, with occurences in seven out of the ten time-brackets. However, pitch-class set 4-20 is virtually absent from the section,similarly to pitch-class set 4-26. This is almost the same for pitch-class set 4-27,except for the last time-bracket where it is prevalent.
We have shown in this paper that a statistical approach of the possible pitch-class set in the Number Pieces of John Cage, drawn from previous studies onsingle time-brackets, allows to analyze at once the distribution of sonic contentduring a performance and to quantify the relative probabilities of occurence ofthe chords in each time-bracket. The analyses proposed in this paper rely onthe hypotheses which were exposed in the methodology section regarding theprocedures used for selecting time marks for the time-brackets. A number ofissues can therefore raised concerning this particular model: • The outer limits of the time-brackets are chosen through a random se-lection within the given intervals. We have discussed in [6] the possibleshortcomings of this approach, among which the fact that it may notrepresent human behavior accurately, since humans are poor random gen-erators and are influenced by the surrounding stimuli as well as previousevents. • The selection of multiple pitches inside a time-bracket is made through asuccession of random time-mark choices. Again, this might not be repre-sentative of human behavior. We have argued in [5] that two conceptionsof time, based either on real time measurements or on time differences(durations), compete in the Number Pieces, and a selection based on du-rations may be more appropriate. • Parts belonging to different players are treated independently. However,musicians are very likely to listen to each other during a performance of20igure 7: Heatmap of the probabilities
P r ( P CS t = i ) over the 49 possiblepitch-class sets (in ordinate) at each time t (in abscissa) in Four , section A. Thecolorbar indicates the corresponding probabilities in pseudo-logarithmic scale(see text). 21igure 8: Heatmap of the probabilities
P r ( P CS t = i ) over the 49 possiblepitch-class sets (in ordinate) at each time t (in abscissa) in Four , section B. Thecolorbar indicates the corresponding probabilities in pseudo-logarithmic scale(see text). 22igure 9: Heatmap of the probabilities
P r ( P CS t = i ) over the 49 possiblepitch-class sets (in ordinate) at each time t (in abscissa) in Four , section C. Thecolorbar indicates the corresponding probabilities in pseudo-logarithmic scale(see text). 23 Number Piece and to take different decisions depending on what theyperceive.The modelisation of the Number Pieces is therefore an open problem, whichwould benefit from a more thorough investigation of actual musician’s behavior.In particular, it would be interesting to see if modifications in the above points(for example taking into account cooperative playing) would influence the resultspresented above and to what extent. Note however that the approach used herehas the advantage of simplicity for computer implementation, and has also beenused for the automated computer generation of performances of the NumberPieces ([12]).Finally, we wish to underline that we have considered here the probabilitydistributions of the possible pitch-class sets without discussing their perceivedconsonant, dissonant or even tonal nature. Parncutt ([13]) has emphasized thefact that pitch-class sets may have tonal implications, drawing from the workby Krumhansl ([14]) on the perceived nature of musical pitch. Since we haveaccess to the instantaneous pitch-class content of the Number Pieces and byusing Krumhansl key-finding algorithm ([14], [15]), it would be interesting totrack the possible keys evoked throughout a performance of a Number Piece, andeven to study the distribution of these keys over a large number of realizations.
References [1] B. J. Weisser, ”Notational Practice in Contemporary Music: A Critique ofThree Compositional Models (Luciano Berio, John Cage and Brian Ferney-rough)” (Ph.D. dissertation, City University of New York, 1998), pp. 82-83.[2] B. Weisser, ”John Cage: ’... The Whole Paper Would Potentially Be Sound’:Time-Brackets and The Number Pieces (1981-92)”, Perspectives of New Mu-sic, 41(2), pp. 176-225.[3] R. Haskins, ”An Anarchic Society of Sounds: The Number Pieces of JohnCage” (Ph.D. dissertation, University of Rochester, New York, 2004), pp.245.[4] J. Cage and J. Retallack, ”Musicage: Cage Muses on Words, Art, Music.John Cage in conversation with Joan Retallack”, ed. Joan Retallack, Hanover,NH: University Press of New England, Wesleyan University Press, 1996, p.108[5] A. Popoff, ”John Cage’s Number Pieces: The Meta-Structure of Time-Brackets and the Notion of Time”, Perspectives of New Music, 48 (1), pp.6584[6] A. Popoff, ”Indeterminate Music and Probability Spaces: The Case of JohnCage’s Number Pieces”, Proceedings of the Mathematics and Computationin Music - Third International Conference, LNAI 6726, Springer, 2011, pp.220-229 247] D. Starr, ”Sets, Invariance and Partitions”,
Journal of Music Theory , 22(1), pp. 142[8] A. R. Brinkman, ”Pascal Programming for Music Research”, University ofChicago Press, 1990, p. 629[9] A. Forte, ”The Structure of Atonal Music”, New Haven and London: YaleUniversity Press, 1973[10] John Cage, ”Five”, New York: C.F. Peters, 1988, EP 67214, performancenotes[11] John Cage, ”Four”, New York: C.F. Peters, 1988, EP 67304, performancenotes[12] B. Sluchin, M. Malt, ”A computer aided interpretation interface for JohnCages number piece Two ”, Actes des Journes dInformatique Musicale (JIM2012), Mons, Belgique, 2012[13] R. Parncutt, ”Tonal Implications of Harmonic and Melodic T nn