Polynomial Functions in Zuraj’s Changeover · 2020. 8. 26. · 1 Introduction When listening to...

Polynomial Functions in Zuraj’s Changeover

Ted Moore

September 1, 2019

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Contents

1 Introduction 41.1 Zuraj on his M-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Analysis 52.1 Similarities in Temporal Densities across Parts and Phrases (mm. 1-16) . . . . . . . . . . . . 52.2 Polynomial Functions across Pitch and Time (mm. 36-59) . . . . . . . . . . . . . . . . . . . . 62.3 Linear Functions Around an Axis of Symmetry (mm. 69-75) . . . . . . . . . . . . . . . . . . . 82.4 Comparing Pitch Space and Frequency Space (mm. 82-86) . . . . . . . . . . . . . . . . . . . . 8

3 (Re)Synthesis of Note Information 9

4 Conclusion 9

References 11

Appendices 12

A Figures 12

B Rendered MIDI Data 42

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List of Figures

1 Screenshot and picture of Zuraj’s M-Matrix software. . . . . . . . . . . . . . . . . . . . . . . . 122 Rhythms mm. 1-16 organized into distinct phrases. . . . . . . . . . . . . . . . . . . . . . . . . 133 Durations between adjacent notes in percussion phrases mm. 1-16. . . . . . . . . . . . . . . . 134 Percussion phrases mm. 1-16 sorted by type. . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Cubic equations fit to graphs in Figure 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Note distance (in beats) from beginning of phrase (percussion phrases mm. 1-16) . . . . . . . 157 Cubic equations for curves fit to graphs in Figure 6. . . . . . . . . . . . . . . . . . . . . . . . 168 Small ensemble groups (mm. 36-59) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Polynomial formulas for curves fit to data in Figure 8. . . . . . . . . . . . . . . . . . . . . . . 1610 Function fit to temporal density curves 2, 4, and 5. . . . . . . . . . . . . . . . . . . . . . . . . 1611 Curve 4a, microtonal descending line in harp and piano. . . . . . . . . . . . . . . . . . . . . . 1712 Curves 4 and 5, microtonal lines in harp and piano. . . . . . . . . . . . . . . . . . . . . . . . . 1813 Duration between notes for all curves in Figure 8 . . . . . . . . . . . . . . . . . . . . . . . . . 1914 All curves in Figure 8 with durations normalized. . . . . . . . . . . . . . . . . . . . . . . . . . 2015 All curves in Figure 8 with durations normalized and reoriented. . . . . . . . . . . . . . . . . 2116 Musical Durations expressed as decimals of a bar . . . . . . . . . . . . . . . . . . . . . . . . . 2217 Function fit to temporal curve in Figure 21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2218 Function fit to pitch space vs. time curve in Figure 22. . . . . . . . . . . . . . . . . . . . . . . 2219 Four of the curves plotted as a measurement of temporal density. . . . . . . . . . . . . . . . . 2320 Duration between notes in Curve 1 plotted by part. . . . . . . . . . . . . . . . . . . . . . . . . 2421 Duration of each note from the beginning of the Curve for each part in Curve 1. . . . . . . . 2522 Pitch space motion of each part in Curve 1 through time. . . . . . . . . . . . . . . . . . . . . 2623 Pitch space distance from fit curve for each note in Curve 1. . . . . . . . . . . . . . . . . . . . 2724 Temporal distance from fit curve for each note in Curve 1. . . . . . . . . . . . . . . . . . . . . 2825 Average duration (measured in bars) from notes in score to the function fit to the collection

of notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2926 Motion through pitch space and temporal density for small ensembles mm. 69-75. . . . . . . 2927 Small ensemble trajectories for mm. 69-75. Dots showing only the first note of each measure. 3028 Small ensemble trajectories as a function of distance from their starting pitch . . . . . . . . . 3129 Small ensemble trajectories as a function of absolute distance from their starting pitch . . . . 3230 Average pitch for each chord confirming axis of symmetry. . . . . . . . . . . . . . . . . . . . . 3331 Half steps between adjacent voices for parts mm. 69-75. . . . . . . . . . . . . . . . . . . . . . 3432 Temporal density of notes mm. 82-86. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3533 Increase in temporal density over time beginning in m. 82. . . . . . . . . . . . . . . . . . . . . 3534 Fit curves in pitch space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3635 Quadratic formulae fit to the data in Figure 34. . . . . . . . . . . . . . . . . . . . . . . . . . . 3636 Fit curves in frequency space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3737 Step sizes between notes for each staff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3838 Step sizes between notes for each staff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3939 Python script used to generate Figure 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4040 Pitch space plot of rendering from Python script (Figure 39) . . . . . . . . . . . . . . . . . . 41

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1 Introduction

When listening to Changeover, an orchestral piece composed by Vito Zuraj in 2011, one hears trajectoriesof motion that span over many measures and often over many different instruments in the ensemble. Thesetrajectories are across the dimensions of time, pitch, and temporal density (i.e., durations between notes). Inthis analysis, I plot and investigate these trajectories in search of a deeper understanding of their structure.Often, what can be clearly heard by a listener is not clearly seen in the score because of the quantization in thetime and pitch domains required for representation in musical notation. Furthermore, the traditional layoutof orchestral scores does not offer a clear time-to-pitch graphical interpretation. This paper analyzes most ofthe material in mm. 1-86 of Zuraj’s Changeover, which is presented here in chronological order. The analysisis limited to these measures as they include the music that has the most audible trajectories. Each trajectoryanalyzed was initially identified through listening. Then, for analysis, each trajectory was abstracted throughmanual data entry into code or spreadsheets, before being processed in Python using the Numpy, Scipy, andMatplotlib libraries. The analysis of each section was driven by the musical materials themselves, pursuedas the author felt they implied further investigation. These analyses first seek to visually clarify the sonictrajectories being created by Zuraj, thereby allowing for a deeper investigation and understanding of theirconstruction. These investigations are often made by fitting polynomial functions to the trajectories heardin the piece and using these functions to theorize how Zuraj used algorithms to compose the trajectories. Itis possible that some of the trajectories heard in the piece were not composed algorithmically, but insteadcreated intuitively by the composer. My analysis seeks to answer how Zuraj may have composed themalgorithmically. Some of the analyses suggest more strongly than others that the trajectories in questionwere composed algorithmically.

1.1 Zuraj on his M-Matrix

Much of the methodology and heuristic choices made in this analysis come from a conversation I had withZuraj about his compositional process. He explained why he uses algorithms when composing, saying,

In the normal process of composing, there are way too many decisions to be made...Some decisionsare secondary, like if there’s an eight note chord and two of the voices have to move, whether itmoves to the violin or the english horn is secondary...The technology is a way to free myself ofthe tons of calculations that would drive my main idea away. When I’m composing, if I have amain idea and then get disturbed by some technical thing, it would get me off road. The computeris a performer of large amounts of basic calculations. [1]

The framework in which Zuraj does his calculations is custom software made in Max/MSP, which he callsM-Matrix (Figure 1) [1]. He says that it, “has the benefits of Open Music1, but in Max”.[1] When usingM-Matrix, Zuraj can input MIDI data through a MIDI keyboard, which then is processed or manipulatedin various ways before being sent to Finale notation software via internal MIDI ports as MIDI data [1].Explaining why this workflow is important to him he says, “I am a pianist, I like to improvise, when Iwanted to play something I had to do it by ear–this training helped me a lot to develop spontaneous flow ofmusic–to control the flow of music”.[1] His interest in real-time, expressive data entry is further expressedin his plan for a future development of M-Matrix: “I wish I had more time to work on voice input into thesoftware. That’s one of the next projects. Every piece I write needs to have one technical improvement”.[1]

During our conversation, Zuraj emphasized the use of randomness in his algorithms saying, “I startedsimple experiments with arpeggios, I found that there are some aspects that work best with randomness.It freed me from some of the decision making that needed to be done...There can also be some side effectideas from the choices that the computer makes.” In addition to randomness he also often works with thetemporal density of notes [1]. In reference to the piece being analyzed below, Zuraj stated, “M-Matrix is thehelper. Something like the pizz clouds in Changeover, it’s much easier to manage the parameters in that”[1].

When I asked whether his use of technology is or is not in dialog with traditions of the past he responded,

1Open Music is the IRCAM-based visual programming language for manipulation of musical materials.

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The early masters of algorithmic composition, like Xenakis–the early machines were very complexto build, you could not have it at home. They would have to wait weeks to get the result. Andwhen it came back it would be numbers, they had to work with numbers. The calculation speednow is enormously faster. Xenakis would make one try and it would maybe be not so good, thenhe would adjust and make a 2nd try and it would be better, but he’d have a deadline so he’d onlyget two tries. I have so many tries...I feel just as connected to contemporary music as traditionalmusic. You can learn so much from that music...Vision combined with Knowledge. This is agoal. Educate myself in both directions. Not only experimental, not only classical. [1]

Hearing how Zuraj uses his M-Matrix to compose led me to consider a few guidelines before beginning myanalyses of the trajectories I heard in Changeover. First, I would focus on their pitch and temporal durationtransformations because he said these are parameters he often works with. Secondly, I would try to fit thedata to lower order polynomial curves, as that seems an appropriate level of conceptual and computationalcomplexity for the tools he is employing in his M-Matrix. Lastly, I would be prepared to find elements ofrandomness in my results and attempt to quantify or qualify that randomness when possible.

2 Analysis

2.1 Similarities in Temporal Densities across Parts and Phrases (mm. 1-16)

The opening of Changeover is a quartet of percussionists playing suspended string drums with an idiosyncraticnotation indicating different locations and techniques for scratching and striking the spring with a spoon.The listener perceives these percussionists as swelling in density and dynamics, and upon looking at the scoreone can see clearly outlined gestures that match crescedos and decrescendos with rhythmic shapes that willsound as though they are accelerating or decelerating. In order to analyze these rhythms I separated theminto distinct phrases (Figure 2). I found that up to the first pause in the music (m. 17), each of the fourpercussion parts had four phrases. I recorded the duration between each of the adjacent notes in each phrasein order to get an understanding of the temporal density of the notes (shorter durations would create higherdensities). My initial plot of this data (Figure 3) was difficult to decipher. I didn’t immediately perceiveany patterns, so I decided to reconfigure the data to try to understand it better. First, I normalized allof the data so the curves these phrases create could all be plotted as the same length and height.2 Also,I roughly sorted these phrases into three types: type 0 represents phrases that begin with long durations,move to shorter durations, and back to longer durations, type 1 represents phrases that begin with shortdurations and move to longer durations, and type 2 represents phrases that begin with long durations andmove to shorter durations (Figure 4). Many of the type 1 gestures seemed to resemble the type 2 gestures,only reversed, so I decided to create a plot which contains all the type 2 gestures and a selected number oftype 1 gestures reversed. Once categorized and plotted in these ways (Figure 4), it became easier to see thatthere does seem to be similarities between some of the shapes. In an effort to quantize these similarities Ifit each of these graphs to a cubic function (Figure 5) and plotted it in black on top of the curves.

Seeing the similarities in these shapes and their relationship to these polynomial functions began tosuggest how Zuraj may have algorithmically generated these percussion phrases. In particular, many of thetype 0 gestures had very similar curves, perhaps suggesting their shared derivation from a root formula. Theidiosyncratic deviations could be explained by the process of rhythmic quantization necessary for musicalnotation and/or the added randomness that Zuraj has discussed. Very few of the type 1 and 2 curves aremonotonic, perhaps demonstrating a feature of Zuraj’s formula. Although these are all fit as cubic functions,one can see by the quadratic shape of type 0 and the linear shape of type 1 (along with the relative smallnessof the function’s higher order coefficients), that the forumlas Zuraj may have used to derive these particular

2I will do this often in my analyses; although this does change the data so that it no longer directly represents the musicit is drawn from, it can be a good way of determining if the curves all come from the same base function, the result of whichcould easily be scaled in any or multiple dimensions. Normalizing is a way to “undo” any scaling that may have taken place,hopefully revealing similarities in the curves.

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varieties of gestures would likely be lower order polynomials, again their deviations explained by eitherquantization or added randomness.

One more way of plotting these phrases was to plot the position of each note from the beginning of thephrase (fig04). These graphs again suggested dividing these phrases in three types (shown by the threegraphs), however they were not precisely the same groupings as suggested by the first set of graphs. Thesegraphs are not normalized, therefore showing a less mediated representation of the music. The similarity ofmany of these curves is interesting, and therefore has potential in describing their construction. In particular,it can be more easily seen that the curves in the top graph all have a slight boost in density between about5 and 7 beats after the start of the phrase (although the cubic function that maps onto these curves is muchless closely related to the curves than the curves are to each other). I consider the cubic functions thatfit to these graphs (Figure 7) as good candidates for explaining the calculations Zuraj may have used tocompose these phrases. Overall, the similarity of certain groups of phrases makes an argument for thembeing derivations a few base curve shapes.

2.2 Polynomial Functions across Pitch and Time (mm. 36-59)

This passage has many audible trajectories through pitch space that can be heard as one long phrase spanningmm. 36-59. These oscillations are reminiscent of a low frequency oscillator often found in electronic music,which modulates a parameter (in this case pitch) slowly and audibly through time. Upon close listeningone also hears the temporal density of notes changing through time, although this is sometimes obscured byother orchestral shapes and timbres.

When plotting this material I chose to use dots to represent short attacks (which comprises most ofthe material in this section) and solid lines for sustained notes (Figure 8). Each note was recorded as itsstart position, duration, and pitch, measured down to the quarter tone and the 100th of a beat (sixteenthnotes increment at 0.25, triplets at 0.33, etc.). The harp and piano that are tuned down a quarter-tonewere entered into the spreadsheet a quarter-tone lower than written (i.e., at sounding pitch). I saw fiveclear curves that warranted further investigation (Figure 8). The most striking result from this plot was theclarity of the curves, especially 1, 2, and 4.3 The polynomial functions that were fit to these curves make avery strong case that these were created algorithmically using the software that Zuraj has described. Onecan easily see how Zuraj would have created an algorithm that would modulate pitch as a function of time.

A few smart orchestrational choices can be observed here. First, all of the notes in curves 1 and 2 areplayed by string instruments, which arguably have more flexibility in playing quarter tones across their entirerange, affording more precision in the realization of his algorithmically produced curves. The second is theuse of a piano and harp tuned down a quarter tone alongside a normally tuned piano and harp. These fourinstruments play all of the notes in curves 4 and 5. The first four notes of curve 4 create a downward shapethat sounds harmonically disorienting through its use of quarter tones (Figure 11). In particular the first fourintervals (all downward) are ¾ tone, ¼ tone, ¼ tone, and ¾ tone. Beginning this larger descent shape (curve 4)with these less common intervals and using them throughout curves 4 and 5 is an effective way to draw thelistener’s ear to directed motion in pitch space and register, rather than harmonic implications. Pairing thesenormally tuned instruments with their quarter tone counterparts enhances this effect by presenting timbresthat are almost always heard in equal temperament playing quarter tones, thereby creating a compositeinstrument seemingly capable of playing equal temperament and quarter tones simultaneously.4

Observing that Zuraj likely created an algorithm that modulated pitch as a function of time led meto wonder how he chose when notes would occur in time. One can see in Figure 8 that there are placeswhere the pitch curves continue to move steadily through time, but the notes get more and more spaced out.Similar to my analysis of the percussion gestures at the piece’s opening, I plotted the durations between thenotes in each gesture to view their temporal density (Figure 13). Curves 2, 4, and 5 all have a relativelyclear shape to them that seems to reflect the steady change in density seen in Figure 8, however, curve 1 has

3Curves 2 and 4 were broken into 2 sub-curves (2a, 2b and 4a, 4b) for fitting their time (measures) vs. pitch functions.4Spatially both quarter tone instruments are on stage left, separated from their counterparts on stage right, thereby possibly

weakening the composite instrument effect. I investigated if Zuraj had considered composing with this spatialization as he doeselse where (see below), but was unable to find any meaningful pattern in the spatialization of piano and harp notes (Figure 12).

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a chaotic shape that does not seem to reflect the change in density seen in Figure 8. Before investigatingcurve 1 further, I decided to determine if there were similarities that could be drawn between the other threecurves. First I normalized the three curves (Figure 14) and then reversed curve 4 (figure 14) to determinehow similar the curves may be. The quadratic curve that fits to these (figure 10) does not align in a waythat makes a compelling argument for it being a common formula from which all of these curves are derived.

A more interesting analysis of these curves can be seen by noticing smaller similarities. There are manymoments where the curve plateaus, indicating that for several notes, the duration between notes does notchange. These are interleaved with moments of ascent, where the durations change. When comparing thesecurves to the notes that comprise them in the score, one can see where these moments of plateau happen.Looking in Figure 13 at the distances between notes (measured in bars) of curve 5, one can see plateausat about 0.12 (eighth notes), 0.08 (eighth note triplets), 0.16 (quarter note triplets), and 0.34 (half notetriplets), which can all be seen in sequence as distances between notes in measures 59-57 (beginning in piano2 and moving up through harp 1). In curve 4 (Figure 13) one can similarly see plateaus at 0.17 (quarter notetriplet), 0.13 (eight note), 0.08 (eighth note triplet), and 0.06 (sixteenth note). It is clearly not a coincidencethat these values are where the plateaus happen; these are artifacts of the temporal quantization inherent inmusical notation. If these curves were able to be calculated and performed without being filtered throughthis notation, they very well may not include plateaus, but instead may align on much more regular curves.This observation demonstrates one difficulty with these analyses: the jitter introduced by quantization makesit challenging to compare curves.

Continuing my analysis of the chaotic nature of curve 1 (Figure 13), I plotted curves 1, 2, 4, and 5 as afunction of time and note number in the curve to see if a clear pattern between the curves could be perceived(Figure 19). Because the nature of this type of graph is necessarily monotonic, these curves look smoother,but one can still see a jitter in curve 1 that is not present in the other curves. Next I chose to separatethe notes in curve 1 by part and plot each separately as a function of note number in the curve and theduration to the next note (Figure 20). This makes for a somewhat clearer picture, but the wide diversity ofrepresentations does not yet suggest an underlying algorithmic approach to the construction of curve 1. Theanalysis that convinced me that the staves in curve 1 all shared a base equation was plotting the time sincethe beginning of the curve against the note number in curve (Figure 21), which clearly shows a manifold ofdata moving through these dimensions (the quadratic fit to this manifold is Figure 17). Upon reflecting onthis curve I decided to also plot each staff as a function of time and pitch space (Figure 22) (as the curvewas originally seen in Figure 21). Unsurprisingly, this also clearly showed a manifold curve with a very wellfit quadratic function (Figure 18). The average distance in pitch space from the fit curve to where a noteappeared in the score was about 0.36 half steps, well under the precision allowed by the quantization of thenotation system Zuraj is using (rounding to the quarter tone, or 0.5 half steps) (Figure 23). The averagedistance in time (measured in bars) from the fit curve (Figure 25) to where the note appeared in the scorewas about 0.032 or about a 32nd note (Figure 24) , a smaller division of the beat than what appears in anyof these curves, implying that these deviations are also within the margin of quantization required for thenotation restrictions Zuraj was using.5

Another interesting observation that comes out of Figures 23 and 24 is the strict sequencing of parts. Allthese notes come from string instruments that are part of smaller groups of instruments spatialized aroundthe audience. The notes cycle through staves 13, 8, 1, 5, 10 (respectively: viola, in group 5; cello 2, ingroup 3; violin 1, in group 1; cello 1, in group 2; and violin 2, in group 4). Comparing the groups of theseinstruments with the spatialization scheme laid out in the score reveals that this sequencing of notes creates arotation of pizzicati around the audience, beginning in the very back behind the audience, moving clockwisefor eight rotations, and ending again in the back of the concert hall. By looking at the change in durationbetween notes for staves in curve 1 (Figure 20), one can see that generally as the curve goes on, the notes getfarther apart (in every part and as a group [Figure 19]) meaning that the rotation gesture Zuraj has createdwith curve 1 not only goes down in pitch through time, but also slows down its rotation through time.

5The largest deviation in pitch from the fit curve was about 1.038 half steps; the largest deviation in time from the fit curvewas about 0.45 beats (both larger that the quantization margin), suggesting that the curve discovered is not exactly the oneused, or that some notes were adjusted based on some other impetus).

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Considering the pitch range of this gesture and the instruments used to create it, it’s worth observing thatZuraj’s may not have programmed the consideration of range into the algorithm (making sure an instrumentis not asked to play a note out of it’s range). The lowest note used in this passage (curve 1) is G3, which isthe lowest note that the highest instrument can play (violin). Perhaps he programmed this note to be thelowest possible note in the curve shape created by these instruments in order to ensure that when the notesbecame low in the curve, none would go out of range of any of the instruments.

2.3 Linear Functions Around an Axis of Symmetry (mm. 69-75)

A flurry of activity (again from the string instruments in the small ensembles surrounding the audience)begins at m. 69. The plectrum pizzicati begin in rhythmic unison for the first two beats before taking onglitchy, stuttery rhythms for the rest of the passage, slowly decreasing in temporal density through m. 75.The most striking sonic observation about this passage is the sense of these plectrum lines moving bothup and down. A closer analysis made by plotting the starting and ending points of these instrumentaltrajectories (Figure 26) immediately shows a symmetry between the lines. It clearly shows three voicesmoving up, three voices moving down, and all voices converging from a six octave range to a two octaverange over the seven measures. The bottom graph of Figure 26 shows the decrease in the rhythms’ temporaldensity over the same duration, with a linear function (black line) fit to the data.

The symmetrical motion in pitch space warranted further investigation, leading to Figure 27, which plotsthe first pitch of each of these measures, and Figure 28 which plots the total distance traveled by each voicein this section (voices that descend over the 7 bars move downward in pitch space and vice versa). In order tobe sure that the six lines are symmetrical to their inverse counterparts, I next plotted the absolute distancefrom the starting pitch (Figure 29). The six symmetrical lines now appear as three groups of two, confirmingtheir symmetry. Figure 30 shows the average pitch of each of the first pitches in each measure. The (verynearly) horizontal line these averages create shows that the axis symmetry is maintained at B3 through allseven measures.

One final angle at which look at these lines is the number of half steps between adjacent voices as eachpart moves through the seven measures. Figure 31 shows that all of the intervals between adjacent voicesbegin at 14 half steps (in m. 69) and linearly decrease to 4.5 or 5 half steps in m. 75. This section strongly tiestogether trajectories across pitch space, different instrumental lines, temporal densities, and time, makingit a clear demonstration of Zuraj’s algorithmic approach and providing a strong insight into the parametersand techniques being used.

2.4 Comparing Pitch Space and Frequency Space (mm. 82-86)

Beginning at m. 82, all of the string instruments in the small ensembles circling the audience begin playingin rhythmic unison with a clear increase in temporal density over time. Plotting just the density of thisrhythm (ignoring pitches for now) shows a clear curve that very nearly fits onto a quadratic function (Figure33) (Figure 32). The maximum deviation of any note from this fit curve is about 0.4 beats, while the averagedistance from the curve is 0.147 beats, or slightly shorter than a sixteenth note triplet.

Plotting the pitches against their temporal position (Figure 34) also shows clear relations to quadraticequations (Figure 35). The exact rhythmic similarity and similar pitch contours in this section present anopportunity to compare the curves in both pitch space and frequency space6, which may reveal one or theother being the more likely domain in which Zuraj’s algorithm was programmed. The frequency space curves(Figure 36) are much less similar visually and do not have closer fits to polynomial functions, making pitchspace the much more likely domain that Zuraj was working in.

Given the similarity of the curves in pitch space, I set out to investigate if these six lines were actuallyrough transpositions of one single line. In order to test this, I plotted the interval between adjacent notesin each line (Figure 37). If these were exact or rough transposition, one would expect to see a similar basicpattern in intervals between adjacent notes across all staves. Instead there is a variety of intervals, with no

6Pitch space being a log2 representation of frequency space.

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clear pattern, indicating that these lines are not all transpositions of one base line. This graph does howeveroffer a different interesting observation: while most of the staves are always increasing by a half step or¾ tone, two of the staves (8 [cello, group 3] and 15 [contrabass, group 5]) never increase by a ¾ tone, andsometimes increase by only a ¼ tone. Logically this means that over the course of their curve those staves willnot travel as far (in pitch space) because their steps tend to be smaller. Because these are two of the lowerinstruments involved and they traverse the least in pitch space, I wondered if the intervals between adjacentvoices of the chords created by the the tutti rhythms changed over time. Figure 38 shows that, although notmonotonic, they all do increase over time. Accounting for their deviations by quantization, these intervalsneatly fit to a linear function (higher order coefficients are not necessary), suggesting that this relationshipdiscovered is not coincidental; this supports the premise that Zuraj used quadratic functions of pitch overtime (this linear function’s integral) to derive the curves in mm. 82-86.

3 (Re)Synthesis of Note Information

As an exercise in further understanding the processes Zuraj is using, I set out to algorithmically render noteinformation that imitates a passage of Changeover. The passage I chose is mm. 69-75: the six part linearmotion around an axis of symmetry. Even though I could have drawn precise measurements from the scoreto recreate the passage, I chose parameters more intuitively to imitate Zuraj’s process (also, being too precisewould verge on data entry, which I’ve already done). The few precise measurements I did use were the axisof symmetry (B3), the number of voices (6), and the general shape of transitioning from a 6 octave rangedown to a 2 octave range over about 7 measures. Using Python (Figure 39) I was able to generate a setof notes that have the same pitch symmetry, and decreasing temporal density over the 7 measures (Figure40). Each dot is a note (in Zuraj’s case would be pizzicato on a string instrument) rounded to the nearestquarter tone and sixteenth note.7

The strategy I used was breaking every measure up into 16 slices and creating for each slice (1) aninterpolated midi value based on the pitch trajectory of each staff and (2) a probability of there being a noteplaced in that slice (as the 7 measures proceed, the probabilities decrease). For each of the 6 staves, I theniterated through each slice and using that slice’s probability, randomly chose whether or not to place a notethere (higher probabilities [i.e., earlier in the section] are more likely to have a note). If a note is placed, Ilookup what midi value it should have based on the pitch trajectory for that staff. The result (Figure 40)looks very satisfactory as it strongly matches the symmetry of pitch trajectories and maintains the decreasein temporal density over the 7 measures. The midi value and temporal location of each note is listed inAppendix B, the data of which could easily be written to a midi file in Python and loaded into a notationsoftware program such as Sibelius.8

4 Conclusion

The experience of listening to Zuraj’s Changeover is marked by following lines aurally through pitch spaceand hearing the densities of notes build and dissipate in smooth transitions. These orchestral shapes echo thealgorithmic composers of past decades, leading one to consider if they were created with similar processes.Zuraj considers himself an algorithmic composer, his main tool being M-Matrix, the custom software hecreated in Max/MSP to support his creative workflow. By plotting and analyzing orchestral momentsin Changeover, one can come to a deeper understanding of the algorithms being used, in particular, thepolynomial curves guiding pitches and temporal densities through time. One of the challenges discovered inthese analyses is the jitter introduced into the data by the quantization in the pitch and temporal domainsnecessary for musical notation. Zuraj’s interest in injecting randomness into his processes may obscure some

7Zuraj’s score also uses triplets; it would be very easy to create a condition in this code that could move some notes to alsocreate triplet groupings.

8The midi protocol cannot represent quarter tones, so a more useful strategy may be to create a music XML file using aPython library such as music21 and labeling quarter tone deviations by adding a “lyric” to those notes indicating quarter toneup or down. Quarter tone symbols would have to be entered manually in the notation software.

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underlying formulas and patterns by creating families of curve shapes that are not similar enough for analysisto thoroughly detect. Overall, the clarity of many polynomial curves in this work, the similarities of manyphrase shapes across different parts, and the successful resynthesis of a passage of Zuraj’s work demonstrateswhere and how Zuraj likely used algorithms in his creative process composing Changeover.

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References

[1] Ted Moore. Interview with Vito Zuraj. Jan. 2019.

[2] Vito Zuraj. Changeover. Music Score. Ljubljana, Slovenia: Edicije DSS, 2011.

[3] Vito Zuraj. Changeover. Audio Recording. Mainz, Germany: Wergo Germany, 2015.

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Appendices

A Figures

Figure 1: Screenshot and picture of Zuraj’s M-Matrix software.

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Figure 2: Rhythms mm. 1-16 organized into distinct phrases.

Figure 3: Durations between adjacent notes in percussion phrases mm. 1-16.

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Figure 4: Percussion phrases mm. 1-16 sorted by type.

Figure 5: Cubic equations fit to graphs in Figure 4.

Type Fittype 0 y = 0.0001592x3 + 0.01117x2 − 0.1841x + 0.8744type 1 y = 1.784× 10−5x3 − 0.0001233x2 + 0.06809x− 0.01543type 2 y = −0.003153x3 + 0.0706x2 − 0.4589x + 1.14type 2 & selected type 2 inverted y = −0.001768x3 + 0.04152x2 − 0.3104x + 1.092

14

Figure 6: Note distance (in beats) from beginning of phrase (percussion phrases mm. 1-16)

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Figure 7: Cubic equations for curves fit to graphs in Figure 6.

Type Fittype 0 y = −0.007903x3 + 0.1614x2 + 0.2893x− 0.2773type 1 y = 0.003042x3 − 0.1208x2 + 1.759x + 0.727type 2 y = −0.003037x3 + 0.08534x2 + 0.03492x + 0.1636

Figure 8: Small ensemble groups (mm. 36-59)

Figure 9: Polynomial formulas for curves fit to data in Figure 8.

Curve FitCurve 1 y = 0.6856x2 − 58.37x + 1296Curve 2a y = −1.59x2 + 151.1x− 3537Curve 2b y = 1.257x− 6.461Curve 3 y = 1.111x + 12.8Curve 4a y = −1.884x2 + 202.6x− 5347Curve 4b y = −29.6x2 + 3333x− 9.375× 104

Curve 5 y = 29.45x− 1654

Figure 10: Function fit to temporal density curves 2, 4, and 5.

y = 0.001633x2 − 0.03087x + 0.1406 (1)

16

Figure 11: Curve 4a, microtonal descending line in harp and piano.

17

Figure 12: Curves 4 and 5, microtonal lines in harp and piano.

18

Figure 13: Duration between notes for all curves in Figure 8

19

Figure 14: All curves in Figure 8 with durations normalized.

20

Figure 15: All curves in Figure 8 with durations normalized and reoriented.

21

Figure 16: Musical Durations expressed as decimals of a bar

Musical Duration DecimalWhole Note 1Half Note 0.5Half Note Triplet 0.33Quarter Note 0.25Quarter Note Triplet 0.165Eighth Note 0.125Eighth Note Triplet 0.0825Sixteenth Note 0.0625

Figure 17: Function fit to temporal curve in Figure 21.

y = −0.1827x2 + 2.445x + 0.1874 (2)

Figure 18: Function fit to pitch space vs. time curve in Figure 22.

y = 0.6768x3 − 57.66x + 1282 (3)

22

Figure 19: Four of the curves plotted as a measurement of temporal density.

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Figure 20: Duration between notes in Curve 1 plotted by part.

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Figure 21: Duration of each note from the beginning of the Curve for each part in Curve 1.

25

Figure 22: Pitch space motion of each part in Curve 1 through time.

26

Figure 23: Pitch space distance from fit curve for each note in Curve 1.

27

Figure 24: Temporal distance from fit curve for each note in Curve 1.

28

Figure 25: Average duration (measured in bars) from notes in score to the function fit to the collection ofnotes.

y = 0.00315x2 − 0.6024x + 64.37 (4)

Figure 26: Motion through pitch space and temporal density for small ensembles mm. 69-75.

29

Figure 27: Small ensemble trajectories for mm. 69-75. Dots showing only the first note of each measure.

30

Figure 28: Small ensemble trajectories as a function of distance from their starting pitch

31

Figure 29: Small ensemble trajectories as a function of absolute distance from their starting pitch

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Figure 30: Average pitch for each chord confirming axis of symmetry.

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Figure 31: Half steps between adjacent voices for parts mm. 69-75.

34

Figure 32: Temporal density of notes mm. 82-86.

Figure 33: Increase in temporal density over time beginning in m. 82.

y = −0.004789x2 + 0.2972x + 82.31 (5)

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Figure 34: Fit curves in pitch space.

Figure 35: Quadratic formulae fit to the data in Figure 34.

Staff Formula1 y = 92.86x2 − 1.546104x + 6.438× 105

5 y = 36.36x2 − 6051x + 2.519× 105

8 y = 26.36x2 − 4383x + 1.823× 105

10 y = 67.84x2 − 1.129× 104x + 4.701× 105

13 y = 49.78x2 − 8285x + 3.449× 105

15 y = 19.59x2 − 3258x + 1.355× 105

36

Figure 36: Fit curves in frequency space.

37

Figure 37: Step sizes between notes for each staff.

38

Figure 38: Step sizes between notes for each staff.

39

Figure 39: Python script used to generate Figure 40.

import matp lo t l i b . pyplot as p l timport numpy as npfrom midi funcs import midiname

a x i s = 59dur = 6.75 # measuress ta r tPos = 69 # beginning o f m. 69divOfBeat = 4startFromAxis = np . f l i p ( [ −1 , −3 . / 5 . , −1 . / 5 . , 1 . / 5 . , 3 . / 5 . , 1 ] , 0 ) ∗ 36 .0 # 3 8 vas up and downendFromAxis = np . f l i p ( [ −1 , −3 . / 5 . , −1 . / 5 . , 1 . / 5 . , 3 . / 5 . , 1 ] , 0 ) ∗ 12 .0 # 1 8va up and downp o s i t i o n s = np . arange ( startPos , s ta r tPos + dur , 0 . 2 5 / divOfBeat ) #0.25 = 1 beat in 4/4probs = np . l i n s p a c e ( 0 . 8 5 , 0 . 2 , l en ( p o s i t i o n s ) ) ∗∗ 1 .2

p i t c h e s = [ ]f o r i , s t a r t in enumerate ( startFromAxis ) :

p i t c h e s . append (np . l i n s p a c e ( s ta r t , endFromAxis [ i ] , l en ( probs ) ) )

notes = [ ]f o r s t a f f in range ( 6 ) :

ns = [ ]p r i n t (”∗∗∗∗∗∗∗∗∗∗∗ S t a f f {} ∗∗∗∗∗∗∗∗∗∗∗\n ” . format ( s t a f f ) )f o r pos , p in enumerate ( probs ) :

i f np . random . random ( ) < p :beats = p o s i t i o n s [ pos ]measure = i n t ( beats )beat = ( beats − measure ) ∗ 4beat = round ( beat ∗ 100) / 100 .0midi = p i t c h e s [ s t a f f ] [ pos ] + a x i smidi = round ( midi ∗ 2) / 2 .0p r i n t (” Note Meausure : {}” . format ( measure ) )p r i n t (” Note Beat Pos : {}” . format ( beat ) )p r i n t (” Note MIDI : {}\n ” . format ( midi ) )ns . append ( [ pos , midi ] )

notes . append ( ns )

f o r num, s t a f f in enumerate ( notes ) :x = [ n [ 0 ] f o r n in s t a f f ]y = [ n [ 1 ] f o r n in s t a f f ]p l t . p l o t (x , y , ” o ” , l a b e l=” s t a f f {}” . format (num) )

p l t . l egend ( l o c =’ best ’ , numpoints=1)y t i c k s = [24 , 36 , 48 , 60 , 72 , 84 , 96 ]p l t . y t i c k s ( y t i ck s , [ midiname (m) f o r m in y t i c k s ] )p l t . y l a b e l (” p i t ch space ”)p l t . x l a b e l (” measure ”)p l t . g r i d ( )p l t . show ( )

40

Figure 40: Pitch space plot of rendering from Python script (Figure 39)

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B Rendered MIDI Data

∗∗∗∗∗∗∗∗∗∗∗ S t a f f 0

Note Meausure : 69Note Beat Pos : 0 . 0Note MIDI : 95 .0

Note Meausure : 69Note Beat Pos : 0 .25Note MIDI : 95 .0



































Note Meausure : 72

42

Note Beat Pos : 0 .75Note MIDI : 83 .5
















∗∗∗∗∗∗∗∗∗∗∗ S t a f f 1










Note Meausure : 70

Note Beat Pos : 0 . 0Note MIDI : 78 .5













43






















∗∗∗∗∗∗∗∗∗∗∗ S t a f f 2


















44













Note Meausure : 71Note Beat Pos : 1 .75

Note MIDI : 64 .5













Note Meausure : 73











∗∗∗∗∗∗∗∗∗∗∗ S t a f f 3



Note Meausure : 69Note Beat Pos : 0 . 5

45

Note MIDI : 52 .0













Note Meausure : 70



























46








∗∗∗∗∗∗∗∗∗∗∗ S t a f f 4































Note Meausure : 71Note Beat Pos : 2 .25

47

Note MIDI : 43 .0













Note Meausure : 73









∗∗∗∗∗∗∗∗∗∗∗ S t a f f 5





Note Meausure : 69Note Beat Pos : 1 . 0

Note MIDI : 24 .0













Note Meausure : 70

48














Note Meausure : 71














Note Meausure : 72














49

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