# STABILITY SPACE AND THE BELOW-N THRESHOLD: A COMPUTATIONAL APPROACH TO MUSIC ANALYSIS

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Page 1 ï~~STABILITY SPACE AND THE BELOW-N THRESHOLD: A COMPUTATIONAL APPROACH TO MUSIC ANALYSIS Mike Solomon University of Florida Department of Music ABSTRACT Pitch-class stability in early-modern music is associated with concepts of musical "force" as defined by Larson [11], Lerdahl [12,13], Schoenberg [7], Baroni [2], and Arnheim [4]. This paper seeks to quantify degrees of pc stability and force by constructing a notion of stability space - a nexus of the time space and pitch space in which stabilizing and destabilizing musical forces occur. Because these forces cannot be delimited by a clear hierarchical structuring mechanism as proposed by Lerdahl [13], each musical moment necessarily belongs to multiple stability spaces of varying temporal and pitch dimensions. The below-n threshold is an algorithm that finds trends between these dimensions by discerning the maximum-sized time interval containing pc sets of average cardinality n in various works. Because this divisional scheme potentially cuts through events and event collections, it does not comport with traditional musical segmentation procedures as defined by Hanninen [9]. Rather, it uses an approach suggested by Mandelbrot in analyzing the correlation between measurement error and the unit of measurement [15]. Case studies into the music of Scriabin and Schoenberg use the below-n threshold to provide an empirical description of the composers' styles and identify interesting outlier relationships in their oeuvre. 1. THEORETICAL FOUNDATIONS 1.1. Survey of stability This section will develop a working definition of earlymodern pitch-class (pc) stability by first surveying various theoretical constructs of pc stability in commonpractice-era music and then showing how these concepts are nullified and qualified in analyses of the earlymodern literature. It is important to clarify at the outset that, although all of these discussions are bound up with questions of prolongation, this paper will only briefly discuss the relationship between prolongation and the concept of stability space as defined in section 1.2. Rather, by assessing a variety of scholarship on prolongation, it seeks to distill a 'lowest common consensus' on what pc stability is. Larson, in a critique of Lerdahl's prolongational theories [12,13], builds the most complete definition of pc stability in common-practice-era music: To hear a note as unstable means to auralize a more stable pitch to which it tends to move and a path (usually involving stepwise motion) that would take it there, displacing its trace. This is a definition of "contextual stability." [11, 106] Larson later expands the notion of "contextual stability" by arguing that "inherent stability," which he equates with intervals that can be rationalized by low integers, is often contextual insofar as it is modified to comport with the temperament of a pitch system. For example, even though the slightly detuned perfect fifth must be rationalized by higher integers than an augmented fourth with the same root, it is perceived as the more stable interval through the context of common-practice conventions and the even-tempered chromatic scale.' Larson then develops a lexicon of "referential collections" to describe gradations of stability in diatonic tonality. These collections transcend the musical surface of a work at any given moment, encapsulating instead the entire collection of pcs to which a pc (or a pc set) may refer. Musical prolongations that access a wide referential collection such as the diatonic scale (unstable) will presumably narrow to a triad, perfect fifth, or tonic scale degree (stable). Prolongations through different zones of stability, then, determine consonance and dissonance instead of the latter being an a priori on which prolongational practices are based. The prolongational work of Lerdahl invests more in the concept of inherent stability, which he calls "sensory consonance" [12,13][14]. He notes that, while Larson's conflation of inherent and contextual stability is applicable to certain musics (most notably post-tonal works), the octave has always retained a modicum of stability regardless of the musical idiom. Furthermore, one rarely sees a harmonic practice that inverts sensory consonance, such as a language that demanded of the listener the substitution of sevenths for octaves. Lerdahl qualifies this statement, however, in saying that he places "more emphasis on the cognitive status of tonal knowledge than does Parncutt (1989), who seeks a direct link between psychoacoustics and musical organization" [12]. Thus, even in pitch languages that are significantly indebted to sensory consonance, the contextual norms in which they operate cannot simply be intuited or anticipated by those lacking a degree of familiarity with the language. A definition of "stability" in chromatic and atonal music is more tenuous. Lacking a tonic or referential collection of pitches against which other collections can sound 1 Lerdahl (1997) contests this claim for all but the lowest audile range.

Page 2 ï~~stable or unstable, analysts diverge on how pc stability and prolongations can manifest themselves. Straus notes that the majority of analysts have "tacitly acknowledged that prolongation has revealed and can reveal little of solid worth about the deeper structural levels of...posttonal [music]" [16]. A reconciliatory approach, favored by Viisili, is to analyze stability in terms of registration [17]. While his analysis of Schoenberg's op. 11/2 shows the potential utility of this method, it is not entirely different from Larson's theory of referential collections. In many works of Debussy, for example, pcs that do not belong to the diatonic referential collection of a root are registrally distant enough to be approximations of its overtone series, allowing one to classify them as part of an extended referential collection. The more distinct theory is that of "salience conditions" as proposed by Lerdahl [12,13], which considers qualities in events outside of pitch class (such as registral placement or metrical weight) to distinguish them as the heads of his well-formed groups. While these may be appropriate for delimiting musical segments in a clear hierarchical structure, they are predicated on the idea of a flat pitch space that is not necessarily tenable. By adopting a different segmentational procedure, such as that which I propose in sections 2 and 3, one may be able to discern meaningful pitch-class topographies in various chromatic and atonal terrains. Despite some significant differences between the abovesurveyed prolongational theories, one sees an emergent consensus that the "stability" of a particular pitch class or sonority can be ascertained by considering the relational lattice(s) in which the pc/sonority is prolonged to varying degrees. One may ask the broader question - what do these relational lattices look like? In asking this, we move from the notion of event-based stability to one of stability space: a temporal field in which the stability of multiple pcs may be assessed. 1.2. Constructing stability spaces I will arrive at a formal definition of stability space via a metaphorical extension of the physical analogies that are ubiquitous in the literature on pc stability. As a brief example, consider the word "gravity" as it is used in a wide array of scholarship on musical practice: * Schoenberg: On the relationship of a tonic with respect to its subdominant and dominant: "[It is like] the force of a man hanging by his hands from a beam and exerting his own force against the force of gravity. He pulls on the beam just as gravity pulls him, and in the same direction. But the effect is that his force works against the force of gravity, and so in this way one is justified in speaking of the two opposing forces" [7]. * Baroni, Maguire, Drabkin: On the linguistic decomposition of a Bach chorale: "Rules of transposition, after having established the chorale's 'centre of gravity' (an imaginary line from which it is not possible to deviate beyond a certain interval), control the conditions of transposing a phrase to the upper or lower octave if this is necessary for the overall balance of the piece" [2]. * Arnheim: Summarized by Brower: "We experience musical tones as subject to both a constant downward pull of gravity, which acts with equal strength at all locations, and a more variable force of attraction exerted by the tonic" [4]. * Larson: "I call gravity...the tendency of an unstable note to descend" [ 11]. The terms "magnetism" and "inertia" have also been used to describe similar pc-stabilizing forces [11 ]. Of course, any dialogue on force presupposes a field in which the force acts as well as some type of forceexerting object. Stability space seeks to furnish a basic vocabulary for describing this field (time) and the agent of force (pitch). I define it formally for work W at time t as the collection of pitch classes HI that comprise the musical discourse during a time-interval T about t. It is not a pitch set, nor is it a time interval, but rather it is a Cartesian product of the two. =60: Figure 1. A stability space with T=4 seconds, Ih= {D#,E,F#,G}= {3,4,6,7}, card(HI)= HI=4. Considered outside of the prolongational theory from which it arises, the ontological simplicity of stability space renders it as basic as defining a physical space in terms of a three-dimensional volume and the matter (if any) that exists within it. This very simplicity is what gives stability space its analytic power in the case studies to follow: rather than presupposing the forces (and by extension, zones of stability) that exist in a certain musical work, stability space only concerns itself with the temporal and pc fields in which these forces manifest themselves. One important methodological distinction between divisional schemes based on stability space and more traditional approaches to segmentation is that the former does not require a clear hierarchical division of the musical surface. In fact, the contrary is true: stability spaces are permitted to intersect, and one moment necessarily belongs to multiple non-nested stability spaces. At one extreme, the stability space covering an entire work will generally contain what Lerdahl identifies as the fundamental basic-space - the chromatic scale [13].' At the other extreme, the stability space of a work at a point in time, or more precisely time interval dt, can be nothing other than the sonority at that moment. Benjamin calls this "the state of the piece" [3]. In between, each dt belongs to an infinite amount of 1 Clearly, not every work that draws from the chromatic pitch space accesses it in full. Some of the most interesting analyses arise from works that intentionally omit certain chromatic degrees, such as Debussy's Voiles.

Page 3 ï~~stability spaces in which it may or may not have musical significance. 1.3. Stability space and prolongational theory Going back to the idea that stability space is a "container" of forces, one may also use it to derive a notion of stable instability. More precisely, an area considered ambiguous or unstable with reference to a collection of competing pcs (e.g. a modulatory passage in the Classical literature that prolongs no one harmony for an extended period of time) has its own larger stability space in which these conflicting pcs are contained. Stability spaces also account for prolongations of harmonies that Larson identifies as "nowhere literally present in the musical surface" [11]. In the example he cites from Benjamin [3],1 the I chord is never sounded even though it is clearly being prolonged. The members of the chord are present, however, in the stability space containing these measures. Furthermore, Benjamin notes the difficulty in finding the Urlinie of musical passages with no one prolonged event (such as the Baroque sequence). Similar to the previous case, stability spaces give us a way to talk about the pitch classes in a sequence without necessarily conferring unto any one of them a higher degree of importance. Most relevant to this study, stability spaces allow for prolongational analyses in chromatic and atonal music. Replacing the idea of a "harmony" with that of a stable pitch convention, such as the use of fixed hexachordal sets, a comparison of stability spaces containing these sets would show the degree of regularity with which they were deployed in time. Stable pc sets share all the qualities of tonally prolonged materials - they can be represented in a reductive analysis and, at a certain point, they leave a "trace" as defined by Larson by virtue of their ubiquity [11]. Such a notion acts as a foil to the set theoretical analytic techniques proposed by Hanson, Forte, and Perle: if the latter seeks to associate pc sets through various transformations and equivalencies, analyses based on stability space are most effective when used to measure the temporal distribution of invariant pc collections in a given work. Even though such a practice is not one of Lerdahl's salience conditions, it certainly comports with his definition of prolongation as "organized collections involving degrees of departure and return" [12]. 1.4. Measurements of stability space The case studies in this paper do not attempt to enumerate subsets of the infinite stability spaces to which a musical moment belongs, but rather inverts this procedure, asking "If one looks for a certain fixed type of stability space in a work, where (if anywhere) can one 1 J.S. Bach, French Suite in E, Allemande, mm. 20-28. find it?" It also ignores the individual pcs that comprise a stability space's constituent pc set (H), focusing instead on the cardinality of that set (|I|). While this approach is potentially hazardous in certain types of music, for the pieces selected in this study, stability spaces in the same work containing pc sets of fixed cardinality will generally draw from a limited repertoire of sets. For example, the majority of stability spaces with IH=7 in a middle-Scriabin work will contain subsets of two diatonic collections related by a chromatic mediant. In early-atonal Schoenberg, by contrast, the majority of stability spaces with 11H-=7 contain pc sets comprised mostly of semitonally adjacent subsets. Even with these rather severe limitations, one can glean significant numerical results by carefully choosing the stability spaces in question. For example, two works that have stability spaces with nearly equal T at II = 7 may diverge radically in T for II > 7. One will also see works that have a consistent correlation between the T and the |IH of their stability spaces. In the final analyses, we will see that these results do not only illuminate binary distinctions between works but also show how similar and how different various works are: that is, one can begin to array stability spaces along a continuous parametric field (such as how T varies over a fixed 1|T|). Of course, the construction of a continuum necessitates a continuous function (or at least a finelyparsed discrete one), for which I have created the belown threshold. 2. BELOW-N THRESHOLD DEFINED Define a pc map P of a work W as a collection of all its pitch events pi. Each pi is represented as a double < ri, Ii >. 2. is the event's duration and i = [tl,,t2 ] c D c T where D is the duration of W (normalized to unity) and T* is the time space [0,oo). iy is the pitch class of the event and E II, where I*H is the chromatic pitch space {C,..., B}. Define a stability mesh S of work W as the set of all stability spaces s =< T, H > such that T = [ti,t2 ]c D andII =proj. ([T x IIH ] n P ). Define a 6 stability mesh S ={Vs1 S I,* (T.) s} where T e s E S as defined above and A* is the Lebesgue outer measure. The set of all S will be indexedS = -{sa }k-1. Define m ~ inf(T8 ). Define a function f: [0, D - 8] -> N taking ms to k where m6k eT andT6k, Sk s6 S ESSS. We

Page 4 ï~~know this is a function because there is one and only one H for each m. Define the below-n threshold of a work B as: n (1D-6 Bn inf - I f(m)dm n o Informally, we may define the below-n threshold as the degree to which one must "zoom in" on a work's time axis before, on average, the musical discourse in the magnified time span contains less than n pitch classes. A higher threshold for any given n, therefore, means that one must use a higher degree of temporal magnification before one begins to see pc sets of cardinality less than n. For this study, a related measure called the "normalized below-n threshold" is used in all of the tables and graphs. This measure takes into account the amount of pitch events in a work under the assumption that, holding all other parameters constant, the time interval T containing a given pc set H with cardinality |H| is positively correlated with the amount of pitch events in B a work. I define this as3 = I AI~I Because of the computational difficulty in producing infinitely many S3 for infinitely many 3, this study is restricted to integral values of -1 and uses finitely many S3 (usually on the order of 102 to 104 depending on the size of the 3) that are randomly distributed throughout the work's time domain (turning B into a Riemann sum instead of an integral). Each Riemann sum is calculated 50 times and averaged, with fan - mean (fn) always yielding a 99% confidence interval that never exceeds Â~0.01 for low n and Â~0.003 for high n. Dividing by the amount of pitch events in a work goes a way towards "normalizing" without completely leveling the playing field. For example, the C Major prelude in Bach WTC Book 1 would have categorically lower fi than its measure-by-measure harmonic reduction in spite of the similar pc content. Another normalization procedure could involve dividing B by the length of a work, although this would lead to divergent results for two works with the same harmonic rhythm over different temporal frames. There is no one filter that completely separates the "signal" of B from the artifacts of analysis, however, the reader will see that the procedure generates data that comports with known scholarship about the works in question (e.g. pieces with similar pitch conventions have similar 13 ), eliminating the need for a different normalization process in these instances. 3. CRITERIA FOR SEGMENTATION 3.1. Hanninen's taxonomy of segmentation methods The stability spaces for the below-n threshold are generated by an equitemporal segmentation method, intentionally suppressing many of the segmentation criteria to which one would ascribe musical meaning. One may question whether it is a valid segmentation procedure at all under Hanninen's tripartite taxonomy of segmentation methods for musical analysis: * Sonic: segmentation according to "psychoacoustic attributes of individual sound-events" [9, 353]. * Contextual: segmentation according to "similarities among groups of sound-events - that is, similarities that arise only at the group level when groups are compared with one another" [9, 353]. * Structural: segmentation according to "transformational relationships with respect to a specific music theory (or theoretic orientation)" [9, 353]. Equitemporal stability spaces are not 'sonic' because they do not necessarily deal with "attributes of individual sound-events" [9, 353]. While 'contextual' initially seems like a better candidate because it encompasses "associations between groupings of sound-events that arise at the group level," [9, 363] its relevancy to this study declines based on Hanninen's assertion that one "must go directly to the music to see how it selects or relates elements within the [contextual] space" [9, 365]. This ascription of agency to a musical work implies a synergistic relationship between musical events and the "specific non-trivial contextual criteria" in which these events are most meaningfully understood [9]. Again, while equitemporality may comport with these norms, it will likely be entirely uncorrelated or detrimentally autocorrelated with them and, even more problematically, will include events that could only be described as partial group members by Hanninen's standards. That is, two segments may contain parts of the same pitch event - a fuzzy type of distinction that is not present in her implicitly axiomatic treatment of segment (and therefore set) construction. An equitemporal divisional scheme is arguably 'structural' because it arises from a "theoretic orientation." However, the latitude of Hanninen's definition narrows when she specifies that such segments must contain "individual groupings in a composition and conceptual representations or interpretations of those groupings supported and rendered intelligible by an abstract conceptual system - a music theory" [9, 355]. Equitemporality is brought further into relief by Hanninen's model theories such as Schenkerian and neoRiemannian analysis - both of which cleave the time domain by using a pitch-based hierarchical divisional scheme. Insofar as equitemporal segmentation fails to capture the types of distinctions that these theories do, it cannot fulfill Hanninen's claim that structural segmentation methods produce "individual segments in a context that allows us to generalize across particulars

Page 5 ï~~within or across pieces" [9, 380]. What is needed, then, is a modification of Hanninen's definition of structural segmentation that admits an equitemporal segmentational scheme while preserving the musically meaningful measurement goals, such as those advanced in the Theoretical Foundations section, on which Hanninen's taxonomy is predicated. 3.2. A Descriptive-Statistical Segmentation Approach One claim upon which stability space is founded is that pitches are understood in the context of those surrounding them in time. While this is related to Hanninen's 'contextual' segmentation procedures, it does not arise from any musical norms that the work portends and thus may be over or under-inclusive with respect to these norms. For example, the two pc sets of cardinality 7 to which one would conventionally ascribe contextual and structural meaning in an 18th-cnetury sonata's exposition are the I (or i) and V (or III) diatonic collections. The below-n threshold, however, counts pc sets of 111H=7 that arise from transitional material. A similar phenomenon occurs at a more local level of analysis - any triads that are being prolonged in a given section are often embellished on the musical surface and thus are not captured by stability spaces of 3 < lii _ 4. What, then, can B say about these spaces? The answer comes by way of a comparison between two Haydn works that access different "referential collections" as defined by Larson [ 11]. The expositions of the first movements from HXVI/12 and HXVI/13 are different in their use of pc sets at both local and global levels of analysis. Table 1 summarizes these differences. HXVI/12 HXVI/13 Global Two sections in the Two sections in the diatonic collections of diatonic collections of E A and E with little and B with a higher transitional material amount of transitional between the two. material and tonal ambiguity. Local Prolonged harmonies Prolonged harmonies are are arpeggiated on the ephemeral on the musical surface with musical surface, with little embellishment, passing and neighboring tone embellishment. Table 1. Description of pitch conventions in the firstmovement expositions of HXVI/12 in A major and HXVI/13 in E major. Because HXVI/12 accesses the diatonic referential collections of I and V for a larger percentage of its duration, stability spaces with |H=7 can cover, on average, a larger T, presumably leading to a lower 13 at n = 7 than the 137 of HXVI/13 (recall that B is a measure of W', so larger T's mean smaller B, and smaller 3). On a more local level, the piece's repetition of triadic referential collections would presumably lead to stability spaces of 11H=4 with larger T and a lower 3n atn 4 than the 134 of HXVI/13. Table 2 compares n to 1n for {nlneZ,44<n<8}, showing the A major sonata's lower 13 for all n. In it, we see how the inclusion of non-diatonic material and non-triadic material in HXVI/13 results in categorically higher 13 than the "purer" collections in HXVI/12. utilization of these n 3n for HXVI/12, fn for HXVI/13, Andante, mm. 1-20 Moderato, mm. 1-30 4.2391930836.3363636364 5.1354466859.1787878788 6.07780979827.09393939394 7.03170028818.03636363636 8.01152737752.01212121212 Table 2. Normalized below-n threshold for {nln e Z, 4 _ n _ 8 } 1in Haydn, HXVI/12, Andante, mm. 1-20 and Haydn, HXVI/13, Moderato, mm. 1-30.2 This comparison illustrates the caveat that must be applied to Hanninen's taxonomy in order to validate the below-n threshold as a segmentation method: a structural segmentation need not yield fruitful lateral comparisons between features of each structural segment (or even segments that have meaning in and of themselves). Rather, the meaning of the segments need only take on shape in the aggregate. In this way, structure can be thought of as a descriptive-statistical phenomenon. Such an idea is not new in the way music is created: Stockhausen and Xenakis (amongst others) use granular synthesis procedures to create conglomerate textures, thereby subsuming the importance of individual contributors into a larger entity. This idea is also prevalent in the sciences: for example, Mandelbrot's seminal paper on the fractal dimension of coastlines ascribes no significance to the individual measured segments, but rather acknowledges that they are approximations and focuses instead on the correlation between measurement error and segment size to generalize across natural shapes [15]. A similar idea is appropriate in analysis: if one accepts Larson's (and, save certain qualifications on inherencies, Lerdahl's) notion of contextual stability as the basis of prolongational practices, it seems to be a precondition that music can be heard in terms of the temporal "weights" of these stable regions and, by extension, the aggregate of these weights as a work progresses. 1 The calculations for this chart and all subsequent analyses were made using Maple 7.0, a product of Waterloo Maple Inc. 2 While this paper does not entertain an exhaustive test of statistical significance for the presented data, in Table 2, one can say with at least 90% confidence that the n values of 4-7 are not equal.

Page 6 ï~~4. CASE STUDIES 4.1. One further note on methodology In addition to the aforementioned qualifications regarding various computational approximations of B, a word must be said about the "significance" of B with respect to traditional hypothesis testing. The idea that null hypotheses about a collection of works can be postulated and verified by B at various levels of statistical significance is problematic for two reasons. First, the data does not generalize about a population of pieces by analyzing a sample, but rather analyzes and describes the entire population. So, the idea of generating a false positive or false negative is irrelevant. Second, because the diversity of the analyzed literature prohibits any general theory governing the musical significance of subtle shifts in B,1 there is a danger in ascribing too much value to any one lateral comparison between works. Thus, the following studies attempt to corroborate and expand previous scholarship by looking at broad trends in B across various n with respect to a collection of works. Each study identifies the impetus for analysis, displays the empirical results in graph form, and analyzes any trends and/or outliers in the results. 4.2. The late preludes of Scriabin The evolution of Scriabin's pitch use from a diatonic system to a language based on octatonic scales and the mystic chord is well documented in studies such as Baker [1], Callender [5], and Chang [6]. This case study focuses on the middle and late preludes of Scriabin,2 analyzing the degree to which his modified harmonic practice is captured at each n of the below-n threshold. the outlier of op. 74/5.3 Figure 2 shows that, for each n, Scriabin's later works yield a higher below-n threshold than his earlier works. Even more strikingly, the data shows that the /3 values of the earlier works consistently lag those of the later works by n=2. Had Scriabin simply substituted an octatonic for diatonic language, one would expect a lag of around n=1 - that is, stability spaces with a fixed T for which 1H=7 in the earlier works would register at |H=8 for the later works. The lag of 2 suggests that, in addition to a change in harmonic language, Scriabin also developed a more adventurous chromatic practice within his new harmonic language. While an exhaustive investigation of this claim is beyond the scope of this paper, analyses of two select passages support its potential interpretive validity. Andante Poetico con delizio Le Figure 3. Chromaticism coupled with the octatonic scale in Scriabin's op. 67/1, m. 1 compared to the extended diatonic language of op. 48/2, mm. 0-1. Figure 2 also proffers a clear outsider - op. 74/5. Looking at the graph, it almost seems to be a direct extension of the earlier works' ft values. The pitch language, however, is decidedly late-Scriabin. Chang notes how this prelude retains the harmonic material of op. 74 while harkening back to the less-semitonal language of the earlier works: "[Op. 74/5] moves to a simpler, minimal application of the [op. 74] harmonic materials. The semitone figure appears occasionally, but the extensively interweaving chromatic texture can no longer be found in this prelude. Conversely, the textural density is greatly attenuated and the harmonic progressions of the Mystic Chord are relatively transparent and straightforward" [6, 69]. I would qualify Chang's statement by adding that the mystic chord is often quartally extended to include a seventh member, aligning it with the earlier works' stability spaces of 1-I=7.: Figure 4. Stability space with |H =7 as a quartally extended mystic chord in Scriabin's op. 74/5, m. 3. 3 n values less than 5 are omitted from the later opuses for the purpose of clarity of comparison with the later works. Additionally, a comparison with the earlier works becomes spurious at that point due to the sprawling range of the data for small n. 0.35 0.3 v 0.25 D s N 0.2..7. i# 61 0.15 L p 0.1 -uop. -OP - -Op, -A- Op. -XOp --Op, -- op.. -X- Op, 45, 3 48, 1 4E, 2 48, 3 48, 4 Si, 2 56, 2 59, 2 67, 1 67, 2 Figure 2. Compare ovals laterally to see the n=2 lag between Scriabin's expanded-tonal (op. 45, 48, 56, and 59) and post-tonal (op. 67 and 74) harmonic language. Note 1 The normalization process as defined in note 4 imposes an even further problem for /3. 2 The earlier preludes are not entertained under the assumption that their harmonic language is similarly (if not less) advanced than the middle preludes. One would assume that they would manifest similar or even more drastic divergences from the later works.

Page 7 ï~~4.3. Schoenberg's serialism(s) It has often been remarked that Schoenberg's twelvetone method was a way to organize certain pitch practices, such as the serialization of hexachordal sets (op. 23/4) and the use of relative complements (op. 23/3), that mark the end of his "contextual" period. One may ask, then, the extent to which this manifests itself through stability spaces via the below-n threshold. Initially, one would suppose that the twelve-tone method yielded a higher degree of chromatic "turnover" - that is, the exhaustion of all twelve tones in a span of time - than the previous works. This leads to the claim that Schoenberg's twelve-tone works diverge from his freeatonal works at n values near 12. permutation of pc sets with lower cardinality [8]. Unlike the Scriabin study, this one proffers no outliers; the two collections of works are clearly separated in 13 at and around n=12. However, we do see one interesting outlier relationship when op. 23/2 is compared to the op. 11 piano works: it seems to belong more to this collection for all n than op. 23. Is there a feature of this work that aligns it with Schoenberg's early atonal style? c 2 3, 0 c Figure 5. Natural log of the normalized below-n threshold for Schoenberg's op. 23 ("contextual") and op. 25 (twelvetone). The twelve-tone op. 23/5 is plotted as a dotted line. Figure 5 is log-normalized on the /3 axis to more clearly show the opuses' bifurcation in the range of 8 < n < 10 before clumping neatly into two groups by n=12, signaling a faster rate of chromatic turnover in the op. 25 collection. If divergence of /3 occurs somewhere between 8 and 10, we may ask: is there a feature of Schoenberg's pitch use that supports this? The most compelling evidence comes from Hyde, which identifies serial devices yielding pc sets of cardinality 8-10 in three op. 23 works: * Op. 23/1: Three pc sets of cardinality 8-10 are recurrent throughout the work either in their complete form or in subsets [10]. * Op. 23/2: Most of the work's material comes from a pc set with cardinality equal to 9 [10]. * Op. 23/4: Entirely comprised of four hexachords which create resultant sets of cardinality 8-10 [10]. What is interesting, then, is that Schoenberg's 13 are undifferentiated in stability spaces of lower pc cardinality, evidencing a unity of approach between these collections despite the different techniques used to compose them. Looking at the manner by which Schoenberg arrived at his op. 25 compositional vocabulary (composed almost simultaneously with op. 23), such unity is not surprising - initial sketches show twelve-tone rows being formed by the combination and Figure 6. Schoenberg's op. 22/2 (plotted in medium-sized dashes) in op. 11? Hyde notes that, unlike op. 23/1, the foundational pc set of op. 23/2 is not repeated in its original order until the canon at the end of the work [10]. In place of this rigidity, we see transformations and repetitions that are akin to his op. 11 practice. As an example, consider the way in which mm. 10-11 of op 23/2 utilizes the work's core pc set {0,1,2,3,4,5,6,8,9}: langsamer beginnend >P Figure 7. Temporally proximal repetitions of 6th and 7th member of pc set {0,1,2,3,4,5,6,8,9} in Schoenberg's op. 23/2, mm. 10-11. Each instance of the set is transposed by a perfect fifth instead of an augmented fourth, thereby avoiding the completion of a twelve-tone row through hexachordal combination. This, in tandem with temporally proximal repetitions of the 6th and 7th member of the pc set, leads to a broader temporal basin of stability spaces containing these pcs at all n. By placing a longer-lasting pc in the bass voice of op. 23/2, Schoenberg is utilizing a common-practice convention from which all three op. 11 works draw. Perhaps the most well known example is the recurrent D3-F3 motif beginning op. 11/2. By contrast, the temporally proximal repetition of pcs in the rest of op. 23 is infrequent and often fails to portend a musically meaningful link between successive and/or concurrent pitch collections. For example, the Bb in m. 1 of op. 23/4 is the only pc employed by all three of the

Page 8 ï~~measure's hexachords. Each instance of Bb serves an entirely different function with respect to its hexachord and the texture in general. Figure 8. Schoenberg's op. 25/4, m. 1. Note the three different uses of Bb identified by *. In summary, while op. 23/2 clearly belongs to Schoenberg's 1920s technique with its serial pitch use, the data suggests an op. 11-like temporal distribution of pc sets that is verified by a traditional set-theoretic analysis. 5. CONCLUSION This paper advances the concept of stability space, a "container" of time and pitch classes, as a tool for music analysis. Because any musical moment necessarily belongs to an infinite amount of stability spaces, most of which portend no traditional analytic value, one must develop a means for discriminating between or describing en masse the temporal and pc contents of these stability spaces. The below-n threshold concerns itself with the latter, ascertaining the average temporal measurements of stability spaces containing pc sets with fixed cardinality n. By applying this algorithm to music by Scriabin and Schoenberg, this paper seeks to provide a convincing empirical description of the composers' styles, support trends in the data with previous scholarship, and identify interesting outlier relationships. Even more importantly, it engages a discourse traditionally confined to questions of type by infusing distinctions of degree through quantitative measurements. Bearing in mind that any computational analysis is only valid insofar as it is predicated on a tenable comparison within or between musical styles, both stability space and the specific way in which I am considering it through the below-n threshold have a wealth of potential by virtue of their definitional simplicity. With these methods (and, more generally, any well-defined empirical interpretive approach), the analyst is not limited by what one can readily perceive from the musical surface but instead is enabled to do what he or she does best - qualify diversities and similitudes along musically meaningful metrics. 6. REFERENCES [1] Baker, James M. 1980. "Scriabin's Implicit Tonality," Music Theory Spectrum 2: 1-18. [2] Baroni, Mario et al. 1983. "The Concept of Musical Grammar," Music Analysis 2/2: 175 -208. [3] Benjamin, William. 1982. "Models of Underlying Tonal Structure: How Can They Be Abstract and How Should They Be Abstract?" Music Theory Spectrum 4: 28-50. [4] Brower, Candace. 2000. "A Cognitive Theory of Musical Meaning," Journal of Music Theory 44/2:323-379. [5] Callender, Clifton. 1998. "Voice-Leading Parsimony in the Music of Alexander Scriabin," Journal of Music Theory 42/2: 219 -233. [6] Chang, Chia-Lun. 2006. Five Preludes Opus 75 by Alexander Scriabin: The Mystic Chord as Basis for New Means of Harmonic Progression. Doctoral Dissertation, University of Texas at Austin. [7] Cherlin, Michael. 2000. "Dialectical Opposition in Schoenberg's Music and Thought," Music Theory Spectrum 22/2: 157 -176. [8] Hamao, Fusako. 1988. The Origin and Development of Schoenberg's Twelve-Tone Method. Doctoral Dissertation, Yale University. [9] Hanninen, Dora A. 2001. "Orientations, Criteria, Segments: A General Theory of Segmentation for Music Analysis," Journal of Music Theory 45/2: 345-433. [10]Hyde, Martha and Arnold Schoenberg. 1985. "Musical Form and the Development of Schoenberg's 'Twelve-Tone Method'," Journal of Music Theory 29/1: 85-143. [l l] Larson, Steve. 1997. "The Problem of Prolongation in 'Tonal' Music: Terminology, Perception, and Expressive Meaning," Journal ofMusic Theory 41/1: 101-136. [12] Lerdahl, Fred. 1997. "Issues in Prolongational Theory: A Response to Larson," Journal of Music Theory 41/1: 141-155. [13]. 2001. Tonal Pitch Space. Oxford: Oxford University Press. [14] Lerdahl, Fred and Ray Jackendoff. 1983. A Generative Theory of Tonal Music. Cambridge, Mass.: MIT Press. [15] Mandelbrot, Benoit. 1967. "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension," Science 156/3775: 636 -638. [16]Straus, Joseph. 1987. "The Problem of Prolongation in Post-tonal Music," Journal of Music Theory 31/1: 1-21. [17] Vfiisfili, Olli. 1999. "Concepts of Harmony and Prolongation in Schoenberg's Op. 19/2," Music Theory Spectrum 21/2: 230-259.