The Effect of Harmony on Melodic Probability 405

TAKING HARMONY INTO ACCOUNT: THE EFFECT OF HARMONY ON MELODIC PROBABILITY

CLAIRE ARTHUR musical examples, as well as acute musical intuitions. Ohio State University A testament to the influence of these theories is the large body of work that has attempted to simplify, quantify, PROBABILISTIC MODELS HAVE PROVED REMARKABLY test, or otherwise expand upon them, most notably: successful in modeling melodic organization (e.g., Cuddy and Lunney (1995), Krumhansl (1995), Larson Huron, 2006a; Pearce, 2005; Temperley, 2008). However, (2004), Lerdahl (2001), Margulis (2005), Pearce and the majority of these models rely on pitch information Wiggins (2006), and Schellenberg (1996, 1997). (For takenfrommelodyalone.Giventheprevalenceof a review of expectation literature, see Pearce & Wiggins, homophonic music in Western culture, however, sur- 2006; Temperley, 2012.) The majority of these models prisingly little attention has been directed at exploring focus on the evaluation of pitch information taken from the predictive power of harmonic accompaniment in the melody alone (e.g., pitch height, interval size, con- models of melodic organization. The research presented tour, etc.). However, while the pitch domain has here uses a combination of the three main approaches received the greatest attention, research has also estab- to empirical musicology—exploratory analysis, model- lished the importance of rhythm and phrasing in con- ing, and hypothesis testing—to investigate the influence tributing to melodic expectation. For example, rhythmic of harmony on melodic behavior. In this study a com- information can help predict the location of phrase end- parison is made between models that use only melodic ings (Jusczyk & Krumhanl, 1993, Krumhansl, 2000; information and models that consider the melodic Krumhansl & Jusczyk, 1990; Palmer & Krumhansl, information along with the underlying harmonic 1987), and pitches located near phrase endings will have accompaniment to predict melodic continuations. A test an increased probability to move towards their note of of overall performance shows a significant improve- resolution (Aarden, 2003; Pearce, 2005). Given the prev- ment using a melodic-harmonic model. When individ- alence of homophonic music (i.e., melody with accom- ual scale degrees are examined, the major diatonic scale paniment) in Western musical cultures, an obvious degrees are shown to have unique probability distribu- avenue for further exploration in models of melodic tions for each of their most common harmonic settings. organization would be that of harmonic context. In That is, the results suggest a robust effect of harmony on other words, in modeling melodic expectancy, it may melodic organization. Perceptual implications and prove beneficial to examine melody not just as isolated directions for future research are discussed. lines, but as lines embedded in a harmonic context. While some scholars have considered the effects of har- Received: June 5, 2015, accepted August 20, 2016. mony on musical expectations in experimental settings, Key words: melodic probability, melodic modeling, it has typically been investigated in isolation, such as melodic expectation, harmony, scale-degree distributions studies examining the ‘‘perceived relatedness’’ of har- monies (Krumhansl, Bharucha, & Kessler, 1982; Bhar- ucha & Krumhansl, 1983; Bharucha & Stoeckig, 1986; USICAL EXPECTANCY IS A FREQUENT TOPIC Bharucha, 1987). One paper that has investigated the of interest in the music theory and music expectations of melody and harmony combined is that M perception literature. Many models of of Schmuckler (1989). Schmuckler’s results are some- melodic expectancy have been proposed over the past what counterintuitive; based on the results from three several decades. Two early theoretical models in partic- studies he proposes that melody and harmony have ular have been highly influential to the fields of music independent effects on musical expectations. However, theory and psychology, specifically those of Meyer as will be further discussed in the conclusion, Schmuck- (1956) and Narmour (1992). Each of these models ler’s unusual finding was never replicated. is highly complex and comprehensive in scope, with Although the processes underlying expectation are insights drawn from observations of hundreds of complex and multifaceted, research has suggested that

Music Perception, VOLUME 34, ISSUE 4, PP. 405–423, ISSN 0730-7829, ELECTRONIC ISSN 1533-8312. © 2017 BY THE REGENTS OF THE UNIVERSITY OF CALIFORNIA ALL RIGHTS RESERVED. PLEASE DIRECT ALL REQUESTS FOR PERMISSION TO PHOTOCOPY OR REPRODUCE ARTICLE CONTENT THROUGH THE UNIVERSITY OF CALIFORNIA PRESS’S REPRINTS AND PERMISSIONS WEB PAGE, HTTP://WWW.UCPRESS.EDU/JOURNALS.PHP?P¼REPRINTS. DOI: https://doi.org/10.1525/MP.2017.34.4.405 406 Claire Arthur

statistical learning plays a substantial role in forming the causal influence between melody and harmony can musical expectations (e.g., Huron, 2006a; Krumhansl, go in both directions: That is, melody might be expected 1990; Pearce, 2005; Pearce & Wiggins, 2006; Temperley, to affect harmony as well as harmony affecting melody. 2007). Given that implicit knowledge can arise from For the purposes of this paper, however, only the prob- probabilistic exposure to sequences and patterns abilistic relationship of harmony on melody will be (Romberg & Saffran, 2010; Saffran, Johnson, Aslin, & investigated, leaving aside the reverse analysis for Newport, 1999), it would be beneficial for researchers another occasion. interested in studying melodic expectancy to under- There are many ways to quantify and dissect melodic stand how various musical features contribute to the information. Many models—in particular those based statistical properties of melody. (For a review of implicit on Narmour’s—tally not only pitch information, but the and probabilistic learning see Reber, 1993.) specific interval and direction from pitch to pitch. The A small but growing body of research has investigated current model is based on ideas of statistical learning harmony from a probabilistic perspective, typically with proposed by Huron (2006a) where only first-order the goal of informing Music Information Retrieval scale-degree information is considered. Using scale (MIR) efforts, such as feature extraction or automatic degree, like pitch class, assumes octave equivalence and music generation (e.g., Ponsford, Wiggins, & Mellish, removes information about the size and direction of an 1999; Quinn & White, 2013; Raphael & Stoddard, interval. This means that the model does not distinguish 2004). Other research taking a probabilistic approach between, for instance, a major second and a minor to harmony includes the work of White (2013), who seventh. However, one should bear in mind that the used hidden Markov models to search for harmonic possible interval and direction between two scale transition information in a large corpus with the aim degrees has only two options (ignoring compound of comparing stylistic and historical trends. White does intervals, which are extremely rare in melodies), and not report the transitional probabilities or the distribu- it is well known that that small melodic intervals are tion of harmonies in the corpus, instead using them as much more common than large ones (Dowling, 1967; hidden factors in his analyses. Similarly, Rohrmeier and Huron, 2001; Ortmann, 1926; Merriam, Whinery & Cross (2008) look at statistical properties of harmony, Fred, 1956; Temperley, 2008). Thus, most scale-degree taking similar aims to the present paper in attempting to successions will represent the smaller of the two inter- provide ‘‘empirical evidence to clarify common assump- vallic possibilities. tions concerning harmony and its perception,’’ and to As will be discussed in further detail in the following ‘‘ground’’ categorizations of harmony found in music sections, ‘‘harmonic information’’ in this case refers to perception literature via a statistical perspective. How- a reduction of the accompanimental texture to a Roman ever, their work only considers the harmonic progres- numeral. This approach has several limitations that sions of the Bach chorales, and like the other harmonic should be acknowledged, such as the reliance on human studies mentioned above, they do not look at melodic (subjective) analysis, the marginalization of voice- probability. leading practices, and the insertion of a ‘‘presentist’’ bias In this paper the role of the supporting harmonic which discards relevant stylistic information (Gjerdin- accompaniment in shaping melodic organization gen, 2014). In this case, using Roman numerals repre- is investigated using a corpus-based, probabilistic sented the simplest method that would generate the approach. A model is constructed that combines har- fewest comparisons, and provided the simplest strategy monic information from the accompaniment with first- for analyzing homophonic music. order melodic information using a digital corpus of It should be stated that this model of melodic proba- encoded musical scores from the common practice bility is not being forwarded as an optimal model of period. This melodic/harmonic model is compared to melodic prediction. Any realistic model of melodic a model that uses solely first-order melodic information expectancy will certainly consider more than first- in order to investigate the effect of harmony on the order scale-degree successions and a crude harmonic predictability of melodic continuations. To anticipate analysis. Rather, the goal is to put a model that relies the results, it appears that melodic prediction is signif- on purely melodic elements in a comparative context, in icantly improved when harmonic information is taken order to gain some measure of improvement to the into account. Following the overall model comparison, predictive power of a model when harmonic informa- an exploratory analysis considers unique interactions of tion is considered. harmony and scale degree in order to examine the The motivation for this research is to better under- specific effect of the former on the latter. Notice that stand melodic tendency as it functions in a (relatively) The Effect of Harmony on Melodic Probability 407

ecologically valid context. While expectations are not first in isolation, and then with the harmony supporting directly tested here, the overarching theory behind this the antecedent scale degree of the consequent tone, in study is that melodic expectations are mediated by the order to investigate whether the harmonic support of harmonic context. Specifically, if expectations are the antecedent note can help predict the melodic behav- largely derived from event probabilities, and the prob- ior of the consequent. Notice that this is a correlational ability of any given melodic event turns out to be largely study, and as such it might be thought that the melody dependent on the underlying harmonic context, then could be equally influencing the harmony. However, this would likely affect melodic expectations in more antecedent information is used to predict consequent ecologically valid musical contexts where melody and information. As such, while an influence of the conse- accompaniment are both present. If true, this bears quent on the antecedent is still possible, it is less plau- implications for current methodologies in aural skills sible. In brief, the results of the study will show that training, as well as general theories of melodic expecta- harmonic context appears to have a strong influence tion. The hypothesis tested here is that harmonic con- on melodic behavior. text plays a sizable role in shaping melodic behavior. This hypothesis is tested using a statistical (i.e., proba- SAMPLING bilistic) approach. It is hoped that this examination of In light of the research hypothesis, a corpus of music harmony’s influence on melodic probability can provide featuring melodies with harmonic accompaniment was necessary groundwork for future perceptual studies on needed. While there are many Western forms of music expectancy and statistical learning. that have rich harmonic traditions—such as rock, pop, and jazz—classical music makes an ideal jumping-off The Corpus Analysis point for this research, as it forms a large body of notated music that is easily accessible. In addition, the OVERVIEW use of classical scores provides a convenience sample, The ultimate goal of the project is to better understand since the pitch information for many works is already the relationship between harmony and melody; more encoded in digital format. Importantly, performances of specifically, to identify possible influences of the sup- classical music are less likely than those of pop and jazz porting harmonic context on melodic organization. The to deviate in pitch from their symbolic representations, approach taken here is a strictly probabilistic one. That making them a better candidate for application towards is, the probabilities of melodic continuation are evalu- perception research. Therefore, this study specifically ated in two conditions: the first examines melody in investigates the statistical properties of melodies from isolation; the second examines melody along with the the common practice period. The corpus thus excluded underlying harmony in the accompaniment. In this way, material typically considered outside of the bounds of if the probabilities of melodic continuation are signifi- the common practice era, such as twentieth century and cantly altered in the latter condition (where underlying Renaissance music, as well as solo and two-part works harmony is taken into account) then one can infer that (e.g., Bach inventions) where harmonies are incomplete the melodic behavior is influenced by the harmonic or implied. Although it can be argued that the common context. To this end, a musical corpus was assembled practice period represents a wide range of harmonic containing melodies set within unambiguous harmonic practices (for instance, some composers included in the contexts. corpus were not yet thinking in terms of triads with To test the main hypothesis, two models were needed: roots), it is nevertheless commonly thought of—and one with first-order melodic succession probabilities taught as—one coherent tonal system. Given the aims calculated using only melodic information, and another of the study, there were several features that were desir- which combined first-order melodic information with able in the corpus: 1) that the voice leading was repre- the harmonic information taken from the accompani- sentative of the common practice period; 2) that the ment of the antecedent scale degree. In order to have melody could be easily determined and, as much as some measure of improvement, a zeroth-order melodic possible, clearly distinct from the harmonic accompa- model was also included. Since it is generally accepted niment; 3) several composers would be represented that first-order models perform better than zeroth- whose works spanned the time frame of the common order models, the zeroth-order model provides a base- practice period; 4) that both vocal and instrumental line against which to calculate each successive model’s styles were included in roughly equal proportions; and improvement. Thus, probabilities were calculated for 5) that homorhythmic and nonhomorhythmic styles melodic antecedent-consequent pairs of scale degrees, were included in roughly equal proportions. Note that 408 Claire Arthur

‘‘proportion’’ is in relation to the total number of mel- of music in the common practice era is not homorhyth- ody notes, not the number of pieces, since that was the mic, this slight imbalance may prove more representa- unit of measure in the study. In assembling the desired tive of melodic expectations in general. sample described above, for practical purposes it seemed preferable to use any preexisting digitally Method encoded scores. A subset of Bach chorales were chosen as a convenience sample, as they had harmonic analyses In assembling the corpus, the goal was to be able to which were already encoded by other scholars.1 This compare adjacent ‘‘slices’’ of the musical texture. Since meant that the works only had to be checked for errors, melodic tendency was the primary area of interest, each rather than encoded entirely from scratch. Similarly, the slice would be one melodic tone (or attack), regardless Schubert lieder had melodic information already of duration. Tones were represented as scale degrees encoded, and only the harmonic analyses had to be in relation to a tonic with enharmonic distinctions added. The remainder of the corpus was supplemented retained (e.g., #2 is different from b3). Two models were by consulting Burkhart’s Anthology for Musical Analysis created that both consider a prior musical ‘‘state’’ to (1994), specifically searching for pieces composed predict the subsequent melodic tone. The first model between nominally Baroque and Romantic eras looks only to the prior melodic note to predict its suc- (roughly 1600-1850), preferring those that had clear cessor; the second model (referred to as the harmonic separation of melody and harmony. In this process, an model) looks to the prior melodic tone and that tone’s effort was made to gather works which represented harmonic accompaniment to predict the melodic suc- a variety of composers, styles, and instrumentation. The cessor. Of course, most melodic tones in the corpus do anthology was an appropriate source since it contains not coincide with an onset in the harmonic accompa- works featuring harmonic accompaniment by many niment. In order to best represent the harmonic infor- composers in different styles and instrumentation dat- mation present in the accompaniment at any given ing from the common practice period. point in a melody, every melodic ‘‘slice’’ is assigned The corpus used in this study yielded close to 10,000 a Roman numeral based on the most recent sounding melody notes and was comprised of 68 pieces, includ- harmony present in the accompaniment. As will be ing: 50 Bach Chorales, 1 Haydn string quartet, 2 Mozart explained in the paragraphs below, determining scale sonata movements, 3 Beethoven sonata movements, 6 degrees and Roman numerals relies on the determina- Schubert Lieder, 3 Schumann piano miniatures, 1 Clem- tion of a key, which was resolved via human analysis (as enti Sonatina, and 2 Mendelssohn Songs Without opposed to automatic methods such as feature extrac- Words. (See Appendix A for complete list of works tion). Apart from the chorales which have clear har- included in the corpus.) All musical information in the monic onsets (i.e., homorhythmic vocal entrances), corpus was encoded in Humdrum format (Huron, the remainder of works in the corpus typically have 1995). All pieces (aside from the Bach chorales) were broken harmonic textures. In these cases harmonic harmonically analyzed and encoded by the author. labels were assigned based on the harmonic rhythm of Details of the analytic procedure are outlined in the the accompaniment. As common with any transcription following section. It should be mentioned that while by ‘‘expert’’ analysts, these methods are necessarily sub- the corpus may appear to be somewhat biased due to jective. However, the level of difficulty posed by the the large number of Bach chorales, the chorales are analysis of the works in the corpus was trivial, suggest- much shorter in length than the remaining pieces, ing that error rates would be minimal. In addition, meaning that, in fact, the chorales do not constitute the although works were not analyzed in duplicate, a recent majority of notes or measures in the corpus (e.g., the collaborative project by the author involving duplicate chorales constitute a total of 726 measures; the remain- Roman numeral analyses on a similar data set suggests ing pieces make up 1306 measures.) This means that, that the level of discrepancy between subsequent anal- bar-for-bar, and, more importantly, note-for-note, there yses would have been minor (Devaney, Arthur, Condit- may in fact be an overrepresentation of nonhomorhyth- Schultz, & Nisula, 2015).2 mic music, where melodies commonly continue over an unchanging harmony. However, given that the majority 2 In fact, the texture in Devaney et al.’s (2015) TAVERN corpus was more complex than the textures in the present corpus, and discrepancies 1 TheBachChoralesusedhereandtheirharmonicanalyseswere between analyses were still minor. The most common discrepancies originally encoded by David Huron in the **harm format, and are involved questions of inversion (typically during complex left-hand pas- freely available from the Kern Scores database at CCARH. sages) and tonicization versus true modulation. The Effect of Harmony on Melodic Probability 409

Since the primary purpose of this study is to investi- was decided that the latter would create discontinuities gate the influence of harmony on melodic behavior, it is in the analysis of sequential progressions. Rests of less imperative to limit the number of factors in order to than one measure make up the majority of rests in the have enough data to be able to make generalizable con- corpus. Rests (in either the melody or accompaniment) clusions. As such, certain decisions were made to sim- greater than one measure in duration were dealt with on plify the harmonic context. For instance, there may be an individual basis. For instance, the majority of longer a difference in melodic behavior for scale degree rest periods were due to breaks in the melodic lines of depending on whether the supporting harmony is in Lieder, such as during introductions, codas, or pauses root position or not (i.e., which note is in the bass). between phrases. If this type of break consisted of a com- However, classifying all harmonies separately based on plete solo phrase in the accompaniment, and did not both Roman numeral and inversion would divide the involve any transition or change of key, then the entire data too much, reducing statistical power and making it phrase was deleted as it would have no impact on the more difficult to discover patterns. Thus, in the analysis melody. Often a break comprised only part of a phrase, all inversion information (i.e., figured bass) was col- where the melodic line was passed to the accompani- lapsed to root position. Although it is possible that the ment, in which case the material was not deleted and the bass tone may be a more accurate predictor than the melody could be traced. Using this procedure, harmonic generic triad, as a preliminary corpus study of homo- and melodic alignment was preserved as best as possible. phonic music, a simpler model using fewer parameters Another challenge was how to handle repeating sec- was preferred over a complex one to start with. In the tions of music. For every section of music that repeats, same vein, rhythmic information, such as the duration should it be encoded (and thus tallied) once or twice? of melodic tones or chords, was removed. Other sim- The most common scenario was that of repeat signs plifications included ‘‘collapsing’’ seventh chords written into the score. However, there were some pieces together with triads. For example, instances of V7 and with repeats that were written out. The procedure for V were all labeled as ‘‘V.’’ It should be noted, however, handling repeats was as follows: ignore repeat signs but that since the scale degrees were tallied and paired with include repeated musical segments that are written out, their associated harmonies, it becomes clear when unless the written-out portion comprises more than a given scale degree is functioning as a harmonic sev- eight measures. Given that most phrases are four or enth. Like any experiment, corpus studies must also eight measures, it seemed that eight measures would operationalize concepts in order to render the hypoth- be sufficiently long to avoid discarding most written- esis testable. However, operationalizing the various out repeated segments. The rationale for ignoring repeat parameters raises a number of complicated issues, signs was that long segments of music that are typically which are described in the following paragraphs. repeated (such as the entire A and B sections of many binary forms, or complete sonata expositions) would THE PROBLEM OF MULTIPLE CONTEXTS occupy a disproportionate volume of the corpus, In this study, scale degrees and harmonies are examined thereby reducing data independence and potentially in their musical contexts. This means both in a vertical biasing the data. context (i.e., which scale degree sits above a given har- mony?), and a horizontal context (i.e., which melodic THE PROBLEM OF MODULATION AND TONICIZATION event precedes another?). This simple scenario is com- In the musical analyses for this study, melodies are plicated by the presence of rests. If rests were to be represented as scale degrees. Since scale degrees rely removed, care must be taken to preserve the synchro- on a reference to a tonic, it is important to reflect, as nization of the melodic and harmonic information. In accurately as possible, the implied key at each moment. order to preserve the original melodic-harmonic align- Of course, in real musical composition, key determina- ment, other musical information would need to either tion can be rather complicated. Music theorists typically be deleted, added, or both. For this study, rests were distinguish three categories of key migration: modula- treated in the following way: Rests were removed when- tion, tonicization, and temporary application of a chord ever they were present simultaneously in both melody (or chords) from outside the primary key area (i.e., and accompaniment. For rests comprising less than one ‘‘applied chords’’) (Laitz, 2008). With modulation, there measure, the melody note prior to the rest was copied, may or may not be an explicitly notated change of key in as if it were present for the remainder of the measure. the score. Furthermore, what sets modulation apart This procedure of extending harmonic or melodic from tonicization is not always clear. Although not all material was preferred over deleting material, since it theorists agree on the criteria for classifying key changes 410 Claire Arthur

under one label or another, for the purposes of this TABLE 1. Representation of Key Changes analysis it was decided that in order for the scale degrees Chord R.N. in key of Dm R.N. used in analysis to reflect most accurately their most appropriate key context and relation to the harmonic accompaniment Dm i i (and to maximize consistency), there should be a sys- Eø7 iiø7 iiø7 tematic procedure in place. Note that since a larger aim A7 V7 V7 Dm i i of this study is to inform melodic perception, this pro- Eø7 iiø7 iiø7 (viiø7) cedure need not align with how a music theorist might *key of F: analyze the segment, but rather it should represent the C7 V7/III V7 scale degree and key as it would most likely be heard by F I/III I a listener. Since research has demonstrated weak mem- C7 V7/III V7 F I/III I ory effects for the perception of the original key after A7 V7 V7/vi a key change (e.g., Cook, 1987), extended segments Dm i i/vi applying to a secondary key area should be notated as BbM7 IV7/III IV7 a key change. Thus, the working definition of key F I/III I change for the purposes of this analysis was as follows: C7 V7/III V7 F I/III I for any segment with four or more harmonies in a row (regardless of harmonic rhythm) that applied to a sec- Note: This table shows how key changes were operationalized for the purposes of ondary key area, a key change was deemed to occur. For encoding the harmonic progressions of complete pieces that included applied chords, temporary tonicizations, and modulations to other keys. When four or more example, Table 1 shows the following arrangement of harmonies in a row belonged to a secondary key area (i.e., encoded as ‘‘x/y’’), they harmonies: The section begins in the original key of D were labeled as having switched into the secondary key. If there were less than four minor, but a C7 harmony appears as a repeated applied chords applying to a secondary key, they would be labeled as applied chords and remain in the existing key. dominant to F (III), after which it appears to go back to D minor (A7-Dm), but then continues in F. According to the protocol for notating a key change, a Roman the melodic nor the harmonic progression at the key numeral analysis for this segment would be represented boundary, since the Roman numerals and scale degrees as shown in the right-most column of Table 1. Note that asterisked do not belong to the same key. To remedy in the example below the first appearance of A7-Dm are this, at the point where the music changes key, the har- given different Roman numeral encodings from when mony immediately prior to the key change is notated as they return towards the end of the passage, where the a pivot chord—regardless of whether it would be inter- same chords are coded as applied chords in F. preted as such by a music theorist—and then given One other issue concerns the process of changing a boundary separation, followed by an additional keys. When pitch classes are represented as scale Roman numeral representation in the second key area. degrees and compared in their horizontal (antecedent- In this way, ‘‘false progressions’’ can be ignored (i.e., consequent) context, a problem arises at key change skipped over) in the process of tallying the first-order boundaries. Consider a scenario represented in the two melodic and harmonic contexts. Take again the follow- illustrations below, with each column representing ing progression from Table 1, where the half-diminished a measure of music, where the third measure (C7) initi- ii chord pivots into a vii chord: ates a key change from D minor to F major. D minor: i iiø7 Melody: A, Bb, A, F, A G, A, G, E, D C, Bb, A, G A *F major: viiø7 V7 I Chords: Dm Eø7 C7 F A side-by-side ‘‘sliding window’’ context showing antecedent-consequent pairs of harmonies or melodic D minor: F major: tones would be represented as shown in Table 2 below. Scale degrees: 5, b6, 5, b3, 5 4, 5, 4, 2, 1* *5, 4, 3, 2 3 Rows showing the pivot chords or tones to be skipped over in the analysis are shown in bold. By preceding key Roman I iiø7* *V7 I numerals: changes with a notated pivot chord, and ignoring antecedent-consequent pairs that cross a key boundary, As indicated by the asterisks (and despite that, by the correct harmonic and melodic contexts are preserved, chance, the resulting Roman numeral progression is and spurious progressions are avoided. Although it is a logical one), this analysis does not accurately represent recognized that this solution causes the pivot chord The Effect of Harmony on Melodic Probability 411

TABLE 2. An Example of “Slices” of Antecedent-Consequent each melodic note, and the overall strength of the model Harmonies and Scale Degree Successions can be determined by evaluating the product of all the Harmony Scale degree probabilities. Since the resulting products are infinites- imally small, the standard procedure is to take the log of Antecedent Consequent Antecedent Consequent each probability, and sum them together to generate i iiø7 5 b6 a log likelihood. In order to be able to compare models iiø7 –b65of differing lengths (as will become relevant in the next viiø7 V7 5 b3 paragraph), the log-likelihood values are divided by the V7 I b3 5 length of the model’s data set (and converted to a posi- 54tive number). The resulting values are labeled as cross 45 entropy scores to be consistent with similar techniques 54 3 42in recent literature (e.g., Temperley, 2007). Lower cross- 21entropy values indicate a better model fit. Using this 1 – methodology the following cross-entropy values were 43obtained: zeroth-order model ¼ 2.22, first-order 32 23model ¼ 1.87, harmonic model ¼ 1.60. The lower cross-entropy score comparing the first-order model Note: Bolded rows indicate a boundary point where there was a change of key. The to the zeroth-order model demonstrates that the for- scale degree or harmony in bold pivots into the scale degree or harmony in the ensuing row. For example, scale degree 1 in the old key becomes scale degree 4 in the mer is more accurate at predicting the melodic contin- new key. This pivot note is not counted as having a consequent note, and thus uation. Notice that the difference in moving from the spurious successions/progressions are avoided. first-order model to the harmonic model is similar to the difference between the zeroth-order model and the to be counted twice—once with each label—the ratio of first-order model. pivot to non-pivot chords in the corpus is sufficiently Of course, a model with more parameters will always small that the chord duplication should not raise cause fit equivalent to, or better than, a model with fewer for concern. Furthermore, this method appears to be the parameters. Thus, there is some possibility that the best way of accurately capturing the antecedent- improved model fit might be due to chance. As such, consequent harmonic and melodic relationships as they it is appropriate to perform a statistical test. A log- relate to the implied key at any given moment without likelihood ratio test was conducted to compare the unnecessarily discarding information. Thus, preservation first-order model to the harmonic model (see Wilks, of immediate key structures was given priority over pres- 1938). This test compares a simpler model with a more ervation of higher-order key interpretations. complex model by evaluating the difference in log like- lihoods between models. This difference is multiplied Results by -2 to produce a value known as the deviance, which is known to be 2 distributed, with degrees of freedom GLOBAL HYPOTHESIS TEST equal to the difference in the number of parameters The primary research question asks: is melodic ten- between the two models. This ratio test produced a devi- dency dependent on harmonic context? In order to test ance of 4,840. Unfortunately, determining the degrees of this hypothesis overall, two models predicting melodic freedom for this test is problematic. One would expect continuations were created: the first uses first-order the degrees of freedom to represent the difference in the melodic information, and the second uses first-order number of parameters in the two models, which in this melodic information plus the harmonic information case is 7,424. (The first order model has 16 x 16 para- from the accompaniment of the antecedent (prior) meters: 16 antecedent possibilities and 16 consequent, melodic note. In order to have some measure of the consisting of the 15 scale-degree representations found in degree of improvement of the harmonic model over the the corpora plus a ‘‘boundary’’ parameter; and the har- first-order (melody only) model, a third model was monic model has 16 x 16 x 30 parameters: the same scale included that used only zeroth-order melodic informa- degrees from the first-order model but with 30 possible tion, since it is well established that first-order models chord types.) However, many of these parameters are tend to be more accurate than zeroth-order models (e.g., Huron, 2006a; Pearce, 2005). In the model tested, all 3 This use of the term cross entropy may not exactly fit the technical scale degrees and harmonies were included from the definition (see Rubinstein, 1997); however, it is consistent with complete corpus. Each model assigns a probability to Temperley’s usage in Music and Probability (2007). 412 Claire Arthur

highly correlated with each other, and furthermore, many harmony and scale degree, and in particular, how the of the parameter estimates are zero (i.e., not every scale former might affect the latter. In order to investigate degree actually appears with every possible chord type). this, the subsequent paragraphs make an exploratory Consequently, this number exaggerates the degrees of examination of the scale-degree distributions in each freedom. Since a theoretical distribution could not be model. calculated, an empirical distribution was calculated through Monte Carlo computer simulation. Five thou- DESCRIPTIVE STATISTICS sand log-likelihood tests were conducted using the first- Mode classification. The complete corpus consists of order model and a ‘‘scrambled’’ version of the harmonic pieces both in major and minor keys. In order to eval- model where the harmony paired with the antecedent uate the summary statistics, a question arises as to scale degree was randomized, producing 5000 null 2 whether to keep the corpus as a whole or divide it into values. The distribution of these 5000 2 values was cen- two parts, with one part comprising the pieces written tered around 2,350 with the top five percent of values in the major mode, and the other part comprising pieces falling within the range of 2,406-2,517. The 2 value of in the minor mode. Of course, major key works can 4,840 observed using the actual data is far above this include brief transitions into minor keys and vice versa, range, allowing us to conclude that the improvement of so classifying entire works based on modality may the model is significantly better than chance, even with- appear to be an arbitrary decision. Moreover, keeping out the precise calculation of a p value. the corpus undivided would preserve statistical power. This procedure follows the traditional approach of However, it is possible that the distribution of harmo- statistical testing, where a critical test applied to the nies and/or scale degrees might differ depending on the complete data set determines the likelihood of a given primary modality. Furthermore, if the corpus remains outcome appearing by chance. For those familiar with undivided, the tallying of all harmonies will produce an modeling techniques, a more common evaluation enormous table that may prove difficult to interpret. For reserves a portion of the data set to test the efficacy of instance, there would be three versions of the submedi- the model using the probabilities calculated from the ant chord: vi, VI, and bVI. This would make it difficult remainder of the data set. Thus, this complementary to distinguish, for example, the proportion of harmo- method was performed as well. The complete data set nies originating in minor key contexts from those that was divided into five portions, where each fifth was are a result of modal mixture. Accordingly, it was rotated through as a reserve set, and tested against the decided to split the corpus into two based on modality. remainder, which acted as the training set. If a parameter Many corpus studies have segregated data based on the (i.e., scale degree þ harmony) in the test set was not modality of the piece (e.g., Albrecht & Huron, 2014; encountered in the training set, the probability for that Rohrmeir & Cross, 2008; White, 2013). However, the parameter was pulled from the higher-order (i.e., first- present approach segregates the data based on the order) probability for the scale degree alone. For each modality of each continuous musical segment. Seg- test set this was only necessary for approximately .05 ments of music that contained major-mode harmonic percent of the data (or 10 in 1,813) on average. Cross- progressions (e.g., vi-ii-V-I) were tagged as ‘‘major,’’ entropy values were calculated for each model on each while segments of music with minor-mode progressions of the five test sets, and an overall value for each model (e.g., VI-iio-V-i) were tagged as ‘‘minor.’’ This method, was obtained by taking the average of all five. The cross- then, places entire sections of music that have modu- entropy values found using this method were: zeroth- lated into a minor key and places them in the minor order model ¼ 2.22, first-order model ¼ 1.88, harmonic corpus rather than keeping them mixed in with the model ¼ 1.67. As can be seen by comparing with the major corpus. While this distorts, for example, the true earlier test, these cross-entropy values (and the differ- proportion of ‘‘minor mode progressions’’ in nominally ences between them) are very similar to those of the first major works, it has the benefit of isolating melodic and test. harmonic idiomaticisms that may only exist in either These global tests show that evaluating a melodic tone modality. With this method, however, one can see along with its underlying harmony allows one to make a problem arises in associating Roman numerals with more accurate predictions about an ensuing melodic modality: major and minor tonic harmonies can be tone than simply considering the first-order distribu- tallied separately (i.e., ‘‘I’’ in a major key; ‘‘i’’ in a minor tions of melody alone. This is consistent with the main key); however, dominant function chords, and a few hypothesis. Of course, this overall test does not tell us other harmonies such as applied chords or Italian sixth anything of interest about the specific interactions of chords, share the same symbols regardless of the mode The Effect of Harmony on Melodic Probability 413

they are employed in. The result of tallying all instances of such chords is that while the tonic harmonies would be subdivided and grouped by modality, the dominant harmonies would all be pooled together, possibly dis- torting the true ratio of dominant to tonic function harmonies. As such, harmonies that traditionally use the same symbol in either mode were specially labeled in the analysis according to the subsequent harmony in the context of the piece. For example, if ‘‘V’’ is followed by ‘‘i’’ then the V chord was marked with an additional symbol in the database to indicate that it preceded a minor chord. These V chords would be tallied sepa- rately from V chords that were followed by a major chord. In this way, common harmonies with labels that could apply to either the major or the minor mode were distinguished so as to most accurately represent the proportion of harmonies in the corpora. In the end, despite dividing the corpus in two, the total number of harmonies found in either part of FIGURE 1.1. Zeroth-order distribution of scale degrees in the corpus. the corpus remained unwieldy. In order to simplify the Bar graphs show the overall proportions of the scale degrees as results, harmonies that represented less than one per- a percentage of the whole that each scale degree makes up in the cent of their respective major or minor corpus were major or minor corpus, respectively. The numeric labels above each omitted from the graphs and tables. Overall, these rare scale degree tally the actual number of instances found for that given harmonies tended to be comprised of: modal mixture scale degree. These tallies demonstrate the under-representation of minor key works in the corpus overall. chords, Neapolitan chords, and less common applied chords. Finally, in an attempt to retain as much musical information as possible, applied chords in the form of leading tone triads were merged with dominant applied chords (e.g., counts of V/vi and viio/vi were pooled together as dominant function applied chords).

Zeroth-order probabilities. The zeroth-order probabili- ties (i.e., the overall distribution) of both scale degree and harmony (independent of each other) were extracted from the corpus in order to determine their relative proportions. These are shown in Figures 1.1 and 1.2. Recall that the corpus was divided into two subcor- pora representing the major mode segments and the minor mode segments, respectively. Thus, the percen- tages and counts shown in the figures below are labeled with regards to their respective corpora totals. The x-axis lists all possible scale degrees (Figure 1.1) or harmonies (Figure 1.2) found in the corpus, and the y-axis repre- sents the proportion of the corpus that is made up by each of those scale degrees or harmonies. As mentioned above, harmonies making up less than one percent of FIGURE 1.2. Zeroth order distribution of harmonies in the corpus. the corpora have been omitted from the figures, and These charts show the overall proportions of the harmonies as seventh chords are collapsed to triads (i.e., V represents a percentage of the whole that each harmony makes up in the major all instances of V and V7). or minor corpus, respectively. The numeric labels above each harmony give a count of the actual number of instances found for that given Not surprisingly, neither scale degree nor harmony harmony. These counts demonstrate the under-representation of follow a simple uniform distribution. Rather, some scale minor key works in the corpus overall. Harmonies making up less than degrees and harmonies occur more often than others. 1% of the corpora were omitted from the graphs. 414 Claire Arthur

Specifically, scale degrees from the diatonic collection appear more often than chromatic scale degrees. Even within the diatonic collection, the first five scale degrees (1 to 5) are more common than 6 and 7. This is finding is consistent with pitch distributions reported by other scholars (e.g., Albrecht & Huron, 2014; Huron, 2006b; Temperley, 2007). Similarly, the use of nondiatonic harmonies is mostly outweighed by diatonic harmonies. Interestingly, the use of dominant and tonic chords greatly outweighs all other harmonies combined. This finding is relatively consistent with the work of Budge (1943), who found that all inversions of I and V made up roughly 45% of her corpus, as well as Rohrmeir and Cross (2008), who found similar counts and proportions for their major mode corpus (translated from pitch class sets to Roman numerals), although their minor mode distributions appear quite different.4 Since the tonic chord supports scale degrees 1, 3, and 5, and the dominant chord sup- FIGURE 2. Visual representation of the likelihood of a consequent scale ports scale degrees 5, 7, 2 (and sometimes 4), one might degree (y-axis), given the antecedent (x-axis). The size of the circle is roughly proportional to its probability (expressed in percent), as shown expect, given this abundance of tonic and dominant in the legend. For example, the probability of scale degree #2 moving to chords, that scale degree 6 ought to be the least common scale degree 3 is 100%, whereas the probability of scale degree 1 moving scale degree. This does appear to be the case in the to scale degree 2 is 19%. Clustering of large circles around a line with minor mode corpus, where b6 and 6 are used less fre- a slope of 1 arises from a strong likelihood of note repetitions and step- quently than nearly all other diatonic scale degrees. wise motion, consistent with previous literature. (The least common diatonic scale degree is b7, which is not surprising given the common practice of raising percent) for a given antecedent-consequent pair, with an b7 to 7 in the minor mode.) However, this is not the case empty space representing 0% and the largest circle in the major mode corpus, where scale degree 7 is the representing 100% likelihood. Note that Figure 2 reveals least used, despite the abundance of dominant chords in certain musical features that may not be evident from the corpus. This suggests that composers may be avoid- examining purely vocal corpora. For example, the ten- ing scale degree 7 in the melodic line. However, what dency for note repetition in these melodies is quite high, appears as ‘‘avoidance’’ may simply be the result of and, to a lesser extent, the tendency for arpeggiation. voice-leading preferences—a possibility that cannot be A line with a slope of 1 represents note repetitions, with ruled out with the information at hand. In any case, this clustering around that line by +1 representing motion finding is worthy of further study. by step. Note also that the scale degrees with the most predictable melodic continuations are those that music First-order probabilities. The conditional first-order theorists would classify as tendency tones (e.g., scale probability for each scale degree was independently tal- degrees #2, #4, and #5 all have 70% or higher probabil- lied. That is, given scale degree x, what scale degree is ities of moving upwards by one semitone). While there likely to follow? For the sake of clarity and brevity, and are relatively few instances of these scale degrees in the to ensure sufficient statistical power, the remaining corpus overall, they do demonstrate highly predictable analyses only consider the data from the major-mode behavior. This high predictability can be seen by looking portion of the corpus. Figure 2 illustrates the percentage at the ‘‘vertical spread’’ of each scale degree. The dia- of the time that a given scale degree proceeds to another. tonic scale degrees show more spread (i.e., more possi- The x-axis shows the antecedent scale degree (i.e., the bilities for melodic continuation) compared with the note of origin) and the y-axis represents the consequent nondiatonic tones that show less spread. Scale degree note. Circle sizes represent the likelihood (expressed in #2, for instance, shows all nineteen occurrences move upwardstoscaledegree3.(RefertoFigure1.1for 4 This is likely due to the fact that Rohrmeir and Cross only consider numeric counts of scale degrees and their distributions). Bach chorales, whereas the present work incorporates other composers’ Finally, this analysis is consistent with existing literature works from later compositional periods. that shows step motion and repetition to be more The Effect of Harmony on Melodic Probability 415

common than motion by leap (Dowling, 1968; Huron, 2001; Merriam et al., 1956; Ortmann, 1926; Temperley, 2008). The trend for step-wise melodic motion can be seen in Figure 2 by the clustering of larger circles around the diagonal. Note also that, aside from the tendency tones that all move upward, there is slightly more clustering below the diagonal than above it, implying that downward step motion is slightly more common than upwards step motion overall, consistent with research on folksong melodies, which exhibit a greater tendency for the melodic line to descend than ascend (Huron, 1996; Temperley, 2007). Notice that the zeroth- and first-order distributions of scale degrees provide useful null distributions against which harmonically informed melodic practice can be contrasted. By comparing the distributions of purely melodic scale-degree continuations with those of scale degrees set in a harmonic context, we can see if the FIGURE 3. Probabilities for both the predicted and observed melodic harmonic accompaniment has any effect on the likeli- continuations for scale degree 1. The predicted probabilities, shown with hood of the ensuing melodic tone. In other words, the shaded bars, are taken from the first-order distribution of melody alone. The observed probabilities, shown in solid bars, are calculated from the zeroth and first order probabilities of melody alone first-order conditional probabilities of scale degree 1 when it is provide a sample of predicted melodic behavior which embedded in a tonic (I) harmonic context. Reported sample size can then be compared, using the actual melodic- represents the total number of instances for the observed condition. harmonic data, with the observed melodic continuation The p value indicates the results of a 2 test comparing the predicted for each scale degree in their respective harmonic and observed distributions (df ¼ 6). contexts. a decrease in probability for continuing to either scale Change in melodic probabilities when harmony is con- degree 7 or 2. In fact, the observed distribution shows sidered. In order to facilitate interpreting the main a roughly equal probability of moving to scale degrees 7, results, Figure 3 shows an enlarged portion of Figure 4. 5, 3 and 2. Here, the predicted melodic continuations for scale Figure 4 shows the predicted and observed continua- degree 1 are compared with the observed continuations tions for all diatonic scale degrees in the major mode. for all scale degree 1s supported by tonic harmony (I). On the far left is a number representing the antecedent The y-axis shows the probability of occurrence that the scale degree. Each row of graphs examines the behavior given scale degree (in this case, 1) will proceed to any of of a single antecedent scale degree in three different the diatonic scale degrees, listed along the x-axis. The harmonic contexts. For instance, the first row of graphs shaded bars represent the predicted, or expected, shows the probabilities of the different melodic trajec- melodic behavior (again, by considering only the first- tories for scale degree 1 when either I, vi, or IV is the order melodic information), whereas the solid bars supporting harmony. For example, scale degree 1 is represent the observed melodic behavior when the given more likely to ascend to scale degree 2 when it is sup- chord (in this case, I) is supporting the antecedent scale ported by IV or vi than when it is supported by a tonic degree. Figure 3 shows that there is a statistically signif- harmony (I). A more dramatic example can be seen in icant difference between melodic probability distribu- the graphs for scale degree 6, where it appears far less tions for scale degree 1 when embedded in a tonic likely to move down to scale degree 5 when supported harmony context compared with the corresponding by submediant harmony (vi). melody-only distribution. The melody-only distribution Some unexpected findings come from examining predicts that scale degree 1 is most often followed by these graphs. For instance, in examining the probable a repeat of scale degree 1, with scale degrees 7 and 2 continuations for scale degree 2, we can see that it is being the second and third most likely consequent scale most likely to remain on scale degree 2 when supported degrees, respectively. When scale degree 1 is embedded by ii. (This is not surprising given the propensity for ii to in a tonic harmony context, however, the probability for move to V, and that scale degree 2 is a common tone to a continuation of scale degree 1 increases, and there is both harmonies.) However, we find this same trend for 416 Claire Arthur

FIGURE 4. A series of graphs illustrating the effect of harmonic context on melodic behavior. Each graph shows the observed and predicted distributions of melodic continuations for a given antecedent scale degree in a given harmonic context. (“Observed” refers to the distribution found given the harmonic context, while “predicted” refers to the probability distribution of the scale degree without regard to harmonic context.) The y-axes show the probability of occurrence that the given scale degree will proceed to any of the diatonic scale degrees, listed along the x-axes. Solid bars represent the observed melodic probabilities (from the melodic-harmonic data), and the shaded bars represent the predicted (using melodic- only data). Each diatonic scale degree is examined in three different harmonic contexts: as the root, third, or fifth of a given harmony. (An exception occurs for scale degree 4 where V replaces vii, since there were too few instances of the latter.) A missing graph indicates that there were fewer than 50 observations in the given context, and so a test was not performed. The counts represent the total number of instances for the observed conditions. p values indicate the results of 2 tests (see footnote 5) comparing the predicted and observed distributions (df ¼ 6). Nonsignificant values are shown in parentheses. The Effect of Harmony on Melodic Probability 417

FIGURE 4. [Continued] scale degree 2 to remain stationary when supported by distributions despite the fact that these chords share the vii. In comparison, when paired with dominant har- same harmonic function and tendency for resolution. monic support (V), scale degree 2 is most likely to However, the observed distributions for scale degrees 2 descend to scale degree 1. Given that vii and V have the and 7 in these dominant function contexts are dramat- same harmonic function (i.e., dominant function), one ically different from each other. In this latter example, might expect the distributions for scale degree 2 sup- scale degree 7 over vii has more than a 60% chance of ported by vii or V to look the same, yet they do not. moving to scale degree 1, the highest probability found Likewise, the consequent behavior of scale degree 7 in the results, whereas scale degree 2 over vii has only supported by either vii or V again show differing about an 8% chance of proceeding to scale degree 1. 418 Claire Arthur

This suggests that for the melodic succession 7 – 1, scale continuations from the melody-only portion of the cor- degree 7 may be more likely to be supported by vii (or pus were compared against those melodic continuations some inversion of vii or viio7) while for the melodic when the underlying harmonic information was taken succession 2 – 1, scale degree 2 may be more likely to into account. be supported with V (or some inversion of V or V7). In order to evaluate the main hypothesis, three mod- However, given the small sample size of notes over vii, els were created to test whether harmonic information the reliability of this finding should be evaluated with contributed significantly to the successful prediction of a larger corpus. melodic continuations. The first model used only A series of 2 tests were performed comparing the zeroth-order information to predict melodic continua- first-order melodic distribution of each diatonic scale tions, the second model used first-order information degree with the first-order conditional distribution of from melody alone, and the third model combined the that scale degree in three different harmonic contexts.5 first-order conditional melodic information with har- The p values are reported on each graph, with nonsig- monic context information. In each model the cross nificant values shown in parentheses. If there were less entropy was calculated. The cross-entropy values than 50 observations for a scale degree in a given har- decreased significantly with each subsequent model, monic context, the graph was omitted and no tests were with the harmonic model showing a substantial conducted. As evident in Figure 4, when the supporting decrease in cross entropy. The improvement in moving harmony is taken into account, the observed melodic from the first-order melodic information to the behavior differs significantly from what was predicted in melodic/harmonic information is roughly comparable 15 out of 19 cases. Of course, the tests themselves are to the improvement between the zeroth-order informa- not particularly important, but rather the graphs were tion and the first-order information, suggesting that meant to provide a visual explanation of how the under- harmony is playing a sizable role in predicting melodic lying harmony can impact the probability of the move- continuations. ment of a given melodic tone. These graphs might be In addition to the model testing the overall effect of thought of as providing scale degree ‘‘profiles’’ for their harmony on melodic behavior, further analyses were most common harmonic settings. Since these tests fol- carried out in order to investigate the specific effects for low the main hypothesis test and are exploratory in each diatonic scale degree. Specifically, a distribution of nature, the p values reported in Figure 4 have not been predicted consequent melodic behavior was calculated corrected for multiple tests. for each diatonic scale degree based on the purely melodic information from the major-only portion of the Discussion corpus. Each of the predicted distributions was then compared against an observed consequent distribution In this paper an empirical study attempted to measure when scale degree x was a melodic tone supported by the effect of harmonic accompaniment on melodic con- one of three potential harmonies, where the antecedent tinuations using a corpus of classical music. A compar- scale degree x is some chord tone (i.e., the root, third, or ison of models that examined melodic continuations fifth of some harmony.) Despite using a relatively small either with or without considering the harmonic corpus—which reduced power—a statistically signifi- accompaniment showed that observing a melodic tone cant effect was found in 15 out of 19 cases. Moreover, within its harmonic context conveys significantly more many of the observed contexts had a small number of information about its continuation than looking at the observations, yet the tests produced very small p values, melody in isolation. This is consistent with musical suggesting again that the effect size may be quite robust intuition as well as rules of music theory. After consid- for certain scale degrees in certain harmonic contexts. ering the overall distribution of scale degrees and Once again, the findings from this study are consistent harmonies, the first-order probabilities for melodic with the hypothesis that melodic organization is not independent of harmonic support. 5 The counts for the melody-only condition far outweighed the melo- Interestingly, the results from the present study con- dyþharmony condition. In order to compute a 2 test, the total counts tradict the perceptual findings from Schmuckler (1989), from each condition must match. The solution was to reduce the counts who claimed that melody and harmony combined addi- for the melody-only condition while retaining its original proportions tively (i.e., they did not interact) in the generation of such that the totals from each condition were matched. For example, if the melody-harmony count was 100, and the melody-only condition was musical expectations. If this were true, given the find- 1000, a hypothetical distribution of 100, 200, 300, 400 would be reduced ings from the present study, it would suggest that sta- to 10, 20, 30 and 40, respectively. tistical or implicit learning (from melodic-harmonic The Effect of Harmony on Melodic Probability 419

schemata) does not contribute to the formation of continuations. Another factor that might influence melodic expectations. Schmuckler claimed that despite melodic expectations is that of enharmonic distinctions. multiple attempts to find an interaction, none were Scale degrees appearing in the model are represented as found to be significant. It seems possible, however, that they were found in the score (i.e., with spelling distinc- Schmuckler’s study may have suffered from low power, tions), which implies that listeners would represent since he only investigated a limited number of harmonic scale degrees in this way. While composers and music settings (probes) from a single musical work. Unfortu- theorists treat them as distinct melodic elements (and nately, the experiment has never been replicated. the present work empirically demonstrates that they Several caveats should be reiterated regarding the have unique melodic behaviors), to a listener their methodology for analyzing the corpus. As mentioned enharmonic identity may be ambiguous. For example, in the methods section, several decisions had to be made given a solo melody, an instance of b6/#5 may go either about how to best encode the musical analysis, given up or down. However, given the same melody with that these are complex works that contain tonicizations, harmonic accompaniment, the harmony may help to temporary applied chords, and modulations to second- disambiguate the spelling (or ‘‘identity’’) of the scale ary keys. The works also contain a fair amount of rep- degree (e.g., if V/vi underlies the ambiguous note in etition, which might be expected to reduce the data question, the note makes more sense as #5). Due to the independence. The methods for analyzing and encoding approach taken in the present study of segregating necessarily involved making decisions such as including enharmonic scale degrees in the model, the predictive or discarding sections that were repeated, or how to power of harmony is likely underestimated. That is, encode and represent harmonies that bordered two dif- since note identity is already explicitly available in the ferent key areas. Of course, these decisions are subjec- purely melodic model, the harmonic model doesn’t add tive and therefore open to question. However, although much information for chromatic tones (i.e., notes out- some decisions were made in order to minimize the side the key). That said, the present model presents segregation of data and maximize power—such as pool- a unique approach to the classification of enharmonic ing Roman numerals regardless of inversion—the scale degrees, and may provide a starting point for majority of decisions were made with the intention of investigating the perceptual representation of scale how to best represent the music as one might hear it in degrees in a listener. real time. It is worth noting that in any large-scale anal- Although music theorists may find the results of this ysis of complex corpora, this decision making process is paper unsurprising, there is a wealth of literature in inevitable, and some interpretive decisions must be whichconjecturesaremadeaboutmelodicexpecta- made in order to reach the final stage of testing the tions, statistical learning processes, etc., where typically hypothesis. the only musical parameter examined is melody in iso- While some studies evaluating scale-degree transi- lation. It is acknowledged that using only one musical tions do not consider note repetitions (e.g., Temperley, parameter makes the task at hand substantially easier. 2008), the actual occurrence of repetitions may be an Furthermore, there are now several accessible corpora important factor for melodic expectations, and there- that are monophonic (e.g., Essen Folksong Collection), fore was included in the present study when considering and therefore offer researchers a convenient sample note transitions. On the other hand, certain factors were from which to test a theory or build a model about not included in this study despite the fact that they melody. This paper hopefully will not only contribute certainly play an important role in forming melodic to the existing body of literature on melodic probability, expectations. For instance, no rhythmic or metric infor- but also support existing research promoting the point mation was included in the models. While this was of view that melodic distributions do not come in ‘‘one omitted to simplify the models, the location of an ante- size fits all,’’ and that factors such as rhythmic duration, cedent harmony carries information to a listener about metric position, phrase position, and even historical when the next harmony will arrive. Since in music from period all contribute information to melodic probability the common practice period harmonies typically (e.g., Aarden, 2003; Albrecht & Huron 2014; Pearce, change on strong beats, a listener’s melodic expectations 2005). In addition to the inclusion of harmonic informa- will likely differ depending on whether they expect the tion, this research has contributed to our knowledge of harmony to change. Thus, knowing the metric position melodic distributions for scale degrees and harmonies, as of an antecedent harmony (as well as the duration of the well as note transition probabilities, using a corpus of melody note that sits above it) would strongly influence common practice era music (as opposed to folksong) the power of the model in predicting melodic on which most perception research on melodic 420 Claire Arthur

expectation is based. Therefore, these probability distri- these analyses, and to the extent that the sample used butions hopefully provide a more appropriate foundation in the corpus is representative of classical music, per- on which to base future studies of melodic expectation. ceptual differences would likely be expected. Given the Given these findings of the effect of harmonic context evidence in support of implicit (statistical) learning on the organization of melody, a logical next step would (e.g., Pearce & Wiggins, 2006), these findings suggest be to consider the relative weightings of factors such as that listeners may be sensitive to the influence of har- bassline (or chordal inversion) and voice leading. Cer- mony on melodic continuations. That is, if a given scale tainly voice leading and chord doubling play an impor- degree regularly and consistently tends to move (in the tant role in governing the behavior of melodic tone context of ‘‘real music’’) in a particular way when succession. For instance, perhaps the finding that scale framed with harmonic support x, but not y, then a clas- degree 7 appears to be avoided in the melodic line sical listener’s expectations could be tested, for example, comes from the voice-leading principle that warns with the use of reaction-time studies. As new corpora against the doubling of tendency tones. Or, take as become available—specifically those with harmonic and another example the finding that scale degree 6 is far contrapuntal information encoded—it will of course be less likely to move to scale degree 5 when supported by beneficial to reexamine the effects of harmony on submediant harmony. If submediant harmony fre- melodic probability with a larger dataset. In addition, quently moves to dominant harmony, the avoidance this study investigated only music from the common of scale degree 5 as a consequent tone in this context practice period. Future work might consider comparing may arise from an avoidance of parallel octaves. Unfor- the harmonic effects on melody with corpora of jazz or tunately, in this study, key pieces of information—such popular music, as well as music from other time periods. as that of chordal inversion—were discarded in the pro- cess of simplifying the musical texture in order to exam- Author Note ine the main hypothesis. As such, questions pertaining to the importance of basslines and voice leading cannot Claire Arthur is now at the Schulich School of Music, be tested here. Nevertheless, the role of voice leading McGill University, Canada. implied by the findings mentioned above warrant future This research was supported in part by a doctoral study. fellowship from the Social Sciences and Humanities Having found these statistical effects, it would be Council of Canada (SSHRC). appropriate to test the perceptual effects of harmony Correspondence concerning this article should be on melodic continuations. As mentioned, the differ- addressed to Claire Arthur, Music Technology Area, ences in probability distributions found in the explor- Schulich School of Music, McGill University, 555 Sher- atory analysis suggest moderate effect sizes for some brookeSt.W.,Montreal,Quebec,H3A1E3.E-mail: scale degrees in certain harmonic contexts. Based on [email protected]

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Appendix A

THE LIST OF PIECES USED IN THE CORPUS

Composer Setting Piece Title Bach Chorale No. 9 Ermuntre dich, mein schwacher Geist Bach Chorale No. 19 Ich hab’ mein’ Sach’ Gott heimgestellt Bach Chorale No. 24 Valet will ich dir geben Bach Chorale No. 28 Nun komm, der Heiden Heiland Bach Chorale No. 30 Jesus Christus, unser Heiland Bach Chorale No. 32 Nun danket alle Gott Bach Chorale No. 46 Vom Himmel hoch, da komm’ ich her Bach Chorale No. 48 Ach wie nichtig, ach wie fluechtig Bach Chorale No. 54 Lobt Gott, ihr Christen, allzugleich Bach Chorale No. 68 Wenn wir in hoechsten Noeten sein Bach Chorale No. 69 Komm, heiliger Geist, Herre Gott Bach Chorale No. 88 Helft mir Gott’s Guete preisen Bach Chorale No. 98 O Haupt voll Blut und Wunden Bach Chorale No. 101 Herr Christ, der ein’ge Gott’s-Sohn Bach Chorale No. 110 Vater unser im Himmelreich Bach Chorale No. 117 Nun ruhen alle Waelder Bach Chorale No. 124 Auf, auf, mein Herz, und du mein ganzer Sinn Bach Chorale No. 136 Herr Jesu Christ, dich zu uns wend’ Bach Chorale No. 153 Alle Menschen muessen sterben Bach Chorale No. 157 Wo Gott zum Haus nicht gibt Bach Chorale No. 158 Der Tag, der ist so freudenreich Bach Chorale No. 165 O Lamm Gottes, unschuldig Bach Chorale No. 176 Erstanden ist der heil’ge Christ Bach Chorale No. 177 Ach bleib bei uns, Herr Jesu Chris Bach Chorale No. 183 Nun freut euch, lieben Christen, g’mein Bach Chorale No. 187 Komm, Gott Schoepfer, heiliger Geist Bach Chorale No. 200 Christus ist erstanden, hat ueberwunden Bach Chorale No. 201 O mensche, bewein’ dein’ Suende gross Bach Chorale No. 217 Ach Gott, wie manches Herzeleid Bach Chorale No. 223 Ich dank’ dir, Gott, fuer all’ Wohltat Bach Chorale No. 224 Das walt’ Gott Vater und Gott Sohn Bach Chorale No. 248 Se Lob und Ehr’ dem hoechsten Gut Bach Chorale No. 255 Was frag’ ich nach der Welt Bach Chorale No. 258 Meine Augen schliess’ ich jetzt Bach Chorale No. 268 Nun lob’, mein’ Seel’, den Herren Bach Chorale No. 272 Ich dank’ dir, lieber Herre Bach Chorale No. 273 Ein’ feste Burg ist unser Gott Bach Chorale No. 276 Lobt Gott, ihr Christen, allzugleich Bach Chorale No. 282 Freu’ dich sehr, O meine Seele Bach Chorale No. 290 Es ist das Heil uns kommen her Bach Chorale No. 299 Meinen Jesum lass ich nicht Bach Chorale No. 303 Herr Christ, der ein’ge Gott’s sohn Bach Chorale No. 306 O Mensch, bewein’ dein’ Suende gross Bach Chorale No. 323 Wie schoen leuchtet der Morgenstern Bach Chorale No. 328 Liebster Jesu, wir sind hier Bach Chorale No. 350 Jesu, meiner Seelen Wonne Bach Chorale No. 354 Sei Lob und Ehr’ dem hoechsten Gut Bach Chorale No. 361 Du Lebensfuerst, Herr Jesu Christ Bach Chorale No. 366 O Welt, sieh hier dein Leben Bach Chorale No. 368 Hilf, Herr Jesu, lass gelingen The Effect of Harmony on Melodic Probability 423

Composer Setting Piece Title Schubert Lied Opus 59, No. 3 Du bist die Ruh Schubert Lied Opus 7, No. 3 Der Tod und das Maedchen Schubert Lied Opus 32, DV550 Die Forelle Schubert Lied Opus 25, DV795 Ungeduld Schubert Lied Op. posth.DV957 Staendchen (No. 4) Schubert Lied Opus 106, No. 4 An Sylvia Schumann Piano Miniature Album for the Young, No. 1 Melody Schumann Piano Miniature Album for the Young, No. 6 Poor Orphan Child Schumann Piano Miniature Album for the Young, No. 8 The Wild Horseman Mendelssohn Lieder ohne Worte Opus 62, No. 3 Trauermarsch Mendelssohn Lieder ohne Worte Opus 19, No. 6 Venetianishes Gondellied Haydn String Quartet Opus 74, No. 3 String quartet in G minor Beethoven Sonata Opus 13, No. 8, Mvt 2 Piano Sonata No. 8 in C minor Beethoven Sonata Opus 13, No. 8, Mvt 3 Piano Sonata No. 8 in C minor Beethoven Sonata Opus 14, No. 1, Mvt 2 Piano Sonata No. 9 in E major Mozart Sonata K333, Mvt 1 Piano Sonata No. 13 in Bb major Mozart Sonata K284, Mvt 3 Piano Sonata No. 6 in D major Clementi Sonatina Opus 36, No. 2, Mvt 1 Sonata No. 2 in G Major

Appendix B

SAMPLE ANALYSIS (SCHUMANN, ALBUM FOR THE YOUNG: NO. 1 Humdrum representation of the melody of Schumann’s **harm **solfa **harm **solfa Album for the Young, No. 1 (‘Melody’) with Roman ¼5:! ¼5:! ¼16 ¼16 Numeral analysis. Dots represent a continuation of the VreVre previous harmony. Equal signs represent barlines. Ido.fa Inversion information is removed, and seventh chords VtiImi ¼6 ¼6 vii/ii so are notated as triads. Vfaii fa ImiVre Vre¼17 ¼17 **harm **solfa **harm **solfa ¼7 ¼7 Imi IV la Vre *k[] k[] ¼11 ¼11 IsoIdo *C: *C: IV la Vfa.ti *met(c) *met(c) IsoImi¼18 ¼18 ¼1- ¼1- Vfa¼8 ¼8 ii la ImiImiVre.do .re¼12 ¼12 .faVti .doii re Imi.re .ti.favii/ii so Ido ¼2 ¼2 Vtiii fa .so ii la .reVre¼19 ¼19 .doIdo¼9 ¼9 IV la Vti¼13 ¼13 ImiIso .reVreVreVfa IdoIdoIdoImi .soVti.ti¼20 ¼20 ¼3 ¼3 ¼14 ¼14 ¼10 ¼10 ii re IsoVfaii la .fa VfaImi.doVti ImiVreVti.re .do¼15 ¼15 .reIdo ¼4 ¼4 IV la Ido¼= ¼= VtiIso.so*- *- V/V la Vfa VsoImi