This study tests the respective roles of pitch-class content and bass patterns within harmonic expectation using a mix of behavioral and computational experiments. In our first two experiments, participants heard a paradigmatic chord progression derived from music theory textbooks and were asked to rate how well different target endings completed that progression. The completion included the progression’s paradigmatic target, different inversions of that chord (i.e., different members of the harmony were heard in the lowest voice), and a “mismatched” target, a triad that shared its lowest pitch with the paradigmatic ending but altered other pitch-class content. Participants generally rated the paradigmatic target most highly, followed by other inversions of that chord, with lowest ratings generally elicited by the mismatched target. This suggests that listeners’ harmonic expectations are sensitive to both bass patterns and pitch-class content. However, these results did not hold in all cases. A final computational experiment was run to determine whether variations in behavioral responses could be explained by corpus statistics. To this end, n-gram chord-transition models and frequency measurements were compiled for each progression. Our findings suggest that listeners rate highly and have stronger expectations about chord progressions that occur frequently and behave consistently within tonal corpora.

The norms and tendencies of how chords tend to progress to one another are foundational to a musical style’s harmonic practice. It is natural to ask how sensitive listeners are to these norms, and the ability of listeners to predict subsequent events given prior events has been the subject of much research using a variety of experimental approaches. In one such approach to melodic expectation, listeners provide spontaneous completions of phrases (Fogel, Rosenberg, Lehman, Kuperberg, & Patel, 2015; Patel & Morgan, 2017), while another involves participants hearing a musical sequence and rating how well the sequence’s final event conforms to their expectations. This latter design has been implemented using single pitch-classes as the final events (Aarden, 2003; Krumhansl, 1979; Krumhansl & Kessler, 1982); however, this approach has additionally been used to test harmonic expectations evoked by chord progressions (Bharucha & Stoeckig, 1986; Patel, Gibson, Ratner, & Besson, 1998; Tillmann, Bigand, & Pineau, 1998; Vuvan & Hughes, 2019). A further approach presents a series of several chords and prompts participants to rate either the tension of some medial event (Bigand, Parncutt, & Lerdahl, 1996) or how attracted some medial event is to its successor (Brown & Tan, 2021). In addition to such behavioral tests, neuro-imaging studies such as event-related potential (ERP) tests have been used to investigate how humans cognitively process harmonic successions, particularly violations of harmonic expectancy (Besson & Faïta, 1995; Janata, 1995; Koelsch, Gunter, Wittfoth, & Sammler, 2005; Patel et al., 1998; Patel, 2010, 2012; Steinbeis, Koelsch, & Sloboda, 2006). Computationally, harmonic grammars have been examined by studying moment-to-moment harmonic successions (Acevedo, 2020; Quinn, 2010; Rohrmeier & Cross, 2008; White, 2014), successions of human annotations (Burgoyne, Wild, & Fujinaga, 2013; deClercq & Temperley, 2011; Tymoczko, 2011), voice leading (Quinn & Mavromatis, 2011; Schubert & Cumming, 2015) and non-adjacent dependencies (Sears et al., 2017b), with methods like Hidden Markov modeling (Raphael & Stoddard, 2004; White & Quinn, 2018) and deep learning (Ju, Condit-Schultz, Arthur, & Fujinaga, 2017) being applied as well.

However, both computational and behavioral studies of harmonic expectation have tended to take simplifying steps in how they represent and analyze musical chords. Indeed, this is out of necessity: successions of chords involve changing sets of pitch-classes, melodic lines, basslines, expressive phrasing, voice leading between chords, the chords’ inversions, their timbres, etc. In order to avoid these potentially confounding parameters, researchers aim to reduce these complexities. Specifically, research into harmonic successions often features Shepard tones (a technique that sounds all octaves of the same pitch-class concurrently, thus reducing the effect of pitch level and voice leading in a musical stimulus: e.g., Brown & Tan, 2021; Krumhansl & Kessler, 1982; among many others) or constructs stimuli using chords in consistently root position (Bigand et al., 1996; Patel et al., 1998; Vuvan & Hughes, 2019). Computationally, similar motivations have led to models that do not consider chord inversion (Rohrmeier & Cross, 2008; White, 2014; White & Quinn, 2018), represent only chord roots (deClercq & Temperley, 2011), use lead-sheet-style notation (Burgoyne et al., 2013), or use only triads (Tymoczko, 2011).

But the parameters excluded by these experiments potentially play important roles in harmonic expectation. Rosner and Narmour (1992) tested listeners’ perception of closure using two-chord progressions that most typically end phrases in Western classical music, finding their participants to rate root-position triads as creating more closure than inverted triads, and particular penultimate chords to endow the final chords with a greater sense of closure. Similarly, Sears, Caplin, and McAdams (2012) and Sears et al. (2017b) show that certain phrase-ending cadences (particularly, cadences involving a root-position tonic and dominant harmonies) are rated as more stable and complete than other endings, with inversions of the penultimate chords affecting these ratings. Wall, Lieck, Neuwirth, and Rohrmeier (2020) additionally demonstrate that musical sequences are perceived as well-formed when both harmony and voice leading adhere to listener expectations, and that deviations in paradigmatic voice leading may be more surprising to listeners than are deviations in harmony. In the same vein, historical research has shown that patterns between outer voices are fundamental to understanding certain compositional styles (Gjerdingen, 2007; Symons, 2017), and music theory textbooks are replete with examples of chords being contextually appropriate only when particular chord members appear as the bass pitch (Aldwell & Schachter, 2003; Laitz, 2012).

These studies show that listeners’ expectations are connected to 1) the pitch-class content of the chord (i.e., the event’s harmonic identity); 2) the linear patterns surrounding some event, like basslines and chord inversion; 3) the presence of some well-known paradigm with a highly determined ending (e.g., a cadence or an expected ending gesture). Our study manipulates each of these parameters to investigate their relative role in ratings of completion, focusing specifically on the attributes of the final chord in a sequence. The current study presents paradigmatic chord progressions in which both the harmony and inversion of the final event are highly constrained and expected within Western European tonal music; participants hear manipulations of the event’s bass note and pitch-classes, allowing us to examine the role of each.

The current study also tests the connections between such harmonic expectation and the properties of musical corpora. Research into musical learning has strongly suggested that harmonic expectations are stylistically grounded and may be learned via passive exposure to a corpus of music over an extended period of time. Most fundamentally, this work has shown that listeners can learn to distinguish between grammatical and ungrammatical tone sequences after relatively short exposure sessions (Creel, Newport, & Aslin, 2004; Krumhansl, 1990; Saffran, Johnson, Aslin, & Newport, 1999), an effect that holds in synthesized chord grammars as well (Loui, 2012). It has been additionally argued that humans learn the complex connections between pitches, scales, chords, and keys implicitly (Bharucha, 1987; Justus & Bharucha, 2001; Krumhansl, 1990), can abstract qualities of musical events through exposure to some repertoire (Arthur, 2018; Huron, 2006), and can develop different expectations for the same chord progression within different musical styles (Hughes, 2011; Vuvan & Hughes, 2019). From a computational perspective, corpus evidence has also supported the idea that musical expectations are tethered to particular corpora. This research has shown that variation in historical criticism tracks with changes in the musical practices surrounding that criticism (Byros, 2013; White, 2014), and that implicit exposure to musical corpora can affect the expectations of participants in a way that can be predicted and modeled by the statistical properties of that corpus (Rohrmeier & Rebuschat, 2012).

This study aims to investigate the connections between participants’ harmonic expectations evoked by chords in particular inversions, and the corpus properties of these inverted chords. Our investigation progresses as a series of three experiments, the first two behavioral, and the last computational. While harmonic expectations are affected by a wide variety of parameters, including short-term memory effects (Bigand, Delbé, Poulin-Charronnat, Leman, & Tillmann, 2014), effects of chord consonance (Bigand et al., 1996) and phrase closure (Tillmann & Marmel, 2013), our study focuses on expectations associated with well-worn successions of chords. To that end, our behavioral experiments use phrase-length chord progressions drawn from harmony textbooks and rely on a participant pool of academically trained musicians, reasoning that they would harbor strong explicit and implicit knowledge of these progressions. In designing these experiments, we replicate aspects of the above-cited context-target designs that test how well certain chords act as completions of a paradigmatic chord progression (e.g., Bharucha & Stoeckig, 1986; Patel, 2012; Tillmann et al., 1998; Vuvan & Hughes, 2019). In these formats, we present a series of chords that paradigmatically end with either a tonic or dominant triad, modify the final chord in that series, and task participants with rating how well that chord completes the series. In order to specifically test the relative roles of chord inversion and harmonic identity, our version of this design modifies the final chord by either varying which of its constituent pitch-classes appears lowest or by varying its pitch-class content. Our final computational experiment investigates the extent to which these behavioral responses correlate to the properties of a corpus of tonal music, testing which corpus-derived statistics might predict participant behavior.

The first experiment selects three harmonic progressions from a standard undergraduate ear-training curriculum, each of which concludes with a triad in root position (i.e., the triad’s root is in the lowest voice) in the paradigmatic version drawn from the textbook. The experiment presents listeners with this progression and varies the inversion of the final triad. Here, participants react not only to the paradigmatic root-position triad, but to that triad voiced such that its third is the lowest sounding pitch (i.e., sounding in first inversion), or with the fifth sounding lowest (i.e., in second inversion). If an event’s inversion influences harmonic expectancy, then the average responses to the different inversions will be significantly different. In addition to the three inversions of the paradigmatic target, this experiment also presents the same progressions now ending with a triad/inversion combination such that the bass note is the same as the root-position completions, but with different remaining pitch-classes. If an event’s bass note is the primary driver of harmonic expectancy, then participant responses should align with the root-position responses. Experiment 2 reproduces the procedures of Experiment 1, but with paradigmatic progressions that end with triads in inversions other than root position. A final computational experiment then attempts to connect our behavioral responses to the properties of a musical corpus, including how often progressions and their completions used in our behavioral experiments appear in the corpus. Given that our design adds complexity to an often-used experimental paradigm, these computational investigations help tease apart the roles played by the various interrelated parameters introduced by our study.

These three experiments rely on only six chord progressions, each of which ends paradigmatically either on a tonic or dominant triad. Although our sample of chord progressions is small, these progressions represent some of the most paradigmatic in Western tonal repertoire, and—with the progressions being drawn from textbooks and our participants being academically trained musicians—represent progressions with which our participants are almost certainly familiar. Furthermore, using well-known, paradigmatic chord progressions also potentially encourages participants to focus on sequential, chord-to-chord expectations rather than simply being drawn to the most-consonant harmony or chords that provide a generic sense of ending, a perennial confound in many context-target musical experiments (Aarden, 2003; Tillmann & Marmel, 2013).

In this experiment, participants heard chord progressions whose possible targets consisted of either a) the expected triad in the expected inversion, b) the expected triad in its different inversions, or c) an unexpected triad with the expected bass note. Respondents were asked to judge how well the last chord completed the progression. The two independent variables—pitch-class content (i.e., whether the harmony was or was not the expected triad) and bass note—allowed us to determine whether the bass note or overall pitch content of the target was the primary driver of these responses.

Materials and Method

Participants

Thirty music majors at the University of Massachusetts Amherst participated in this study on a voluntary basis (n = 30 approximates the number of participants in many of the above-cited chord-based context-target experiments). Participants all indicated that they were musicians and reported a mean of 12 years of music training (with a minimum of 4 and a maximum of 20 years; SD = 4.66; median = 11). All participants had completed at least one semester of formal music theory study.

Procedure

Each participant took a 12-question survey hosted on the website SurveyMonkey. These questions asked participants to listen to short (˜10 second) harmonic progressions. The three progressions were drawn from the UMass Amherst undergraduate ear training curriculum and appear in multiple undergraduate music theory textbooks. Progressions and the textbooks from which they are drawn are presented in the Appendix. Two of the progressions (1 and 3) concluded with a root-position tonic triad in their paradigmatic versions, while the third (Progression 2) paradigmatically ended on a root-position V triad. The context portions of the progressions were between 5 and 7 chords. Progressions of this length were chosen because they we judged to be ecologically valid (i.e., they represented a chord progression a student might encounter in their ear-training or theory classes), they approximated the length of the contexts used in Koelsch et al. (2005) and Patel et al. (1998), and they provided substantial context to form judgements about the completion of sequential events (Bharucha & Stoeckig, 1986; Janata, 1995; Pearce & Wiggins, 2006).

Participants heard each of these progressions with four possible target endings: the paradigmatic concluding triad, the first and second inversions of that triad, and a harmony with the paradigmatic bass note but with unexpected pitch-class content. For the last of these, we use the second inversion triad whose bass note matched that of the paradigmatic root-position ending. (In what follows, we refer to such target conditions as the “mismatched” endings, reflecting the misaligned pitch content of these solutions.) The progressions and their four possible final events are shown in Table 1. The three progressions with four possible completions resulted in twelve total questions, and each participant heard and rated all twelve. The questions’ ordering was initially randomized in the design but was held constant between respondents. The ordering of survey questions can also be found in the Appendix.

Table 1.

Chord Progressions Sorted by Target and Inversion Type

Chord ProgressionFinal ChordInversion
Progression 1 I-V4/3-I6-ii6-V6-5/4-3- I Root 
I6 First 
I6/4 Second 
IV6/4 Paradigmatic bass note, mismatch remaining scale degrees 
Progression 2 I-V-vi-iii-IV-I-IV- V Root 
V6 First 
V6/4 Second 
I6/4 Paradigmatic bass note, mismatch remaining scale degrees 
Progression 3 I-vii06-I6-IV-V- I Root 
I6 First 
I6/4 Second 
IV6/4 Paradigmatic bass note, mismatch remaining scale degrees 
Chord ProgressionFinal ChordInversion
Progression 1 I-V4/3-I6-ii6-V6-5/4-3- I Root 
I6 First 
I6/4 Second 
IV6/4 Paradigmatic bass note, mismatch remaining scale degrees 
Progression 2 I-V-vi-iii-IV-I-IV- V Root 
V6 First 
V6/4 Second 
I6/4 Paradigmatic bass note, mismatch remaining scale degrees 
Progression 3 I-vii06-I6-IV-V- I Root 
I6 First 
I6/4 Second 
IV6/4 Paradigmatic bass note, mismatch remaining scale degrees 

These progressions were encoded using standard four-part chorale voicing and were sounded on a synthesized MIDI piano patch (as implemented in Sibelius, version 7.1.3). Each progression was presented in C major at 70 beats per minute, with each chord given a 1 beat duration (a 857 ms interonset interval). To indicate that the passage was nearing completion, a 2-beat rest (1.71 seconds) was inserted before the final chord of the progression. This format was communicated to participants, and they were prompted to rate how well they thought the last chord completed the progression using a 7-point Likert scale, with 7 indicating the best completion and 1 indicating an inappropriate completion.

After the experimental questions, participants were asked to indicate their years of musical experience and their familiarity with music theory. Responses were self-paced; participants had the option of completing the survey in multiple sittings, saving their work between sessions. The average survey duration was approximately 7 minutes.

Results

A two-way ANOVA with participant responses as the dependent variable and the progression type and target chord type (root-position, first inversion, second inversion, or mismatch) as independent variables returned the target chord type as significant, F(3, 359) = 52.55, p < .001, η2 = .28, as well as progression type, F(2, 359) = 28.35, p < .001, η2 = .10, and showed an interaction between the factors, F(5, 359) = 42.87 p < .001, η2 = .13. Figure 1a shows the averaged results of each target chord and Figure 1b shows the average results for each progression. Table 2 quantifies the differences in these averages using two-sided t-tests. Table 2a shows that responses to root-position triads were significantly higher than those of the remaining two inversions, while responses to the first and second inversions were not significantly distinguishable from one another, and all inversions were rated significantly higher than the mismatched triads. Table 2b shows that responses to each of the three progressions were all significantly different from one another. Table 2c and 2d further divide the responses to target chords by progression type. Responses to Progression 2 were consistently lower overall than the other two progressions.

Figure 1.

A) Average participant ratings for each target in Experiment 1; B) average ratings for each progression.

Figure 1.

A) Average participant ratings for each target in Experiment 1; B) average ratings for each progression.

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Figure 2.

Average participant ratings for each target in Experiment 1, divided by progression type.

Figure 2.

Average participant ratings for each target in Experiment 1, divided by progression type.

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As a post hoc investigation of the role of pitch-class content within these data, Figure 3 shows the mean responses to each of the three progressions, comparing the three inversions of the paradigmatic target chord with the mismatched chord. Each of the targets with the expected pitch-class content were rated significantly higher on average than the average assessments of the mismatched target in Progressions 1, t(118) = 13.00, p < .001, d = 2.65, and 3, t(118) = 13.64, p < .001, d = 2.74, while no significant differences were found between any responses to Progression 2, t(118) = .23, p = .821, d = .05.

Figure 3.

Responses to final (mismatched) chords in Experiment 1, overlain with the range of responses from other targets’ inversions (root position, first inversion, or second inversion).

Figure 3.

Responses to final (mismatched) chords in Experiment 1, overlain with the range of responses from other targets’ inversions (root position, first inversion, or second inversion).

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Table 2.

Comparison by Two-sided t-test

A) Average Responses to Each Inversion Overall

RootFirst SecondMismatch
Root  t(178) = 4.10 p < .001 d = .60 t(178) = 2.25 p = .025 d = .34 t(178) = 11.57 p < .001 d = 1.73 
First   t(178) = 1.57 p = .118 d = .23 t(178) = 7.19 p < .001 d = 1.07 
Second    t(178) = 8.59 p < .001 d = 1.28 
RootFirst SecondMismatch
Root  t(178) = 4.10 p < .001 d = .60 t(178) = 2.25 p = .025 d = .34 t(178) = 11.57 p < .001 d = 1.73 
First   t(178) = 1.57 p = .118 d = .23 t(178) = 7.19 p < .001 d = 1.07 
Second    t(178) = 8.59 p < .001 d = 1.28 

B) Average Responses to Each Progression Overall

Prog.1Prog. 2Prog. 3
Progression 1  t(238) = 12.36 p < .001 d = 1.60 t(238) = 4.82 p < .001 d = .62 
Progression 2   t(238) = 8.73 p < .001 d = 1.13 
Prog.1Prog. 2Prog. 3
Progression 1  t(238) = 12.36 p < .001 d = 1.60 t(238) = 4.82 p < .001 d = .62 
Progression 2   t(238) = 8.73 p < .001 d = 1.13 

C) Average Responses to Each Inversion and Progression in Experiment 1

Prog. 1 FirstProg. 1 SecondProg. 2 FirstProg. 2 SecondProg. 3 FirstProg. 3 Second
Prog. 1 Root t(58) = 3.84 p < .001 d = .99 t(58) = 2.29 p = .026 d = .59     
Prog. 1 First  t(58) = 1.72 p = .09 d = .44     
Prog. 2 Root   t(58) = 1.89 p = .063 d = .49 t(58) = 1.57 p = .121 d = .41   
Prog. 2 First    t(58) = 0.27 p = .789 d = .07   
Prog. 3 Root     t(58) = 3.72 p < .001 d = .96 t(58) = 1.51 p = .138 d = .39 
Prog. 3 First      t(58) = 1.84 p = .071 d = .48 
Prog. 1 FirstProg. 1 SecondProg. 2 FirstProg. 2 SecondProg. 3 FirstProg. 3 Second
Prog. 1 Root t(58) = 3.84 p < .001 d = .99 t(58) = 2.29 p = .026 d = .59     
Prog. 1 First  t(58) = 1.72 p = .09 d = .44     
Prog. 2 Root   t(58) = 1.89 p = .063 d = .49 t(58) = 1.57 p = .121 d = .41   
Prog. 2 First    t(58) = 0.27 p = .789 d = .07   
Prog. 3 Root     t(58) = 3.72 p < .001 d = .96 t(58) = 1.51 p = .138 d = .39 
Prog. 3 First      t(58) = 1.84 p = .071 d = .48 

D) Average Responses to Each Inversion and Progression in Experiment 1

Prog. 1 MismatchProg. 2 MismatchProg. 3 Mismatch
Prog. 1 Root t(58) = 14.33 p < .001 d = 3.70 Prog. 2 Root t(58) = 1.10 p = .323 d = .26 Prog. 3 Root t(58) = 12.54 p < .001 d = 3.24 
Prog. 1 First t(58) = 8.79 p < .001 d = 2.27 Prog. 2 First t(58) = 0.91 p = .366 d = .24 Prog. 3 First t(58) = 8.31 p < .001 d = 2.14 
Prog. 1 Second t(58) = 11.41 p < .001 d = 2.95 Prog. 2 Second t(58) = 0.61 p = .541 d = .16 Prog. 3 Second t(58) = 9.81 p < .001 d = 2.53 
Prog. 1 MismatchProg. 2 MismatchProg. 3 Mismatch
Prog. 1 Root t(58) = 14.33 p < .001 d = 3.70 Prog. 2 Root t(58) = 1.10 p = .323 d = .26 Prog. 3 Root t(58) = 12.54 p < .001 d = 3.24 
Prog. 1 First t(58) = 8.79 p < .001 d = 2.27 Prog. 2 First t(58) = 0.91 p = .366 d = .24 Prog. 3 First t(58) = 8.31 p < .001 d = 2.14 
Prog. 1 Second t(58) = 11.41 p < .001 d = 2.95 Prog. 2 Second t(58) = 0.61 p = .541 d = .16 Prog. 3 Second t(58) = 9.81 p < .001 d = 2.53 

Discussion

In this experiment, we manipulated the inversion and pitch-class content of the final chord of three standard tonal progressions, presenting participants with the progressions’ original root-position ending as well as that same triad in its first and second inversions, and a different triad with the expected bass note but unexpected pitch content. We then asked participants to rate how well those triads completed the progressions. An ANOVA found that the factors of inversion and progression significantly contributed to participant ratings, and that there existed an interaction between these factors, along with a reasonable effect size. Subsequent tests showed that participants rated root-position triads higher on balance than other inversions, but also that the second progression elicited significantly lower responses than the first and third progressions. This suggests that bass notes do indeed influence participants’ assessments (as shown in Rosner & Narmour, 1992), and that there is a preference for root-position triads (as shown in Rosner & Narmour, 1992, and Sears et al., 2012); however, these preferences change with different target chords (as shown in Sears et al., 2012, and Sears et al., 2017). Additionally, in several instances it was clear that pitch-class content played a strong role in participant responses: all inversions of Progressions 1 and 3 were rated higher than the mismatch condition, suggesting that pitch-class content plays a leading role in participants’ assessments of these stimuli.

While there do seem to be certain larger trends present in these data, there are also some particular responses worth noting. For instance, in Progression 1, the second-inversion ending yielded higher average ratings than did the first inversion. Indeed, while standard textbook pedagogy teaches that second-inversion chords are more unstable than the other two inversions (Beach, 1967; Parncutt, 2011; Rosner & Narmour, 1992), it seems musically plausible that such a progression could indeed continue with a second-inversion tonic triad (i.e., extending the cadential sequence). Particular connections between participant responses and the frequency with which specific inverted chords end these sequences is the focus of Experiment 3.

In sum, there are four factors at play in our participants’ responses to the target: its bass note, the triad’s pitch-class content, its inversion, and the target chord’s identity. To the last of these, while the paradigmatic endings of other progressions were root-position tonic triads, Progression 2 (the lowest overall rated progression) ended in a non-tonic chord. It is therefore possible that our findings only show that our participants prefer root-position tonic endings over all other options. To identify whether this is the case, the next experiment repeats the methods of Experiment 1, now using progressions whose paradigmatic targets are not root-position triads.

This experiment repeated the procedures used in Experiment 1. Paradigmatic progressions whose concluding events were not root-position tonic triads were selected, and our experiment presented listeners with three new progressions. As before, they ended with either the expected paradigmatic target, or one of three additional targets: the final triad inverted to another position, or a (“mismatched”) triad that shared the paradigm’s final bass note but with the remaining pitch-classes altered.

Materials and Method

Participants

A group of thirty music majors at the University of Massachusetts Amherst participated in this study on a voluntary basis. The participant pool was different from that of the previous experiments, but participation in Experiment 1 did not preclude participation in Experiment 2. Participants all indicated that they were musicians and reported an average of 12 years of music training (with a minimum of 4 and a maximum of 21 years; SD = 4.41; median = 12). All indicated that they had at least a basic understanding of music theory.

Procedure

The procedure and stimuli were identical to Experiment 1, but now used three new progressions, as shown in Table 3. Here, paradigmatic endings appear in each progression’s highest row, and examples of each appear in the Appendix. As before, these progressions were harvested from textbooks of tonal theory (again, see the Appendix). Because we were interested in distinguishing between participants’ preferences for expected basslines and potential predispositions for root-position triads, our progressions were all paradigms that ended on an inverted tonic triad (i.e., not in root-position). In what follows, these three new progressions will be referred to as Progressions 4, 5, and 6, in order to distinguish them from the prior experiments’ three progressions. In Progression 4, the target chord was a first inversion tonic triad, a paradigmatic continuation of the penultimate V4/2 harmony. Progression 5’s target chord was a second inversion tonic triad (generally referred to as a “cadential 6/4” in contemporary theory pedagogy) and instantiated a resolution of raised scale degree four frequently used in 18th-century Western European Classical music (Byros, 2009). Progression 6 also ended with a second-inversion tonic triad and used a textbook cadential progression that illustrates that chord’s typical usage. All mismatched triads were root-position major triads that followed the precepts of the prior experiment: mismatched chords share the bass note of the paradigm, but no other pitches. (It should be noted that mismatches were produced by simply following this procedure, and produced root-position dominant chords in Progressions 5 and 6, chords that also intuitively seem to provide reasonable endings for these progressions. Connections between participant responses and the contextual appropriateness of each target chord will be considered in Experiment 3.)

Table 3.

Chord Progressions Sorted by Target and Inversion Type in Experiment 2

Chord ProgressionFinal ChordInversion
Progression 4 I-vi-IV-ii-V-V4/2- I6 First 
I Root 
I6/4 Second 
III Paradigmatic bass note, mismatch remaining scale degrees 
Progression 5 I-vi-IV-viio7/V- I6/4 Second 
I Root 
I6 First 
V Paradigmatic bass note, mismatch remaining scale degrees 
Progression 6 I-vi-ii6-ii- I6/4 Second 
I Root 
I6 First 
V Paradigmatic bass note, mismatch remaining scale degrees 
Chord ProgressionFinal ChordInversion
Progression 4 I-vi-IV-ii-V-V4/2- I6 First 
I Root 
I6/4 Second 
III Paradigmatic bass note, mismatch remaining scale degrees 
Progression 5 I-vi-IV-viio7/V- I6/4 Second 
I Root 
I6 First 
V Paradigmatic bass note, mismatch remaining scale degrees 
Progression 6 I-vi-ii6-ii- I6/4 Second 
I Root 
I6 First 
V Paradigmatic bass note, mismatch remaining scale degrees 

As before, participants interacted with an online survey hosted on SurveyMonkey, and answered 12 experimental questions (one for each progression type and chord ending); these questions took approximately 7 minutes to complete. The ordering of the questions was initially randomized in the survey design, but all participants took the same survey; ordering can be found in the Appendix.) Participants were prompted to rate how well they thought the last chord completed the progression and did so using a 7-point Likert scale.

Results

A two-way ANOVA with participant responses as the dependent variable and the progression type and target chord type as independent variables returned the chord inversion as a significant factor, F(3, 359) = 3.12, p =.027, η2 = .02, as well as progression type, F(2, 359) = 11.7, p < .001, η2 = .06, but showed no significant interaction between the factors. Figure 4a shows the average responses for each chord inversion, and Figure 4b shows the average results for each progression. Table 4 tests the differences between these averages using two-sided t-tests. Table 4a shows that responses to each inversion condition were not significantly different; Table 4b, however, shows Progression 4 to have elicited significantly higher overall responses than the latter two progressions.

Figure 4.

A) Average participant ratings for each target in Experiment 2; B) average ratings for each progression.

Figure 4.

A) Average participant ratings for each target in Experiment 2; B) average ratings for each progression.

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Table 4.

A) Comparisons of Average Responses for Each Inversion By Two-sided t-test

RootFirstSecondMismatch
Root  t(178) = 2.05 p = .042 d = .31 t(178) = 1.22 p = .226 d = .18 t(178) = 0.38 p = .703 d = .10 
First   t(178) = 1.12 p = .264 d = .11 t(178) = 1.72 p = .087 d = .41 
Second    t(178) = 0.87 p = .384 d = .28 
RootFirstSecondMismatch
Root  t(178) = 2.05 p = .042 d = .31 t(178) = 1.22 p = .226 d = .18 t(178) = 0.38 p = .703 d = .10 
First   t(178) = 1.12 p = .264 d = .11 t(178) = 1.72 p = .087 d = .41 
Second    t(178) = 0.87 p = .384 d = .28 

B) Comparisons Between Progressions in Experiment 2

Prog. 4Prog. 5Prog. 6Mismatch
Progression 4  t(178) = 5.08 p < .001 d = .76 t(178) = 4.12 p < .001 d = .61 t(118) = 0.86 p = .34 d = .73 
Progression 5   t(178) = 1.28 p = .204 d = .19 t(118) = 0.28 p = .777 d = .05 
Progression 6    t(118) = 1.09 p = .279 d = .14 
Prog. 4Prog. 5Prog. 6Mismatch
Progression 4  t(178) = 5.08 p < .001 d = .76 t(178) = 4.12 p < .001 d = .61 t(118) = 0.86 p = .34 d = .73 
Progression 5   t(178) = 1.28 p = .204 d = .19 t(118) = 0.28 p = .777 d = .05 
Progression 6    t(118) = 1.09 p = .279 d = .14 

To investigate the effect of chord inversion further, Figure 5 divides results both by progression type and by inversion. Comparing averages within each progression and within each inversion using two-sided t-tests results in only four significant differences, each involving the first inversion completion to Progression 4. On the one hand, this progression’s first inversion target was judged significantly higher than its root-position ending and its mismatched ending, t(58) = 3.14, p < .001, d = .79 and t(58) = 2.28, p < .001, d = .42, respectively. On the other hand, this target was also significantly higher than first inversion responses to Progression 5, t(58) = 4.47, p < .001, d = 1.15, and Progression 6, t(58) = 4.63, p < .001, d = 1.2. Outside of responses being generally higher to Progression 4, no other overall trends were found in these divisions.

Figure 5.

Average participant ratings for each inversion of the final chord in Experiment 2, divided by progression type.

Figure 5.

Average participant ratings for each inversion of the final chord in Experiment 2, divided by progression type.

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Discussion

In this experiment, we manipulated the final chord of three standard tonal progressions whose paradigmatic concluding event was an inverted tonic triad. We presented participants with the expected completion in each triadic inversion, along with a mismatched triad that shared the bass note but no other pitch-classes with the paradigm ending. As before, we asked participants to rate how well those triads completed the progressions. Progression 4 elicited higher responses than other inversions, with listeners’ responses to the first inversion paradigmatic ending being significantly higher than responses to most other targets and to responses to first inversion targets in other progressions.

These findings suggest three main conclusions. First, the significantly higher responses to Progression 4’s paradigmatic ending suggest that the expected basslines/inversions can affect participant responses, at least in some contexts. This indicates that participants are not simply drawn to root-position tonic triads regardless of context but are instead potentially drawn to a progression’s paradigmatic completion. These findings reflect the modeling of Ashley (2019), in which local cues (e.g., the expected completions of certain scale degrees, the tension of particular chord structures, and the completion of harmonic schemata) give rise to harmonic expectations.

However, second, this effect was only seen in Progression 4. Similar to our observations in Experiment 1, these findings indicate that participants are not equally sensitive to each paradigmatic progression. Looking again at Progression 5 and 6’s data in Figure 4a, we might note that participants did not exhibit any clear preferences in these progressions. It would seem that not all cases elicit a sensitivity to a non-root-position inversion of the target.

Finally, it is notable that progressions that elicited higher average responses were also those in which participants had stronger differences between their more- and less-preferred targets. Over these two experiments, the three progressions in which chord inversion elicited significant differences between responses (Progressions 1, 3, and 4) were also rated higher on average (4.92, SD = 1.99, N = 360) than progressions in which participants responses did not differ significantly by inversion (Progressions 2, 5, and 6; mean = 3.85, SD = 1.92, N = 360), and this difference is significant, t(718) = 7.36 p < .001, d = .62.

Therefore, there seems to be some connection between how high participants rate a progression overall and their sensitivity to the inversion of its endings. These findings suggest a provocative link to the “mere exposure effect” (Krumhansl, 1990; Temperley, 2007; Zajonc, 1968), in which humans develop preferences for things with which they are more familiar and to which they have been more frequently exposed. Additionally, research has pointed to musical continuations being rated as more pleasant and/or satisfying when listeners are more acclimated to expect that particular continuation (Huron, 2006; Meyer, 1956). It is therefore possible that Progressions 1, 3, and 4 were both rated more highly and also with more sensitivity to inversion because they represent sequences with which participants are more familiar and whose endings are more determined. To test this as well as other variations in participant responses between progressions, the next experiment quantifies expectancy using several corpus-derived models. These models assess the probability of our target chords given the context from a corpus perspective. We then observe the extent to which these probabilities correspond to the behavioral responses documented in Experiments 1 and 2.

Given the above-cited connections between learning, musical expectations, and the properties of the corpus to which a listener has been exposed, the current experiment explores the extent to which probability distributions drawn from a corpus of tonal music align with our behavioral results, focusing particularly on whether events that occur more frequently and with greater determinacy are rated higher. In our initial discussion, we manipulated three parameters that previous research had identified as contributing to harmonic expectation: 1) the pitch-class content of the chord, 2) the basslines/inversions manifesting within the progression, and 3) the use of a familiar/determined paradigm. In what follows, we derive a suite of probabilistic descriptors of the six chord progressions used in the previous experiments from the Yale Classical-Archive Corpus and compare those modelings to our previous results. Each of our models addresses one of these parameters: if one of these corpus-derived models correlates to participants’ responses, then that model’s parameters might be seen as playing a leading role in how our participants processed our stimuli.

Procedure

Probabilistic models were designed, all involving n-gram models trained on a corpus of tonal music. These models will capture what we will call the conditional probability, the frequency, and the determinacy, with the first of these representing how probable our progressions’ target events are given the preceding chords, the second representing how often the progressions occur overall, and the third capturing how certain a progression is to conclude with its paradigmatic ending.

The Corpus

All models were based on the Yale-Classical Archives Corpus (YCAC), a corpus of tonally-analyzed Western European art music, 1650–1900 (White & Quinn, 2016). This corpus was chosen because a) as a corpus of so-called “canonical” music from the Western European tonal art music tradition, it represents a reasonable approximation of the style on which contemporary musical theory and aural skills curricula are based, b) the piano timbre used in our stimulus has been shown to evoke harmonic expectations associated with this style (Vuvan & Hughes, 2019), and c) the corpus contains information about both chord type and inversion. (Importantly, we are not claiming that this corpus represents the aggregate musical experience of our participants—such a corpus would, for instance, include genera like pop and rock music. Rather, this corpus is better imagined as an approximation of the students’ pedagogical, performance, and listening experience with a particular repertoire and tradition. We will consider potential connections between our pedagogically oriented methods and the distinction between implicit and explicit learning in our general discussion.)

The corpus represents this music’s surface events, specifically showing every moment a pitch is added or subtracted from the texture (i.e., from every note of an Alberti bass to every turn of a trill). One fallout of this surface-oriented compilation method is that the corpus’s constituent vocabulary contains a large diversity of chord structures (including passing and neighboring dissonances), and structures that are considered equivalent in a music theory classroom are distinct within its annotations (for instance, a complete dominant seventh and a dominant seventh with the fifth omitted would be considered equivalent in a music theory classroom, but inequivalent within the YCAC). However, in constructing our models, we retrieved only events that exactly matched our search criteria; due to the size of the YCAC, a sufficient number of these successions were present in the corpus.

To ensure against any peculiarities of the YCAC, each model and test were additionally run on the music of a selection of Western common-practice composers in the music21 corpus (Carl P. E. Bach, Johann S. Bach, Ludwig Beethoven, Arcangelo Corelli, George Handel, Joseph Haydn, Wolfgang Mozart, Clara Schumann, Robert Schumann, Giuseppe Verdi, and Carl Weber). The vectors produced for both corpora in the methods described below were reasonably correlated, giving us relative confidence in the performance of the YCAC to capture generic tonal trends. While not reported in this paper, results from the music21 models can be found in our online supplementary materials, posted at chriswmwhite.com/HarmonicExpectancy.

N-gram Models

In order to quantify how often each target ended each progression, we constructed corpus-based computational models of each progression. N-gram models were compiled using a standard Markov-chain approach. Our approach calculates the probability of a target event conditional on its two prior context events. We focused on groupings of three chords (i.e., trigrams) to avoid problems with data sparsity that arise when examining groupings based on larger orders of Markov chains.

We constructed three such models. The full-chord model used the same chord sequences (i.e., chords indexed by their inversions) and considered all possible completions present in the corpus when calculating probabilities. This model approximated an approach that employs both pitch-class content and linear bassline patterns. The bass-only model used sequences of bass pitches with probabilities again derived from the entire corpus. This model approximated an approach that only employs linear bass-line patterns. Finally, the pitch-class model tracked the sequences of pitch-class sets. This model approximated an approach that only employs pitch-class content. Information for each trigram appears in Table 5.

Table 5.

Results From the YCAC as Examined in Experiment 3, Sorted by Progression, Inversion, and Bass Note; the Paradigmatic Inversion Type Appears First

Chord / Bass trigramFinal Chord & InversionTarget Probabilities (Chord)Target Probabilities (Bass)Target Probabilities (Pitch)
Progression 1Chord & inversionn = 2,900 Bassline onlyn = 6,473 Pitch-class setn = 19,271 V6-4/5-35^5^… I (1^) Root .12 .20 .294 
I6 (3^) First .011 .011 
I6/4 (5^) Second .065 .70 
IV6/4 (1^) Mismatch .002 (.20) .0015 
Progression 2Chord & inversionn = 4,394 Bassline onlyn = 4,115 Pitch-class setn = 16,457 I-IV… 1^4^… V (5^) Root .037 .10 .026 
V6 (7^) First .003 .008 
V6/4 (2^) Second .0002 .0005 
I6/4 (5^) Mismatch .043 (.10) .333 
Progression 3Chord & inversionn = 1,026 Bassline onlyn = 944 Pitch-class setn = 2,177 IV-V… 4^5^… I (1^) Root .16 .13 .186 
I6 (3^) First .025 .04 
I6/4 (5^) Second .03 .65 
IV6/4 (1^) Mismatch (.13) .046 
Progression 4Chord & inversionn = 2,732 Bassline onlyn = 1,476 Pitch-class setn = 24,079 V-V4/25^4^… I6 (3^) First .171 .104 .139 
I (1^) Root .012 .15 
I6/4 (5^) Second .001 .377 
III (3^) Mismatch (.104) .0007 
Progression 5Chord & inversionn = 61 Bassline onlyn = 61 Pitch-class setn = 138 IV-viio7/V… 4^–♯4^… I6/4 (5^) Second .26 .689 .268 
I (1^) Root .033 .098 
I6 (3^) First 
V (5^) Mismatch .23 (.689) .116 
Progression 6Chord & inversionn = 752 Bassline onlyn = 752 Pitch-class setn = 2,044 ii6-ii… 4^2^… I6/4 (5^) Second .08 .34 .114 
I (1^) Root .007 .035 
I6 (3^) First .015 .10 
V (5^) Mismatch .016 (.34) .141 
Chord / Bass trigramFinal Chord & InversionTarget Probabilities (Chord)Target Probabilities (Bass)Target Probabilities (Pitch)
Progression 1Chord & inversionn = 2,900 Bassline onlyn = 6,473 Pitch-class setn = 19,271 V6-4/5-35^5^… I (1^) Root .12 .20 .294 
I6 (3^) First .011 .011 
I6/4 (5^) Second .065 .70 
IV6/4 (1^) Mismatch .002 (.20) .0015 
Progression 2Chord & inversionn = 4,394 Bassline onlyn = 4,115 Pitch-class setn = 16,457 I-IV… 1^4^… V (5^) Root .037 .10 .026 
V6 (7^) First .003 .008 
V6/4 (2^) Second .0002 .0005 
I6/4 (5^) Mismatch .043 (.10) .333 
Progression 3Chord & inversionn = 1,026 Bassline onlyn = 944 Pitch-class setn = 2,177 IV-V… 4^5^… I (1^) Root .16 .13 .186 
I6 (3^) First .025 .04 
I6/4 (5^) Second .03 .65 
IV6/4 (1^) Mismatch (.13) .046 
Progression 4Chord & inversionn = 2,732 Bassline onlyn = 1,476 Pitch-class setn = 24,079 V-V4/25^4^… I6 (3^) First .171 .104 .139 
I (1^) Root .012 .15 
I6/4 (5^) Second .001 .377 
III (3^) Mismatch (.104) .0007 
Progression 5Chord & inversionn = 61 Bassline onlyn = 61 Pitch-class setn = 138 IV-viio7/V… 4^–♯4^… I6/4 (5^) Second .26 .689 .268 
I (1^) Root .033 .098 
I6 (3^) First 
V (5^) Mismatch .23 (.689) .116 
Progression 6Chord & inversionn = 752 Bassline onlyn = 752 Pitch-class setn = 2,044 ii6-ii… 4^2^… I6/4 (5^) Second .08 .34 .114 
I (1^) Root .007 .035 
I6 (3^) First .015 .10 
V (5^) Mismatch .016 (.34) .141 

The full-chord model combined two pieces of information available in the YCAC: each moment’s unordered scale degree set, and that moment’s bass notes. For example, I6/4 would be represented as scale degrees <1^, 3^, 5^> with 5^ as the lowest pitch. Given that the YCAC’s “chords” are aggressively surface events, we ignored adjacent subsets of cardinalities less than 3. That is, if a tonic triad was followed by a single tonic pitch (a singleton subset of the former), which in turn was followed by a dominant triad, the succession would register as I-V rather than I-<1^>-V. The unordered scale degree set served as a proxy for the “chord,” while the bass note served as proxy for “inversion.” The YCAC was queried for successions that matched the requisite trigrams for each of our six progressions. (Full-chord trigram probabilities were therefore calculated as the count of times within the corpus that the target chord followed the progressions’ penultimate pair of chords, divided by the number of times the penultimate pair of chords was followed by any chord.) The bass-only model was similarly constructed but used only scale degrees from the progressions’ bass pitches. The pitch-class set model proceeded identically to the full-chord model but used only pitch-class sets from the YCAC, and ignored inversion.

Each of these models could then assess the conditional probability of the trigram by calculating how often each final event occurs given the prior events. We then also calculated the progression’s frequency and its determinacy. For the former, we tallied the frequency of occurrence of all trigrams under consideration, yielding the probability that the trigram would occur in the overall corpus. We then used the probability of the most probable target chord (always the paradigmatic ending) to model the determinacy of each chord progression (with relatively more probable endings being more determined). Lists of these statistics can be found in our online supplementary material, at chriswmwhite.com/HarmonicExpectancy.

Behavioral Comparisons

These computational models were measured against three aspects of our behavioral responses: 1) the average responses to each target in each progression, 2) the average rating of each progression, and 3) the strength of participants’ preference for a single target in a progression, approximated by the difference between the average responses to the highest-rated and lowest-rated targets.

Each aspect was compared to its analogous corpus-derived statistics. Recalling that conditional probabilities have been linked to harmonic expectation (Bharucha & Stoeckig, 1986; Patel, 2012; Tillmann et al., 1998; Vuvan & Hughes, 2019), average responses to each target in each progression were compared with the conditional probabilities associated with each target, and were derived from each of the three n-gram models’ probability assessments (full-chord, bass-only, and pitch-class models). (In the case of the bass-only models, average participant responses to the three inversions of the paradigmatic completion for each progression were used, and mismatch targets were excluded.) Both a standard Pearson correlation and a Spearman rank-order correlation were used to compare the average participant responses over all progressions in each experiment to the corresponding probabilities assessed by a given model for the respective progressions and completions. (For Spearman rank-order calculations, if averages were not significantly distinct, they were assigned the same rank; if a rank was occupied by more than one average, the following rank was skipped).

Additionally, recalling that listeners often demonstrate stronger and more specific expectations associated with highly certain and very familiar outcomes (Krumhansl, 1990; Meyer, 1957; Rohrmeier & Neuwirth, 2015; Sears, 2015), the corpus-derived frequency and determinacy values for each chord progression were both compared to the average rating for each progression (showing how highly each progression was rated overall) and the difference between the highest- and lowest-rated target for each progression (showing the strength of preference for the paradigmatic target). Comparisons were undertaken using Pearson correlations.

Given that our motivation in this experiment is exploratory in nature and our objective is simply to investigate general corpus properties and their potential connections to behavioral data (and not, for instance, to test whether a particular parameter correlates better than chance), confidence values and significance testing were not undertaken for these computational/behavioral correlations (Huron, 1999).

Results

Figure 6 represents the conditional probability of the paradigmatic target as assessed by the full-chord, bass-only, and pitch-class models. Additionally, the figure overlays average participant responses from each progression, along with average responses to the paradigmatic target (always the highest rated target), shown with x-marked dotted lines. The chart also shows the Pearson correlation coefficients (r) and Spearman’s rho (ρ) for each individual target chord’s rating and probability for both the Full Chord (F.C.) and Bass Only (B.O.) approaches within each progression. Both correlation approaches return positive coefficients for Progressions 1-4, with r values always at or exceeding .60. The average correlation between full-chord model probabilities and highest rating was .58. This finding indicates that the YCAC’s chord/inversion conditional probability distribution tracks the overall contours of participant behavior in these cases. In contrast, the average correlation between bass-only probabilities and highest participant ratings was .01. This suggests that probabilities resulting from a bass-only trigram model does not align with participant results. Finally, Pearson correlations for the pitch-class (P.C.) model are shown as well; however, given that this model binned targets into only two categories (targets either included the expected pitch-class content or did not), these values are either 1 or -1 (participant responses either aligned with these two categories, or did not). The pitch-class only model aligned relatively well, with an average r of .33.

Figure 6.

Conditional probabilities (using the paradigmatic target) for each model and each progression used in Experiment 3, compared to the average rating and highest average rating for each progression; Pearson and Spearman correlations between each target chord’s rating and probability (Full Chord and Bass Only) are shown for both below.

Figure 6.

Conditional probabilities (using the paradigmatic target) for each model and each progression used in Experiment 3, compared to the average rating and highest average rating for each progression; Pearson and Spearman correlations between each target chord’s rating and probability (Full Chord and Bass Only) are shown for both below.

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Table 6 investigates the relationships between progressions’ determinacy, and participants’ overall ratings and preferences in response to those progressions. The average ratings for each progression are shown in the chart, as are the differences between the highest-rated and lowest-rated inversions that participants heard at the end of each progression. These vectors are then correlated with all target chords’ conditional probabilities for the full-chord model and pitch-class model (the bass-only model was excluded from this analysis due to its poor performance outlined above).

Table 6.

A) Representations of Determinacy, Preference, and Variation in Each Progression

Progression% prob of paradigmatic ending in F.C. model (determinacy)% prob of paradigmatic ending in P.C. model (determinacy)Average ratingsAverage ratings (excluding mismatch)Difference between highest- and lowest- rated targetDifference between paradigmatic and mismatched P.C. sets
1 12.0 29.4 5.79 5.80 4.36 3.53 
2 3.7 2.6 3.48 3.48 0.90 0.09 
3 16.0 18.6 5.52 5.52 3.94 3.69 
4 17.1 13.9 5.25 5.33 1.23 0.33 
5 26.0 26.8 3.90 3.88 0.60 0.12 
6 8.0 11.4 4.14 4.24 0.50 0.41 
Progression% prob of paradigmatic ending in F.C. model (determinacy)% prob of paradigmatic ending in P.C. model (determinacy)Average ratingsAverage ratings (excluding mismatch)Difference between highest- and lowest- rated targetDifference between paradigmatic and mismatched P.C. sets
1 12.0 29.4 5.79 5.80 4.36 3.53 
2 3.7 2.6 3.48 3.48 0.90 0.09 
3 16.0 18.6 5.52 5.52 3.94 3.69 
4 17.1 13.9 5.25 5.33 1.23 0.33 
5 26.0 26.8 3.90 3.88 0.60 0.12 
6 8.0 11.4 4.14 4.24 0.50 0.41 

B) Correlations Between These Data

Correlations(r)
Chord Determinacy and avg ratings .18 
Determinacy and difference .003 
Difference and avg ratings .83 
PC Set Determinacy and avg ratings .52 
Determinacy and difference .52 
Difference and avg ratings .79 
Correlations(r)
Chord Determinacy and avg ratings .18 
Determinacy and difference .003 
Difference and avg ratings .83 
PC Set Determinacy and avg ratings .52 
Determinacy and difference .52 
Difference and avg ratings .79 

Finally, Figure 7 investigates how the frequency of each progression interacts with other parameters. The figure plots each progression by its frequency and its determinacy (i.e., the paradigmatic target chord’s conditional probability in the YCAC). The figure shows three additional statistics: 1) the average rating of a progression is shown by the surrounding shading, with darker shades indicating higher ratings; 2) the correlation between participant ratings and determinacy is indicated by circle size; and 3) the strength of preference is shown by the thickness of the circle’s outline, with greater differences between the highest and lowest ratings shown with thicker outlines. Progressions 1, 3, and 4 cluster to the higher sides of the two axes (i.e., these progressions are frequent, and their targets are probable), and they also sport the largest, darkest, and thickest circles. These progressions feature both relatively high probabilities and frequencies, and participants respond to these progressions with high ratings, and clear preferences that correlate highly to a tonal corpus. Conversely, lower and more ambivalent ratings arise when these progressions are either infrequent in the corpus (as is the case with Progression 5), when their completions are relatively uncertain (as in Progression 2), or when there exists a combination of infrequency and indeterminacy (as in Progression 6).

Figure 7.

Interaction of determinacy and frequency for each progression, where larger circle size indicates a greater correlation coefficient between behavioral and corpus data, darker shading shows higher average participant ratings, and thickness of the circle’s outline designates the range between highest- and lowest-rated completion.

Figure 7.

Interaction of determinacy and frequency for each progression, where larger circle size indicates a greater correlation coefficient between behavioral and corpus data, darker shading shows higher average participant ratings, and thickness of the circle’s outline designates the range between highest- and lowest-rated completion.

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Discussion

This computational experiment demonstrated several quantifiable aspects of how the progressions under consideration behave in a corpus of tonal music, and how those characteristics might interact with the participant responses of previous experiments. First, participant ratings of progressions correlated more strongly with trigram probabilities drawn from bass-oriented pitch-class sets (the full-chord model) than with those drawn from simple bass-note progressions (the bass-only model). Furthermore, the pitch-class model corresponded fairly well, but, on average, did not correspond to participant responses as much as the full-chord model.

Second, our corpus data illustrated that some of our mismatched stimuli were themselves plausible completions of Progressions 2, 5, and 6. This plausibility was illustrated by the alignment of relatively high participant ratings of the mismatched targets in each of these progressions. This is particularly notable given that the paradigmatic and the mismatched targets were root-position dominant triads and second-inversion tonic triads in each of these cases. Connections with music theory, particularly the role of the cadential 6/4 as an elaborating dominant, will be explored below.

Third, the probabilities surrounding the targets of these progressions appeared to have some further connection to participant ratings on two fronts: as evidenced in the correlation between average ratings and determinacy, the more probable a progression’s paradigmatic target, the higher participants rate that progression on average, and as evidenced by the correlation between determinacy and the difference between participants’ most-liked and least-liked targets, the more probable a progression’s target, the clearer the preference expressed by participants.

One other notable facet of our analysis is the consistently high probabilities assigned by the bass-only models. Note that, in Figure 6, the probability is always higher than that of the full-chord model. On the one hand, this is not surprising: the universe of 12 chromatic scale degrees is much smaller than the universe of possible chords, and therefore there are fewer opportunities to divide the probability mass of possible completions. However, the fact that there are more potential options in one model than another does not require the larger model have a more random distribution and lower determinacy. It is particularly interesting, then, that participants do not seem to rely on these more constrained and determinate aspects of this corpus for their assessment, but instead seem to favor an approach that incorporates the complexities, difficulties, and computational expense of a mixture of scale degree information and bass notes.

These three experiments presented behavioral and corpus data associated with six chord progressions drawn from music theory textbooks and used in American conservatory-style tonal pedagogy, each of which featured paradigmatic (expected) final events. Experiment 1 presented participants with the first three of these progressions and asked them to rate how well various targets completed these progressions. Targets included the paradigmatic triad in its standard root-position, as well as the triad’s other two inversions, and a mismatched triad which contained the paradigmatic bass pitch but changed the remaining harmonic content. This first experiment found that participants not only often rated the paradigmatic targets significantly higher when the progression ended with that target (in Progressions 1, 3) but rated all targets higher on average for these progressions compared to the remaining progression (Progression 2). We found that the mismatched targets that used the paradigmatic bass pitch but altered other pitch content were rated significantly lower than the paradigmatic inversion of the expected triad and were even rated significantly lower than all inversions of the expected triad. However, this finding only held with Progressions 1 and 3, something we speculated may have been due to a combination of participants’ familiarity with the progressions and the level of determinacy associated with the paradigms’ endings.

To test participant responses to progressions that paradigmatically end on non-root-position triads, Experiment 2 replicated the processes of the prior experiment, now using Progressions 4, 5, and 6, all of which ended on first or second-inversion tonic triads. Here, only responses to Progression 4 showed a significant preference for the paradigmatic ending. To investigate potential explanations for participant responses, Experiment 3 constructed several corpus-derived probability models associated with each progression and found that many of these correlated to participant responses, suggesting that listeners were most sensitive to progressions that were both relatively frequent and whose endings were relatively determined in a corpus of tonal music. In what follows, we outline some larger topics with which these findings interact.

The Mosaic of Harmonic Expectation

The above findings suggest that harmonic expectation can be affected by a variety of interacting musical parameters, particularly pitch-class content and chord inversion. In other words, the act of listening to a harmonic progression is a complicated activity that is difficult to reduce to any one musical parameter. When participants were presented with targets that provided paradigmatic bass pitches but non-paradigmatic pitch-class content, their ratings of that completion decreased; similarly, an n-gram model relying solely on bass-note progressions predicted participant behavior worse than did a model that also accounted for pitch-class content in addition to bass note/inversion. Further, a model that used only pitch-class content (and ignored inversion) predicted participant responses relatively well, but less reliably overall than a model including inversion. These findings suggest that while a chord’s pitch-class content may be a primary driver of a listener’s expectations, a chord’s inversion can also play an important role in expectation.

Yet, while participants did often seem to be sensitive to the inversion of the paradigmatic targets, they often did not generally distinguish between the other inversions of that target. Indeed, especially in Progressions 1 and 3, responses can be bundled into three groups, with the expected inversion of the paradigmatic triad yielding the highest responses, the other inversions of the paradigmatic triad yielding the next highest, and finally the mismatched stimulus yielding the lowest ratings. These three tiers of preferences suggest the relative weight that participants seem to give to different musical parameters when forming their expectations, with pitch-class content playing a leading role, followed by bass note/inversion.

However, this dynamic does not hold in all cases. In Progression 4, participants rated the root-position version of the paradigmatic triad as lower than the mismatched triad, suggesting that listeners give the bassline particularly strong weight in this instance. Aligning with the findings of Wall et al. (2020), this likely demonstrates strong and specific expectations associated with the bassline pattern of this particular progression (i.e., the tendency of the bass of a V4/2 chord to resolve downward by step). Additionally, we did not observe participants harboring general preferences for particular chords and inversions. For instance, participants did not consistently rate tonic triads higher than other triads, nor did we see a reliable preference across all progressions for root-position events. In fact, the mismatched targets of Experiment 2 were all in root position, and we observed no particular preference for these targets.

Many of these observations might be a feature of these stimuli being progressions rather than isolated events (Bigand & Parncutt, 1999; Parncutt,1997; Wall et al., 2020): if we had provided less progression-oriented stimuli or instructions, we might well have observed some different preferences in our participants (as argued in Tillmann et al., 1998). Additionally, there are many musical parameters that contribute to the feeling of connection and attraction between two chords, including common-tone retention, half-step motion, chord hierarchy, etc. (Bigand et al., 1996; Brown & Tan, 2021; Krumhansl, 1979). As such, an approach that incorporates other such parameters may add further complexity and explanatory power to how listeners experience harmonic expectation.

Cadence and Tonal Function

These findings also suggest several conceptual connections with the notion of “cadence.” In the Western European tonal style, authentic cadences (conclusive motion from a root-position dominant harmony to a root-position tonic triad) are both highly constrained and very frequent. Indeed, it has been suggested that the definition of cadence is intrinsically interwoven with high frequency and high determinacy, yielding an ability to satisfy very strong and specific predictions (Acevedo, 2020; Gjerdingen, 2007; Meyer, 1957; Narmour, 1990; Pearce & Wiggins, 2012; Rohrmeier & Neuwirth, 2015; White & Quinn, 2018), and that these cadential moments of high predictability signify a feeling of ending (Arthur, 2018; Bigand et al., 1996; Huron, 2006; Sears et al. 2017). It is not surprising, then, that the progressions that we found to exhibit high frequency, high determinacy, and high/consistent ratings were also those progressions that constitute paradigmatic cadences in this style.

Somewhat speculatively, our findings may also interact with notions of the “Tonic Function,” the category of chords that most often begins or ends phrases, and is associated with qualities of stasis, resolution, and finality (Huron, 2006). Researchers have noticed that the pitches of the tonic triad tend to receive higher ratings of completion or belonging (Krumhansl, 1979), are responded to more quickly in timed tasks (Aarden, 2003), are more consistently grouped together into the same similarity categories by participants (White & Schwitzgebel, 2019), and are rated as more stable than other pitches in the diatonic scale (Bharucha & Krumhansl, 1983; Bigand et al., 1996; Parncutt, 2011). Our findings align with these previous studies insomuch as those progressions that ended with root-position or first inversion tonic triads elicited ratings from our participants that were both relatively high and also exhibited a clear preference. We would speculate that there exists some connection between these reliably high ratings and the tonic’s associated qualities.

Finally, the notion of the “Dominant Function” (the harmonic category that tends to precede the Tonic function and is associated with the dominant triad and seventh, White & Quinn, 2018) is also evoked by our findings. In particular, we see high participant ratings often elicited by second-inversion triads when scale degree five is in the bass (i.e., I6/4 triads). In many cases (Progressions 1, 2, 6) we saw these high ratings corresponding to the predictions of a corpus-based model. However, each of these instances were situations in which scale degree five was highly favored in the bass-only corpus models and were also moments in which a dominant triad would also have been a plausible continuation. In other words, these second-inversion tonic chords occur in the same context as—and elicit comparable reactions to—dominant triads. Indeed, much mainstream theory pedagogy teaches this chord as having a dominant function (e.g., Aldwell & Schachter, 2003; Clendinning & Marvin, 2016; Laitz, 2012).

Similarities Between Behavioral Responses and Corpus-derived Properties

Roughly a decade ago, Rohrmeier and Rebuschat (2012) lamented, “despite the plentitude of models, not much cognitive work has yet been done bridging behavioral and computational approaches to implicit learning of music” (p. 537). This study joins in a trend of studies that have answered this call, specifically suggesting some such bridges between corpus properties and harmonic expectations connected to chord inversion. Our computational experiment used probability models to explore how these six progressions manifested within a musical corpus, and showed ways that these manifestations align (and misalign) with our participants’ responses. Overall, participant ratings were reasonably predicted by the probabilities produced by an n-gram model of pitch-class sets and bass-oriented scale-degree sets trained on a corpus of tonal music, with participants rating more probable targets consistently higher.

We identified three corpus properties—frequency of progression’s occurrence, determinacy of the progression, and probability of the target given the context—that appeared to correspond to three aspects of the behavioral data: how highly participants rated a progression’s target on average, how correctly their ratings were predicted by the behaviors of a tonal corpus, and the strength of their preference for a particular completion relative to other completions. As summarized in Figure 7, above, our study suggests that progressions that maximized these corpus parameters elicited the most certain and highest-rated responses in our participants. This suggests a specific overlap between musical learning and corpus properties: the strongest harmonic expectations appear to be associated with progressions that appear often and with consistent and highly predictable completions.

Pedagogical Learning Versus Implicit Learning

While much of the above-cited literature connecting corpus properties and statistical learning relies on the concept of implicit learning (i.e., internalizing expectations through passive exposure to some repertoire), our study’s materials potentially also evoked participants’ explicitly learned expectations. Our progressions were drawn from music theory textbooks, and our participants were recruited from a pool of students who explicitly studied identical or similar materials. Naturally, there is no bright line between expectations garnered from classroom learning versus those learned through a music student’s exposure to repertoire (indeed, music curricula are generally designed such that one reinforces the other!). However, there can be differences between the materials present in a textbook and the tendencies of the repertoire on which that textbook is focusing (White, in press). While the pedagogically derived examples in the current study did indeed evoke responses that aligned with corpus-derived statistics in a way that suggests connections with implicit-learning theories, the potential role that explicit learning may have played in our participants’ expectations should be acknowledged.

Future Research Directions

One future direction of this research might investigate the kinds of brain activity elicited by the different musical expectations identified by our study. Music researchers have used ERP methodology to better understand the ways that the brain reacts to violations and fulfillments of harmonic expectation (e.g., Fogel et al., 2015; Koelsch, 2005; Koelsch et al., 2005; Patel, 2010; Patel & Morgan, 2017; Võ & Wolfe, 2013), and further studies might adapt these designs to investigate such reactions to chord inversion and basslines.

The current study suggests a view of harmonic expectation that is multifaceted and complex, and we can easily imagine various ways to expand our current approach to include further complexities. Previous work has investigated how parameters such as “roughness” and measures of harmonic distance (such as distance on the circle of fifths) interact with musical expectations (Bigand et al., 1996; Brown & Tan, 2021; Krumhansl, 1979), as well as how the metrical positioning of a harmonic event effects the cognition of that event (Dawe, Platt, & Racine, 1995; London, 1990, 2012; Mirka, 2009; Prince, Thompson, & Schmuckler, 2009; Rosenthal & Hannon, 2016; White, 2017). Incorporating more potential parameters and endings into future behavioral and computational designs would provide insights into possible interactions between inversion, these parameters, and listeners’ harmonic expectations. Future research might also modulate the language used to prompt participants’ answers (i.e., completion, continuation, target, etc.) to better observe how listeners are responding to musical stimuli.

Additionally, while the correlations between corpus models and participant ratings suggest connections between statistical exposure, familiarity with a progression, and preference for certain completions, more work is required on this front. Future studies might, for instance, incorporate an exposure session into the experimental design to test how listeners learn to associate chord inversions with their preferred harmonic completions, or might include a population exposed to different musical styles to more robustly test the role of music training in our results.

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Appendix

Figure A1.

Progression 1, where the labeled final chords 1-4 are from Experiment 1. Paradigm drawn from Aldwell and Schachter (2003, p. 133), Clendinning and Marvin (2016, pp. 288, 278), Sánchez-Kisielewska (2017).

Figure A1.

Progression 1, where the labeled final chords 1-4 are from Experiment 1. Paradigm drawn from Aldwell and Schachter (2003, p. 133), Clendinning and Marvin (2016, pp. 288, 278), Sánchez-Kisielewska (2017).

Close modal
Figure A2.

Progression 2, where the labeled final chords 1-4 are from Experiment 1. Paradigm drawn from Koskta, Payne, and Almén (2013, p. 107), Clendinning and Marvin (2016, p. 398).

Figure A2.

Progression 2, where the labeled final chords 1-4 are from Experiment 1. Paradigm drawn from Koskta, Payne, and Almén (2013, p. 107), Clendinning and Marvin (2016, p. 398).

Close modal
Figure A3.

Progression 3, where the labeled final chords 1-4 are from Experiment 1. Paradigm drawn from Aldwell and Schachter (2003, p. 133), Clendinning and Marvin (2016, p. 352).

Figure A3.

Progression 3, where the labeled final chords 1-4 are from Experiment 1. Paradigm drawn from Aldwell and Schachter (2003, p. 133), Clendinning and Marvin (2016, p. 352).

Close modal
Figure A4.

Progression 4, where the labeled final chords 1-4 are from Experiment 2. Paradigm drawn from Koskta, Payne, and Almén (2013, p. 212), Clendinning and Marvin (2016, pp. 256, 265). As noted in these textbooks, the 7th of a V7 chord must resolve down by step, making the root-position tonic chord resolution incorrect.

Figure A4.

Progression 4, where the labeled final chords 1-4 are from Experiment 2. Paradigm drawn from Koskta, Payne, and Almén (2013, p. 212), Clendinning and Marvin (2016, pp. 256, 265). As noted in these textbooks, the 7th of a V7 chord must resolve down by step, making the root-position tonic chord resolution incorrect.

Close modal
Figure A5.

Progression 5, where the labeled final chords 1-4 are from Experiment 2. Paradigm drawn from Koskta, Payne, and Almén (2013, p. 266), Clendinning and Marvin (2016, p. 441). Each of these textbooks mandate the step-wise resolution of the leading tone in secondary chords such as this; the root-position tonic triad is thus an ineffective resolution because of its tritone leap in the bass and its non-resolution of both the leading tone and the tritone.

Figure A5.

Progression 5, where the labeled final chords 1-4 are from Experiment 2. Paradigm drawn from Koskta, Payne, and Almén (2013, p. 266), Clendinning and Marvin (2016, p. 441). Each of these textbooks mandate the step-wise resolution of the leading tone in secondary chords such as this; the root-position tonic triad is thus an ineffective resolution because of its tritone leap in the bass and its non-resolution of both the leading tone and the tritone.

Close modal
Figure A6.

Progression 6, where the labeled final chords 1-4 are from Experiment 2. Paradigm drawn from Koskta, Payne, and Almén (2013, p. 136). Here, it would be syntactically incorrect to resolve a root-position supertonic chord to a root-position tonic chord due to the likelihood of parallel fifths or octaves.

Figure A6.

Progression 6, where the labeled final chords 1-4 are from Experiment 2. Paradigm drawn from Koskta, Payne, and Almén (2013, p. 136). Here, it would be syntactically incorrect to resolve a root-position supertonic chord to a root-position tonic chord due to the likelihood of parallel fifths or octaves.

Close modal
Instructions for Participants

“For each question you will click a link to hear a short chord progression followed by a brief pause, and then the final chord of the progression. You will then be asked to rate from 1-7 how well this final chord completes the progression (1 = Worst, 7 = Best).”

Table A1.

Ordering of Survey Questions Sorted by Survey, Experiment, Progression Number, and Question Number

QuestionExperimentProgression #Progression
1 I-V4/3-I6-ii6-V6-5/4-3-I 
I-vii06-I6-IV-V-I6 
I-V-vi-iii-IV-I-IV-V6 
I-vii06-I6-IV-V-I6-IV6/4 
I-V4/3-I6-ii6-V6-5/4-3-I6/4 
I-vii06-I6-IV-V-I 
I-V4/3-I6-ii6-V6-5/4-3-IV6/4 
I-V4/3-I6-ii6-V6-5/4-3-I6 
I-V-vi-iii-IV-I-IV-V 
10 I-V-vi-iii-IV-I-IV-I6/4 
11 I-vii06-I6-IV-V-I6/4 
12 I-V-vi-iii-IV-I-IV-V6/4 
2 I-vi-IV-ii-V-V4/2-I6/4 
I-vi-IV-viio7/V-I6/4 
I-vi-ii6-ii-I6/4 
I-vi-IV-viio7/V-V 
I-vi-ii6-ii-I 
I-vi-IV-ii-V-V4/2-I6 
I-vi-IV-viio7/V-I 
I-vi-ii6-ii-V 
I-vi-IV-ii-V-V4/2-I 
10 I-vi-ii6-ii-I6 
11 I-vi-IV-ii-V-V4/2-III 
12 I-vi-IV-viio7/V-I6 
QuestionExperimentProgression #Progression
1 I-V4/3-I6-ii6-V6-5/4-3-I 
I-vii06-I6-IV-V-I6 
I-V-vi-iii-IV-I-IV-V6 
I-vii06-I6-IV-V-I6-IV6/4 
I-V4/3-I6-ii6-V6-5/4-3-I6/4 
I-vii06-I6-IV-V-I 
I-V4/3-I6-ii6-V6-5/4-3-IV6/4 
I-V4/3-I6-ii6-V6-5/4-3-I6 
I-V-vi-iii-IV-I-IV-V 
10 I-V-vi-iii-IV-I-IV-I6/4 
11 I-vii06-I6-IV-V-I6/4 
12 I-V-vi-iii-IV-I-IV-V6/4 
2 I-vi-IV-ii-V-V4/2-I6/4 
I-vi-IV-viio7/V-I6/4 
I-vi-ii6-ii-I6/4 
I-vi-IV-viio7/V-V 
I-vi-ii6-ii-I 
I-vi-IV-ii-V-V4/2-I6 
I-vi-IV-viio7/V-I 
I-vi-ii6-ii-V 
I-vi-IV-ii-V-V4/2-I 
10 I-vi-ii6-ii-I6 
11 I-vi-IV-ii-V-V4/2-III 
12 I-vi-IV-viio7/V-I6 

*Note: Progressions’ target chords are bolded.