BACKGROUND: HOW RARE ARE RARE INTERVALS? HOW MAJOR IS THE MAJOR MODE?

THIS article deals with a particularly elusive object for observation: the skeletal tones associated with Robert Gjerdingen's (1988, 2007) galant schemata. I cautiously write "skeletal tones associated with" the schemata rather than referring to "the" schemata, since the schemata are multi-parametric and include metric, harmonic (or contrapuntal), and ornamental features beyond their soprano-and-bass skeletons. For the purpose of this article, I am particularly interested in the soprano skeletal threads that undercut the melodic surface of eighteenth-century music, as well as the intervals that they form with the skeletal bass. Gjerdingen defines his outer-voice skeletons as scale degrees respective to a local tonic: historical approaches to tonality suggest that every tonal shift that we might think of as a "tonicization" was conceptualized as a wholesale shift to a different tonal center (Lester, 1992; Gjerdingen, 2007; Byros, 2009; Holtmeier, 2011). By analyzing the musical surface, Gjerdingen abstracts a soprano-and-bass contrapuntal skeleton and identifies local key contexts. For instance, the soprano skeleton of the Fonte schema (Example 1, Appendix C) is annotated as [ii]: ^4-^3[♭] followed by [I]: ^4-^3, not monotonally (^5-^4-^4-^3). Thus, Gjerdingen's annotations represent both an abstraction of an outer-voice skeleton from the musical surface as well as the local key context or key contexts. Though my present inquiry analyzes Gjerdingen's expert annotations of a corpus of 14 pieces, my assumption is that Gjerdingen reacts to melodically salient events in the style and their typical configurations, such as the Fonte schema of Example 1. In other words, I assume that the present inquiry is not merely a study of the idiosyncrasies of a single scholar's analytical technique, but rather has potential implications for understanding the growing prevalence and salience of tritone resolutions in eighteenth-century music.

Though this is an armchair theoretical inquiry with some pilot results, I will conclude by proposing hypotheses that can be operationalized and tested in the future by evaluating the musical surface directly. After presenting some theoretical considerations, I will provide a pilot account of schemata based on the expert-annotated corpus of full-piece analyses in Gjerdingen (2007). To my knowledge, this is virtually the only corpus annotated by an expert analyst that includes all schemata, rather than a corpus analysis focusing on an individual pattern (cf. Gjerdingen, 1988; Byros, 2009; Mitchell, 2016; Aerts, 2017). The corpus in Rabinovitch (2015) represents my methodological biases and is therefore not useful here. Gjerdingen's (2007) sample has allowed him to examine the first-order transitional probabilities between schemata (Gjerdingen, 2007, p. 372, Figure 27.1). My analysis of Gjerdingen's annotated corpus provides tentative results regarding the central role of rare intervals in the abstracted skeletons. By detecting common skeletal patterns of the style, Gjerdingen de facto marks certain surface events as salient, which amounts to a form of pitch reduction (Rabinovitch 2019). Despite proposals for formalizing schema or pattern detection (Symons, 2017; Sears, 2017; Finkensiep, Neuwirth, & Rohrmeier, 2018), schemata are not yet observable features of the musical surface, making this expert-annotated corpus valuable at the present stage.

Gjerdingen proposes galant schemata as a reconstruction of musical communication between expert musicians and enculturated listeners around 1720-1780. Gjerdingen (1988) situates the peak of one particular schema, the ^1-^7…^4-^3 in the 1770s based on a corpus study and estimates that schemata in general peaked around 1765 (Gjerdingen, 2007). Byros's (2009) corpus shows that an additional schema peaked in the 1790s. Gjerdingen eschews "macrotheoretical" and "macrohistorical" generalizations and is interested in providing rich, descriptive "microhistories" and "microtheories" of individual patterns. Yet the proximity in time between the peak of the ^1-^7…^4-^3 in the 1770s and the estimated peak of schemata in general around 1765 based on his career-long study of the style seems to suggest, prima facie, that Gjerdingen's research captures general stylistic trends that peaked somewhere in the latter half of the eighteenth century. Like Gjerdingen's skeletal schemata, the major mode peaked somewhere during the latter half of the eighteenth-century in terms of the proportion of pieces composed in it. The major mode has a single tritone between ^4-^7, as opposed to the (harmonic) minor mode, which additionally has a prominent ^2-^6♭ tritone that creates some tonal ambiguity. (The presence of the ^2-^6♭ tritone in minor may explain the frequent modulations to the relative major in minor-mode pieces.) Schema types over-emphasize tritone resolutions in the outer voices (Rabinovitch, 2018), and their tacit, time-span-reduction-like analytical process (Lerdahl & Jackendoff, 1983) implies high priority for tritone resolutions within metric segments (Rabinovitch, 2019). Since schemata are closely related to outer-voice tritone resolutions, and since the major mode has a unique tritone (Browne, 1981), the musical "ecosystem" of the latter half of the eighteenth century seems particularly favorable to the abstraction of tritone resolutions as salient outer-voice skeletal events. Abstracting outer-voice tritones also reveals the localized key context on which the scale-degree identities of the schemata redundantly rely. Needless to say, the coincidence between the peak of schemata and the peak of the major mode does not imply causation, but only allows us to reflect on some aspects of the style and of the evolving tonal system.

I have referred to the "peak" of the major mode in the latter half of the eighteenth century. This means that major-mode pieces are particularly prevalent proportionally during this half century. This trend is more marked in the latter half of the eighteenth century in comparison with the rest of the common practice narrowly defined (1700-1899), as discussed in musicological and empirical literature to be cited below. In fact, throughout this article I will focus exclusively on the prevalent major mode and will largely ignore the minor mode. Hence, all of my statements about schemata below should have the caveat "when embedded in the major mode…" in front of them, not only since the "ecosystem" of the major mode seems ideally suited for the properties of galant schemata, but also due to its sheer prevalence during the period 1750-1799.


My hypothesis is the following:

H1: The usage of the schemata prioritizes skeletal tritones as vertical intervals at the expense of their diatonic generic counterparts, P5 and P4. Thus, by emphasizing a rare harmonic interval of the diatonic-set template (Browne, 1981) through musical usage as a skeletal interval, the schemata create clarity of local key context. By abstracting characteristic skeletal patterns of the style out of the musical surface, Gjerdingen's analytical process also clarifies localized tonal contexts. I had initially hypothesized that skeletal schemata skew the balance in favor of rare soprano melodic semitones as compared to common whole tones, but this is more problematic to evaluate even in the context of the current, preliminary pilot (see below).

Elsewhere, I have showed that Gjerdingen's implicit reductive decisions move from the musical surface to a skeleton by giving priority to tritone-resolutions (or resolution of local ^4 over ^5) within a metric segment implied by his annotations, and otherwise prioritizing the first available vertical consonance within the segment: this heuristic, which amounts to a type of time-span reduction (after Lerdahl & Jackendoff, 1983), approximates Gjerdingen's analytical choices of skeletal soprano tones within a segment at about 85% (Rabinovitch 2019). The possibility of reconstructing an aspect of schema reduction according to this heuristic suggests exploring the role of tritone resolutions within schemata from additional angles, as I attempt to do here. The apparent priority given to tritone finding in the pitch-reductive process brings to mind the hypothesis that the rare intervals of the diatonic set are crucial for key finding (Browne, 1981; Butler, 1989; Butler & Brown, 1994). If schema finding is intertwined with high priority for tritone finding, the processes of pitch reduction and the discovery of the local key context of the schema seem intertwined. Since schemata as typical scale-degree structures rely on a local key-context for their definition, there seems to be some redundancy built into the abstraction of salient pitches and localized key contexts. I recognize, of course, that the rare-interval hypothesis had been seriously questioned if not altogether rejected in favor of distributional models (Krumhansl, 1990). However, we should bear in mind that the improved success of key-finding algorithms (e.g., Albrecht & Shanahan, 2013) does not necessarily represent the psychology of key finding, which is still an open question (e.g., Temperley & Marvin, 2008; VanHandel & Callahan, 2012; Farbood, Marcus, & Poeppel, 2013; Anta, 2015). Indeed, some of these experiments suggest that scale-degree frequencies by themselves are not sufficient tonal cues for listeners. The current inquiry, of course, cannot give us any information about the minds of listeners, dead or alive. Rather, it is a preliminary exploration of schemata and what they might tell us about the evolving usage of tonally unambiguous tritones within common-practice tonal music. The connection between Meyer and Gjerdingen's ^1-^7…^4-^3 schema (later renamed Meyer) and rare intervals was briefly suggested by Spitzer (2004). The tritone-resolution sub-schema in several schemata and its key-defining aspect is discussed by Jan (2013, 2015) and in more detail in Rabinovitch (2018). The idea that the diatonic ^4-^7 tritone is central in tonality and its evolution is, of course, an old one (Fétis, 1840, 1844), but the changing usage of the interval within a subset of the common practice still requires some exploration: the peak of the schemata in the latter half of the eighteenth century may reveal something about a basic, evolving tonal feature.

What exactly does it mean that an interval is "rare"? The interval vector of the diatonic set is <254361> (Browne, 1981), which creates two pairs of "rare" and "common" intervals with the same generic size: six P5+P4 (ic5) vs. one tritone (ic6), and five whole tones (or minor sevenths, ic2) vs. two semitones (or major sevenths, ic1). In both cases, the "common" interval (P5+P4, whole tone) is tonally ambiguous, while the "rare" one (semitone, tritone) is unambiguous, at least in the context of a tritone resolution without enharmonic reinterpretation, which is generally what is relevant for eighteenth-century music. I had initially hypothesized that skeletal galant schemata skew the balance in favor of the "rare" intervals of the template in both cases: tritones are made more common—or at least are overemphasized—in respect to perfect fifths or fourths as vertical skeletal intervals between the outer voices; semitones are overemphasized at the expense of the common whole tones as melodic motions in the skeletal soprano. The former hypothesis seems to be reflected in the small corpus of complete-movement analyses in Music in the Galant Style (Gjerdingen, 2007); the latter is problematic. The question of common vs. rare intervals is intertwined with a distinction between two types of "rarity":

  1. Rarity in the diatonic-set template, to which I will refer below as "rarity in template."
  2. Rarity in musical usage in a musical corpus, to which I will refer below as "rarity in usage."

Skeletal tones are the "core tones" of Gjerdingen's (2007) schemata: though Gjerdingen has been averse to contrapuntal reduction of any type, his analytical annotations de facto create pitch reductions (Rabinovitch, 2019). When thinking about musical usage, we have to distinguish between two things:

  1. The usage of intervals on the musical surface (surface textures such as those from which Gjerdingen's skeletons are derived);
  2. The usage of intervals in the skeletons—melodic intervals in the soprano skeleton and vertical intervals respective to the skeletal bass.

The distinctions above raise a question regarding the relations between the diatonic-set template, surface usage, and skeletal usage. To my knowledge, this question has been somewhat downplayed in the polemics on rare intervals. The proponents of the rare-interval hypothesis were interested both in the rare intervals of the diatonic template and in the temporal ordering of these elements that eliminates ambiguity (as in a tritone resolution) or, conversely, increases it (Butler, 1989). However, the actual rarity of rare intervals in musical usage seems to me to have been somewhat neglected in those old polemics on key finding. One might ask:

  • Are the rare intervals of the diatonic template actually rare in musical usage? Is the ratio in the template between M2 and m2 or P5+P4 vs. tritone retained in musical usage, or is it skewed in musical usage in either direction? Are rare-in-template intervals made more or less rare in usage respective to the abstract diatonic template?

In the context of galant schemata, another pertinent question arises:

  • What is the relation between surface musical usage and schematic skeletons? Is the abstraction of an outer-voice contrapuntal skeletons associated with a high frequency of tritone resolutions, which makes local key contexts unambiguous?

The ratios in the abstract diatonic-set template are 2.5 for whole-tones vs. semitones (or, more precisely, ic2/ic1, which may also represent major sevenths and minor sevenths), and 6 for perfect fifths and fourths (P5+P4) vs. tritones (after Browne, 1981). Of course, the interval vector of the diatonic set represents not intervals per se, but rather interval classes. For the skeletal vertical (soprano-bass) intervals that are central in my discussion (P5+P4 vs. tritone spelled as aug. 4 or dim. 5), this makes less difference, since tritones are either generic fifths (diminished) or fourths (augmented): the pair of ics that compete are ic6 (tritone, i.e., aug. 4 or dim. 5) and ic5 (P5+P4). For melodic whole and half steps, there are several potential mitigating factors for the fact that ic1 and ic2 encompass both seconds and sevenths:

  1. The cyclical, octave-neutral nature of scale degrees, which are the entities captured by Gjerdingen's scale-degree annotations.
  2. The rarity of large intervals such as sevenths as melodic intervals, which is cross cultural, a statistical universal if not a universal (Huron, 2006; Savage, Brown, Sakai, & Currie, 2015).
  3. The fact that skeletal scale-degree transitions in galant schemata tend to be stepwise, somewhat similar to Schenker's notion of melodic fluency (Pastille, 1990; Metz, 2017).

Huron's (2008) corpus analysis of the two-part inventions by J. S. Bach—while stemming from different theoretical motivations—allows us to calculate the (P5+P4)/tritone ratio in usage within a 2-voice eighteenth-century corpus as 3.42. Thus, in a 2-voice corpus from the early eighteenth century, the surface usage of tritones makes them proportionally less rare than their share in the abstract diatonic template, yet they are still 3.42 times less likely to occur as surface vertical intervals than their diatonic counterparts, P5 and P4. Of course, it would be preferable to calculate the surface vertical intervals of the corpus of pieces analyzed in Gjerdingen (2007), which would be a way to strengthen the current pilot. In the meantime, Huron's (2008) data give us an approximation from a corpus that is close in time and has the advantage of being a pure 2-voice corpus. If expert analyses into Gjerdingen's schemata by the main proponent of this theoretical position tentatively suggest a further skew in favor of the tritone, it would seem that this abstraction of salient skeletal events coincides with greater clarity of localized key contexts.

It is less clear how to compare frequencies of whole tones and semitones. David Temperley (personal communication, 2017) indicates that the whole-tone/semitone ratio (henceforth: WT/ST ratio) in the first violin parts in the corpus of Haydn and Mozart's string quartets is ca. 1.08, in comparison with the WT/ST ratio of 2.5 of the diatonic template. In other words, whole tones and semitones are almost balanced in their usage. Eighteenth-century music has many chromatic embellishments. The task of manually recording every melodic surface interval in the principal, surface soprano melody of eight of the pieces analyzed in Gjerdingen (2007) or excerpts thereof had proved unwieldy. However, it suggested a WT/ST ratio of 1.34 as a pilot approximation across the intervals recorded—certainly closer to the Haydn and Mozart corpus data than to the diatonic template.

Finding an equal ground for comparison is indeed problematic: Gjerdingen's skeletal schemata rely on a hyper-local sense of key, in which every brief tonicization changes the contextual scale-degree identities of the skeletal events involved, as we have seen above in the discussion of Example 1. The scale degrees available in soprano skeletons (Gjerdingen, 2007) are the diatonic scale degrees and one step in the sharp and flat direction on the line of fifths (Temperley, 2000), ^7♭-^4-^1-^5-^2-^6-^3-^7-^4♯. As we shall see below, ^4♯ and ^7♭ are infrequent in Gjerdingen's annotations. Comparing galant skeletons to Temperley's Haydn and Mozart calculation is problematic: the principal semitones in skeletal galant usage are local ^7-^1 and ^4-^3, whereas the Haydn and Mozart corpus involves chromatic motions, with a larger repertoire of semitones available.

Another possible corpus for comparison is the Essen Folksong Collection (Huron, 2006). Though this is a corpus of European folk melodies, not "art" music, it has some advantages in speculating about the relations between the diatonic template and the usage of "rare" intervals in some variety of Western tonal music. In this corpus, the musical usage is overwhelmingly major, strongly tending towards major-diatonic usage. For instance, by comparing the ratios of mode-dependent transitions, we can see that minor-mode usage is negligible. For instance, the ^4-^3 to ^4-^3♭ ratio is ca. 49.13; the ^6-^5 to ^6♭-^5 ratio is ca. 173.42. While the corpus does contain some chromaticism, it is largely major-diatonic. The WT/ST ratio for diatonic-major stepwise motions within this corpus can be calculated as follows: ("^1-^2" + "^2-^1" + "^2-^3" + "^3-^2" + "^4-^5" + "^5-^4" + "^5-^6" + ^6-^5" + "^6-^7" + "^7-^6") / ("^1-^7" + "^7-^1" + "^4-^3" + "^3-^4") = (0.02806 + 0.04190 + 0.03282 + 0.04865 + 0.01712 + 0.03653 + 0.02076 + 0.03642 + 0.00854 + 0.01327) / (0.02321 + 0.02025 + 0.04127 + 0.02644) = 2.55527 ≈ 2.5. Thus, if we isolate the major-diatonic stepwise transitions from the Essen corpus, the WT/ST ratio in usage remains almost balanced respective to the ratio in the diatonic-set template, despite the fact that the transitions ^6-^7 and ^7-^6 are particularly infrequent. In other words, there exists a tonal corpus in which the abstract rarity of semitones in the diatonic collection is nearly mirrored in musical usage.

If we were to add the two chromatic scale degrees that are available in Gjerdingen's skeletal soprano scale-degrees, hence, add ^4♯-^5, ^5-^4♯, ^7♭-^6, ^6-^7b from the Essen data to the denominator, the WT/ST ratio would be 2.41. The question of the WT/ST ratio seems even more problematic than that of the (P5+P4)/tritone ratio, and the results in Gjerdingen's analyses are not favorable to my initial hypothesis that skeletal usage overemphasizes semitones. As we shall see, however, the results below indicate that skeletal melodic semitones are very frequently annotated as the locally-diatonic ^3-^4 and ^4-^3 as well as ^7-^1 and ^1-^7, highlighting again the affinity between schema finding and the detection of a localized tonal center.

I have cited above the peak of schemata somewhere in the latter half of the eighteenth century (Gjerdingen, 1988, 2007; Byros, 2009). Musicologists have observed the gradual shift from a relatively balanced distribution of major- and minor-mode compositions at the beginning of the eighteenth century to a sheer prevalence of major-mode pieces in its latter half (Riley, 2014). The prevalence of major-mode pieces in this half century respective to the nineteenth century is also clearly reflected in Horn and Huron's (2015) corpus. The latter half of the eighteenth century is thus the relative peak of usage of major-mode compositions within common-practice tonality narrowly defined, 1700-1899. A very rough sketch of the modal history of Western music might be useful here (cf. Lester, 1989; Albrecht & Huron, 2014; Tompkins, 2017):

  1. From modes to a major-minor system (up to 1700 or before);
  2. Major and minor modes are relatively equally prevalent (ca. 1700-1749);
  3. Major mode as the predominant mode, taking a particularly large share of compositions (ca. 1750-1799);
  4. A relative rise in the usage of the minor mode, 1800-1899.

Horn and Huron's (2015) data for 1750-1899 and Albrecht and Huron's (2014) data for 1700-1749 reflect the sharp fluctuations in the usage of the major and minor modes, as represented in Table 1 by the ratio of major-mode pieces to minor-mode pieces for each half century:

Table 1. Ratio of major-mode/minor-mode compositions through the common-practice era (calculated after Albrecht & Huron, 2014; Horn & Huron, 2015)
Time periodRatio of major mode/
minor mode
1700-17491.27
1750-17994.88
1800-18492.33
1850-18991.77

Albrecht, Horn, and Huron situate each composer's work in the decade in which the composer turned 25; hence, their results do not reflect work chronology per se. However, their data reflect clear generational trends that mark the latter half of the eighteenth century as the heyday of the major mode. The first half of the eighteenth-century has a fairly balanced distribution of major- and minor-mode pieces, followed by a rise in prevalence of the major mode. After its heyday, the major mode significantly declined throughout the nineteenth century. Though the minor mode did not subsequently overthrow the major mode from its "lead" (nor did it even regain the relative balance of 1700-1749), its share grew in the nineteenth century, especially in its latter half. 3

When the majority of pieces are cast in the major mode, it is likely to serve as a prototype for both modes. Parncutt (2012, p. 121) writes: "Because major was more common, minor was perceived as a variant of it, rather than the reverse: minor became 'the Other' of the major-minor system." This "otherness" is, of course, particularly relevant surrounding the peak in prevalence of the major mode. One is reminded of Tversky's (1977) classic discussion of the asymmetry in perceived similarity: instead of Tversky's cold-war example of Cuba and the U.S.S.R one might suggest to millennial students that Academia.edu is more similar to Facebook than Facebook is similar to Academia.edu. In the present context, the minor more is more similar to the major mode than the other way around. The roughly joint peak of galant schemata and the major mode allows us to examine the properties of galant schemata when they are embedded in a major-mode context as a central stylistic tendency in the latter half of the eighteenth century.

If one were to imagine a favorable musical ecosystem in which tritone resolutions are associated with skeleton abstraction (Rabinovitch, 2019) and with identifying the hyper-local key context on which the schemata's scale-degree identities rely, then a musical reality dominated by the major mode, with its single tritone, seems ideal and clearest. As stated, the minor mode has two prominent tritones, between ^7 and ^4 as well as between ^2 and ^6♭. Again, the joint peaks of the major mode and the schemata do not suggest causality, but rather tell us something about the history of a tonal feature. Gjerdingen (2007, pp. 16–19) proposes the schemata as "cognitive archaeology" or a reconstruction of historical musical communication. The rare-interval hypothesis was a proposal for a cognitive mechanism for key finding in living listeners. Without making direct claims about musical minds, dead or alive, it seems that this joint peak may tell us something about the evolution of the tonal system and the prevalence and salience of tritone resolutions, which opens avenues for future inquiry to be discussed in the conclusion section.

Before I turn to analyzing Gjerdingen's annotations, I would like to take a quick look at the skeletal melodic aspect of Gjerdingen's schemata. The skeletal soprano lines of galant schemata from Gjerdingen (2007) are represented in Diagrams 1a and 1b (see Appendix D) alongside additions by other scholars: Byros (2009, "le-sol-fi-sol"), Rice (2014, "Heartz," 2015, "Morte"), Mitchell (2016, "Volta"), and Aerts (2017, "Svago," similar to Mitchell's Volta but proposed independently). Looking at the list of prototypes may be somewhat similar to looking at the diatonic-set template: without information about usage, it is difficult to assess what this means. However, this representation shows that the diatonic semitones, ^4-^3 and ^1-^7, emerge as central junctures of melodic activity, ending many of the patterns. By annotating the musical surface and segmenting it into typical patterns of the style, Gjerdingen highlights patterns ending on a diatonic semitone respective to a highly localized key center.

A PRELIMINARY ASSESSMENT OF GALANT SKELETAL USAGE

This section offers pilot analyses of skeletal vertical intervals and skeletal melodic intervals in Gjerdingen's analytical annotations. I have included in this survey the full-movement analyses in the following fourteen chapters in Gjerdingen (2007): 5, 8, 10, 12, 15, 17, 19, 21, 22, 23, 24, 26, 28, 29, with pieces by Giovanni Battista Somis, Carl Ditters von Dittersdorf (*2), Joseph Haydn, Christoph Willibald Gluck, Baldassare Galuppi (*2), Johann Christian Bach, Simon Leduc, Leonardo Leo, Niccolò Jommelli, Wolfgang Amadeus Mozart, Johann Joachim Quantz, and Francesco Galeazzi. All fourteen pieces are in the major mode. In the case of an aria da capo, the repetition of the big A section (not re-notated) was not taken into account (only events in the notated portions were recorded). From the Theme and Variations by Haydn (Chapter 10), only the theme was represented, since including all of the variations would have inflated the proportion of tritones and given my hypothesis an unfair advantage. Due to the very small size of the corpus and some of the constituent pieces, and due to the corpus's stylistic uniformity, I have decided to tally skeletal intervals across the entire corpus. The Haydn quartet movement analyzed in the theoretically central Chapter 27, "Il Filo", has been left out of the present survey due to sketch-study issues (Rohringer, 2015).

Unfortunately for present purposes, the analytical annotations in Gjerdingen (2007) do not provide an exhaustive two-voice skeleton for each movement in its entirety: Gjerdingen does not abstract a skeleton from each and every measure of the music, though he annotates almost all of the musical surface. In analyzing the vertical intervals, I had to devise a method that would create some transparency and allow others to critique my reading of Gjerdingen's annotations, listed in Appendix A (comparing Appendix A to Gjerdingen's annotations would show interested readers that my interpretative freedom was indeed limited). I have used the following principles in analyzing Gjerdingen's annotations:

  1. Only explicitly marked soprano (or principal melodic line) skeletal scale-degrees were recorded. If the bass was not marked, it was reconstructed by me based on the score. If the soprano was sustained or immediately re-articulated, the soprano was counted again against a new bass when this was deemed meaningful.
  2. If the same interval was implied again by the markings, it was counted again (e.g., C5 over C3 repeated twice in Gjerdingen's annotations). This stems from the assumption that the re-annotation implies markedness of this musical event for Gjerdingen as an analyst and listener.
  3. Small-circle markings, typically associated with auxiliary features such as the high-^2 and high-^6 drop, were not included in the survey. However, when features such as high-^2 and high-^6 drops were marked with a normal-sized circle, they were recorded in Appendix A.
  4. The Cudworth surface elaboration, ^7-^6-^5-^4, annotated inconsistently by Gjerdingen, was taken as ^4 only, followed by a ^3-^2-^1 cadential string, (see Rabinovitch, 2018, 2019), regardless of which of the markings for ^7, ^6, ^5, were present.

The listing of intervals is provided in Appendix A: the ratio of perfect fifths and fourths to tritones (P5+P5)/tritone in it is 1.82 (213 vs. 117 occurrences). This is considerably lower than the surface ratio in Bach's 2-part inventions (after Huron 2008), which is stylistically close. By abstracting a skeleton from the musical surface, Gjerdingen seems to be overemphasizing tonally unambiguous cues. While the tritone is not "ubiquitous," it seems to be prioritized above its "share" in the template and usage in a stylistically proximate corpus. Once again, a calculation of surface soprano-bass intervals in the pieces analyzed by Gjerdingen proper would improve the current, preliminary results.

I have discussed above the problems with comparing surface and skeletal whole tones and semitones, as well as a preliminary survey of eight pieces from Gjerdingen (2007), which suggested a 1.34 surface soprano WT/ST ratio as an approximation, as well as Temperley's data for a related corpus, which indicates 1.08 WT/ST surface ratio. In order to examine the properties of skeletal melodies in a preliminary fashion, I recorded the soprano scale-degree transitions in the same corpus of 14 analyses from Gjerdingen (2007), shown in Appendix B. The following method has been taken:

  1. Only record normal-sized soprano circle annotations;
  2. If there is a tonal skip (shift in scale-degree identities due to change of tonal center of reference), leave a space between strings that will not figure in transition counts. If there is an (intuitive) disjunction in the annotation between schemata or groups of schemata, leave a blank space that will not figure in transition counts.
  3. Differentiate major-diatonic and minor 6, 3, 7, (vs. 6♭, 3♭, 7♭) as well as 4 and 4♯, so that the chromatic identity of scale degrees is reflected clearly. (Thus, a context like Example 1 would be taken as 4-3♭ then 4-3).
  4. Ignore immediate repetitions of the same scale degree, unless it is reinterpreted in a new tonal segment, in which case it will be represented twice: once at the end of a segment within one key and then once at the beginning of a segment in a different key after a blank space.
  5. Same amendment discussed above (under bullet point 4 in the previous list) for the Cudworth;
  6. Quiescenza schema taken to be a combination of tritone resolutions (this affected Leduc's piece only, where a soprano line can be reconstructed for the Quiescenza when it is missing, cf. Rabinovitch, 2018, 2019).

Thus, the resultant representation of Appendix B contains scale-degree segments of varying lengths, which are segmented by tonal shifts (on a local level) or what was intuitively interpreted as gaps in the annotation. They give us an assessment of scale-degree frequencies in the annotation as well as a tentative and preliminary sense of scale-degree transitions within individual schemata or within successions of several adjacent schemata. (Again, interested readers are welcome to scrutinize the scale-degree successions listed in Appendix B and observe that my interpretative freedom was limited, despite the intuitive decisions on points of disjunction, in which I segmented melodic events into separate strings).

The frequencies of individual skeletal scale degrees are unusual (see Table 2), inter alia since they emphasize ^4 and suppress skeletal ^5 in comparison with surface properties of melodies (cf. Huron 2006, especially pp. 147-153), though we should keep in mind that these are scale-degree identities respective to a highly localized tonal center. Note the relative rarity of ^4♯ and ^7♭, which suggests that Gjerdingen generally interprets skeletal semitones as locally diatonic.

Table 2. Frequency of individual skeletal scale degrees as reflected in the analysis of Appendix B.
Skeletal scale degree^1^23♭^3^4^4♯^56♭^6^7♭^7
Frequency2251363823923661831913810109

The frequencies of stepwise transitions within the annotations, represented in Table 3A and Table 3B, have interesting properties as well:

Table 3A. Frequencies of whole-tone skeletal transitions according to the analysis of Appendix B.
Scale-degree transitionsFrequency
^1-^210
^2-^167
^2-^317
^3-^298
^3-^4♯2
^4-^52
^5-^4115
^5-^614
^6-^589
^6-^719
^7-^67
^1-^7♭10
^3♭-^42
^4-^3♭31
Total483
Table 3B. Frequencies of semitone skeletal transitions according to the analysis of Appendix B.
Scale-degree transitionsFrequency
^1-^770
^7-^152
^3-^411
^4-^3174
^3♭-^26
^2-^3♭3
^4♯-^52
^5-^4♯2
^4♯-^44
^6♭-^513
^5-^6♭4
^7♭-^610
Total351

The WT/ST ratio in this analysis is 1.376, which does not seem to support a special status for semitones in the abstracted skeleton. However, the major-diatonic semitone transitions, ^4-^3 and ^3-^4 or ^7-^1 and ^1-^7, account for 87.46% of skeletal semitone transitions in the annotations. Notice, also, that ^4-^3 is the single most common stepwise transition in the annotations. Again, Gjerdingen's analytical process involves both marking salient events as part of typical schemata as well as finding a local key context, which may be intertwined.

The limitations of the present pilot are obvious: Gjerdingen does not annotate the complete musical surface of all 14 pieces within this small corpus, the analysis of the annotations left some limited room for intuitive judgment calls rather than an entirely consistent procedure, and events were tallied across a small corpus with pieces of uneven length. Moreover, it would have been preferable to compare the results of skeletal annotations to the musical surface of the pieces analyzed themselves, rather than to other eighteenth-century musical corpora serving as proxies. Nevertheless, these preliminary results seem to support the notion that Gjerdingen's abstraction of a contrapuntal skeleton provides considerable emphasis to rare intervals—in particular vertical tritones.

CONCLUSION AND HYPOTHESES FOR FUTURE TESTING

This article has tentatively suggested possible connections between Gjerdingen's galant schemata, the rare intervals of the diatonic collection, and the peak of the schemata and the major mode in the latter half of the eighteenth century. Whether or not this is related to the clear, redundant, and delightful communication among stylistic insiders in the eighteenth century (Gjerdingen, 2007) or to key finding in present-day listeners, is of course a question that cannot be answered here. Gjerdingen proposes his schemata based on a close study of historical repertoires and teaching methods as well as on his expertise in theory and cognition. The extent to which schemata are learnable given an idiomatic corpus still requires testing (but see Symons, 2017): if Gjerdingen's reconstruction of patterns of musical communication has some validity, his annotations of conventional patterns represent stylistic trends and salient melodic features of eighteenth-century music. Might these be a way to operationalize and evaluate such trends through observable features of the musical surface as well, bypassing the expert annotations?

Working through a different lens, White (2014) reports that his Bach, Handel, Telemann, and Vivaldi cluster uses inversion-neutral I-V-I and V-I-V sonorities more frequently than I-V7-I and V7-I-V7, while the situation is reversed in his first-Viennese cluster (of Haydn, Mozart, Beethoven, and Schubert). White's results almost certainly indicate a rise in the sheer prevalence of surface tritone resolutions in moving from early- to late-eighteenth-century music. If the period 1750-1799 is in fact unique within common-practice tonality, the importance of the ^4-^7 tritone should manifest itself on the musical surface as well. I would like to propose two hypotheses for future testing:

  1. During the period 1750-1799, tritone resolutions or dominant-tonic resolutions that situate ^4-^3 and ^7-^1 in the outer voices are most common; this tendency is more marked in the period 1750-1799 compared to the rest of the common practice, narrowly defined (1700-1899). This can be assessed by testing the percentage of dominant-tonic resolutions containing these strings in the outer voices in comparison with the other chordal/contrapuntal inversions in each 50-year period.

If this hypothesis is correct, it would suggest an increasing tendency to situate an unambiguous tonal cue in the perceptually salient outer voices. In fact, the specific arrangement of ^7-^1 in the bass and ^4-^3 in the soprano seems paradigmatic in Gjerdingen's schemata (Rabinovitch, 2018, 2019): this is almost equivalent to saying that "V6/5-I" with soprano ^4-^3 is the most prevalent localized dominant-tonic resolution in the period respective to the home key or to a secondary key area. In order to operationalize this hypothesis and test it on a corpus, it would be necessary to perform harmonic analysis respective to a local tonic and find an appropriate way to capture a ^4-^3 soprano succession—either as a contiguous pattern or within a certain window.

My analysis above suggests that the skeletal melodic transition ^4-^3 respective to a local tonic is central in Gjerdingen's annotations. On the musical surface itself, I hypothesize that this would also manifest itself in the following way:

  1. During the period 1750-1799, descending soprano scalar flourishes of 8 eighth-notes or sixteenth-notes are most frequently of the type ^4-^3-^2-^1-^7-^6-^5-^4 respective to a local center,or ST-WT-WT-ST-WT-WT-WT in interval sizes: this rotation is more common than any other rotation of the diatonic set as a surface, stepwise element. These flourishes are most common in the time frame 1750-1799 in comparison with other parts of the common-practice period narrowly defined (i.e., 1700-1899).

The ^4-^3-^2-^1-^7-^6-^5-^4-(^3) descending flourish encodes the motion ^4-^3 in a skeletal fashion between two successive strong beats and also—following the flourish—as an adjacency across a metric boundary (cf. Creel, Newport, & Aslin, 2004; Symons, 2017; Rabinovitch, 2018). If this hypothesis is true, it would further emphasize the importance of skeletal soprano ^4-^3 respective to a local tonic, whose tonal identity would likely be supported by skeletal bass ^7-^1 (or ^5-^1).

Gjerdingen offers a rich and impressive description of the galant style, based on his expertise in music theory, history, and cognition. He eschews theoretical generalizations and favors "microtheories" and "microhistories" of individual, multi-featured prototypes. While Gjerdingen's rich descriptions of individual patterns have considerable merit and enrich our understanding of eighteenth-century style, his patterns also seem to point to some general stylistic trends. The rise of Gjerdingen's schemata may point to the growing prevalence and salience of "rare" tritone resolutions, which—as controversial as they have been in empirical scholarship—are unambiguous indicators of a local tonal context. Since the peak of the schemata is somewhere in the second half of the eighteenth century, it also coincides with the half-century of tonal music most strongly dominated by the major mode, with its unique tritone. I hope that this article will participate in dialogs about schemata, rare intervals, and tonality among "armchair" scholars like myself and more empirically-minded scholars.

ACKNOWLEDGMENTS

I am grateful to David Temperley and David Sears for their help with this project. This article was copyedited by Niels Chr. Hansen and layout edited by Diana Kayser.

NOTES

  1. The title of this article alludes to Perlman's (2004) book, which traces the emergence of theoretical abstractions regarding melodic skeletons in Javenese gamelan musicians' ways of theorizing about their musical tradition. While cross-cultural studies do not yet allow us to assess the relations between skeletons and surface patterns in the musics of the world and the continuum of fixity and flexibility in composition, improvisation, and performance (cf. Jeffery, 1992), the examination of skeletal frameworks like Gjerdingen's might ultimately enrich discussions of melodic skeletons and surface activity in a variety of musical traditions.
    Return to Text
  2. Correspondence can be addressed to Gilad Rabinovitch, Florida State University College of Music, grabinovitch@fsu.edu.
    Return to Text
  3. The asymmetry between the major and minor modes is also responsible for the "markedness" of meaning associated with the minor mode (Hatten, 1994). Beethoven's works, which are Hatten's focus, are early nineteenth-century pieces that are heard at the backdrop of the spike in prevalence of major-mode pieces. In fact, Beethoven was ideally situated in history to exploit the rarity and markedness of the minor mode: he was active after the peak in prevalence of the major mode, making the minor-mode effect in some of his works particularly stark and fresh. In contrast, for a listener in the 1890s, say, the mere use of the minor mode might already have sounded more normative and trivial. De-automatizing our casual experience of minor-mode romantic music is thus part of our task if we can ever hope—like Gjerdingen and Byros—to uncover historical modes of listening.
    Return to Text

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APPENDIX A:ANALYSIS OF GJERDINGEN (2007), VERTICAL INTERVALS

Chapter 5 - Somis
Measure numberPosition
(a,b,c…)
Intervalcomments
1AP8
1Bm6
2Am3
2Bm3
3AM3
3Bm3
4Am3
4Bm7
4CM3
5AM3
6Am3
7Am3
8AM3
10Am3
12AM3
14AM3
15Am3
16Am3
17AM3
18AP5
Chapter 8 - Dittersdorf String Quartet
Measure numberPosition
(a,b,c…)
IntervalComments
1AP8
2Am3
3Am3
4AM3
5AM3
5Bm3
6Am3
7AM3
9ADim5
10AM3
11ADim5
12AM3
12Bm6
13AAug4
13BM3
13CM6
14AP4
14BM3
Chapter 10 - Haydn variations (theme only)
Measure numberPosition
(a,b,c…)
IntervalComments
1Am6
2AM6
3ADim5
4AM3
5AM3
6Am3
7AP8
10ADim5
10BM3chromaticized
12ADim5
12BM3
17Am6
18AM6
19ADim5
20AM3
21AM3
22Am3
23AP8
Chapter 12 - Gluck
Measure numberPosition
(a,b,c…)
IntervalComments
1AP8
1Bm6
2Am3
2Bm6
2CP5
2DM6
2Em7
3AM3
3BM3
4AP5
4Bm3
5AM3
6AM3
6BM3
6CP8
6DM3
6EM3
7Am3
7Bm3
7Cm3
7DM3
7EM6
7FP5
8AP8
8BM6
8CP5
8DP8
9AAug2
9BM3
9CAug4
9DM6
10AAug2
10BM3
10CAug4
10Dm6
10EP8
11Am6
11Bm6
12AM3
12Bm3
13Am3
13BM3
13CM3
13Dm3
13Em3
14AM3
14BM3
14Cm3
14Dm3
14EM3
14FP4
14GM3
15AP8
15BM6
15CP5
15DP8
15EM3
15Fm3
15Gm3
16AM3
16BM6
16CP5
16DP8
Chapter 15 - Galuppi grave sostenuto
Measure numberPosition
(a,b,c…)
IntervalComments
2Am7
2BM6
4AP4
4BM3
5AM3
6Am3
7Am3
8AM3
9Am2
9BP8
9Cm7
10Am3
11Am2
11BP8
11Cm7
12AM3
13Am7
14Am3
14Bm7
15AM3
16Am3
17Am3
18AM3
19Am2
19BP8
19Cm7
20Am3
21AM2
21BP8
21Cm7
22AM3
24AP5
24BM6
25AP8
26AP8
27AP8
27BP8
27CP8
28AM6
28BP5
29AP8
29BP5
29CM6
29DM7
30AP8
30BP5
30CM6
30DM7
31AP8
31Bm3
31CM3
31DAug4
32Am6
32Bm3
32CM3
32DM6
33AP8
34Am2
34BP8
34Cm7
35Am3
36Am2
36BP8
36Cm7
37AM3
37BP8
37CP8
38AP5
38BP5
39AP8
40AM6
42AP8
42BP8
42Cm6
42DP5
43AP8
44AM6
45AP8
47Am7
47BM6
49AP4
49BM3
51Am6
51BP5
52AM3
53AM3
54Am2
54BP8
54Cm7
55Am3
56Am2
56BP8
56Cm7
57AM3
58Am2
58BP8
58Cm7
59Am3
60AM2
60BP8
60Cm7
61AM3
63AP5
63BM6
64AP8
65AP8
66AP8
66BP8
66CP8
67AM6
67BP5
68AM3
70AM6
70BP5
71AP8
Chapter 17 - Dittersdorf Quintet
Measure numberPosition
(a,b,c…)
IntervalComments
1AP8
1BP5
2AM3
3AP8
4AP5
5AP5
6AM3
7AP8
8AP5
9AP5
10AM3
11AM3
12Am3
13Am3
14AM3
14Bm6
15AM3
15BP8
15Cm7
16AP5
16Bm6
17AM3
17BP8
18AP8
23ADim5
24AM3
25Am6
25BDim5
26AM3
26BP8
26CM6
27Am6
27Bm6
27CDim5
28AM3
30AP4
30BM3
33AP4
33BM3
34AP8
47AP8
47Bm3
47Cm3
47DM3
48Am3
48Bm3
48CM3
49Am6
50AP5
51AP4
52AM3
53Am6
54AP5
55AP4
56AP5
67Am6
68AM3
69Am6
70AM3
71Am6
72AAug4
73ADim5
74AAug4
75ADim5
76AP5
76Bm7
77AM2
77Bm7
78AP5
78BM3
78CP8
79AP8
79BP5
80AM3
81AP8
82AP5
83AP5
84AM3
85AP8
86AP5
87AP5
88AM3
89AM3
90Am3
91Am3
92AM3
92Bm6
93AM3
93BP8
93Cm7
94AP5
94Bm6
95AM3
95BP8
96AP8
96Bm6
97AM3
97BP8
97Cm7
98AP5
98Bm6
99AM3
99BP8
100AP8
105ADim5
106AM3
106BP8
106CM6
107Am6
107Bm6
107CDim5
108AM3
108BP8
108CM6
109Am6
109Bm6
109CDim5
110AM3
113AM3
113BP8
113CM6
114Am6
114B m6
114CDim5
115AM3
115BP8
115CM6
116Am6
116Bm6
116CDim5
117AM3
Chapter 19 - J.C. Bach
Measure numberPosition
(a,b,c…)
IntervalComments
1AP5
1BP8
2Am3
2Bm7
3AP5
3BP8
4AM6
4BP5
4CM3
5AP8
6AM6
6BP5
6CM3
7AP8
8AM3
9Am3
10Am3
10BP4
10CM3
11AM6
11BP5
12AP5
13AP8Bass ^1 reconstructed
13BM6
13CDim5
14AM3
16AM7
16BM6
16CDim5
17AM3
18AP5
19Am3
19BP8
20AM6
20BP4
20CM3
21AP8
22AP5
23Am3
23AP8
24AM6
24BP4
24CM3
25AP8
26AM3
27AP5
28AP8
28BM3
29AP5
29BP8
30AM3
30BP5
31AP5
33AP5
34AP8
34BP5
34CM6
34DM7
34EP8
34FP5
34GM6
34HM7
35AP8
36AP5
37Am3
37Bm7
38AP5
38BP8
39AM6
39BP5
39CM3
40AP5
40BP8
41AM6
41BP5
41CM3
42AP8
43Am2
43BP8
43Cm7
43Dm2
43EP8
43Fm7
44AM6
45AM2
45BP8
45Cm7
45DM2
45EP8
45Fm7
46AM3
47AM3
48Am3
49Am3
49BM3
50AP5
52AM6
52BP5
53AP5
54Am3
54BP8
55AM6
55BP4
55CM3
56AP8
57AM3
58Am3
59ADim5
60AM3
60BP5
61Am3
61BP8
62AM6
62AP4
62AM3
63AP8
63BP8
64AM3
65AP5
66AP8
66BM3
67AP5
67BP8
68AM3
68BP5
69AP5
70AM3
70BP5
71AP5
72AP8
72Bm7
72CM6
72DM7
73AP8
73Bm7
73CM6
74AM7
74BP8
Chapter 21 - Leduc
Measure numberPosition
(a,b,c…)
IntervalComments
1AUnannotated until m. 6
2A
2B
3A
3B
4A
4B
5A
6AM3
7Am3
8AM3
8BDim5
8CM3
9AM3
9BDim5
9CM3
10AM6
10BP8
10CM3
10DDim5
11AM3
12AP5
13Am3
15AP4
15BM3
16AM3
17AM3
17BP8
17CM6
18AM3
18BP8
18CM6
18DM3
18EP5
19AM3
23AM3
23BP8
24AM6
25AM3
25BM6
26AM3
26BM6
27AM3
28AP8
28BM6
28CDim5
29AM3
32AP5
33AP4
33BM3
36AM6
36BP5
37AP8
42AM3
42Bm6
43AM3
43Bm6
44ADim5
45AM3
45BP8
45CM6
45DP5
46AP8
47AUnmarked until m. 54
48A
48B
49A
49B
50A
50B
51A
54Am3
54BDim5
55AM3
59Am6
59BDim5
60Am3
60BP8
60CM6
61AM6
61Bm6
61CDim5
62Am3
62BP8
62CM6
63AM6
64AP4
64Bm3
66AP4
66BM3
67Am3
68Am3
68BDim5
69AM3
70AM6
71Am6
72AM6
72BP8
72CM3
72DDim5
73AM3
76AAug6
77AP8
78AP8
78BP8
78CM3
78DP5
78Em7
78FP8
79AP8
79BUnmarked until m. 84
80A
80B
81A
81B
82A
82B
83A
84AM3
85Am3
86AM3
86BDim5
86CM3
87AM3
87BDim5
87CM3
88AM6
88BP8
88CM3
88DDim5
89AM3
95AM3
95BM6
96AM3
96BM6
97AM3
103AP4
103BM3
106AM6
106BP5
107AP8
112AM3
112Bm6
113AM3
113Bm6
114ADim5
115aM3
115BP8
115CM6
115DDim5
116Am3
116BM6
116CP5
117AP8
Chapter 22 - Leo
Measure numberPosition
(a,b,c…)
IntervalComments
2AM6
3AP5
3BP8
4AM3
4Bm7
5AM6
5BP5
6Am6
7AP5
8AP4
9Am3
9BDim5
10AM3
10BM2
10Cm3
10Dm3
10EM3
10Fm3
12AM3
12BP8
12Cm7
12DM3
12Em3
14AM3
14BP8
14Cm7
14DM6
14EP5
15AM6
15BP5
16AP8
19AM3
20Am3
20BP8
21AM3
21Bm7
22AM3
23AM3
24AP5
24BP8
25AM6
25Bm3
25CM3
26AM6
26BP5
27ADim5
28AM3
29ADim5
30AM3
32AP4
32BM3
33Am6
34AP5
35AP4
36Am3
36BDim5
37AM3
38AM3
38BP8
38Cm7
38DM6
38EP5
40AM3
40BP8
40Cm7
40DM6
40EP5
42AM3
42BP8
42Cm7
42DM6
42EP5
43Am6
44AM3
44BM2
44CP8
44DM6
44EP5
45AP8
46AP8
47AM3
47Bm7
48AM6
48BP5
50ADim5
50BM3
51ADim5
52Bm3
53ADim5
54AM3
55AM3
56Am3
58AM6
58BDim5
59AM3
60ADim5
61AM3
62ADim5
63AM3
64ADim5
65Am3
66Am3
68AM3
69Am6
71AM3
72Am6
73AP5
74AP4
75Am3
75BDim5
76AM3
77ADim5
78AM3
79AM3
79BP8
79Cm7
79DM6
79EP5
81AM3
81BP8
81Cm7
81DM6
81EP5
83AM3
83BP8
83Cm7
83DM6
83EP5
86AP5
87AP8
87BP8
88AM3
88Bm7
89AM3
89BM6
89CP5
90AP8
91AP8
92AP5
93Am3
94Am3
95Am3
97Am3
97BP8
97Cm7
97Dm6
97EP5
99Am3
99BP8
99Cm7
99Dm6
99EP5
102Am6
102BP5
103AP8
105AM3
106Am3
106BP8
107AM3
107Bm7
108AM3
108BM6
108CP5
109AP8
Chapter 23 - Galuppi aria
Measure numberPosition
(a,b,c…)
IntervalComments
1AP5
2Am3
2Bm3
3AM3
3Bm3
3Cm3
3DM3
4AM6
4BP5
7AM6
7BP5
8AP8
9AP5
10Am3
10Bm3
11AM3
11Bm3
11Cm3
11DM3
12AM6
12BP5
14AM3
14Bm3
14CDim5
15AM3
15BP8
15CM6
15DP5
16AP8
17Am3
17Bm3
18Am3
18BM3
19AM3
19Bm3
19Cm3
19DM3
20AM6
20BP5
20CM3
21AM3
21Bm7
22Am3
22BP5
23AM3
23Bm7
24AM3
25Am7
25Bm6
25CDim5
26AM3
28AM3with reconstructed bass ^4
29Am3
29Bm3
30AM3
30BP8
30CP8
31Am6
31BM7
31CP5
32AP8
33AP8
33BM6
33CP5
34AP8
Chapter 24 - Jommelli
Measure numberPosition
(a,b,c…)
IntervalComments
1AP8
2Am3
2Bm6
3AM3
3BP5
4Am7
4BM6
4CP5
5AP8
6AM3
7Am3
7BDim5
8AM3
8Bm7
9AP5
9BP8
9CM6
9DP5
10AP8
11AP8
12Am3
12Bm6
13AM3
13BP5
14Am7
14BM6
14CP5
16AM3
17Am3
17BDim5
18AM3
18Bm7
19AP5
19BP8
19CM6
19DP5
20AP8
20BP5
21AM6
22AP5
22BP5
23Am6
24AP5
24BP5
25Am6
25BM7
26AP8
26BP5
28AM3
29ADim5
30Am3
31ADim5
32Am7
32BP5
32Cm7
33AP5
33BP8
33CM6
33DP5
34AP8
34BP5
35Am6
35BM7
36AP8
36BP5
36CP5
37Am6
37BM7
38AP8
40AM3
41ADim5
42Am3
43ADim5
44Am7
44BP5
44Cm7
45AP5
45BP8
45CM6
44DP5
49AP8
49BM6
49CP5
52Am7
52BP5
53AP8
53BM6
53CP5
54AP8
56AM3
56BP4Missing note, cf. m. 57
56CM3
57AM3
57BP4
57CM3
58AP8
59ADim5
60Am3
61ADim5
62AM3
62BP8
63Am3
63Bm6
64AM3
64BP5
65Am7
65BM6
65CP5
66AP8
66BP8
67Am3
67Bm6
68AM3
68BP5
69Am7
69BM6
69CP5
70AP8
70Bm7
71AM3
71Bm7
72AM3
72BM3
73AM3
73Bm7
74AP5
74BM3
75AM3
75BM3
76AM3
78AM3
79ADim5
80Am3
81ADim5
82Am7
82BP5
82Cm7
83AP5
83BP8
83CM6
83DP5
84AP8
84Bm7
85AM3
85BDim5
86AM3
92AM3
93ADim5
94Am3
95ADim5
96Am7
96BP5
96Cm7
97AP5
97BP8
97CM6
97DP5
101AP8
101BM6
101CP5
106AP8
106BM6
106CP5
107AP8
108AM3
108Bm6
108CDim5
109AM3
109BM3
109CP4
109DM3
110AM3
110BP4
110CM3
111AP8
112AP8
112BM3
112CP5
113AP5
113BM3
113CP5
114AP5
114BM3
114CDim5
114DM3
115AM6
115BP5
117ADim5
117Bm3
117Cm7
118AM3
118BP5
118Cm7
118DM3
118EP5
118Fm7
119AP5
119Bm7
119CM3
120ADim5
120Bm3
121ADim5
121BDim5
122ADim5
122BM3
123Am3
123BP8
123CM6
123DP5
123EP8
124AP5
124Bm7
124CM3
124Dm7
125AM3
126AP5
126Bm7
127AP5
127Bm7
127CM3
128ADim5
128Bm3
129ADim5
129BDim5
130ADim5
130BM3
131AP8
131Bm7
132AM3
132BP8
132Cm7
132DM3
137AM6
137BP5
138AM6
138BP5
139AP8
140Am3
140Bm6
140CM3
140DM6
141Am6
141BP5
143Am3
144AP8
144BDim5
Chapter 26 - Mozart
Measure numberPosition
(a,b,c…)
IntervalComments
3AM3
3Bm3
4Am3
4BM3
5AM3
6Am3
7Am3
8AM3
9AM6
9BM6
10AM3
10Bm6
10CP8
10DDim5
11AM3
14Am6
15AAug4
16Am6
17AAug4
18AM3
19Am3
20Am3
21AM3
22AM6
22BP8
22CM3
23AM6
24AM6
25AP5
26AP8
31AP5
32AM3
32Bm7
33Am3
35AP4
36Am3
36BM6
37AM6
37Bm3
38AM3
39Am3
40Am3
44AM3
44Bm3
45Am3
45BM3
46AM3
47Am3
48Am3
49AM3
50AM3
51Am3
52Am3
53AM3
54AM6
54BM6
55AM3
55Bm6
55CP8
55DDim5
56AM3
59Am6
60AAug4
61Am6
62AAug4
63AM3
64Am3
65Am3
66AM3
67AM6
68Am3
69AM6
70AP5
71AP8
Chapter 28 - Quantz
Measure numberPosition
(a,b,c…)
IntervalComments
1AP8
1BM3
1Cm3
2Am6
2BDim5
2CM3
2DM3
3Am3
3Bm3
3Cm7
3DM3
3EM3
4Am3
4Bm3
4Cm7
4DM3
5AM3
5BAug4
5Cm6
6AM3
6BAug4
6Cm6
6Dm3
7ADim5
7BM3
7CP5
7DP8
8AM6
8BM6
8CP5
8Dm3
9Am3
10Am3
11ADim5
11BM3
11CM6
11DP5
11EDim5
12Am3
12BP4
12CM3
13Am6
13BDim5
13Cm3
14Am6
14BDim5
14Cm3
15Am3
15BP5
15CM3
15Dm3
15EM3
15Fm3
16Am3
16BM6
16CP5
16DP8
17ADim5
17Bm3
18ADim5
18BM3
19AM3
19Bm3
19Cm3
19DM3
20AM6
20BP8
20CM3
20DDim5
20EM3
21AP8
21BM3
21Cm3
22Am6
22BDim5
22CM3
22DM6
23Am7
23Bm6
23Cm7
23Dm3
24Am7
24BM3
24CM7
24DM6
24EP5
25AAug4
25Bm6
25CP8
25DM6
25EP5
25FP8
25GP8
26AM3
26BDim5
26CM3
26DP8
27AM3
27BDim5
27CM3
27Dm6
28AP5
28BM6
28CP5
28DM6
29AP5
29Bm6
29CDim5
29Dm6
29EDim5
30AM3
30BDim5
30CM3
30DM6
30EP5
31AM3
31Bm3
31Cm3
31DM3
31EM6
31FAug4
31Gm6
31Hm3
32AM3
32BP8
32CM6
32DP5
32EP8
Chapter 29 - Galeazzi
Measure numberPosition
(a,b,c…)
IntervalComments
1AP8
1BM3
2AP5
2BM3
2Cm6
3AM3
3BAug4
3Cm6
4Am6
4Bm7
5AP5
5BP8
5CM6
6Am3
6BM3
6CP8
7AM3
8AP4
8BM3
9AP8
10AM3
12Am3
14Am3
15AM3
17AM3
18AM3
18Bm6
18CDim5
19AM3
20AM3
20Bm6
20CDim5
21AM3
21Bm3
23AP8
23BM6
23CP5
24AP8
24Bm7
25Am6
25BM7
26AP8
26Bm7
27AM6
27BM7
28AP8
30AAug2
30BM3
30CAug4
30DM6
32AAug2
32BM3
32CAug4
32Dm6
33ADim5
34AM3
36AAug6
37AP8
38AAug4
39AM6
40Am6
40BP5
41AP8
42AP8
42BM3
43AP5
43BM3
43Cm6
44AM3
44BAug4
44Cm6
45Am6
45Bm7
46AP5
48AP8
48Bm6
48CDim5
49AM3
50Am6
50Bm3
50Cm7
51AM6
51BM6
51CP8
51DM3
51EDim5
52AM3
53AM3
54AM3
54Bm6
54CDim5
55AM3
56AM3
56BM2
56Cm6
56DDim5
57AM3
57BP8
57CP8
58AP8
58BP8
59Am6
59BP8
59CM6
59DP5
60AP8
60Bm7
61AM6
61BM7
62AP8
62Bm7
63AM6
63BM7
64AP8

APPENDIX B:ANALYSIS OF GJERDINGEN'S (2007) SOPRANO SCALE-DEGREE ANNOTATIONS

CHAPTER FIVE SOMIS

15156543 6543 3♭ 3 65432

CHAPTER EIGHT DITTERSDORF

123 6543 43 43176217

CHAPTER TEN HAYDN SONATA (theme only)

1743 654 43 43 1743654

CHAPTER TWELVE – GLUCK SONATA

151743 6543 671671 654321321 71 7151654365436543171321654321

CHAPTER FIFTEEN GALUPPI GRAVE SOSTENUTO

17436543 6b543♭ 6543 43b 43 543 6b543♭ 6543 671 51432156715671 56715671 6♭543♭ 6♭543 5143♭21 71 1743 543 3 6♭543♭ 6543 6♭543♭ 6543 671 514321 321

CHAPTER SEVENTEEN DITTERSDORF QUINTET

1231231236543165431651 43543171543 17 171 5246427 5 53 53 5 7 4 7 4 5 24642751231231236543165431651 165431651 43171543171543171543171543

CHAPTER NINETEEN - J.C. BACH ANDANTE

5432173217165432 51743 3217 5432171 5432171351351352156715671 543217532171 6♭546♭43♭ 6546543 65432 32 54321716543543217135135135235217♭6717♭671

CHAPTER TWENTY-ONE LEDUC CANTABILE

34343654♯434♯4324617 543642642617 3232323217 617 321 316434321 3♭43♭43♭ 617 543♭171543♭171 43♭ 43 65432424617 4♯57245 134343654♯434♯4324617 3 32323 17 321 3164343214321

CHAPTER TWENTY-TWO LEO

65174321 43 65432 65432 65432321 651743 6517432 43 43 431 43 65432 65432 654321 654321 17432 43 43♭ 4365 217 43 43 43♭ 6♭5 431 43 4365432 65432 65432 2174321 123♭ 6♭ 6♭543♭2 6♭543♭2 3♭21 65174321

CHAPTER TWENTY-THREE GALUPPI ARIA

51565432 321 51565432 654364321 242765432 1743♭ 1743 3217 65431413214321

CHAPTER TWENTY-FOUR JOMMELLI

51654321 6543654321 5165432 6543654321 5656♭56♭715 3217♭6565432156♭7156♭71 3217♭6565432 432 654321 6176171 43♭ 435165432151654321 43 43 6743♭1717 3217♭65654321 43 43 3217♭6565432 432 432165436176171 13513553432 43♭ 43543543543 43♭ 4354321 54343 543543 43♭ 43543543 32321 1462 21 4

CHAPTER TWENTY-SIX MOZART

65436543264617 171765432462321 1743♭ 1743♭ 6♭543♭ 6543 6543 6543264617 1717654326321

Note: the Meyer schema in mm. thirty-five to thirty-seven was interpreted as moved to the bass (invertible counterpoint), in parallelism with measures thirty-one through thirty-three.

CHAPTER TWENTY-EIGHT QUANTZ

13243 654365436716714321 123♭ 23♭ 432 43♭17 543♭ 543♭ 6♭176♭543♭21 43♭ 43654324617 13243654321714321 1743174312345434326543271434321

CHAPTER TWENTY-NINE GALEAZZI

135317146531316171 6543 3754375431 43217♭6717♭671 71 71 17 4♯5 71 3♭21 1353171465 6543 6424617 37543765431642143217♭6717♭671

APPENDIX C:MUSICAL EXAMPLES

The Fonte schema depicted in musical notation. More description below.

Example 1. The Fonte schema. Gjerdingen annotates as [ii]: ^4-^3, [I]: ^4-^3, not monotonally. The schema, like many additional schemata, serves as an outer-voice contrapuntal skeleton for surface musical activity.

APPENDIX D

Diagram 1a. Skeletal soprano scale-degree transitions in galant schemata in a major-mode context
SCHEMA^7^1^2^3^4 (♯4)^5^6(7♭) ^7^1
Romanesca*right arrow
|…
left arrow
Comma|bolded left arrow*(left arrow)
Sol-Fa-Mi|bolded left arrowleft arrow*
Fonte|bolded left arrowleft arrow*
Prinner|bolded left arrowleft arrowleft arrow*
Meyerbolded left arrowbolded left arrow*|bolded left arrow
Fenarolibolded right arrow|bolded left arrowbolded left arrow*
Pastorellaleft arrowleft arrow* |bolded left arrow
Aprilebolded left arrowbolded left arrow* |left arrow
Jupiter*right arrow right arrow | bolded left arrow
Mod. Prin.|bolded left arrowleft arrow left arrow*
Montebolded left arrowleft arrow* |bolded left arrow
Converging 1|bolded left arrow left arrow* (left arrow)
Converging 2|bolded left arrowbolded left arrow(left arrow)
Indugio†|bolded left arrow left arrow* (left arrow)
Jommelli| left arrow*
[Heartz-Rice]*right arrow
|
left arrow
Mi-Re-Do /
Cudworth
|left arrowbolded left arrowbolded left arrow*(left arrow)(left arrow)(left arrow)
Do-Si-Do right arrow left arrow
|
left arrow*(left arrow)
Grand Cadence|left arrowleft arrowleft arrow….left arrow*
HC 1|left arrow*
HC 2|bolded left arrow*
Quiesc. 1bolded left arrow left arrow*
bolded right arrow
|
Quiesc. 2*right arrow right arrow bolded right arrow |
[Volta/
Svago-
Mitchell/
Aerts]
|*♯right arrow
left arrow
bolded right arrow
Clausula V.(right arrow)*bolded right arrow|
Passo I.(right arrow)*bolded right arrow|
Do-Re-Mi*right arrow right arrow |

*=start of motion

()=optional prefix before start of motion

|=end of motion

…=move on to next …

♭/♯=sharp or flat version of a scale-degree

| / bolded right arrow [bolded] = scale degrees involved in a semitone motion (in a major-mode context)

†The Indugio adds a polyphonic-melody diminution respective to the Converging Cadence; only the underlying stepwise (Converging Cadence) motion is represented here.

Bolded schema name=associated with ^4-^7 tritone resolution; Prinner may be articulated by ^7-^1 bass (or ^5-^1 bass) at its end, hence assimilated in some sense to the tritone-resolution group of schemata. (Even with bass ^2-^1, the figured-bass sonorities 7-6 or variants lead to a matched tritone).

[]=schema not originally proposed in Gjerdingen (2007).

Diagram 1b. Skeletons of schemata that necessitate elements from the parallel minor when embedded in a major-mode context
SCHEMA^7^1^2^3^4 (♯4)^5^6(7♭) ^7^1
Aug. 6*♯right arrow|
[Morte-Rice]*♭right arrow right arrowright arrowright arrow |
[Le-sol-
fi-sol-
Byros]
| left arrow*
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