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date: 24 May 2018


Abstract and Keywords

This article defines cadence as a convention of musical ending realized through counterpoint. It encodes the principle of imperfect sonorities resolving to perfect sonorities. The authentic cadence is unambiguously enacted in a five-part texture, and its constituent parts can be rearranged through invertible counterpoint to create provisional endings, labeled clausulas. Analytic demonstrations of these cadences and clausulas are given in music by Bach and Telemann. The plagal cadence is defined in dualist terms as the counterpart of the authentic, while the half cadence is defined as an authentic cadence that stops at the penult. Cadences can be altered by rhetorical schemes of continuation or emphasis, and specific schemes of these kinds are named and defined.

Keywords: authentic, plagal, counterpoint, clausula, invertible counterpoint, dualist, rhetoric

How music starts and stops is one of the first things students of the art learn. Of the two, stopping—or pausing, articulating, and so forth—is the more conventional and therefore easier to name and thence to categorize. Most students learn to perform, hear, and recognize stopping conventions well before they find out their names, though these are among the first items learned in elementary theory and composition. Some names seem self-explanatory, such as the fade-out ending of many popular song recordings, and perhaps are never really taught. But the techniques involved in producing them certainly are, as any recording engineer knows. Others, particularly those in the Western art tradition, have ancient roots and complicated family trees. These are the focus of this essay.

The Contrapuntal Background

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Figure 1 Mandatory ending of major sixth to perfect 8ve in strict counterpoint a2 with counterpoint above the cantus firmus. Curved lines mark semitones; angled lines mark whole tones. (a) applies to cases in which a diatonic semitone is below the tonic. It’s transposable to C. (b) shows a case with no diatonic semitone below the tonic; a chromatic one is used instead. It’s transposable to A, G. (c) shows an instance in which a diatonic semitone is above the tonic as well as in the cantus firmus.

Taken together, the stopping conventions are described as types of cadence (Ger. Schluß; It. cadenza; Fr. cadence). In strict counterpoint, the most rudimentary polyphonic procedure, they are defined as a mandatory pair of ending intervals, illustrated in Figure 1. They show situations in which the given part (cantus firmus) descends by step to the tonic, which is Johann Joseph Fux’s influential practice. Because these successions work under invertible counterpoint at the octave, it is immediately apparent that they could apply to cases in which the cantus ascends to tonic or the counterpoint is below the cantus. These will prove consequential in the development that follows. For now, most of the remaining illustrations use the model at (a), with important exceptions noted as needed.

While it’s usual to begin work in composition and theory with this kind of axiomatic beginning—given: three cadence types (a), (b), (c)—putting these in historical and theoretical context repays effort.1 It explains that stopping is as conventional as it is because it encodes metaphysical ideas about finality (worked out expressly by theologians and philosophers). It also provides more sensitive analytic instruments especially suited for nonstandard repertory. And if this essay has a point, it is to increase the reportorial range of any “theory of cadence” and thereby (hopefully) encourage less compression of the topic in textbooks and treatises.

Initial bearing is taken from the observation that the models in Figure 1 musically enact medieval ideas about completion, finality, closure, and perfection. These have permanently settled into the nomenclature of intervals, in which the perfect consonances reflect their symbolic value—defined classically by monochord arithmetic and later explained by acoustics and psychology—as the best possible ending sonorities. The basic structure of Figure 1, then, is a symbolic ending sonority (the perfect octave or unison) preceded by an imperfect one a step away in each voice. Moreover, steps are taken in contrary directions, underscoring the difference between the two sonorities. That is, in the change from one to the other, neither voice remains the same, and neither voice moves the same way. Contrary motion maximizes difference and distinctness and also symbolizes unity-of-opposites harmony.

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Figure 2 A motion to perfect fifth, a possible cadence between circa 1330 and 1430.

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Figure 3 Schematic approaches to concluding perfect consonance a3, mid-fourteenth century. “T” = tenor. (a) The “double leading tone” combines stepwise approaches of Figures 1 and 2; (b) the “leaping octave” from below the penult tenor gives perfect fifth to both sonorities; and (c) the “authentic” has no fifth.

The models in Figure 1 are what remain of quite a large range of cadential possibilities in late medieval music. Figure 2 shows one type, also using a semitone and whole tone in contrary motion.2 Adding a third voice to the contrapuntal texture permits new possibilities. Those shown in Figure 3 reconcile the requirements of concluding perfection with some kind of differentiated motion.3 These characterize widely used formations, named by modern scholars and still learned by specialists in the repertory. In light of the illustrations at (b) and (c), the requirement of a perfect sonority at the conclusion overruled, for a time, the preference for proximate voice-leading in all parts. By the later standards of strict counterpoint, illustrations (a) and (b) are noncompliant and thus not available as models. Illustration (c), as its more familiar name suggests, remains relevant.

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Figure 4 Schematic of formal cadence a4; a fifth (quintus) part can diverge from either the alto or tenor stream to reach a harmonic triad. The tenor divergence, TQ, ensures a complete chord at the expense of melodic closure in a basic structural part. The alto divergence, AQ, permits the tenor to resolve at the expense of a complete final triad; the longer distance AQ traverses is filled with a passing tone.

This is so, I argue, not because it is innately superior to (a) or (b)—though the leaping octave is difficult to perform—but because it scales up perfectly with an increase in default texture from three to four voices, which is the scaffolding for polyphonic art music from the sixteenth through the nineteenth centuries. In particular, between the tenor and soprano is room for a contratenor alto on G4, which eliminates the need for a leaping octave and introduces oblique motion into the mixture. Figure 4 shows all this (and more). One can appreciate immediately how the lines in the basic four-part SATB framework move by every motion type except parallel: contrary between S and T/B, oblique between A and other voices, and similar between T and B. In terms of individual activity, S and T move by step, B moves by leap, and A holds its place.

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Figure 5(a) Ockeghem: Missa L’Homme Armé; conclusion of first Kyrie. (b) Mozart: Requiem; conclusion of Kyrie.

Figure 5(a) shows a typical example from a model repertory, a beautifully elaborated close of a mass movement by Ockeghem, in which all parts lock in for the cadence on the last quarter of the penultimate measure and then proceed according to rule. The same can be said of (b), though it is a studied bit of sonorous “antiquing” by Mozart that brilliantly underscores the solemnity of the movement. Hereinafter, I call this kind of cadence—strict SATB a4 as modeled in Figure 4—the “All-Perfect Authentic Cadence” (APAC). Taste for it changed over the course of centuries—from being a staple of Ockeghem’s and even Dufay’s technique, to becoming an increasingly disused (and therefore old-fashioned) option in the later Ars Perfecta, to becoming a special effect in Mozart’s practice. These differences notwithstanding, the intuition of finality and completion predicated on the all-perfect final sonority can, I think, be located in well-tuned performances in which spectral overlap fuses the individual tones nearly into a single timbre. Thirds being comparatively difficult to locate (because of conflicting intonational ideals), their absence in an all-perfect chord makes this fusion more prominent. Listening intently to such a sound, especially if under a fermata, is to hear the techniques of spectralism avant la lettre.

The more consequential aspect of Figure 4 is additional voice-leadings, AQ and TQ (shown together on the quintus staff). These target an imperfect, mediating third between a perfect fifth, making the final chord a harmonic triad. The lines from A and T—in keeping with the values enacted in the other parts—reach the third by the closest approach. Doing so is an easy upward step for T, but A requests a dissonant passing tone to smooth its way.

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Figure 6 Adapted from Rameau ([1722] 1971); illustration of perfect cadence. (Cf. ex. II.1.)

The collecting of the two alternate A and T voice-leadings into one notional part is to assert that five-part texture—SATB plus one from Q—is sufficient to ensure a cadence from one complete harmonic triad to another. It is the textural “sweet spot” of all-triadic progression under parsimonious voice-leading in general and is ideal for cadences in particular. J. P. Rameau’s well-known cadence model, revoiced and adapted in Figure 6, repays reflection on this point.4 The final chord is a complete triad a5, with the telltale trace of QA—the minor seventh dissonance—leading the way. The disposition of perfect intervals over the lowest voice is clearly of importance to Rameau, as is identifying the essential imperfect note of the penultimate chord as the “leading tone.” Rameau’s bass line is precisely that which works contrapuntally, as we have seen, but what distinguishes it from the others is the labeling as scale degrees instead of by interval membership.

While using both Q lines (for a texture a6) is possible, the imperfect doubled thirds in the final chord outweigh the single perfect fifth, undercutting its closing powers, and are not a favored part-writing choice. Adding even more voices in homophonic writing proves to be difficult under laws of strict counterpoint, as more disjunct motion is needed to avoid parallels or dissonances. In this way they behave like interior bass lines, jumping from one note to the next, doubling any existing consonances.

In contrast, composing triadically in a thinner, a4 texture is possible, but it cannot be done strictly—that is, as fully voiced chords accessed by traditional voice-leadings. This can be quickly grasped by considering the two a4 adaptations of Figure 4 that maintain the SB outer-voice frame but use one of the Q lines. SAQTB is an attractive choice, because its passing dissonance takes over the role previously performed by imperfect consonance in the penultimate chord of the strict setting (M6 over T), thus reinscribing the general rule of succeeding from a less to a more perfect sonority when making a cadence. Yet the cost for this is an incomplete final triad, missing a perfect fifth over T—that is, no G, one E4, and a tripled C3,4,5. This missing interval contributes to stability and therefore to the perfecting power of the cadence, and its absence prevents the final chord from even being a triad, strictly speaking. On the other hand, choosing SATQB permits A to hold station at the fifth, but the ersatz T declines to descend properly to the tonic and is deflected upward into the third instead. The final chord is thus complete (i.e., one G4, one E4, and a doubled C3,5), but at the cost of the requisite (even primordial) contributions of T to the cadence. With AQ, a chord tone goes conspicuously missing; with TQ, a voice part wanders off track.

These kinds of solutions to composing triadically a4 are precisely those that shape the “free” counterpoint of the eighteenth century, distinguishing it from the strict observance illustrated thus far. The first freedom—tolerating imperfection in the final chord of a cadence while maintaining its prerogative to signal closure—models no longer an All-, but rather a mostly, “Perfect Authentic Cadence” (PAC). This familiar textbook construct is the polestar for the rest of this essay. But to understand the terrain ahead, we should zoom out from the cadential moment and survey its approaches and exits.


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Figure 7 Time plot of a cadence.

Figure 7 does this by placing the cadential breve-and-semibreve pair of Figures 14 in an extended metric flow, symbolized by filled noteheads. Chords are labeled according to distance from the cadential resolution, C. According to this figure, the cadential moment studied so far involves chords C and –1.

The a6 counterpoint between a pair of chords—S, A, T, B, and two Qs—is collected into various sets called pathways, which are given the label i. In this way, the cadential mandate is found in the set i between –1 and C, formally (yet awkwardly) notated as –1 i C. Better is a default shorthand iC; when analytic interests suggest, the synonym –1i might be preferred.

Figure 7 suggests ways in which listeners can register cadence besides noticing the final iC move. Contrapuntal convention requires that parts have to be voice-led into position at the head of the final pathway to C, a process that takes time depending on how far away the voices are when they are summoned for cadential duty. As a result, cadences can signal their approach from as far away as –3, if not farther,5 with a noticeable loss of the freedom that characterizes musical flow in midphrase as the voice parts make a beeline toward –1 so as to perform the final mandated move, iC. With respect to chords, these operations define the harmonies at C and –1, but they also affect –2 to a large degree, –3 to a lesser degree, and even earlier in some cases. The cadential moment is thus the predicate of antecedent objects and moves, chords and pathways, arranged syntactically to terminate at C.

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Figure 8 Cadence a5 SAQTB. Four approaches: (a) soprano suspension prepared from tonic (b) soprano suspension prepared from pre-dominant; (c) a particular soprano and tenor suspension creating cadential six-four chord; (d) soprano and alto suspension creating six-five chord.

This apparatus allows us to analyze the context around cadence, which work is begun in Figure 8 with a survey of the pathways around –2.6 These show the preparations for suspending the soprano—a customary treatment that serves to signal the ending of the approach phase of the cadential moment and the beginning of its execution. In cooperation, the tenor is contrived to move by step if not repeat, as is appropriate for a cantus firmus.7

If alto, tenor, and bass parts are in their proper position underneath the soprano near –1, two kinds of i–1 pathways from –2 suggest themselves; they are sketched at (a) and (b). Taking a harmonic functional perspective, we will identify them as the tonic and the pre-dominant approaches, respectively. For tonic, one of C or E can take the bass, with the tenor then sounding the other. The alto can maintain G for both tenor/bass configurations, but can also take an A over a bass C, as the dotted line highlights. More typical accompaniments for A are shown at (b), the pre-dominant approach. There, a bass F and tenor D make the smoothest counterpoint, and the alto has two options, F and E, should A3 be sounded in the bass.

Figure 8(c) works out one additional elaboration of (a) by suspending the tenor along with the soprano. Elaborating multiple parts like this allows for a certain amount of “chord reification” in the analysis. Here, –1½ is singled out as the point at which the bass and alto have reached dominant but the two other voices have not. This cadential six-four chord is thus a functional mixture somewhere between –2 and –1. The same idea for a pre-dominant approach is tried out at (d), where the alto’s A is suspended and an additional elaboration is worked into the bass, which opens descending stepwise passage for the tenor. Additional elaborations and diminutions can easily be imagined, which the reader is invited to explore.8

As the cadential time plot in Figure 7 suggests, the situation prior to –2 becomes more difficult to predict because the general entropy of the musical unit increases with distance from C. The same can be said for activity after C, but for different reasons, to be adduced below. For now, we note the invitation to more rigorous statistical analysis of these larger contexts and resume focus on the cadential moment by detailing its contrapuntal possibilities.

Cadence and Clausula

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Figure 9 Annotated adaptation from Werckmeister and Mongoven (2013, 300).

We can make an important connection from strict composition to thoroughbass harmony by way of invertible counterpoint. The site was identified as early as Viadana’s time (ca. 1602), but a crucial transformational development was undertaken by Andreas Werckmeister in 1702.9 The basics are quickly grasped in Figure 9, an adaptation of his original diagram. The top staff names four cadential moves that are characteristic of particular voices in strict counterpoint, from –2 to C. Werckmeister’s choice—apparently conventional—to describe these as verbals makes for no less awkward renderings in English than in German, but the idea that the soprano “discantusizes,” the tenor “tenorizes,” and so forth, is clear and is the basis for the annotations added to the original: the voice-leadings of S, AQ, T, and B (with AQ denied an opportunity for the dissonant passing tone).10

Werckmeister intends his adjectives to describe cases in which a cadence is made, but the upper parts are not in their accustomed reigstral slots. Holding the bass constant—in order to maintain the formality of the cadence—he shows various examples, which he realizes by means of a triple-counterpoint permutational subgroup transforming the upper parts. Measure 1 is familiar from previous figures, especially 8(a). Measure 2, however, shows an “alto-ized” cadence—that is, the soprano is behaving like an alto.11 Measure 3, likewise with the tenor. In mm. 4 and 5, Werckmeister swaps TQ (and A) for A(Q) and shows the beginning of the relevant permutational structure.

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Figure 10 Invertible counterpoint and the perfect cadence.

Of the three arrangements Werckmeister shows, those in mm. 1, 3, and 5 correspond to current definitions of the PAC—wherein the top and bottom voices are both on the tonic. Measure 2, an alto-ized cadence, ends imperfectly, as does measure 4. These are accordingly forms of the “Imperfect Authentic Cadence’ (IAC), in which A or one of the Q lines is in the top voice.12 But more noteworthy is the observation that the PAC has two types of approaches: (1) a strict, “by the book” and regular setting of the S in the top voice, and (2) a cantional setting with a “tenorized” top voice.13 Ramifications of this observation can be appraised in Figure 10, with the top staff working out all the permutational arrangements of the APAC, and the lower two staves doing the same for the PAC (tracking both Q approaches). The basic differences between them all, not surprisingly, come down to chord spacing and register, with those arrangements having a “low” quintus most noticeable.

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Figure 11 Two settings of “Old Hundredth”: (a) a regular by setting W. Parsons, 1563. Number 1 in Havergal (1854, 54). (b) a cantional thoroughbass setting by C. Müller, 1703; transposed, rebarred, and inner parts realized. Number 13 in Havergal (1854, 58). Rhythmic variance in the last phrase is original.

The most consequential distinction, however, remains that between the regular setting dictated by strict counterpoint, in which a structural tenor can be counterpointed both above and below, building texture from within, and the cantional setting, in which texture is built from below toward a melodic primordial structure featured at the top.14 Extending our purview beyond the cadential moment and considering the distinction as a generalized approach to polyphonic composition, the two settings register important differences in linear function and purpose. These are displayed in arrangements of a famous psalm tune shown in Figure 11. In setting (a), three out of the four cadences are made with voices in their proper slots (at the end of mm. 1, 2, and 4). The last is an APAC, and the previous two are PACs, made by swapping TQ for T in m. 1, and likewise with A(Q) in m. 2. The cadence in m. 3 is inverted (TQ-ized, if we must), with S sent to the T range and A and B staying in their regular lanes. The cantional setting at (b) can be analyzed in much the same way, even in m. 2, where the phrase ends at –1 instead of C (i.e., a half cadence, discussed below); even so, we can describe tenorizing “in progress,” with S and A being displaced downward by one slot and B holding its station. The difference in harmonic progression—an increase in tonal directionality—seems noticeably stronger in the cantional setting.

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Figure 12 Examples of clausulas (cadences in which nonbass parts are inverted into the lowest voice), showing characteristic top voice accompaniments: (a) tenor clausula with S. (b) soprano clausula with T. (c) alto clausula with B.

Werckmeister’s invertible counterpoint procedures have so far resulted in one new cadential type, the IAC, and have also introduced a useful distinction between the regular and cantional PAC. But they also create options for other arrangements in which B participates rather than being fixed in the lowest voice. To mark the significant difference in stopping power when B is not the lowest voice in iC, I identify the resulting structure as a clausula rather than a (formal) cadence. This term is adapted from Werckmeister and his contemporaries, who used it as something of a synonym for cadence. I intend it more precisely to describe formations like those illustrated in Figure 12, from which the reader can derive other permutational arrangements along the lines of Figure 10, though with certain limitations, as we will see.

Clausulas (a) and (b) are arranged so that C has the same outer-voice perfection that characterizes the PAC. The former, a tenor clausula with S at the top, is sufficiently standardized that some authors identify it with the name clausula vera.15 The soprano clausula at (b) is voiced as something of a cantional counterpart to (a), as the outer voices are inverted. But the differences in the inner parts are telling. Tenor clausulas cannot properly support B and A lines because of the fourths they make with the tenorized lowest voice. Hence, detours or shortenings have to be imposed on A pathways to make thirds rather than fourths, and B cannot be accommodated at all. In contrast, the alto clausula at (c), which leads to an inverted and imperfect C chord, can compensate for this with B in the top voice, as shown.

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Figure 13 Order of precedence for arrangements of contrapuntal lines. Left column shows outer-voice perfection at chord C; right column shows imperfection.

At this point, as the proliferation of cadence and clausula arrangements threatens to obscure the basic closing convention, we collect and abbreviate many of these into the diagram shown in Figure 13, arranged according to an order of precedence. The most conclusive arrangements are in the upper left quadrant, and the least in the lower right. The abbreviations for cadences are familiar: APAC, PAC, IAC. Clausulas are labeled according to the pathway traced by the lowest voice followed by a lowercase “c.” Some clausulas are also cross-referenced with the parenthesized terms used in Gjerdingen (2007). Both cadences and clausulas are subdivided according to the top-voice pathway, marked with a pair of slashes. (Thus, for example, PAC//T signifies a cantional arrangement of a PAC.) Most differences between various top-voice arrangements are not enough to call for distinctions in precedence, so they are separated by commas on the same row. The difference between IAC//Q (either TQ or AQ) and IAC//A seems rather clearer, however, because of the driving voice-leading movement of Q, so//Q ranks higher than//A.

A Brief Analytical Application

The development so far has moved quite a distance from the PAC, which for some is the only structure with power to close a proper musical utterance in the same way that, for instance, a period can end a sentence. What can be closed, then, by the IAC, or for that matter, by any kind of clausula?

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Figure 14 Analysis of cadences and clausulas in Bach, Sinfonia 1 in C, BWV 787. The first column denotes measures, and the second identifies the local key in force. The third column is for cadences, of which there is only one. Additional columns to the right show clausulas, with Tc taking precedence over Sc. The broken line separates clearly articulated clausulas from those that have suspensions that stagger the arrival of pitches or have lines that are broken off before reaching the goal (shown in brackets).

Figure 14 attempts to answer this question with an analysis of Bach’s C-major Sinfonia, BWV 787. This well-known elementary study in fugal writing has but one cadence, a PAC//S in its final measure. But it clearly has internal divisions and lesser articulations marked by clausulas of various kinds, shown in the figure.16 The closer a structure is to the identified local key, the greater its formal precedence. The heavy grey lines show the sectional divisions between an opening exposition, a modulating middle, and a closing section based on the tonic key. A Tc//S at m. 7, well-marked by a cadential trill, unquestionably closes the exposition. Another one at m. 17 does the same for the middle, though with telling differences in register and duration of pitches in the T pathway. These make the clausula at m. 17 less of a formality than that at m. 7. Nevertheless, sectional conclusion is still effected. Of the remaining tenor clausulas, that at m. 13 is labeled to bring out its comparative imperfection: S is submerged by AQ. This keeps the middle section going, as it were, while bringing to a conclusion the move into minor-mode areas. This move was initiated by the deceptive Tc in m. 10, which sets up in GM (in parallel with the events at m. 7) but ends up activating Em, collapsing space between a potentially well-articulated dominant area and a move into minor-mode regions that normally would follow.

This transformation—altering the expected flow of a line—is a rhetorical scheme that plays with (and therefore relies on) expectations of cadential closure. We turn to the important topic of schemes in the next section, but an introduction can be made here by considering the clausulas shown to the right of the broken line in Figure 14. These have various kinds of linear disruptions that maintain flow and prevent the goal chord of the cadence from setting up. At mm. 4 and 5, suspensions are applied to the upper voices, offsetting their arrival with respect to the bass. In the remaining cases, suspensions may be accompanied by S lines that are diverted from their expected pathways. (For example, on the downbeat of m. 6, note the skipping out to G4 from the expected C5.) All of these schematic transformations are applied to situations in which the Sinfonia subject is in the lowest voice, and they work to dampen functional discharge upon its completion. In contrast, the first two clausulas of the piece are cleared to discharge so that the subject might be defined unambiguously; after that, schemes maintain musical flow through and past the endings, preventing them from chopping the music into subject-sized pieces.

The C-major Sinfonia is particularly clear about the division of formal labor between cadence and clausula, with the former being reserved solely for the absolute formal close.17 Clausula articulations are well suited to fugues and other contrapuntal genres (e.g., imitative motet) where themes move around in the musical texture. But even when clausulas do not close conventional, recognized formal sections, they clearly can close various kinds of smaller musical utterances. Reviving the grammatical metaphor that began this discussion, we note that if a formal cadence marks the end of a sentence (though, by extension, also a paragraph, section, chapter, etc.), then a clausula can mark the end of the musical equivalent of a subordinate clause, phrase, fragment, and the like. All of these bits are based on centers of tonal gravity, and clausulas inherit the power of cadences to identify these centers, no matter how local or passing. In short, cadences and clausulas mark the conclusion of processes that create centers. Determining the relationship of these centers to the larger structure is a task for formal analysis.

Adaptations for Plagal and Half Cadences

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Figure 15 Hypothetical contrapuntal background for plagal analogue to authentic cadence.

The main line of historical development of cadential structures, as recounted in the opening of this article, is clearly oriented toward the authentic variety. The plagal cadence, “oddly controversial”18 despite being a well-known textbook verity, cannot be satisfactorily explained, as might be hoped, as a kind of retrogression from C to –1 (with the latter considered a tonic, making the former a subdominant), since this would also retrograde the imperfect-to-perfect motion that characterizes the plagal as well as the authentic cadence. Instead, the plagal cadence can be theorized as the harmonic and contrapuntal dual of the authentic, as illustrated in Figure 15. In place of the major dominant chord at –1, the plagal employs a minor subdominant chord. Both S and B preserve their characteristic motions, though with reversed directions. The pathway-types of A and T are exchanged, so that T maintains a common tone while A moves stepwise. The Q pathways into the final chordal third are dualized as well; the superior perfection of the major triad compared to the minor permits a major (Picardy) third at C.19

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Figure 16 Giovanni Gabrieli, “Hodie Christus natus est,” C. 40 (1597), plagal cadence at conclusion. Two choirs (SSATB and ATBBB) have been condensed onto one system.

While the pathways of plagal cadence are not supported by the same kind of historical evidence adduced for the authentic, being generated theoretically, they are nonetheless as real and are traversed wherever plagal cadences are performed. A particularly clear illustration of all the pathways in operation—thanks to ten-part texture—is offered in Figure 16, the conclusion of a motet by Giovanni Gabrieli. The passage begins after a regular-setting PAC on an A-major chord, which places S on the pitch A5. As the minor subdominant is elaborated in the following measures, the voices move into position to discharge in regular plagal order. Although the thick texture means some duplication of pitch classes, the layout generally conforms to expectations: S and B are at the extremes, T is over B, AQ is over T, and A is over T(Q). (The second tenor, on F3 in the penultimate measure and thus doubling S, is constrained to skip away to A3 at the cadence to avoid parallel octaves. The first tenor eschews the passing-note opportunity TQ affords, as the parenthesis indicates.)

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Figure 17 Plagal cadence fused onto a preceding authentic cadence.

The Gabrieli excerpt also illustrates another normative characteristic of plagal cadences: their placement immediately after an authentic cadence as a signal of a theologically inflected ultimate finality, being the conventional setting for “Amen.”20 (Indeed, its association with sacred music is so strong that the influential music theorist A. B. Marx called it the “church cadence” [Kirchenschluß], which name persists in German discourse to this day.21) Figure 17 constructs short, elementary pathways that enact this fusion of cadential types. The contrapuntal outlines of many codas, both short and lengthy, are easily discerned here, and even more can be brought into view if a cantional transformation mapping T into the top line is imagined, with other parts distributed into other arrangements. The widespread applicability of this model suggests that the plagal cadence is primarily a four-stage event, with the –1 chord of the authentic cadence being –4 of the plagal.

Because of the special conventions around the plagal cadence, it does not regularly exhibit the same kind of clausula derivatives as the authentic, though the not infrequent use of a tonic pedal point supporting plagal activity in parts above—putting the subdominant into six-four position—could be described as a plagal tenor clausula.

The preceding discussion suggests a surprising conclusion about the plagal cadence: that it is in fact superior in stopping power to the authentic. It is a declaration of ultimate conclusion, ne plus ultra. It is eminently suited for essential structural closure in strophic forms, like hymns. That it is consigned to a particular place in compositional layout, strongly identified with sacred matters, and not subject to denaturing invertible-counterpoint transformation seals it as a special effect, which has been used topically to symbolize spirituality, soulfulness, transcendence, and so forth.22 This is certainly the meaning of its employment in secular genres of the nineteenth century by composers of as different aesthetics and techniques as Wagner (e.g., the endings of Tristan and Parsifal) and Brahms (e.g., the endings of the second movements of the First and Third Symphonies).23

Unlike the plagal cadence, which preserves closing perfection as the dual of the authentic cadence, the half cadence is inconclusive and pointedly imperfect—in contradiction to the values and aesthetics that support the idea of cadence in the first instance. Although examples of phrases and sections ending on half cadences or other similarly inconclusive progressions can be found in sixteenth-century music,24 these events could not be recognized as cadences per se before the middle of the seventeenth century, when theorist Conradus Matthaei expanded the purview of perfection by shifting interest from the final product—all-perfect chord C, with tolerable thirds—to the process that leads to C. In other words, this shift considers perfection in relation to the time plot of Figure 7. Mutch (2015b) recognizes this as the first proposal for an “entirely new meaning for the familiar imperfect-perfect binary,” mentioning but not inferring anything from Matthaei’s use of vernacular rather than Latin terms, which may have opened the way to the shift. While Matthaei’s notion of imperfection includes cases in which C is different than expected (such as in deceptive cadences), it also primarily covers the case in which process stops at –1, the “penult.”25 Thus, the half cadence is an attempt at an authentic cadence that stops prematurely.

From this perspective, the counterpoint involved in making half cadences is nothing other than that already encountered in Figure 8, illustrating authentic cadential approaches from –2, but now with –1 moved into the metrically strong position of C and the preceding chords adjusted accordingly. This highlights an important yet undernoted feature of textbook definitions of the half cadence: it does not have a single mandatory progression to the goal like the authentic dominant to tonic, but two: pre-dominant or tonic to dominant. In other words, the final chord in a half cadence can be specified (since –1 is specifically dominant), but its predecessor is as variable as –2.

This greater degree of freedom is responsible for the many and diverse approaches half cadence can have. As its possibilities were explored during the eighteenth century—reflecting “the increasingly important role of mid-section rhetorical articulations in music of the era”26—some of these achieved schematic status as conventions in their own right. As a result, the half cadence has recently been the object of remarkable attention in studies of galant music.27 The observations, however, put no small pressure on Matthaei’s granting cadential title to an imperfectly executed authentic cadence. It is fundamentally a contradiction in terms that can be papered over by relegating it to specific locations in larger forms (like the plagal cadence, though much less heavily constrained), with the express purpose of functioning as a separating rather than a stopping event.

The loss of potential finality entailed in this assignment gives the half cadence a tentative and (as Burstein 2014 puts it) “slippery” quality, which clearly underwrote the prepending of “so-called” to its name in the eighteenth century.28 Admitting it to the set of cadences is to enlarge the meaning of cadence to cover generalized harmonic punctuation, not just the mark that ends the musical equivalent of clauses. This move toward recognizing different degrees and types of closure was prepared in the discussion of clausulas above and advanced to some degree in the discussion of the plagal cadence. Conferring recognition on the half cadence crosses a border into the large province of rhetoric from the smaller and more easily defined jurisdiction of contrapuntal syntax. Some confusion and dilution of meaning attend this move, which are monitored and corrected as warranted in the remainder of this article. But the enrichment of musical expression that rhetorical treatments of cadence permit is an attractive development that invites further investigation.

Schemes of Cadential Rhetoric

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Figure 18 Types and sites of continuation schemes applied to cadence.

Viewed as a rhetorical effect, the half cadence belongs to a set of continuation schemes, which cause the stopping power of the cadence to fall short to some noticeable degree, with the music going onward toward another goal, which may even be another run at C. The most important schemes of this type are shown in Figure 18, with the half cadence shown as a reassignment of C to the chord at –1. Another well-known entity in elementary harmony was encountered (in Tc arrangement) in the Bach Sinfonia: the so-called deceptive cadence. In its simplest form, the bass motion in iC is deflected into an ascending step, failing its customary fifth leap in a SAQTB cadence. The effectiveness of this single change in the collective pathway is best appreciated by noticing how it neatly “reverses” the perfection of the final chord tones: in regular order, C has perfect intervals between bass, tenor, and soprano, with the AQ sounding the imperfect third; when this deceptive scheme is applied, the bass makes an imperfect third with tenor and soprano, while AQ is made a perfect fifth.

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Figure 19 Mozart, Vesperae de Dominica, K. 321, Magnificat. Chorus and continuo of mm. 8–12.

Conventional theory has unfortunately claimed this just-described transformation as the deceptive cadence. While likely to be statistically justified as the most common (and efficient) version, I suggest that the entire class ought to be brought into consideration—effects in which C is unexpectedly different from what was anticipated. More particularly, as shown in Figure 18, the action takes place at iC, where one or more irregular pathways are taken, leading to a markedly different kind of C chord, noted as ~C. The actual chord that shows up for ~C is, properly speaking, anything other than root-position tonic as predicated by voice-leading—with the submediant being a statistically normal and default choice. But as a remarkable passage from Mozart’s K. 321 demonstrates, much more can happen in iC than a single nudge on the bass line (see Figure 19). The chord it delivers in ~C is not I but II♯ (alias V/V), practically a “joke” chord in the context of the absolutely clear regular SATQB cadence being set up from pre-dominant –2. We could possibly call this some kind of irregular deceptive cadence, but I prefer extending the writ of the unmodified term to cover any and all changes in iC that produce ~C.29

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Figure 20 Sprung PACs. For S in cantional cadences, a jump to the A position at C in m. 2; a jump to Q in m. 4.

To nuance matters of pathway alteration, we should briefly consider another form of cadential deception that does not change C but does change its expected voicing by swapping in pitches from another part. This is common in four-part cantional settings in which a complete triad at C is desired, but –1 is not set up to deliver one. Figure 20 shows paradigm cases in which S—inverted into the interior of a chord—has its leading tone “sprung” from its proper pathway and into that of another part. In the first illustration, S takes a short jump into A, and top-voice T covers the pitch that was originally targeted. In the second, parts are set up for an APAC, and S jumps—farther and more noticeably—into Q in order to provide a chordal third.30 The identity of C is unaffected by these changes in iC; indeed, C is sonorously strengthened by them. Thus, there is no cause to propose them as deceptive cadences per se. But performers assigned to a sprung line certainly experience the characteristic disruption associated with a deceptive cadence, and all the more so since the leading tone—a mandatory element of the originary cadence—is involved. It is suggestive, I think, to compare the operations of the classic deceptive cadence with these sprung cadences: both alter a single pathway at iC, but the change in B from leap to step changes the aspect of the chord as a whole, while the change in S from step to leap does not. Pathways in both are altered unexpectedly—and those performing the moves know this—but only the B alteration registers significantly on chordal structure.

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Figure 21 Examples of cadential evasion: (a) motion into deceptive cadence; (b) substitution of AQ for B; (c) cadential inversion in a cantional setting using A(Q) in the lowest voice and a broken T path in the top voice.

Figure 18 identifies another continuation scheme, evasion, in which a noticeable change occurs at –1. A chromatic motion in B into V6/5 of VI, as illustrated in Figure 21(a), is a typical case (and could well be described as prepared deception). A more formally significant application is shown at (b), in which what appears to be an Ac comes off as evading the consequences of the events in B leading up to –1, at which point a kind of “pivot” operation occurs that swaps AQ for B. This scheme is a basis for what Janet Schmalfeldt calls the “one more time” routine, which can be deployed into elaborate cadential deferrals and multiple looped-back approaches that are frequently found in galant music.31 This routine can also be activated using Werckmeister’s A(Q), in which the telltale passing tone is suppressed at –1 and a direct skip is made from ^5 to ^3 in iC, resulting in an expectedly voiced C chord in inversion. More dramatic forms of the inversion scheme affect other pathways at iC if the top voice in a cantional or other inverted setting leaps in away from its target in contrary motion with the lowest voice, as shown at (c).

Finally, the scheme of suspension, already encountered in the analysis of the C-major Sinfonia (Figure 14), acts on pathways by delaying their progress toward resolution.32 The continuation involved here is somewhat different from the other schematic treatments, in that the continuation past the cadential point predicates no additional or new music, only a wait for remaining voices to finish traversing their pathways. This is clearest when suspensions are used in a final or other section-ending cadence. But when combined with evasions, inversions, and other schemes, suspensions can boost the effect of continuation noticeably at the expense of cadential recognition, an important compositional feature of the Sinfonia.

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Figure 22 Types and sites of emphasis schemes applied to cadence.

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Figure 23 Corelli, Chamber Sonata, op. 4, no. 2, Allemanda. Measures 18 to end, showing cadential loop in m. 21.

Some schemes of emphasis are suggested in Figure 22. Unlike continuation schemes, which produce music after the cadence in order to make up for some loss of stopping power, emphasis schemes produce postcadential music in order to dissipate kinetic energy built up over a stretch of music leading up to the cadence, extending the point of cadential resolution into a temporal area. The simplest technique is elongation, symbolized by a fermata, in which C is markedly longer than the preceding chords, a length that may well be extended beyond what the notation suggests by a preceding rallentando. Elongation of this kind is, of course, most suitable for final cadences, where it signals an ultimate finality.33 Another signal of similar function is (multiple) repetition of the final C chord, a technique that became a commonplace in the faster movements of galant music and was developed to extraordinary lengths by some later nineteenth-century composers. The loop emphasizes the cadence by rearticulating it, either minimally—by going back to the dominant at + 1 (thus, + 1 = –1) and resolving to C at + 2—or at some greater remove. Figure 23 shows a typical case of Corelli’s practice, in which an approach to a cadenza doppia in m. 20 is repeated more softly, with + 1 looping back to –3.34 Loops may not involve literal repetition, as is the case here, but instead use different chords and arrangements when reapproaching, as will be seen in the next example. Finally, a hybrid figure of elision is recognized, which has properties of both emphasis and continuation. As the figure shows, elision assigns to the concluding C an initiating function, “0.” Thus, while the cadence has adequate power to close the previous musical unit—and thus does not stand in need of any remedial continuation—the overlapping of a phrase beginning onto the cadential moment emphasizes it by an instantaneous re-energizing of musical flow. The emphasis, however, is forward looking and leaves the just-achieved cadence behind as the composition continues into new areas.

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Figure 24 Telemann, Sonata Methodiche (1732) TWV41:c3 III, analysis of cadential rhetoric.

A brief passage from the conclusion of a Telemann sonata, shown in Figure 24, can serve to illustrate various rhetorical techniques of cadence in combination. Four cadential modules are identified, terminating in mm. 35, 38, 39, 40, and 41. All but the last receive schematic treatment. Measure 35 is a clear instance of inversion, in which both outer voices are subjected to surprising detours in expected pathways, resetting the music back at –6 for a reapproach to cadence. This is broken off in m. 36 due to an internal expansion of the figuration at –3. The gesture is rejoined at the end of m. 37, but an evasion in the following measure forces another reset toward –2. Surprisingly, the inversion performed in m. 35 is revisited in m. 39, forcing yet another reapproach. A PAC is finally achieved in m. 40, and a subsequent loop provides space to absorb the impact of the previous inversions and evasions as well as rearranging the cantional PAC into a regular setting.

Formal Functions

Much recent theoretical work on cadences—and there has been a lot—centers on the galant, classical repertory, extending and exploring ideas developed initially by William Caplin.35 Cadences play a crucial, form-defining role in Caplin’s theory and the analyses that depend on it. The success of the project and the intensive activity taking place around it have certainly fostered innovative thinking about cadence, which had not enjoyed serious theorizing for some time and had become a commonplace of the harmony textbook. But the project also has shown signs of making cadence a creature of formal function in a particular repertory and stylistic practice. That is, a broadly general theory of cadence as a musical effect, convention, and structure seems in danger of losing out to a particularist theory of cadential function in the galant style. We can see effects of this in Caplin’s relegation of the plagal cadence to the status of “myth,”36 a move that may be excused by its limited functional uses in the classical style, but which is a bald overstatement if the practices of previous and subsequent repertories are taken into account.37

Putting cadence into a dependent relationship with classical formal function leads to impoverishment of recognized cadence types, which this article has highlighted by following the opposite course, choosing proliferation of types in order to register a very large range of closural effects. In this, it somewhat follows the Ramist method of definition and division that influenced late seventeenth-century theorists of cadence such as Wolfgang Caspar Printz.38 And it also struggles with the attendant terminological inflation that bedevils this style of theorizing, a feature found in many early-modern sources both learned and pedagogical, and with various levels of synonymy.39

But at this point, an enriched system of cadence types is a necessary corrective to the assumption that later eighteenth-century styles are normative for tonal theory, as attractive as it is on account of the highly schematic construction, preference for periodic phrase rhythm, simplified modality, and melody/accompaniment default texture. The implication of such a norm is that the music of the later eighteenth century is a culminating point of tonal composition, a place where antiquated artifices of earlier styles had finally been expunged, but not yet affected by the chromatic complications to come in the nineteenth century. It is an exceptionally pellucid music that admits broad-spectrum analytical light and supports robust theoretical generalizations. Yet from another point of view, it can also be regarded as a mannerist simplification of compositional techniques perfected in the early eighteenth century, analytically unproblematic compared to its flanking repertories, and therefore not the best basis on which to make general claims about common-practice tonality. The armamentarium for tonal analysis needs to be larger than that which serves late eighteenth-century music, with the cadence particularly benefiting from a wider range of treatments.

The Bach Sinfonia analyzed in Figure 14 makes this point clear. Form-functional theory has no purchase on the work.40 Because it has one “classical” cadence, a regular PAC//S, and that only at the end, it could be claimed that it is internally undifferentiated and unarticulated. The broken line in the figure might then be sited just to the left of the Tc//S events in the second column, marking the separation between formally “significant” and “insignificant” articulations. But the analysis was designed to show the formal significances—registrations of tonal centers—of the various Tc events in the piece, as well as the one Sc. Even the effects of the several AQ clausulas to the right of the broken line were noted, though their contributions to formal articulation were deemed too weak to take much note of. It is thus true that the fine-mesh net for cadential effects catches more than conventional techniques do, which increases the overhead of analytical operations (e.g., where to draw the broken line of structural significance). Yet it is also true that instruments built to specifications of individual styles will have trouble when tried with others. On balance, I prefer the work of filtering too many results to the worry about having too few.

Contemporary Developments

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Figure 25 Two PACs in cantional settings: (a) normal movement order of parts in the eighteenth century; (b) possible movement order in the nineteenth century.

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Figure 26 Mahler, Symphony no. 2, mvt. 5: Rehearsal 48–4; reduction.

The contrapuntal basis developed so far, along with the sketch of rhetorical transformations, can support and explain most conventional closing gestures in Western art music up through the common-practice era. As convention was reassessed as cliché in the nineteenth century, the authentic cadence came under pressure to freshen its sound while maintaining contact with traditional practice. One noteworthy effect—associated with Chopin’s style but by no means limited to him—deforms the standard approach by altering the order of part movement through the pathways. Figure 25 shows the basics, with the well-established eighteenth-century cantional norm shown at (a) and the nineteenth-century transformation at (b). The essential change is delaying the descent of T to tonic until after AQ moves into its passing seventh. Thus, in the third chord of the measure, two harmonic sevenths dissonate noticeably: the normal minor seventh from bass G to passing F, and the deformational major seventh from F to delayed E.41 The dominant thirteenth resulting from the combination of the two sevenths was a durable deformation that continued to be used even into the last decade of the century, as the excerpt from Mahler’s Symphony no. 2 illustrates (Figure 26).

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Figure 27 Functionally mixed cadence, plagal features predominating.

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Figure 28 Florent Schmitt, Piano Quintet, op. 51, first mvt., condensed (Casella and Rubbra 1964, ex. LXVIII, extended by one and a half measures).

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Figure 29 Hindemith, Organ Sonata III, third mvt., final cadence.

The cadential thirteenth chord is a small but telling piece of evidence for a general increase in nontonic dissonance throughout the nineteenth century. In cadential situations, other signs of this increase include freely mixing authentic and plagal pathways to create functionally mixed chords at –1, one example of which is illustrated in Figure 27.42 In general, as harmonic variety and richness increased (and periodicity as a formal norm decreased), contrapuntal cadential approaches became harder to detect before –1. This can be quickly apprehended in the later examples collected in Casella and Rubbra (1964), in which cadences can come on very suddenly. One is reproduced in Figure 28, from about halfway through the first movement of Florent Schmitt’s estimable Piano Quintet of 1908. Although –1 is a traditional dominant seventh (in a regular setting, to boot), it is completely unpredicated by the preceding harmonic activity, with –2 being a nontraditional passing verticality and the chords before giving no hint about an upcoming cadence in G-flat.43 A similar kind of surprise cadence from later in the twentieth century is shown in Figure 29. The cadence here is plagal rather than authentic, with –1 being the Leittonwechsel of the minor subdominant triad (i.e., the Neapolitan sixth chord in A-flat major. The preceding chords work through conjunct pathways toward –1, producing nontraditional, pentatonically inflected passing sonorities on the way.

By the late nineteenth century, the increased importance of secondary parameters in shaping musical process and structure permitted new forms of articulation that did not rely on the traditional cadence and its clausula derivatives. Leonard B. Meyer was loath to admit that structural closure could be effected by such means, since “music based on them can cease, or end, but cannot close,” adding in a footnote that “termination created by secondary parameters will be referred to as cessation to distinguish it from closure.”44 Such effects can certainly be found in Debussy’s music and in that of other impressionists. Similarly, the “emancipation of the dissonance” made distinctions between a priori imperfect and perfect sonorities nugatory, necessitating innovative contextual forms of articulation and ending. Casella and Rubbra’s collection documents these developments quite clearly. Some theorists could want to extend the franchise of cadence to them, but I am reluctant to do so. Instead, I aver that cadence was a shared convention of many musical styles—born from counterpoint, schematized in harmony, and deployed according to rhetorical need. Developed in the Middle Ages from venerable ideas about perfection, it proved to be extraordinarily durable yet capable of remarkable transformational elasticity. It is perhaps the longest-lived convention of Western musical composition, but it is not a transcendent category. It is finite, and its pathways of development do, in the end, find their end.


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(1) Valuable guidance for this effort was provided in Moll (1998), Cohen (2001), Bain (2003), Taruskin (2005, vol. 1), Schwind (2009), Neuwirth and Bergé (2015), and Mutch (2015b), with Schenker (1987) acting as a consultant.

(2) The (extravagant) range of possible moves to perfect consonances in early contrapuntal theory is shown in Coussemaker (1931, 3:72–73). Figure 2 is adapted from illustration 72,1.

(3) See Mutch (2015b, ch. 2), for further background on the theoretical development of cadence for polyphonic textures.

(5) In music with regularized, deep metric hierarchies, cadential anticipation can be triggered via hypermeter and thus rises in direct proportion to phrase length, starting from chord +1.

(6) Eberlein and Fricke (1992, 70–71) draw a similar picture, but at a lower level of generality and theoretical detail.

(7) Schenker (1987, 1:103–4).

(8) The way toward complex cadential elaborations, such as the cadenza doppia of the partimentists (Menke 2011), should be clear from this point.

(9) Werckmeister and Mongoven (2013, 301). Arnold (1965, 1:12–14) details the differences between Viadona and Werckmeister. Mutch (2015b, 142–43) notes that the basis for this move can be traced to the “Cologne school” theorists of the sixteenth century. Werckmeister’s illustration and its usefulness in analyzing Bach’s music (especially fugal subjects) are discussed in Deppert (1993, 73ff.). See also Byros (2015, ex. 4).

(10) Mutch (2015b, 167) points out that Werckmeister inherited this style of vocabulary, citing Johann Andreas Herbst (Musica Poetica, 1643) and Conrad Matthaei (Kurtzer Bericht, 1658).

(11) This is just one way to compose the effect: giving singers on one part moves that properly belong to another part. But it can also be done with voice crossing: the singers keeping their moves but in different registral slots.

(12) Werckmeister pointedly does not illustrate a strict “alto-ized” arrangement that would result from transforming m. 5 into a m. 6. This perhaps speaks to an assumption that such an arrangement might not satisfactorily perform cadential work, possibly because the comparatively uninteresting (straight and narrow) A path is overly exposed in the top voice.

(13) Cantional is a term originally associated with sixteenth-century Lutheran hymn settings, which have the tune in the top voice instead of the (interior) tenor.

(14) The observation that Schenker’s Ursatz is thoroughly cantional is pertinent in this connection. The disqualification of regular-setting upper-voice structures may be related to his rejection of other “outmoded” concepts like the “church modes,” but this is entirely conjectural. But every Schenkerian analyst has encountered the unfolding of ^2 to ^7 at the cadence (also in reverse, though less often), a move that helpfully enlists the discharge energy of both T and S lines to tighten the close.

(15) Gjerdingen (2007, 164ff.). In note 21 (pp. 490–91), Gjerdingen points out that this seemingly archaic phrase is perhaps of nineteenth-century vintage.

(16) The analytic interests here, developed independently, are entirely consonant with those propounded at greater length and detail in Byros (2015).

(17) The other Sinfonias in the collection have internal cadences as well as clausulas, though the former are often IACs. See, for example the sinfonias in A major and A minor. Those in E major and E♭ major end with an IAC.

(18) Mutch (2015a, 69).

(19) Eberlein and Fricke (1992, 63) also informally propose the plagal as the dual of the authentic.

(20) In “Hodie Christus natus est,” Gabrieli sets the word “Alleliua” to the cadence, which preceding music also set. But the association of the plagal cadence with “Amen” is so strong—and executed normally in many of his other works—that it is easily legible underneath “Alleluia.” Note also the same feature at the conclusion of Handel’s “Hallelujah Chorus” from Messiah.

(21) Marx (1846, 299). See also Louis and Thuille (1913, 12), and Amon (2005, 220). On other authors’ views of the association of the plagal cadence with church music, see Mutch (2015a, 69).

(22) Amon (2005, 100).

(23) See Meyer (1989, 285–91) for more examples and discussion of nineteenth-century usage. Meyer’s argument against my “surprising conclusion” is that the plagal cadence “is not a substitute for strong syntactic closure but a sign confirming prior closure” (286). One might turn the terms around and claim that a plagal cadence performs a stronger syntactic closure after a provisional closure of a prior authentic cadence.

(24) See, for example, the extraordinary chord progression that marks the conclusion of an uninterrupted span, mm. 1–45, in Thomas Tallis’s “Dum Transisset Sabbatum.” A simplistic analysis could label it a deceptive cadence with a Picardy third, but the effect is rather more ineffable than the description suggests. A more conventional-sounding half cadence marks the conclusion of the following, shorter section, mm. 45–58. The final cadence of the piece sounds almost equally inconclusive, though an obscured plagal basis can be uncovered analytically. Burns (1994, 47–50) discusses sixteenth-century theories about such “irregular endings.”

(25) Mutch (2015b, 152–53). Burstein (2015), following Burns (1994), cites Johann Andreas Herbst (Musicae Poëtica [1643]) for first illustrating an irregular ending with a half cadence constructed as an incomplete authentic cadence. (See Burns 1994, 51, ex. 3.)

(27) Caplin (2004), Gjerdingen (2007, 153–54, 160–62), and Burstein (2014, 2015).

(29) This suggestion agrees with the general program of Neuwirth (2015).

(30) J. S. Bach’s chorale settings provide multiple instances of both treatments, with the first being more common. For examples of the second, see number 3 (Ach Gott, vom Himmel sieh’darein) at m. 4, and number 179 (Wachet auf, ruft uns die Stimme) at m. 5.

(31) Schmalfeldt (1992). Her examples 1c and 7a (both from Mozart’s music) clearly illustrate this technique.

(32) The presence or absence of suspensions in cadences gave rise early on to a variety of descriptors, helpfully summarized in Neuwirth (2015, 126). However, these are concerned only with the situation at –1, not at C, which is where the suspension scheme is applied in the current development.

(33) A related scheme of understatement is occasionally employed, especially in Bach’s music, in which the final chord is remarkably and perhaps even uncomfortably short. See, for example, the eighth-note final C chords of the E-minor fugue from WTC 1, BWV 855.2, and that of the B-minor organ prelude, BWV 544.1.

(34) The particular form of this cadenza doppia is found in Menke (2011, ex. 5, col. 3, row j).

(35) Among many related writings, Caplin (1998, 2004) provide deepest context; his contribution in Neuwirth and Bergé (2015) develops his ideas in dialogue with Gjerdingen (2007). Cadence is also an important category in the related work of Hepokoski and Darcy (2011).

(36) Caplin (2013, 56).

(37) Mutch (2015a) expertly analyzes a mid-eighteenth-century attempt to theorize the plagal cadence, emphasizing its links to Phrygian modality. Burns (1995, 43–50) discusses in detail plagal effects in Bach’s chorales. Meyer’s views on nineteenth-century plagal cadences have already been cited in n. 21, above.

(38) See Mutch (2015b, 154ff.) for detailed background on the proliferation of cadence types according to Ramist principles.

(39) Diergarten (2015, 63–70) sketches the problem of the terminological complexity in a variety of sources and proposes a modern reconciliation. Mutch (2015b, 169) gives an overview of Printz’s unique and highly ramified terminology.

(40) Judging from Caplin (2009), contrapuntal genres and their formal types seem generally to be illegible. Note their absence in the comprehensive summary presented in Caplin’s table 1.1 (2009, 33).

(41) See Narmour (1991, 98–111) for discussion and background on Chopin’s use of this sonority, though not necessarily in cadential settings. Liszt, among others, favored it; see, for example, Sonetto 123 del Petrarca, m. 74.

(42) This particular formation is disassembled and analyzed in Harrison (1994, 64–68), illustrated with examples from Richard Strauss and Max Reger. Unlike the dominant thirteenth, which is a particularly nineteenth-century innovation, functionally mixed cadences can be found throughout the common-practice era. See, for example, the conclusion of J. S. Bach’s E-major Prelude from the Well Tempered Clavier, book 1.

(43) The surprise effect of this cadence contrasts with the much more traditional cadential effect at Rehearsal 9, the seam between the slow introduction and the main body of the movement.

(44) Meyer (1989, 209 and n.184).