As does Lampe, Rameau on occasion analyzes a harmonic succession with reference to bass pitches rather than fundamentals [6.2]. A single chromati-cally inflected pitch – Fs in place of F – prompts a reassessment of the tonal center: G becomes a tonic note. The brevity of the example reinforces Rameau’s assertion that what follows the G chord is irrelevant to its inter-pretation. It is solely “by means of the difference between the progression of a tone and a semitone ascending to the note bearing the perfect chord (accord parfait)” that one forms an analytical response. Once that perfect chord sounds, the composer may choose to – but is not compelled to – resume composition in the original key, “since after a perfect chord we are free to pass To be in a key means to segregate the twelve pitch classes into two cate-gories: seven pitch classes residing within the key, the other five residing outside. Each pitch class is a member of fourteen of the twenty-four keys.
Though the exact roster of keys for each pitch class may vary from analyst to analyst depending upon exactly how the minor keys are formulated (7 being especially contentious), each pitch class is both member and non-member in the same proportion. The pitch class C is a member of most of the most common keys, while Fs/Gb is a member of most of the least common keys. For example, C’s membership list is C Major, C Minor, Bb Major, Bb Minor, A Minor, Ab Major, G Major, G Minor, F Major, F Minor, E Minor, Eb Major, D Minor, and Db Major.
Once a key is established, the occurrence of a pitch from outside its diatonic realm will have a novel effect. The analyst is faced with the task of justifying its presence within the composition. Certainly the most facile explanation will rely upon the fact that the pitch in question belongs to fourteen diatonic keys, one of which may be called into service as a temporary tonic. Yet at the moment of its first sounding, the listener does not know what the event heralds. Does it launch a major thrust in a new key? Or is the occurrence merely episodic – a ruffle within the expanse of the principal key?
Analytical responses to such pitches fall into two broad categories.
Either they trigger a shift (either temporary or for a longer duration) of the tonal center, or they are absorbed as a separate class of pitches within the original key. We explore in turn how both of these perspectives were practiced, beginning with that in which the key shifts as needed to keep each chord within a diatonic context.
wherever we desire.” Though Rameau’s analytical notation does not convey both interpretations simultaneously, his prose acknowledges the chord’s dual meaning: “we may still continue after this dominant (which would then appear to be a tonic note) in the original key.”2
A more extended example later in the Traité displays clear boundaries between keys [6.3a]. The C chord of measure 1, analyzed in C Major (Ut), lies outside the domain of the following dominant region (Sol), while the G chord of measure 2 plays no role in the restored tonic region that follows.
Lampe’s analytical practice is suppler [6.3b]. His E6 chord (measure 2), unlike Rameau’s second chord, relates both backward to the key of C and forward to the key of G.
Daube’s analytical response to the raised fourth scale degree conforms to Rameau’s.3In an example that begins and ends in D Major, Daube interprets 6.2 Rameau: Traité de l’harmonie (1722), p. 208 [Gossett, p. 228].
“The composer is thus free to make the bass proceed by a tone or by a semitone, even if he should be in a key in which the semitone is not appropriate; for since the dominant can be treated as a tonic note, it may be approached using all those sounds which naturally precede a tonic note.”
6.3a Rameau: Traité de l’harmonie (1722), p. 253 [Gossett, p. 272].
Rameau’s brackets delineating regions in C Major (Ut), G Major (Sol), and again C Major admit no overlap. At one later point in the example Rameau interprets the same chord successively in two keys (writing whole note G twice within the same measure): the first G closes a region in G Major; the second opens a region in D Minor.
Gs in the context of A Major [6.4], just as Rameau had interpreted Fs in G Major [6.2]. Later, in the first volume of Der musikalische Dilettant (1770), Daube inaugurates the practice of numbering his “three chords” using the digits 1, 2, and 3 [6.5].4Though only one digit appears beneath each bass note (or group of notes bound by a slur), his commentary is more nuanced.
In fact his explanation concords with the notion of Mehrdeutigkeit (multi-ple meaning) that would dominate later accounts of key shifts.
Koch explores the implications of approaching the dominant via the raised fourth scale degree in the first volume of his Versuch einer Anleitung zur Composition (1782–93). Once this chromatically targeted dominant arrives, the composer is at a crossroads (Scheideweg). If the original tonic is restored, then the modulation is classified as “incidental” (zufällig) or “a brief discretionary modulation” (eine kurze willkührliche Ausweichung).
6.3b Lampe: A Plain and Compendious Method of Teaching Thorough Bass (1737), plate 17 (near p. 31).
The notion of what we now call a “pivot” chord – the same event interpreted simultaneously in two keys – is amply demonstrated in Lampe’s analysis. Observe also that Lampe employs the symbol “7th.” for pitches such as E and B, and (“7th.”) for pitches such as Fs and Cs. The stroke through the 7 reflects a mere happenstance of music notation and not a distinction between diatonic and chromatic. That is, each 7th is diatonic in the key in which it functions as the seventh scale degree.
6.4 Daube: General-Baß in drey Accorden (1756), pp. 70–71.
“In [this] example one observes how through the addition of a single s this chord of the fourth scale degree in D Major [G-B-D-E, a chord with added sixth] can be transformed into the chord of the fifth scale degree in A Major [inversion of E-Gs-B-D], and how one may reattain D Major by applying a 7 to the A-Major tonic chord.” [“Bey ersterem Exempel siehet man, wie durch Beysetzung eines einzigen s dieser 4ten-Accord von D dur könne in den 5ten-Accord von A dur verwandelt werden: und wie man durch die 7 auf dem Grundtons-Accorde A dur wiederum in D dur zurück gelangen könne.”] Recall from chapter 1 that Daube pursues a functional analysis focused on just three chords, all of which are invoked in this explanation.
6.5 Daube: Der musikalische Dilettant: Eine Abhandlung des Generalbasses (1770–71), pp. 89–91.
This example is a continuation of 1.9. The numbers 1 through 3 refer to Daube’s three principal chords, introduced in chapter 1, above. The + or m symbol placed beside a number indicates interpretation in another key. The label “1+” for the second chord of measure 11 is misleading.
Either “1” or “2+” was perhaps intended. Though it is tantalizing to speculate that Daube’s manuscript read “1 2+” and that this novel juxtaposition was botched during publication, it is unlikely that Daube actually created such a “pivot-chord” numerical analysis: had he, he would have labeled the G chord in the second half of measure 12 as “1+ 3.” Daube’s commentary reads as follows: “The numbers below the staff marked by a plus sign (+) denote the modulation into the most closely related key, here G Major. This occurs on beat two of measure eleven. Here one finds the 1 chord, namely the tonic C chord, and since this perfect concord functions also, as explained above, as the 2 chord of G Major, the modulation into another key is most opportunely accomplished at this point. The 3 chord of G Major follows, after which the melody returns again to C Major after a few notes. Now at this point this chord must once again sustain a dual role: namely, it should simultaneously also function as the harmony of the 3 chord of F Major, as we have put forward and amply demonstrated above. Now the melody is in the second most closely related key to C Major, namely in F Major. This is indicated by the double m symbol. But here it goes immediately back again into C Major, which occurs over stationary bass F, a member of the 3 chord in C Major, and since it appears here in the bass, the 3 chord of C Major is in third inversion.” [“Die untenstehende Ziffern, die neben sich ein einfaches + haben, bedeuten die Ausweichung in die nächst anverwandte Tonart, welches hier G dur ist. Diese geschiehet im eilften Tackt bey der zweyten Viertelsnote. Hier ist der erste Accord, nämlich der herrschende Accord C, und weil diese ganze Harmonie zugleich auch, nach der vorhergehenden Erklärung, im zweyten Accord von G dur befindlich ist; so findet die Ausweichung in eine andere Tonart die beste Gelegenheit auf dieser Stelle. Auf diesen Accord folget der dritte Accord von G dur, worauf diese Melodie nach etlichen Noten sich wider zurück in C wendet. Hier muß nun dieser Accord aufs neue eine zweyfache Stelle vertretten, nemlich er soll zu gleicher Zeit auch die Harmonie des dritten Accords von F dur versehen, wie wir dieses oben mit mehrern erwiesen und vorgelegt haben. Nun steht die Melodie in der zweyten anverwandten Tonart von C dur, mithin in F dur. Dieses wird durch ein doppeltes m angezeiget. Hier aber geht sie gleich wider zurück in C dur, welches durch die Liegenbleibung des Basses geschiehet, als welches F auch im dritten Accord von C dur sich befindet, und da es hier im Baß erscheinet; so stellt es die dritte Umwendung oder Verkehrung des dritten Accords von C dur vor.”]
But if the succeeding progression reinforces the dominant key, then the modulation is “essential” (notwendig). In 1787 Koch revises his terminol-ogy and expands to three classifications: “incidental” (zufällig), “passing”
(durchgehend), and “structural” (förmlich).5In examples that follow Koch employs a chromatic pitch to reinforce chords on the second, third, fourth, fifth, and sixth scale degrees of C Major, asserting that these alterations are pertinent not to the foundational key itself, but instead to these “related keys” (verwandten Tonarten).6Ernst Wilhelm Wolf reaffirms Koch’s selec-tion of “auxiliary keys” (Nebentonarten) for a major tonic. He goes astray in minor, however. Though he correctly asserts that the third, fourth, fifth, sixth, and seventh scale degrees constitute the Nebentonarten of a minor tonic,7his analysis of a passage from Händel’s Alexander’s Feast proceeds from B Minor ultimately to Cs Minor, a turn of events for which he pro-vides no rationale beyond a comment that passing cadences may target goals that are generally avoided.8Fortunately this and other aspects of his analysis come into better focus by taking a broader view of the work’s tonal context [6.6].
The nineteenth century thus inherited both a perspective concerning which keys were most suitable as goals for modulation and a notion that some modulations are more consequential than others, judging from how long the alternative key persists and the means by which it is confirmed (its leading tone being a principal factor in the latter). In the early nineteenth century Antoine-Joseph Reicha, a Bohemian who lived for a time in Bonn and Vienna before settling in Paris, distinguishes between “conventional”
(régulière) modulation and “modest transitory modulations” (petites modulations passagères) in his Cours de composition musicale [ca. 1816]. A ten-measure example in C Major touches on most of the same keys that Koch employs. Reicha provides brief commentary below the score, as follows:
A passing modulation from C Major into D Minor.
Return [to C Major].
From C Major into F Major.
From F Major into G Major.
Return to C Major.
From C Major into A Minor.
From A Minor into D Minor.
Return to C Major.
and comments that “these transitory modulations are so brief that the ear does not lose the impression of the key of C Major, and they have moreover
6.6 Wolf: Musikalisicher Unterricht (1788), plate 28, ex. Qqq, with commentary paraphrased from pp. 51–52.
Wolf ’s abbreviated score for a passage from Händel’s Alexander’s Feast includes figured-bass numbers instead of written-out chords. His harmonic analysis (paraphrased below the score, above) is rendered within a textual commentary. (The original German is provided below.) In measure 4 he omits a chordal seventh E above Fs, present both in Händel’s score and in a printing of an excerpt in Johann Friederich Reichardt’s Kunstmagazin (vol. 1, 1782, p. 140), to which Wolf makes reference. He offers three different labels for the cadence that, in his view, occurs at that point: half, Phrygian, and plagal.
Two questions on key choice arise from a study of Wolf ’s analysis. First, why does Cs Minor occur prominently in the context of B Minor? And second, why does Händel not include a modulation to G Major? Händel’s key choices would make better sense if Wolf had indicated that the aria from which the passage is extracted is in A Major. As the chart below reveals, Händel’s key choices within this passage (measures 72 through 82, about two-thirds of the da capo aria’s middle section) are more closely aligned with A Major than with B Minor.
the benefit of enlivening a musical phrase which, without them, would often turn out to be routine.”9
Though Gottfried Weber maintains that “many digressive modulations are so very transient that they scarcely deserve the name”10and that “the ear, after imperfect digressive modulations, is inclined of its own accord to resume again its state of attunement to the yet scarcely quitted principal key,”11the analyses of block-chord progressions in his Versuch einer geordneten Theorie der Tonse[t]zkunst emphasize key shifts at the expense of continuity.12 Chromatically reinforced chords will inevitably appear to be the most significant in his analyses. In one example he calculates the intervals between adjacent tonicized keys: C Major to D Minor is an ascending major second (2•); D Minor to F Major is an ascending minor third (•3); F Major to C Major is an ascending perfect fifth (5•); and so on [6.7]. Thus C–d–F–C is highlighted at the expense of G (the eighth chord), which, though lacking a preparatory Fs, plays a vital role within the progression – arguably more vital than either d or F.13
Weber’s analytical system had a wide influence. It became routine for nineteenth-century analysts to extract a chord containing an altered pitch from its tonal context and to assign it a temporary home key for diatonic interpretation.14This preoccupation with diatonic identity – matching the components of every chord to one of the twenty-four pitch collections of the tonal system – proliferated at the expense of connectedness. In a pro-gression that begins in C Minor, Jadassohn interprets Fs-A-C-Eb in G 6.6 (cont.)
Tonic and auxiliary keys of B Minor: b D e fs G A
Tonic and auxiliary keys of A Major: b cs D E fs A
Händel’s key choices: b cs D e fs A
Wolf ’s commentary reads as follows: “So wie es hier steht, geht es mittelst des Quartensprungs im Basse zuerst in h moll; beym dritten Viertheil im ersten Takte macht es mittelst des kleinen Septimenenakkords auf der Dominante von fis eine durchgehende Kadenz, oder einen Gang in fis moll; im zweyten Takte, vermöge des kleinen Septimenakkords auf der Dominante von e, eine unterbrochene und durchgehende Kadenz in e moll; im dritten Takte mittelst des
Septimenakkords auf der Dominante von d, eine durchgehende Kadenz in d dur, und alsdann bis zum vierten Takte, eine halbe – oder phrygische Kadenz in die Dominante von h moll; (diese Kadenz wird sonst auch eine plagalische Kadenz genennet;) im fünften und sechsten Takte, mittelst der Quartensprünge, eine durchgehende Kadenz in h moll, eine in a dur, und im siebenden Takte, mittelst eines Sextensprungs und eines Septimenfalles im Basse, eine Ausweichung in die Dominante von fis, die sich vermöge des kleinen Septimenakkords in die zwote Verwechselung des Dreyklangs von fis moll auflöst; im achten Takte ergreift der Baß den Unterhalbenton von cis, und modulirt durchgehend in cis moll; im neunten Takte eine Fortschreitung in die Dominante von cis, und schließt im eilften Takte in cis moll.”
Minor15even when the resolution is to G-Bn-D (which, in any event, he ana-lyzes in neither G Major nor G Minor) [6.8]. And a surprising tierce de Picardie is rendered humdrum by his shift to C Major before even the preceding dominant. Jadassohn seems blithely indifferent concerning the progression of two consecutive diminished seventh chords and the appar-ent unconvappar-entionality of the dissonance treatmappar-ent.16The progression sits at the boundary between what he regarded as cogent and what was considered to be beyond concrete appraisal from a harmonic perspective. Concerning a longer progression of diminished sevenths, Weber comments: “If more-over a really equivocal chord be followed by still others which are themselves also equivocal . . . the ear must at last entirely lose the thread of modulations so very complicated and can really no longer know where it should be, but is obligated as it were to fluctuate hither and thither between several keys to which the different harmonies occurring might belong.”17
6.8 Jadassohn: Die Kunst zu moduliren und zu präludiren (1890), p. 175.
Compare Jadassohn’s treatment of two successive diminished seventh chords here with that of Momigny [6.11, mm. 12–13]. Though Momigny does not analyze the harmonic progression, he marks the chromatic elements (in the context of C Major) with filled-in noteheads and asserts that the progression occurs entirely within the key of C Major. Jadassohn instead indicates a diatonic context for each chord, neglecting to provide a rationale for the succession from one chord to the next.
6.7 Weber: Versuch einer geordneten Theorie der Tonse[t]zkunst (1817–21, 31830–32), vol. 2, p. 193 [Warner, p. 411].
Weber indicates interval quality by the left/right positioning of a bullet beside an Arabic numeral:
minor/major for imperfect intervals, perfect/augmented for fourths, and diminished/perfect for fifths. Chord quality is indicated by the size of the Roman numeral.