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Musical examples illustrative of unequal temperament derived from Part I of »Das wohl temperirte Clavier«, played on a modern piano

(SECOND PART OF THE TEXT)

Before going on to the next examples, let us take note again of a feature which the first three of the preludes that we have discussed have in common when performed in the kind of unequal temperament at issue. In the first few bars of each of those preludes, the pitch-class intervals are the ones represented in the right halves of the following diagrams:





— and thus the major 3rd in the Subdominant triad is, in each of those preludes, tempered a little less than is the one in the Tonic triad, which is tempered a little less than the one in the Dominant triad, which in turn is tempered a little less than the one in the Dominant of the Dominant. These features are notable also in the E-major prelude, as we will hear presently.

All this will change, however, when we then come to some preludes with a few flats in their key-signatures and thus involving mainly the intervals represented in the left half of those diagrams. In such keys the major 3rd in the Subdominant triad is tempered a little more than in the one in the Tonic triad, which is tempered a little more than the one in the Dominant triad, which in turn is tempered a little more than the one in the Dominant of the Dominant. It is especially this latter fact (in effect: that the leading tone to the Dominant sounds slightly "lazy" or "cautious" after one has heard the leading tone to the Tonic) which seems paradoxical, since modulating from the Tonic to the Dominant, if done in a straightforward way, is inherently a moment of tonal "uplift". (And that kind of modulation is indeed done in a very straightforward way in Bars 3–5 of the G-major prelude and in the second half of Bar 3 in the A-major prelude.) We will soon see how this paradox is manifested in the A♭-major prelude, and also how the composer has evidently exploited the fact that in A♭- and B♭, the Dominant triad is tempered a bit less than the Tonic, but the Subdominant more.

But first let us consider the major-mode prelude with four sharps in its key-signature.


The E-major prelude

This piece sounds quite good in equal temperament:

Track 48: The first 3 1/2 bar, played in equal temperament.





In equal temperament the qualities of the intervals would be exactly the same melodically if the music were played in F- or F♯-major as in E-major, and virtually the same with regard to the beating of the consonant intervals:

Track 49: In equal temperament, the first 1/2-bar played in E, then transposed up a semitone to F, then transposed up another semitone to F♯.





In the unequal temperament, G♯ is tuned nearly the same as in equal temperament, then D♯ slightly higher (i.e. in relation to B and E; I think this begins to become perceptible in the last beat of Bar 2), then A♯ yet higher (in relation to F♯ and B), and finally E♯ yet higher (vis a vis C♯ and F♯); but the potentially noxious effect of this remarkably high E♯ is avoided by not having, in Bars 10–12, an acoustically prominent C♯-major triad:

Track 50: The first 12 1/2 bars.





This series of successively more incisive sharps is a virtue, the value of which is then increased by the "tapering off" of incisiveness in Bars 13–14 when the harmony reverts (as it were) to D♯ and then G♯, and then, best of all, to the deliciously moderate (in this context) C♯ in Bars 15-17:

Track 51: From Bar 11 through the first half of Bar 17.





Notice how sweet it is then to regain in Bar 17 the slightly more "edged" D♯ (i.e. slightly more so in relation to E and B than is G♯ in relation to A and E):

Track 52: From Bar 14 through the first half of Bar 19.





The entire quasi-Schubertian "recapitulation" (starting at Bar 15 with the statement in the Subdominant of the theme that had established the Tonic at the outset of the piece) thus gains in the unequal temperament some highlights of luminosity which enhance the inherent beauty of the harmony – including that of the plagal cadence re-evoking the gentle Subdominant in the next-to-last bar:

Track 53: Bars 12-24.





The beginning of the B♭-major prelude

The main triads in the first phrase are those on B♭, G, E♭ and F:

Track 54: The first 2 1/2 bars (plus an extra note below the range of the piece).





Upon hearing them as block chords, one can readily observe that the trochaic macrorhythm is enhanced by the fact that whereas the tempering of the pitch-class interval between B♭ and D is more exposed in the major 10th in the first half of Bar 1 than in the minor 6th in the second half, the E♭ triad in Bar 2 is tempered altogether more, and then the F-chord less:

Track 55: The first five main chords of the piece.





The performer can use nicely this aspect of the tuning. The "extra" bit of tempering in the E♭ triad adds a slight dose of emphasis at Bar 2 which helps (a) to keep the music driving forward and thus (b) to prevent the sequence from sounding merely academic:

Track 56: Again the first 2 1/2 bars (plus the extra low note).



(Whereas the Subdominant chord occurs here on a strong beat of the second bar, in the analogous series of chords heard in Track 6 the Subdominant chord was on the weak beat of the first bar. I would associate this difference with the fact that whereas this Subdominant triad in B♭-major is tempered more than the Tonic and Dominant, in G-major it's the other way around.)

If the music is transposed down a semitone, then the sonority of the Subdominant (D-major) triad doesn't provide that nice extra touch of forward drive:

Track 57: The same 2 1/2 bars (plus the low note) transposed down a semitone to A-major.






Interlude: a theme by Beethoven in A

This example shows that (1) if a piece in A♭-major begins with a Tonic triad including baritone's low A♭, the beating of that bass-note with C, at 5.5 per second, is like an expressively fast vibrato and is nicely shy of intermittence, and then (2) if there is soon afterwards a straightforwardly lyrical use of the leading-tone to the Dominant, then that note, D, tuned as it is lower in relation to B♭ and E♭ than C is tuned in relation to A♭ and D♭, will lead up to the E♭ in a modestly quasi-rallentando way rather than in a notably striving way:

Track 58: The first 8 bars of the second movement of Beethoven's Opus 13 piano sonata (the "Pathetique").





(More examples in regard to this piece are to be found in Tracks 23–29 of the CD accompanying the book, Beethoven's Opus 34. Genesis, Structure, Performance, published for the Institute by Schott in 2007.)


The A♭-major prelude

The nuances of the unequal temperament are helpful throughout this piece in one way and another.

The intermittent beating (at 11 per second) of A♭-C in the initial triad is like a slap in the face at the outset. That such was Bach's intention seems to me to be implied by the fact that the chord consists of quarter-notes rather than half-notes as one would ordinarily expect for the rhythm in question; it would have been easier and more natural to write in each bar a half-note chord followed by a quarter-note rest, rather than a quarter-note chord followed by two rests. (In the following track, notice also the unusual open-5th sonority at the beginning of Bar 5. I will refer back to this later.):

Track 59: The first 5 bars.





This piece is a quasi-concerto movement conceived for keyboard. But whereas Bach's "Italian Concerto" for harpsichord is in F-major and begins with a thumping big chord in the bass, in this prelude an analogous effect is due to the "slapping" quality – due in turn to the tempered tuning – of the smaller initial A♭-major chord. This fact by itself may provide sufficient explanation for why Bach wrote such piece in the key of A♭ (which he would never do in a real concerto, as it would be an unduly awkward key for the orchestral players) rather than, say, in G-major. Compare the sound of the left hand's first two chords plus an additional A♭-major chord (to make a complete mini-progression) with the sound of the same three chords transposed down a semitone to G-major:

Track 60: Three chords in A♭-major, then transposed down to G, then in A♭ again.





As in some other preludes discussed previously, the nuances of the temperament help project a trochaic macro-rhythm in the first few bars. Since the leading-tone, G, is tuned a little less high in relation to E♭ and A♭ than is C in relation to A♭ and D♭, the V-chords in Bars 2 and 4 sound a little less virile than the Tonic chords in Bars 1, 3 and 5. (This difference supports the macrorhythm implied contrapuntally as the G's in Bars 2 and 4 are mere neighbor-notes to the A♭'s in Bars 1, 3 and 5.) If the music were transposed up a semitone to A-major, the reverse would become the case: the V-chords would be tempered a little more saliently than the Tonic triads. The overall amount of "spicing" might be adequate in the A-major transposition, but to have relatively more of it in Bars 2 and 4 than in Bars 1, 3 and 5 is contrary to the trochaic macro-rhythm which can be discerned analytically at the beginning of the piece and which ought, in my opinion, to be conveyed by the performer. It is graceful.

Track 61: The first 5 1/2 bars transposed up a semitone to A-major, then played in the originally intended key of A♭.





Notice that the trochaic pattern is achieved by using Tonic and Dominant chords only. The Subdominant D♭ chord, the most heavily tempered major triad in this kind of tuning, is held in reserve, as it were, for Bars 32 and 43. It seems to me that the complexity of the phrase-structure in this piece is due in part to the fact that if four-bar phrases were as prevalent here as in some of the other preludes, the backing-&-forthing between Tonic and Dominant chords would become insipid.

There is an additional possible reason for Bach to have written with quarter-notes the chords in Bars 1 and 2: to allow the right hand's notes on the second beat of each of those bars to be heard more saliently than if each chord was held over into the second beat, and thus to allow the stepwise ascent in the tune in Bars 1–3 from A♭ to B♭ to C to be heard clearly. (The melodic need for the C at Bar 3 explains why the A♭-major triad there is indispensable to the musical design, unlike at the beginning of Bar 5.) However, this second possible reason doesn't render the other one less likely. People often do things for two (or more) reasons at once.

Observe also that, as we have heard in a very different aesthetic context in the theme by Beethoven, the D♮ in Bars 9–10 makes a rather "laid back" leading-tone to the Dominant:

Track 62: The first 10 1/4 bars plus a dyad.





Before going on to the next examples, let me comment on some of the lineaments of the harmonic counterpoint in the first half of the piece:       





In the top line, the high A♭ which is arrived at in Bars 5–6 is not pulled down to G by the B♭ in the harmony in Bar 7. (One of Bach's pupils, Kirnberger, in his copy of the piece evidently changed the high A♭ in Bar 7 to a G. The logic for that "correction" is obvious, but I think it weakens the music.) Instead, the contrapuntal obligation to resolve down to G is deputized to the lower A♭ at the end of Bar 7, whereas the high A♭ is succeeded in Bar 8 by a B♭ which initiates a line going down stepwise to the E♭ in Bar 13 (and ultimately 18). It is during the descending sequence in Bars 9–12 that the modulation to the Dominant is signaled by the uses of D♮ instead of D♭; so, there is no need for the tuning to make salient the fact that this new leading tone will presently lead upwards to the Dominant; it can perfectly well be slightly "lazy," and indeed this aspect of it fits well the immediate context of the descending sequence. But then there is an ascending sequence in Bars 13–15. The Bach-style tuning helps make salient the fact that the pivot between the two sequences is the dyad at the beginning of Bar 13, as that dyad is one of the most strongly tempered major 6ths (here a compund major 6th) in this kind of unequal temperament:

Track 63: From Bar 6 to the beginning of Bar 19.





(In my opinion, the only major 6th or minor 3rd which should be tempered more than one between F and A♭ is the one between D♭ and B♭, and indeed it is this latter kind of interval which will at Bar 30 serve the same kind of pivotal function in the second half of the piece, i.e. in the analogous modulation from the briefly evoked Subdominant at Bar 26 to the Tonic at Bar 35.)

When the music is transposed up a semitone, then in Bar 9 the leading-tone (D♯) to the Dominant sounds a little more edgy than does C♯ in the Tonic triad (because D♯ is tempered more vis à vis B than is C♯ vis a vis A), and then at Bar 13 the pivotal major 6th, A-F♯, sounds banal because it is tempered in an inadequately incisive way for its function at that point in the music:

Track 64: Bars 6–14 transposed up a semitone.





Here is the same comparison in reverse, i.e. first the transposed version and then the notes that Bach actually wrote:

Track 65: Bars 6–14 transposed up a semitone and then Bars 6-22 not transposed.







Notice that whereas the rather unusual use of an open-5th sonority on the downbeat at Bar 5 had avoided a repetition of the "shocking" effect of the initial downbeat-A♭-major triad, the analogous sonority at Bar 20 is a complete E♭-major triad.

Before going on to a few examples from the end of this piece, let us consider the following description of its overall tonal design:       


This description shows (among other things) that the Subdominant triad in the next-to-last bar is important in the overall tonal design of the piece.

In Bars 39–42 the trochaic macrorhythm of the first few bars is evoked, and meanwhile the flurry of constant 16th-notes for both hands in Bars 39–40 helps set a context of vigor for the very heavily tempered - and therefore itself suitably vigorous - D♭-major triad in the next-to-last bar:

Track 66: The last 6 bars.





When this material is transposed up a semitone to A-major, the Subdominant triad becomes slightly stodgy and "academic" – not the right kind of sound for this moment in this piece:

Track 67: The last 4 bars transposed up to A-major.





Here is the comparison in reverse:

Track 68: The last two bars transposed up a semitone and then not transposed.






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