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

The following examples are from 14 of the 24 preludes in Part I of Das wohl temperirte Clavier. Nearly all of them (the exceptions are indicated) were recorded on two Bösendorfer grand pianos brought to the Institute for a lecture on this topic in October 2009: (1) an "Imperial" model tuned in the following unequal temperament:

— and (2) a smaller model tuned in equal temperament. All the examples that are not tagged as being played in equal temperament are played in the unequal one.

Whenever I mention that a note in the unequal temperament is tuned relatively high or relatively low in relation to the note a major 3rd below it, I mean implicitly that it is also tuned relatively high or low to the note a major 6th below.

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The G-major prelude

Let us begin (as in the musical part of the lecture) with some very subtle distinctions. The leading-tone, F♯, is tuned slightly higher in relation to D and G than is B in relation to G and C, but meanwhile E, in the second half of Bar 1, is tuned slightly lower vis a vis C; so, when F♯ is heard in Bar 2 it sounds slightly and pleasantly "salted":

Track 01: Bars 1-2.

Let us hear these nuances in block chords:

Track 02: Four chords.

When we go on to the next few bars, the fact that C♯, the leading tone to the Dominant, is tuned slightly higher in relation to A and D than is F♯ in relation to D and G is significant. Such a C♯ is well suited to the music because it augments nicely the sense of "lift" in the harmony at Bar 3:

Track 03: Six chords.

Track 04: Bars 1-4 (plus a dyad).

When I transpose down a whole-tone to F-major, the corresponding nuances differ; the leading-tone, E, sounds dull (vis a vis the way F♯ sounded when the music was played in the key that Bach composed it in), and then B, the leading-tone to the Dominant of F, is tuned no higher in relation to G than was A in relation to F:

Track 05: The first 4 bars (plus a dyad) transposed down to F-major.

Here is the comparison in terms of block chords – and notice how the first set of chords implies a suitably trochaic macrorhythm (i.e. with the first half of each bar bearing a more stress than the second half) better than does the second set (i.e. the set transposed down a whole-tone):

Track 06: Two sets of six chords.

If you agree that it is a virtue for the F♯ in Bar 2 to be "spiced" subtly more than the B and E in Bar 1, and then for the C♯ in Bar 3 to be spiced again slightly more (i.e. more than F♯), then you may also find it felicitous (as I do) for G♯ in Bar 6 to be spiced a bit more, and D♯ in Bars 8-9 yet more – and then to have a tapering off of these effects as the music reverts to C♯ in Bar 10 and F♯ in Bar 11:

Track 07: The first 12 1/4 bars.

This aspect of the way in which the nuances of the tuning are related to the harmonic structure of the piece is reflected in the fact that nearly all the semitones and major 3rds in the music are represented in the right halves of the following diagrams:

My rubatos etc. may have been exaggerated, and maybe the piece should be played piu mosso; but still I think that to play it truly veloce in this kind of temperament is to waste the nuances of what I consider to have been Bach's preferred style of harpsichord tuning when he composed it:

Track 08: The first five bars or so, played veloce.

Such playing misconstrues the genre. This was originally intimate music, played by Bach for only one or two students at a time. The music at this fast tempo is just as comely in equal temperament as in the unequal one:

Track 09: The first five bars or so rendered in equal temperament by a midi-synthesizer.

Please go back now and listen again to Tracks 1–7 before we go on to consider whether, since at a moderate tempo the piece sounds a little better in G-major than in F, it might sound even better in A than in G.

In the tracks comparing between the music as composed in G and as transposed down to F, the interesting differences began to occur in Bars 2 and 3. There was no difference of nuance within the first half of Bar 1:

Track 10: The first half-bar in G, and then transposed down to F; and then the first bar in G.

But now let us add A-major to the comparison:

Track 11: The first half-bar transposed down a whole-tone to F, then in the original key of G, then transposed up a whole-tone to A, then again in the original key of G; then the first 4 bars gain transposed up to A, and then the first 7 bars in G.

In the A-major transposition, C♯ sounds nervous in the first bar of this piece as it beats intermittently (distinctly more than six times per second) with A and E.

(My statements as to how fast certain intervals are beating are based on assuming that concert A is at 440HZ. If it were a semitone lower, then the beats for each interval would be about 6% slower.)

The average of the amounts of tempering for all the major 3rds has to be the equal-temperament amount (7 schismas), and the amount for A-C♯ should be just a little less than that average. (In the diagram near the beginning of this Webpage the amount assigned to it is slightly more than 6 schismas.) However, in equal temperament the analogous amounts would be identical (at 7 schismas) for A-C♯, G-B and F-A, and therefore the initial chord of this piece would have nearly the same nervously intermittent quality regardless of whether it were played in G-, A- or even F-major:

Track 12: In equal temperament, the first half-bar transposed up to A, then in G, then transposed down to F.

— whereas in the unequal temperament, B in the initial G-major chord is easy-going, and then the successively more sprightly qualities of F♯ in Bar 2 and C♯ in Bar 3 are a blessing:

Track 13: Bars 1-4 plus a dyad.

Yet I would not say that given equal temperament, one might just as well render the music as in Track 8. First-class dynamics and timing will make their good effect even if the style of the tuning is insipid in relation to the music.

The A-major prelude

The same C♯ that sounded uncomfortably nervous in the first bar of the G-major prelude when it was transposed up a whole-tone sounds perfectly OK in this inherently more sciolto kind of music. Here is the very beginning:

Track 14: From Bar 1.

Notice also how appropriate to this piece are the nuanced temperings of G♯, tuned as it is a little higher vis a vis E than it would be in equal temperament, and of D♯, which is distinctly higher than it would be in equal temperament:

Track 15: The first 5 1/2 bars plus a dyad.

When the music is transposed down a whole-tone to G, the resulting B in Bar 1 sounds slightly tepid (as if the player were dull!):

Track 16: The beginning in A, then transposed down to G, then again in A.

And the same is true (when the music is transposed thus a whole-tone down) of the F♯ in Bar 2:

Track 17: The first 2 bars (plus a dyad) transposed down a whole-tone to G.

The sprightliness of the corresponding G♯ in the original key is more appropriate to the context:

Track 18: The first 5 1/2 bars (plus a dyad).

If the piece sounds thus better in A-major than in G, would it sound even better in B than in A?

Track 19: The first 3 1/4 bars transposed up a whole tone to B-major.

No, the nuances in the original key are the right ones, with neither too much nor too little "spice."

The B-major prelude

The same D♯ and A♯ that sounded uncomfortably high in the A-major prelude when it was transposed up a whole-tone sound appropriately expressive in this quite tender music:

Track 20: The first 5 bars plus a dyad.

Not only the remarkably high tempering of A♯ but also the even higher tempering of E♯ is OK here. And, the player can, because of the rather high tempering of the D♯ at the outset, treat that note effectively as if it were an appoggiatura to the following E (instead of treating it as a stable consonant note and the E as a dull passing-note). This nice but "ungrammatical" treatment of the D♯ is aesthetically compatible with treating the A♯ at Bar 2 and the first three E♯'s as accented neighbor- or passing-notes:

Track 21: The same music.

The kind of playing which these nuances prompt one to adopt is appropriate also in equal temperament even though the result in that tuning is not quite as expressive:

Track 22: The same five bars in equal temperament.

When the music is transposed up a semitone to C-major, the appoggiatura at Bar 2 loses some of its edge (in the unequal temperament):

Track 23: The first 1 1/2 bars (plus a note) in the original key of B, and then the first 2 bars transposed up to C.

And, the Dominant at Bar 5 sounds almost ecclesiastical when the piece is transposed up to C-major, but properly expressive when the notes are those which Bach composed:

Track 24: Bars 1–5 (plus a dyad) transposed up a semitone, and then again but not transposed.

The last chord of the piece is in an unusual disposition. If it had included middle D♯ as well as the B just below middle C, then the beating of that major 3rd would have been intermittent (at nearly 11 per second), whereas the beating of the major 3rd an octave lower, at about 5 1/2 per second, is just shy of intermittence and is thus more like a lively vibrato:

Track 25: The last 3/4 bar, featuring the major 3rd between tenor's low B and the D♯ a major 3rd above.

Observe the difference between that nicely expressive beating (i.e. nice for the last chord of this piece) and the more laid-back beating, at 3 per second, of the major 3rd a semitone higher, C-E:

Track 26: The major 3rd between tenor's low C and E, and then the one a semitone lower.

In large part because of this difference (yet also because the leading-tone is more expressive in B- than in C-major) it seems to me that the ending of the piece is more satisfying in B-major than when it is transposed up a semitone to C:

Track 27: The last 2 bars in B and then transposed up a semitone to C.

In this musical context, that particular chord in C-major makes a curiously insipid effect, doesn't it? Here is the same comparison the other way around:

Track 28: The last two bars transposed up to C-major, and then in the original key of B-major.

The C-major prelude

The remarkably moderate (but not altogether dull) tempering of C-E is just right for the beginning of this piece:

Track 29: The first 1 1/4 bars.

In this context of moderate tempering, the "slightly salted" quality of F♯ is a benefit in bars 6 and 10:

Track 30: The chords that are (in Bach's composition) arpeggiated in Bars 1-11.

Here is the first of those chords and then the last four of them, played in equal temperament:

Track 31: In equal temperament, the chords which are arpeggiated in Bars 1 and 8-11.

In the last of those chords, the beating in equal temperament of the major 10th, G-B, at about 7 3/4 per second makes an intermittent effect, whereas in the unequal temperament the effect of the beating at less than 5 1/2 per second is not noxious:

Track 32: The same chords in the unequal temperament.

Also rather stark is the comparison between C- and D♭-major in the unequal temperament. Here it is for the beginning of the piece:

Track 33: The chords in C-major that are arpeggiated in Bars 1-4, and then the same chords transposed up a semitone to D♭-major.

And here it is for the end:

Track 34: The last two bars in C-major and then transposed up to D♭.

The difference between melodically sweet and sour major 3rds is perhaps even more salient here than the different qualities of their beating as harmonic intervals.

The C♯-major prelude

The same E♯ and B♯ that were too piquant in the previous track are OK in this sprightly piece:

Track 35: The first 14 bars.

The extreme sharps sound more appropriate in this music than do the corresponding naturals when the music is transposed down a semitone to C-major:

Track 36: The first 6 bars plus a dyad in C♯, then the same material transposed down a semitone to C, and then the first 13 bars in C♯.

In the moderately tempered key of D-major, the theme could perfectly well be simplified to the following melodically straightforward version:

Track 37: The first 6 bars (plus one dyad) of a more natural-sounding version (not used by Bach) in D-major of the same piece.

In a preliminary draft of the piece he was even more cautious than in the final version about the use of E# in the first bar:

Track 38: The first 6 bars (plus one dyad) of an earlier draft.

The piquant effect of the E♯ was here minimized at the outset, but at the cost of making the right-hand part sound, in my opinion, a little too "artificial" melodically for a piece of this overall character.

When played in equal temperament, the first bar sounds equally OK (though not remarkably scintillating) in C-, C♯- or D-major:

Track 39: In equal temperament, Bar 1 transposed down to C, then played in C♯, then transposed up to D, then again played in C♯, and then again transposed down to C.

The last chord of this piece is notable in somewhat the same way as that of the B-major prelude, inasmuch as the only major 3rd is in the low-tenor range. On the harpsichord, full chords mean forte playing. Here at the end of the C♯-major prelude, the thickness of the chords is complemented by the intermittent beating (at 8 per second) of the major 3rd:

Track 40: The last 8 bars.

If these bars are transposed down a semitone, the concluding C-major triad seems banal to me (mainly because the major 3rd, beating 3 times per second, sounds insufficiently aggressive for this particular musical context):

Track 41: The last 8 bars transposed down a semitone to C-major.

Here is the comparison more succinctly put:

Track 42: The last 1b bars, then the same 1b bars transposed down a semitone to C, and then heard again in C♯.

You may recall that I said, when discussing the G-major prelude, that the entire set of pieces in The well-tempered clavier belongs to an intimate genre and that Bach played them for just one or two students at a time. I would reconcile that premise with the robust quality of the music at the end of the C♯-major prelude (and indeed in many other places in the set) by saying that very often one genre will imitate or evoke another. (But of course the big chords at the end of this prelude shouldn't be played like, say, the ones at the beginning of the Grieg concerto.)

The F♯-major prelude

This is an unusual piece in that the right-hand part is full of syncopations and clever echappes and cambiatas. The nervous effect of the notably high tempering of A♯ and E♯ can be exploited to foster a sense of delicate frolic which the music is (in my opinion) obviously intended to convey:

Track 43: To the beginning of Bar 7.

The opening motif sounds inherently a little less scintillating when transposed down or up a semitone than when played in the intended key of F♯:

Track 44: The first 1/2-bar in the original key, then transposed down a semitone to F, then up a semitone to G, then played again in the original key (F♯).

Try as I might to play the first few bars expressively in F-major, some of the notes that ought to shimmer sound pale and tentative instead:

Track 45: Bars 1–6 (plus a note) transposed down a semitone.

The originally intended nuances of tuning give the notes the right qualities:

Track 46: Bars 1–10 plus a dyad.

If you find the rubatos etc. overdone in my playing, I readily admit it. The best I can do as a performer is to suggest where they should be made, not exactly how much.

The opportunities for them continue throughout the piece, but at the end it tapers off to a single eighth-note and thus avoids a "shimmering" last chord:

Track 47: The last 6 1/2 bars.

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