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


Please note, in the following reading of the whole A♭-major prelude, that (a) the D♮'s in Bars 9 and 11 sound "gentler" (because of their modest tempering) than the dyad at the beginning of Bar 13, and the modest tempering of the D♮ in Bar 19 helps give to that bar a macro-rhythmically weak character vis a vis Bars 18 and 20; (b) in Bar 22 the low intonation of the D♭ helps give a sense of impetus to the modulation back from E♭- to A♭-major (and that impetus is beneficial here in the middle of the piece as the harmony is about to move on into Subdominant territory), and then (c) the low intonation of the D♭'s in Bars 26–33 help to project a trochaic macro-rhythm whereby Bars 26, 28, 30 and 32 are stronger than 27, 29, 31 and 33.

Track 69: The whole prelude.

I have dwelt upon this piece because my ideas about interpreting it in performance are rather different from those of Hans Bischoff (in his invaluable edition of the '48') and of many other pianists who have played it only in Equal Temperarment and never in the style of tuning which I think Bach in the early 1720s liked more than he did Equal Temperament. I agree that the tempo should be Allegretto or moderately Allegro, but not that the overall character should be dolce. I think that most of the notes should be played sciolto, and that the tradition of playing the piece legato and calm is due to the fact that pianists have been ignorant of the nuances of temperament discussed here and have been influenced instead by the general history of A♭-major as a key hosting tender rather than vigorous affects. Bach would indeed readily compose tender pieces in E♭-major (some illustrations of such tender music in a Bach-style unequal temperament are available in the Webpage of examples of organ music), but we have seen also that for the major key a semitone above that of C♮, he favored a vigorously frolicsome rather than a tender prelude and he wrote it in C♯- rather than in D♭-major. One reason may perhaps have been that he felt that the "prickly" look of sharps and double-sharps would be better suited to the character of the piece than the look of flats. In any case he was willing to have for that piece a key signature of seven sharps rather than of five flats. However, to notate the A♭-major prelude (and/or the fugue) in G♯-major instead of A♭ would have been out of the question as it would have entailed having a double-sharp in the key-signature; so, I think one shouldn't be deterred by the sight of those flat-signs from adopting the more interesting, virile interpretation – especially since it is supported by the nuances of an historically correct style of tuning – rather than the less interesting dolce interpretation.

Again: I do not claim that my playing is extremely good (I am not a professional performer), but only that my ideas as explained here are well founded.

An ancillary reason for discussing this piece and the E-major prelude is that one of the main differences between Bradley Lehman's scheme and my concept of Bach-style unequal temperament is that he tempers A♭-C significantly less than E-G♯ whereas I think it should be the other way around. The musical value and interest of the heavily tempered A♭'s seem to me quite clear in this piece and very clear in some of the examples in the Webpage of organ music.

Let us consider now some of the preludes in minor keys.

The C-minor prelude

Most pianists treat this as an etude in fleeting finger-technique. The aesthetic effect of a mechanical rendering in equal temperament is akin to that of an insensitive pianist's performance in the nuanced temperament.

Track 70: Bars 1–5 rendered in equal temperament by a midi-synthesizer.

Track 71: Bars 1–5 played fast (by a pianist) in the unequal temperament.

However, in the unequal temperament the dark E♭'s and A♭'s and the rather dull leading-tone (B) support the view that Glenn Gould's and Wanda Landowska's concepts of how to play this piece were correct. It is indeed a toccata, but it should not be played veloce at the outset; it should start big and heavy (like Bach himself), and probably not faster than this:

Track 72: Bars 1–6 played rather slow.

When this same material is heard transposed up (in the unequal temperament) to D-minor, the third and sixth degrees of the scale and the leading-tone sound inherently less expressive and more nondescript and are therefore less able to sustain interest at a slow tempo. The following track stops at the beginning of Bar 5 in order to bring out how unexpressive (and thus inadequate to the moment) is the right hand's high F there:

Track 73: The first 5 bars plus a dyad, transposed up a whole-tone to D-minor.

You may recall the "curiously insipid effect" and the "insufficiently aggressive" sound of the C-major triads at the end of transposed versions of the B- and C♯-major preludes. Those unsatisfactory results were due to the mildly vibrato-like quality of the of the major 3rd C-E beating at 3 per second. At the end of this prelude, however, the same kind of beating in the major 10th between tenor's low C and middle E is appropriately expressive:

Track 74: The last 1 1/2 bars.

When this material is transposed up a whole-tone to D-minor, the beating of the final F♯ at about more than 4 per second, though by no means ugly, seems nonetheless ill-at-ease when compared to that of C-E in the untransposed version. Notice also how in Bar 36 the D♭ in the music as Bach composed it sounds inherently more expressive than the corresponding E♭ in the transposition; this is of course related to the fact that D♭-C is a slightly smaller semitone than E♭-D:

Track 75: The last 4 bars transposed up a whole-tone, then untransposed.

The D-minor prelude

This piece begins in an almost happy, "flowing" mood (and later builds up to a strong conclusion):

Track 76: The first 1 1/4 bars.

When this beginning is transposed down a whole-tone to C-minor, the resulting E♭ and A♭ sound inherently darker than their counterparts (F and B♭) in D-minor, since E♭ and A♭ are tuned lower in relation to C and G than are F and B♭ in relation to D and A:

Track 77: The first 1 1/4 bars transposed down a whole-tone to C-minor.

I think the transposed version would be more suitably performed in a more sombre tempo:

Track 78: The same transposed 1 1/4 bars played slower.

In the untransposed version, the mood of happy flowing is then confirmed by dwelling on the relative major key for several bars, and is helped by the mild tempering of the F-major triad:

Track 79: From the beginning until the first part of Bar 6.

When the music is transposed down a whole-tone, the relative major sounds less euphonious because the beating of middle G with middle E♭ is intermittent (at more than 7 per second):

Track 80: From the second half of Bar 1 to the beginning of Bar 3, transposed down a whole-tone.

Listen again to the last part of that transposed version:

Track 81: The last beat of Bar 2 and first beat of Bar 3, transposed down a whole-tone.

The transition from the fairly calm mood of the beginning of the piece to a more extraverted one at the end of it begins with the modulations in Bars 6–9. F♯ as the leading-tone to G is introduced in Bars 6–7 (along with E♭ as the sixth degree of G-minor) and then in Bars 8–9 G♯ as the leading-tone to A - and this new leading-note is more incisive as such than F♯ has been in relation to G, partly because G♯-A is a smaller semitone than F♯-G (though partly also, of course, because of the stepwise-ascending sequence):

Track 82: From Bar 6 to the beginning of Bar 10.

The amount of tempering in the D-major chord at the end of the piece is enough to sound suitably "alive" (not too pure), but not so much as to sound nervous or shrill:

Track 83: The last 2 1/2 bars (plus three notes before).

When the same material is transposed down a whole-tone, the new last chord sounds curiously insipid because the tempering of the major 3rd C-E is too tepid for this particular context:

Track 84: The last 2 1/2 bars (plus the three preceding notes) transposed down a whole-tone.

Here is the comparison more succinctly:

Track 85: The last four chords and then the same chords transposed down a whole-tone.

The F-minor prelude

This key is notable, in terms of tuning nuances, for the extent to which the Tonic triad is heavily tempered, the leading tone is droopingly lazy, and the semitone between the fifth and sixth degrees of the scale is remarkably small. These nuances are (in my opinion) largely accountable for the fact that F-minor has been traditionally described as "melancholic":

Track 86: The first 3 bars plus a dyad.

If this piece were transposed down a semitone to E-minor, the resulting high leading-tone might be considered expressive in a certain edgy way, but still there would be net loss of expressivity already in Bar 1 because of the very moderate tempering of the Tonic triad (and then later also because of the large semitone between the fifth and sixth degrees of the scale):

Track 87: Bar 1 transposed down a semitone to E-minor.

Looking ahead now: F-G♮ as a melodic whole-step is featured, thematically, not only in the first half of Bar 1 but also again in the second half Bar 2, and therefore the melodic semitone F-G♭ in Bar 4 will be even more "surprising" than the C-D♭ in Bar 3. The surprise would be there in any reasonably presentable tuning, but is especially pleasing in this kind of temperament because of the remarkably small size of F-G♭ and because the small semitone sounds all the more sneaky down in the bass.

And then when the harmony finds its way at Bar 6 to the relative major – A♭-major – the bass line walks up an octave (in Bars 6–8), and here the lavish tempering of the 3rds in the Tonic and Subdominant triads can be exploited to project a luminosity complementing the pathos of the sections in F-minor between which this section is nested:

Track 88: From the middle of Bar 2 to the last beat of Bar 12.

It would be something of a waste to perform such music academically "straight":

Track 89: The first 2 1/4 bars played faster.

(Some pianists play Chopin with as much rubato as their sense of the music prompts them to make, but feel that avoiding rubato (except at the end of a movement) is essential to achieving "Bach style." However, since the harpsichord and organ provide far less opportunity for dynamic nuance than the piano does, intelligent use of rubato is an important expressive device for performers of instrumental compositions on those instruments.)

The beginning of the B♭-minor prelude

In this key the Tonic and Subdominant triads are tempered even more darkly than their counterparts in F-minor; and, the leading-tone, though not quite as droopy as the one in F-minor, is still notably lazy. These qualities can be well exploited in the opening bars of this prelude:

Track 90:

Even if the leading-tone is played louder halfway through Bar 2, it will still droop. And then halfway through Bar 4 the piercing smallness of the left hand's harmonic semitone F-G♭ seems to help impel the tune up to high C (the top note on keyboard instruments used routinely by Bach):

Track 91: To the beginning of Bar 5.

When this material is transposed down a semitone to A-minor, the resulting C's and F's are less expressive than the original D♭'s and G♭'s:

Track 92: Those same bars transposed down to A-minor.

The greater poignancy of the chord halfway through Bar 4 in the original key than in the transposition is due to a combination of facts: (1) F-G♭ is some 15% smaller than E-F, and (2) in the A-minor version, F-A is tempered by only about 2/3 as much as is E-G♯, whereas (3) in the B♭-minor version, G♭-B♭ is tempered by nearly twice as much as is F-A:

Track 93: A pair of chords, played four times: first in the transposition down to A-minor, then in B♭-minor, and then again in each of those keys.

The A-minor prelude

This is not a sad piece. The right hand's first three notes could make a sad motif, but should not be so interpreted. A correct (i.e. not sad) interpretation is served by the fact that the third degree of the A-minor scale is tempered very moderately (i.e. not very low). Also well suited to the context of this theme are (1) the somewhat but not very high leading note – high enough to be strong, but not so high as to be tenderly "yearning," and (2) the rather high and therefore not pathos-laden F♮ (which in the tune in Bar 3 does not go down to E but directly up to the leading-tone):

Track 94: The first nearly 3 1/2 bars.

When transposing the theme up a semitone up to B♭-minor, I sense a different mood already with the D♭ in Bar 1 (morose-sounding, because it is tempered so low); and this feeling is sustained by the pathetically "lazy" sound of the right hand's A's in Bar 3.

Track 95: The same material transposed up a semitone.

In Bars 5–8 the fact that D♯-E is a smaller semitone than G♯-A, and then A♯-B yet smaller, serves to help give the key of E-minor an "icy" but not sad aura, appropriate to the rhetorical context. Then, after a "tapering off" (from such acutely tuned semitones) in the descending sequence with its F♯-G and C♯-D in Bar 9 and E-F in Bar 10, the music begins in Bar 11 to revel for six bars in the warmth of the moderately tempered key of C-major (the relative major of A-minor):

Track 96: Bars 1–13.

The warmth is due is part to the fact that C-major has been approached from E-minor and by a descending sequence (rather than by a sequence ascending from A-minor). But still, the substantial role (in creating this effect of warmth in C-major) of the tuning nuances can be demonstrated by hearing the modulation from E- minor to C-major performed without those nuances:

Track 97: Bars 9–14 in equal temperament, then Bars 7–16 in the unequal temperament.

If the modulation to C-major is transposed up a semitone so that it becomes a modulation to D♭-major instead, here is how it sounds:

Track 98: Bars 9–11 and the first dyad of Bar 12, transposed up a semitone.

The section in C begins and ends with alto's high E in the top line (at Bars 11 and 16), and within that section the right hand's statement of the I-IV-V-I thematic material starts (at Bar 13) with middle E. This pitch-class is not only inherently the most "tuneful" member of the C-minor triad but also the overarching link between the intermediate key of C-major and main key of A-minor. However, the harmonic path back to A-minor involves in Bars 17–22 a succession of three diminished-7th chords, each with a leading-note that is slightly more incisive than the immediately preceding one. (That is, F♯-G in Bars 17–18 is a smaller semitone than B-C in Bars 15–16; then C♯-D in Bars 19–20 is slightly smaller than F♯-G; and then G♯-A in Bars 21–23, and ultimately 27–28, yet smaller.)

Track 99: Bars 13–28.

The nuances of the tuning thus make the whole piece sound interesting and graceful.

As a kind of postscript I would like to add the following remarks (having nothing to do with nuances of tuning) putting into a broader analytical context my comment about F♯, C♯ and G♯ in Bars 17–21:

A feature of the right-hand part in Bars 1–4 is the stepwise ascent from E in Bar 1 to F in Bar 2, G♯ in Bar 3 and then A in Bar 4. Notice, by the way, that the same set of pitch-classes which make a Subdominant chord in Bar 2 make a Dominant chord in Bar 3 (with the left hand's A acting as a pedal-point). This is a witty exploitation the fact that one of the four steps up in the right-hand part is the augmented 2nd from F to G♯.

There is an ensuing stepwise descent in Bars 5–8, with the G♮ in Bar 5, the F♯'s in Bar 7, and the E in the last beat of Bar 8 – and another such descent in Bars 9–11, but this time with F♮ (rather than F♯) in Bar 10.

The E in Bar 11 recurs at Bar 16 still supported by C in the bass but now hinting implicitly – one might almost say "shouting implicitly" – that the music is about to return to A-minor. (A nice detail is that the right-hand part in Bar 15 is exactly the same as in Bar 2 except for the difference between G♯ and G♮; the harmony must now somehow restore G♯ as the leading-tone to A.) But the return to A-minor is attenuated until Bar 21. While the F♯ at Bar 17 serves as a leading-tone to the G at Bar 18, and then the ensuing C♯ as a leading-tone to the D at Bar 20, there is, in the right-hand part, a stepwise descent from the E at Bar 16 to the middle F a Bar 20: this is a matter of E♭ and D and the last of the three C's in Bar 10, and then B♭ and A and the last of the G's in Bar 19. The middle F thus reached the beginning of Bar 20 is then replicated an octave higher (and thus a step up from the E in Bar 16) on the last beat of that bar, and then the ensuing G♯ evokes the ascending augmented 2nd of Bar 3.


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