Now that I'm back at my home computer, library, and hpsi and clavi, let's
see if I can dot all the i's and cross all the t's of my argument. All
this past week (traveling on business) I've been writing from borrowed
computers and without the reference works.
Here's my train of thought on all this, summing up and adding some new
things. It's not entirely organized, but I don't think it's utter nonsense
FACT: In meantone temperaments, the three specific augmented seconds F-G#,
Bb-C#, and Eb-F# are usable. Particularly in 1/4 comma meantone, those
intervals are nearly pure! Not as normal 6:5 minor thirds, but as 7:6
minor thirds. This phenomenon allows composers to use the key of F minor
even if the keyboard is tuned with G#, C#, and F#.
FACT: In meantone temperaments with G#-C#-..-Bb-Eb, the crucial interval to
worry about is the "wolf" G#-Eb: a diminished sixth but looking like a
fifth on the keyboard.
FACT: The augmented sixths Bb-G# and Eb-C# are similarly special cases in
meantone, being close to pure minor 7ths (very flat to modern ears). G#
and C# function well as Ab and Db in such a harmonic context.
AXIOM: Bach grew up in an environment where meantone temperaments and their
variants were the norm. He became acquainted with well temperaments at
some time in his career after he had already formed his understanding of
how all the keys sound in meantone.
AXIOM: A keyboard composer/improviser in a meantone world will develop
habits tending to favor the extreme flat keys more than the extreme sharp
keys, given a choice. Why? Because in flat keys the dominant triad has a
major third that is not too sharp. (The keys of e minor, b minor, f#
minor, c# minor, and E major are all flawed in this way.)
OBSERVATION: Bach himself demonstrated all the usable keys in
meantone-flavored temperaments: his inventions and sinfonias cover exactly
that set. There are seven flat keys (c, d, Eb, F, f, g, Bb), C, a, and six
sharp keys (D, E, e, G, A, b).
AXIOM: Some chord spacings with the "wrong" enharmonic notes still work OK
in practice; for example, F-G#-C# as a Neapolitan in C major/minor, and
C-Eb-G# as a Neapolitan in G major/minor. The augmented fifth and
augmented second (or minor third) interact in a way that the ear does not
find unpleasant, even though the notes are "wrong." This is true even in a
regular meantone, no alteration.
FACT: All the regular meantone temperaments (1/3 comma, 2/7, 1/4, 2/9, 1/5,
1/6, ...) give the same effect, same set of keys, as long as the same set
of notes is chosen. The difference is only a matter of degree: stability
of the fifths, major thirds, and minor thirds. The narrower the fifth (up
to about 1/3), the stronger the perceived sense of key center, but the less
the notes can substitute enharmonically for one another.
AXIOM: Choice of which regular meantone temperament to use is a matter of
taste and preference, centuries ago and now. So is any insistence that the
temperament be regular. Few composers (for example, Ligeti) specify
exactly which keyboard temperament they want and demand explicitly that it
HYPOTHESIS: Most composers, players, improvisers, and tuners likely did not
count beats with mathematical precision (like Jorgensen); they simply
relied on experience and taste to set the fifths and thirds "as the ear
will bear" according to the music to be played. We have no assurance that
temperaments in practical use on clavichords, harpsichords, or organs were
ever mathematically perfect, or that mathematical perfection was even a
goal *in practice*...only for theorists. Practical musicians simply made
the instrument sound as good as possible within their tastes and
abilities. It was an expressive art. We have no assurance that such
musicians set absolutely regular temperaments. Such an assumption of
perfection says more about our modern scientific age than about practical
musicianship. I suspect every musician had his/her own temperaments, all
with a basic meantone flavor but altered deliberately or accidentally to
suit their tastes. It could have been different from day to day. This is
what I mean when I say "vernacular" temperaments: generated by some
sequence of tempered (or maybe some pure) fifths, to taste.
AXIOM: If one is going to start modifying a regular meantone temperament to
improve commonly needed keys, the most crucial notes to mess with are Eb,
Bb, and G# (those at the end of the temperament). Lowering those flats
improves the B major and F# major triads (dominants to E and B). Raising
the G# helps to tame the wolf, and improves the G#-C diminished fourth.
HYPOTHESIS: People did this fudging as regular practice, to taste and by
experience, according to the music to be played. It was not necessary to
write this down. They simply learned where to tweak things for desirable
results, particularly in the Eb and G#.
AXIOM: There are compositional tricks in chord spacing allowing even a
noticeable G#-Eb wolf to be dodged. For example: the notes can be struck
at different times, or the triad can be inverted, or the spacing of the
voices can be wide, or the fifth of a (notated) Ab major chord can be
HYPOTHESIS: John Bull was an example of a composer who knew those tricks;
his chromatic "Ut re mi fa sol la" shows them in practice. These tricks
are not earth-shattering gnostic knowledge...any reasonably sensitive
composer/improviser playing in meantone-flavored temperaments picks them up
by experience. They can also be found in Kuhnau's Biblical sonatas. It
would not be surprising if Bach knew them all as well: if not as a
checklist, at least by trying things in practice.
HYPOTHESIS: Bach's toccata in f# minor, if transposed to f minor, displays
those tricks! For example, all the major cadences into Ab major omit the
fifth. Other occurrences of Ab major triads happen on weak beats, or with
non-simultaneous attackes, or with the C in the bass. This might be
evidence of an f-minor genesis of the piece. Or it could be coincidence.
OBSERVATION: We have no hard evidence of any specific temperament Bach ever
used on any given day of his career. We do have a record that, at least at
some point in his career, he could tune an entire harpsichord in fifteen
minutes. This does not allow much time for scientific precision counting
HYPOTHESIS: Bach, being a practical musician, composed music that can sound
good when played by other people, not only by himself. It should sound
good even if the player's temperament is different than the composer's. A
practical composer should therefore stay mainly within the keys available
on other people's instruments, unless that composer is specifically out to
make a point (as in Fischer's "Ariadne musica" or Bach's WTC). If a
composition turns up in a strange key, it is logical to assume that the
strange key is due to a pedagogical situation or perhaps a transposition
from a common key.
OBSERVATION: I have tried Buxtehude's f# praeludium in f#, f, g, and e
(playing through on harpsichord for the sound of the temperament). My
opinion (agreeing with other commentators) is that it works best in e: the
lowest note is C, and requires only the A# and D# to be reasonable as
"wrong" enharmonics. It does not work in f: there are too many accented Ab
major and Db major triads where the wolf G#-Eb and wide C#-F are
prominent. G minor is also a plausible key for this piece, by temperament,
but it requires the pedals to have a high eb' available. The manual parts
go up to c''' in that case.
OBSERVATION: There are already Bach "manualiter" toccatas in g and e. And
(speaking broadly) within any given genre it is rare for Bach to repeat
himself, writing multiple pieces in the same key while other keys have not
HYPOTHESIS: The f# minor toccata works well in meantone-flavored
temperaments in either g or f (as I have noted over the past weeks). And
it does not work well in f# in those temperaments. It is reasonable to
assume that this *might* be a transposed piece, especially if such a simple
transposition solves more than one practical problem elegantly.
OBSERVATION: Errors of accidentals would be evidence in favor of a
transposition. As I pointed out last week, bars 89-92 indeed have problems
that suggest a transposition. So do bars 58, 64, and 99.
OBSERVATION: While making my own transposed edition (into f minor) this
week, working at a desk with white-out and copies, I made about half a
dozen transposition errors myself in the accidentals. I then corrected
those at the first play-through test, but the point is that it's easy to
make such errors even when being careful.
AXIOM: The musical ear tolerates out-of-tune minor thirds (generally
anything between 6:5 and 7:6) more than it tolerates out-of-tune major
thirds. Therefore the musical usability of any temperament for
tertian-harmony music is determined mostly by the quality of the major
thirds and the fifths.
OBSERVATION: In the f# toccata several sections end with chord spacings
where the major third is on top, the most exposed position...and they are
G# major and F# major. Another section ends in C# major with a wide
spacing; the E# is very prominent in the middle. These are all odd keys in
any known temperament before 1700, even Werckmeister III.
HYPOTHESIS: If we assume Bach was musical and wanted his music to sound
good, he "would not" have spaced these chords with such a prominently
raunchy sound so early in his career. Of course, he could have done so for
the shock effect. But if the entire piece is transposed down a half step
or up a half step, all those sectional cadences are in good meantone
keys. If the piece is in f, those three cadences are in the best keys of
all (G, F, and C), giving a notable repose.
OBSERVATION: The "best keys" in well temperaments are C, G, and F: these
triads have their major thirds nearest to pure. This is also true in
meantone temperaments where the naturals are generated by narrowed fifths,
and the accidentals are fudged (if anything is fudged). I believe it is
safe to assume that most musicians c1705 expected the keys of C, G, and F
to sound the best regardless of the temperament (meantone, modified
meantone, or well). The triads of D, A, and G are also of course good, but
less pure than C/G/F (unless the temperament is completely regular).
OPINION: There is something nicely dramatic about a piece that is in the
remotest plausible key (f minor) but which has its sectional cadences in
the purest-sounding keys.
OBSERVATION: Bar 129's downbeat is a dramatic spot in either f minor
(Bb-Db[C#]-Gb[F#]) or g minor (C-Eb-Ab[G#]). As I noted above, these
Neapolitan sixths with wrong enharmonics are a special case, usable. Bach
uses this same effect at several places in the g minor toccata
(915). Always it comes as a surprise, and the music then relaxes as this
shock resolves into better-tuned intervals. In the f# toccata (910) this
effect is backwards: the sudden Neapolitan is *in* tune (B-D-G) and
resolves to a chord that is out of tune (C#-E#-G#-B).
HYPOTHESIS: Either this piece was not originally in f#, or young Bach is
experimenting with a backwards universe where key relationships are
reversed. Did he really intend resolutions and major cadences to be
higher-tension (more out of tune) than the surprising moments?
OBSERVATION: We know that Bach was already fooling around with the
expressive limits of f minor before 1707: the Capriccio in Bb (probably
modeled on the Kuhnau sonatas). The sections are in Bb, g, f, and
Bb. This piece works well in this same style of meantone-flavored
"ordinaire" temperaments we've been talking about: slightly lowered Eb and
slightly raised G#, to taste.
HYPOTHESIS: Young Bach did know what he was doing, using the
"out-of-tuneness" of the temperaments for drama. He did not sling the keys
around haphazardly. At least in this Bb capriccio: in-tune keys mean
comfort and happiness, g minor (used chromatically) means a warning of
foreign danger, and f minor (used chromatically) means intense sorrow, a
lament. The f# toccata was probably similarly in f or g, with the
temperament aiding the drama instead of working against it.
QUESTIONS: As for writing a toccata in f#, "What was he
thinking?" Obviously, nobody really knows. Perhaps if he originally wrote
it in g he transposed it so there would not be two toccatas in g? Or if he
originally wrote it in f he transposed it so it could be played on
instruments whose lowest note is C? Or perhaps someone else initiated the
transposition, for one of those reasons or some other reason (such as "just
for fun"?). Or perhaps he really did write it in f# just to see what would
happen. A wild experiment.
OPINION: As a player, composer, improviser, and fifteen-minute-tuner
myself, when I play this piece in f minor it feels right. In f# minor it
feels wrong; in g minor it is OK but feels like only a "maybe." Obviously,
some of that opinion comes from the sum of facts, axioms, observations, and
hypotheses above; some of it also comes from intuition. I cannot "prove"
that intuitive sense. But I know when it speaks to me. And I know that I
am not going *only* on feeling or hope: the feeling comes from seeing the
facts line up plausibly, not the other way around.
As I asked earlier: is there any place for intuition and feeling in
musicology? Often it seems there is not, and that is (I feel) a sad state
of affairs. Is not music an art more than a science? An art of
interpretation and finding meaning?
p.s. Ibo, I'm especially enjoying your contributions to this discussion also.
Bradley Lehman, Dayton VA
home: http://i.am/bpl or http://www-personal.umich.edu/~bpl
clavichord CD's: http://listen.to/bpl or http://www.mp3.com/bpl
trumpet and organ: http://www.mp3.com/hlduo
"Music must cause fire to flare up from the spirit - and not only sparks
from the clavier...." - Alfred Cortot