Claudio wrote:

 > Scores of pages have been written and published on the temperament 
for > Bach, especially in recent years...
 > - by yourself supporting your own Bach temperament proposal based on
 > the WTC title-page
 > - by Lindley and Ortgies strongly arguing that your proposal is
 > fundamentally flawed

Claudio, I hope you've read my detailed rebuttals to the latter, and to 
O'Donnell's.  They're here:
http://www-personal.umich.edu/~bpl/larips/lindleyortgies.html
http://www-personal.umich.edu/~bpl/larips/odonnell.html

And the notes of my one-hour lecture from October 2008:
http://larips.50webs.com/

To me, on the topic of Bach temperament, any argument worth its salt has 
to go into the way the temperament handles the notes actually called for 
in his compositions.  Not just superficial observation of major 3rds, 
but the way all the named notes fit into scales and music.  Especially 
important is the notion of enharmonic equivalence.  18 of the 
prelude/fugue units from WTC book 1, and all 24 of book 2, call for more 
than 12 notes.  (And of the six in book 1 that have only 12, only one of 
them sticks to the central Eb-Bb...C#-G#.  The other five use a 
different 12 notes.)  That's using the implicit assumption that one is 
supposed to play both a prelude and fugue together without retuning 
anything between them; and probably also to play any and all of the book 
without retuning.  (Bach played the whole thing straight through for 
student Gerber....)

Here's a list of the WTC requirements for enharmonic equivalence, across 
both books.

Ebb and D: 2
Bbb and A: 3
Fb and E: 5
Cb and B: 8
Gb and F#: 7
Db and C#: 8
Ab and G#: 5
Eb and D#: 8
Bb and A#: 9
F and E#: 7
C and B#: 8
G and Fx: 7
D and Cx: 9
A and Gx: 9
E and Dx: 4
B and Ax: 1

That is: 8 of the prelude/fugue pairs need a Db/C# at the same time, 
because the music modulates so far that it needs both.  8 of them need a 
C/B#.  Etc....  The whole collection needs an A at such a well-balanced 
place that it can also serve as either Bbb or Gx.  B has to sound like a 
decent Cb or Ax.  F has to sound like a decent E#.  Etc.  The fugue 
subject of the B minor book 1 requires both B# and C natural, and only 
one bar apart, within the melody!  Bach's music lays out the problem 
clearly: tuned pitches needing to be balanced very carefully to serve 
all their multiple functions, melodically and harmonically.

And then, in my hypothesis, Bach's drawing solves the problem he has 
laid out: it's a diagram showing exactly the type of hands-on 
adjustments needed, listening closely and working by 5ths, proceeding 
from each note to the next.  Put down all the C major scale first 
(F-C-G-D-A-E-B), continue into the remaining five notes (F#-C#-G#-D#-A#) 
with the appropriate adjustments, and then proceed to play music using 
every possible major or minor scale.  It is straightforward hands-on 
work, not calculation.

Enharmonic reckoning is important for other contemporary music, too, in 
choosing a workable temperament.  For example, to play all eight of the 
preludes in Couperin's "L'art de toucher le clavecin", we need all the 
notes Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, A#, and E#. 
That's 16 notes, and if we assume no retuning between pieces, we have to 
have an Ab/G#, Eb/D#, Bb/A#, and F/E# all at somewhat intermediate 
positions so they can serve both ways.  We also have to have the basic 
tempering unit considerably lighter than 1/4 comma, and closer to 1/6 
comma on average, to be able to connect the circle at all for these 
enharmonic requirements.  The music itself demands it.

My basic method, for all this, is to try to keep the notes of the C 
major scale as close to regularity as possible...and to adjust the 
others minimally until they're good in their dual functions.

- In music that requires extended flats such as Fb/Cb/Gb/Db, we might 
have to break the regularity of the C major scale at its last note (B), 
sharpward, to get the chain of sharps high enough to serve as flats.

- In music of extended sharps, requiring A#/E#/B#, we might have to 
break the regularity of the C major scale at its first note (F), 
flatward, to get F-Bb-Eb low enough to serve as sharps for all their 
musical contexts.

It's a process of adjustment by irregular 5ths, stretching them wider 
than the "downtown" regular 5ths: listening to and checking scales 
hands-on at the harpsichord, NOT doing calculations.  Every note has to 
work well in all its enharmonically-required contexts.  Ye got a 
keyboard that needs to play a bunch of notes outside the old-fashioned 
set?  Here's how you adjust it until it all works out smoothly and you 
can play anything.  The first fugue of WTC book 1 lays out all six of 
the regularly-spaced notes (F-C-G-D-A-E as the C-D-E-F-G-A hexachord). 
Everything else is tastefully adjusted off that regularity so you can 
finish that prelude/fugue (which happens to need both Ab and G#), and 
the rest of the book.

That first fugue subject is pretty amazing, all by itself.  It walks up 
from Ut to hit Fa strongly on the first downbeat.  It then decorates the 
expressive descent from Fa to Mi.  (Witness Bach's later canon that 
asserts in its description: "Fa-Mi et Mi-Fa est tota Musica", playing 
with the way Fa and Mi interact during modulations.)  Having established 
Ut, Fa, and Mi as the three fixed points in the chain, the subject then 
shows us how to zigzag by 4ths and 5ths to generate La, Re, and Sol.  Fa 
and Mi are the endpoints of the hexachord, generated Fa-Ut-Sol-Re-La-Mi 
by 5ths/4ths, and laid out as the scale Ut-Re-Mi-Fa-Sol-La.  The longest 
note of the subject is Sol, before falling with decoration down to Mi.

And this fugue (about melody?) is coupled with a prelude (about 
harmony?) that showed us how to handle triads and modulations.  Bar 1 of 
the prelude gave us Ut, Mi, and Sol.  Bar 2 adds Re, Fa, and La. 
There's the complete hexachord already.  Bar 3 gives us the seventh 
note, Ti (Si), completing the diatonic scale.  The first foreign note, 
F#, shows up a bit later and we diddle around with that for a while, 
having modulated via the F# (replacing F) to the G major scale.  The 
introduction of both Bb and C# together lurches us into the scale that 
contains them both: D minor.  The same two bars then happen again, a 
step lower, juxtaposing Ab against B, and taking us back to C major. 
When the next Bb appears (replacing B), it sends us to the nearest scale 
that contains it: F major.  The extraordinary motion in the bass, with a 
melodic diminished 3rd of F# to Ab, decorates our move into a long pedal 
point on dominant G.  When that thing finally gets resolved to C, a Bb 
intrudes immediately: really surprising.  On our way back to home C 
major, we have to deal with the subdominant F major and dominant G major 
once more, each.

And, if my hypothesis is correct: once we get off the regularity of the 
C major hexachord and into the irregular notes (B/Ti, and the sharps and 
flats), each of them by their careful irregularity signals the ear that 
we're heading into a modulation.  Whenever a flat or a sharp intrude 
into the plainness of C major, they sound like a relatively large 
surprise, because they're off-spot.  (They break the regularity of the 
55-note division of the octave, the one Sauveur wrote in 1707 was the 
common practice of musicians.)  The off-spot-ness emphasizes the new 
scale that we're going to.  Any intruding accidental is an intense 
event, an irritant, a break of expectations.  More things will happen 
before the music can relax back into plainness and regularity.  And the 
really weird stuff, such as the F# to Ab motion in the bass, comes 
through as doubly intense.

That's what I believe Bach was illustrating in this prelude and fugue. 
The first piece in the collection shows us how to handle harmony and 
melody in this home key, before we set off to music that requires more 
exotic notes: a 27-note enharmonic gamut, all the way from Bbb to Ax, 
inclusive.  The title page in front of that gives a diagram, showing how 
to start from the notes of the C major hexachord as Fa-Ut-Sol-Re-La-Mi, 
and then adjust the other six properly with less tempering.  The whole 
book demonstrates the system in action, by musical example (Bach's 
preferred manner of instruction) rather than in words or any numbers. 
No calculation is required; just DO THIS and then play anything, in any 
and every scale.  It goes through every "Ut Re Mi" and every "Re Mi Fa" 
possible on a 12-note keyboard.

Brilliant, brilliant music, whether I'm right about the tuning scheme or 
not.

Book 2 has a different 27 notes: it loses the Ax off the top, but gains 
the Ebb at the bottom.  And, as I mentioned above, every one of its 24 
preludes/fugues requires more than 12 notes.  These pieces all force the 
player/tuner to grapple with enharmonic compromises.

=====

Claudio also wrote:

 > Brad clearly stated that he would "review" not just Barnes but my
 > WHOLE book here.

To be clear: I didn't promise to review the whole book here.  I only 
said that "I'll reserve review until I've read the whole thing."  That 
is, I won't write evaluative comments about it UNLESS I've invested the 
time to read the whole thing FIRST...which is the way I wish people 
would treat my work, too.  Read the whole argument, and listen to any 
proposed solutions hands-on and ears-on in real music, before 
formulating an evaluation.

Now, please, let's discuss those reasons why Barnes's method yields 
skewed and ultimately meaningless results.  If Barnes's article is (at 
best) a shaky foundation, anything built upon it is not going to be 
solidly convincing either...at least, not to me.  If we're going to tune 
for, say, Couperin, we can look at all the enharmonic notes required; 
or, we can look at only the major 3rds in only one organ mass.  The 
latter approach seems superficial, to me.  Major 3rds don't tell us 
enough, by themselves, about what music actually does.  If I'm playing 
straight through the fairly short Ordre 25 by Couperin, with its pieces 
in Eb major, C major, and C minor, I have to have a temperament that 
makes the instrument sound non-silly through all the notes Db, Ab, Eb, 
Bb, F, C, G, D, A, E, B, F#, C#, G#, and D#.  Statistical methods 
applied to small samples aren't going to tell me what I need to know, to 
accomplish that.

I'm curious about the reaction of PP's alter ego to this Barnes stuff, too.


Cheerio,
Brad Lehman