One more posting in this thread for me (originally "an experimental 
temperament for B & B"), and then I retire:

Tom Dent wrote:
> Look at the musical context of G# in Boehm: it's the root of a
> _diminished_ chord G#-B-F with E as a tiny passing note. So much for
> an 'E major triad'! I don't think it's going to be so bad if E-G# is
> slightly *wider* than Pythagorean, leaving Ab-C as slightly better and
> certainly nothing to worry much about. So we can try 11 notes of
> carefully executed quarter-comma, plus Ab tuned as a mini-wolf down
> from Eb, leaving C#-Ab as the teenage werewolf.

Look, Tom, if you're going to construct such an argument, at least get 
the musical analysis right (and thereby avoid red herrings), and look at 
the bigger picture too.  That couple of bars in the Bohm example I 
presented (26-27 in verse 2 of "Christe, der du bist") is music that's 
obviously in A minor, at the time.  The F is diddling up and down as a 
neighboring tone, but we're still "in" A minor.  The G# occurs on a 
downbeat, creating an E major dominant, and then we go right back to A 
minor again.  Yes, there's an F suspended temporarily through there (in 
a 5-6 progression), but it resolves back down immediately to the 
expected E.  Furthermore...G# and Ab both get used again in the next 
variation.  The G# in there, again, is serving a normal E-major function 
as the music diddles through A major and D minor for a couple of bars.

While we're on that same line of music, bars 26-27 of variation 2 (on 
the scanned pages I supplied), look ahead to bar 28.  It modulates back 
to D minor, and there's a C#-F sonority going into the cadence.  It's a 
diminished 4th, and it's correctly spelled here according to the voice 
leading of the music.  This is a spot where (in my opinion) it would 
*not* be any virtue to have C#-F or Db-F be anywhere near 5:4.  It's a 
dissonance, a diminished 4th.  Meantone-ish intonation here is a VIRTUE 
against near-equal, because that diminished 4th is allowed to bite.  In 
this piece by Bohm, it looks pretty obvious from usage all through it 
that we should expect both F-A and A-C# to be "better" major 3rds than C#-F.

As for Ab and C played together, where (according to you in your desire 
for 1/4 comma), you're quite welcome to assert that it's "nothing to 
worry much about."  An Ab-C that wide sounds like #*&%#% on my 
instruments, but obviously, you and I hear differently.  Again I draw 
your attention to the spots in bars 34 and 45 where I drew the boxes 
around these Ab-C 10ths.  Go right ahead and put a pure C-E on your 
harpsichord, and let both E-G# and Ab-C be lousy if you want to.

I also wish you'd stop using the straw-man argument against me that I 
allegedly prefer equal or equal-ish temps, because I simply don't.  I 
prefer things to be as meantone-ish AS POSSIBLE, because I really do 
like major 3rds that are better than equal can deliver.  That "AS 
POSSIBLE" is the kicker, and I've already explained it in another recent 
posting: if we go all the way down to 1/4 comma for some major third(s) 
on the naturals, giving a pure 5:4, we're stuck.  The other major thirds 
above and/or below it will *both* on average have to be terrible: 
Pythagorean, a full comma off of 5:4, give or take a schisma.  When I 
set up temperaments, I listen to the way all twelve major 3rds are 
behaving, according to the music I'm going to play.  If I'm really not 
going to be playing one or more of them, I do favor the other two above 
and below.  (Example: in music that has no Eb-G in it anywhere, I make 
both G-B and B-D# better than Eb-G.)  If I'm playing music that really 
does need all three major 3rds of such a stack to be decent, I CANNOT 
make any one of them as small as 5:4, because Pythagorean and wider 
major 3rds sound so repulsive to me.

If they're played functionally as major 3rds, they'd better sound like 
it.  And if they're played functionally as diminished 4ths, a wideness 
(or at least a substantial distinction against better major 3rds) is 
still a virtue.  That's one of the reasons why I love unequal 
temperaments in the first place: finding that practical balance in the 
music I care about.  The better a temperament matches the functional 
behaviors of music, the more vivid and directly communicative the music 
sounds, to me.  I really dislike equal temperament, because it bores the 
socks off me when I'm listening to or playing tonal music.  Equal 
temperament is just a flat and inflexible board, doing nothing.  Got 
that point now?  Good!  :)

As for the more-than-once whining about the way you don't like the sound 
of my quickie recording equipment: point noted, months ago, but I'm 
still not in a position to do anything about it.  Sorry.  My money goes 
to family care and other fine places, not to the buying of new 
electronic devices for myself.  I don't even LIKE electronic devices 
very much.  If I choose to throw together a quick recording session in 
my own house, for my own reference in comparing temperaments or 
whatever, and then to share that work with other people who MIGHT also 
enjoy or learn something from it...well, it's "free and worth at least 
every penny".  Treat them as a pencil-and-paper sketch, as opposed to a 
full-color matted and framed photograph under glass.  If you don't like 
them, well, you're free and encouraged to make your own, in colors as 
nice as you prefer, and with as much effort as you're willing to put 
into it.

=====

Now...let's take a step out to a wider perspective.  This is for 
everybody, and not so specifically to Tom.

In this discussion thread over the past week or so, my aim was:

- To present an easy and entirely by-ear temperament that I feel works 
nicely in playing a swath of 17th century Germanic repertoire, without 
having to stop and retune any accidentals.  (Froberger, Reincken, 
Buxtehude, Bohm, Kuhnau, Pachelbel: have a play.  Play the real music, 
not merely sitting around speculating about what sounds good *in 
general* in music that has nearly-empty key signatures.)

- To present my musical reasoning, giving some lists of the notes 
actually used in the music, and explaining why I believe they should be 
in moderate positions: because two notes sharing the same key lever get 
used in the same composition, so frequently, and (axiomatically) should 
sound decent as either one.

- To present some listening examples (here it was a pair of videos), so 
those not inclined to go do this stuff hands-on themselves can at least 
hear it.

- To present a reasonable temperament for playing this music TODAY, 
without addressing the issue of how the old guys "might have done it" 
then.  I didn't put up a historical argument, other than pointing out 
the basic precedent for 1/5 comma and its environs; 1/4 comma WAS NOT 
the only game in town, no matter how much some people may like it today. 
   (And, as I said, I used to play mostly in 1/4 comma systems myself; I 
changed my mind, for reasons I've explained.)  Anybody is free to use 
whatever temperaments they choose to, for whatever reasons are 
compelling to them.

- To present my reasoning *process* as I work out ways to play music I 
care about, with an intonation scheme that sounds good to my ears: by 
analyzing the music for the notes and intervals actually used, and then 
trying to make the most-frequent intervals (those on the naturals) the 
best ones...while letting all the others be usable, too.

Owen wrote, presenting his own straw-man version of me: "You read my 
three hundred pages of numbers lists and grapple with each 'graf of it, 
or, be damned to you, you admit that I am the most fearfull disputante." 
  Well, that's kind of funny, but again it severely misunderstands what 
I'm about on here.  My presentation has been the OPPOSITE of "numbers 
lists"; it's been a straightforward and easy by-ear temperament to do 
without calculating a single thing first.  My point is to play the 
lovely musick on my severall harpsichords, to have it please me and 
those who hear me play live, and then to share this pleasure in a tuning 
layout that other fine keyboard-ticklers MIGHT ENJOY trying...if 
inclined to try out things they may not have tried before.

Now please, enjoy.  If y'all don't care to enjoy, but would rather 
dispute, well, I'm done.


Cheers,
Brad Lehman