Here's a simple temp I've formulated and have been messing about with 
for a while, as an all-purpose circulating scheme that takes about 
two to four minutes to set up in the bearings area.  It emphasizes 
listening to interval quality, rather than counting anything, and it 
all has basically one size of 5th.  No notes need to be dinked around 
(like neighboring "half-tempered" 1/12 comma 5ths) after they've been 
put into place.

What should I call this, "Lehman 2"?


- C from fork, and over to middle C.

- Set E from C, somewhere near the quality of regular 1/6 comma (i.e. 
with the familiar sound from Vallotti, or slightly sharper than in 
Werckmeister 3).  If it happens to be a little high or low to taste, 
that's OK.  (*)  We just want to stay out of the range where it's so 
high that C-E turn into a blur, or so low that there's no room left 
to work with in the following steps.

- Fit the G, D, and A into this so C-G-D-A-E all have the same 
quality as 5ths or 4ths.  (**) Whatever regular size we're setting 
here, this is our basic unit.

- From E, pure B, pure F#.  G-B should sound slightly "harder" or 
brighter than C-E, and then D-F# even more so.

- From C, pure F.  F-A has the same character as G-B.  The C major, F 
major, and G major triads are our three best.

- From F, temper Bb as a narrow 5th (or wide 4th) of approximately 
the same quality as the others above.  Listen also that Bb-D is 
similar quality to the F-A already available; and that F#-A# is high 
but acceptable-sounding.

- From Bb, pure Eb.  Confirm that Eb-G and D-F# have the same quality 
as one another, both sounding much like they do in equal 
temperament.  Confirm also that B-D# has approximately the same 
character as F#-A#, and the whole triad B-D#-F# is quite good, owing 
in part to the pure 5th.

- From Eb, make Ab a narrow 5th or wide 4th of approximately the same 
quality again, or perhaps a little bit gentler to taste.  Test that 
E-G# is high and bright, but not quite as wide as Pythagorean.  Ab-C 
should sound very slightly wider in character than Eb-G does, but 
still a good complete triad Ab-C-Eb.

- From F#, make C# a narrow 5th or wide 4th similarly.  Check that 
A-C# makes a nice transitional character between D-F# and E-G#.  Also 
check that Db-F has a character resembling that of B-D#.  Finally, 
note that our leftover interval C# to G# is probably slightly *wide* 
as a 5th (or narrow as a 4th), but that the character of C#-G# sounds 
the same as Eb-G#.  They simply happen to be tempered in opposite 
directions, but a similar amount, not that anyone would notice during 
the playing of music.


Play suitable music in a variety of keys, to test that everything 
works nicely. (***)

=====

There's an odd set of properties that come up if the initial C-E is 
too wide and venturing into the blur-range.  (Value judgments, of 
course; others' mileage may vary....)

- The natural keys sound as if they're reluctant to relax, noticeable 
especially in F major; and the Bb major triad gets difficult to set 
with any reasonable quality.  C, F, G, and Bb are still the best 
four, but we've lost a good bit of their resonance.  (And if we're 
going to do so, why not just use equal or something pseudo-equal, 
instead?)  The D major triad also emerges sounding rather hectic.

- Around the service entrance at the back of the building, the 
leftover C# to G# 5th (or C# to Ab) turns out narrow.

- The triads of Db major, B major, and F# major turn out much too 
consonant, ahead of the qualities of D/Eb, A/Ab, and E.  It's as if 
the shape flips itself inside-out: making the key of C# major sound 
fantastic, but at the expense of too much "nervous tension" everywhere else.

=====

(*) Somewhere in the quality range of 1/5 syntonic, 1/6 Pythagorean, 
1/6 syntonic, or about 1/7 Pythagorean.  Somewhere in there, perhaps 
somewhat differently from instrument to instrument, there is a happy 
medium where we'll end up with a quite good C major, F major, and G 
major.  Taste and experience!  For those who "must" count beats: put 
C-E somewhere between 3.5 and 6 beats per second, and this is easier 
to hear if playing a major 10th from middle E down to the tenor C.

(**) A method to do this accurately is at
http://www-personal.umich.edu/~bpl/larips/tetrasect.html
...or just guess it out so it's close enough, with those four 
5ths/4ths all sounding similar.

(***) One of my favorite half-hour test sets is as follows...play as 
much of this as necessary to be satisfied with temperament quality 
and usability -- melodic/harmonic integrity, and enough character:

- Bach's four Duetti (for various examples of enharmonic swappery, 
and chromatic motion within the basic characters of common keys);

- The following major-key preludes of WTC book 2, all spaced a minor 
3rd apart: F major (for smoothness and big resonance in an "overheld" 
harpsichord texture), Ab major (for chromatic adventures and the 
character of flat keys), B major (for a cycle through the sharpest 
keys, with some fairly wide spacings), and D major (for a strong and 
vigorous sound that projects well);

- Flip through the last three _Ordres_ 25-27 of Couperin, for 
spot-checking into these remote areas similarly.  Anything that 
sticks rather closely to meantone doesn't do so well, in these pieces.


Brad Lehman