Le 12/01/2023 16:50, Bradley Lehman via The Jackrail écrit :
[bpl] bpl https://jackrail.space/u/bpl Bradley Lehman
January 12Continuing the discussion from Lehman Bach temperament
https://jackrail.space/t/lehman-bach-temperament/1077/6:So, if we are considering only “how many cents wide of pure are the
major thirds”, it would be:Note major3rd
Eb 14
Bb 12
F 6
C 6
G 10
D 14
A 18
E 20
B 18
F# 16
C# 18
G# 16The F-A and the C-E are the same as they are in 1/6 Pythagorean comma
meantone (the 55-division).The widest major third is at E-G#, and it is smaller than Pythagorean.
So, there are no full commas of error in any major third anywhere. It
is smoother than the widest thirds we get from Vallotti, Young 1 and
2, the Kirnberger temperaments, the Werckmeister temperaments,
Barnes-Bach, Kellner, and many others. I showed all of this in the
2005 article. I explained again in 2022 why it is important to avoid
all Pythagorean thirds, according to 18th century expert taste (citing
Sorge, Marpurg, et al).
Thanks for these figures. I now recall that what bothered me at the time
was the “bad” E-G# third. Certainly it is much more frequent and useful
than the four following thirds in your table, which are all better. What
is the justification for placing the worse third on E? Another objection
I had at the time is the fact that this temperament favors the flat
side, where all the thirds with the same number of accidentals are
better. In my opinion a good temperament should favor the sharp side,
because minor keys need the good thirds of the parallel major keys.