Re: Two fine questions from Joe Pehrson

Hello, there, Joe Pehrson, and maybe I can give some reply to two
questions you've raised in recent threads.

First, from reading Mark Lindley, it's my understanding that viols are
typically fretted more or less for 12 equal semitones to the octave,
but with a good deal of leeway for adjusting each note -- a kind of
"adaptive tuning."

Lindley remarks that any Renaissance viol ensemble or the like with a
fair degree of skill can achieve some near-just sonorities, and
suggests that in the 17th century, a skillful violist (for example in
France) could accord well with a meantone keyboard.

As for 27:25 as a "large limma," this usage seems a bit curious to me,
but possibly I can explain it by analogy, inviting feedback from
people who know the terminology of 5-limit JI better than I.

A limma, most properly, means to me a diatonic semitone in
Pythagorean, 256:243 or ~90.22 cents. It comes from the Greek for
"remainder" -- what remains in a diatonic tetrachord when you take 4:3
fourth (~498.04 cents) and subtract the two 9:8 whole-tone steps of
~203.91 cents each, together adding up to a ditone or major third of
81:64 (~407.82 cents). That is, it's the difference between a fourth
and a major third.

More generally, the term "limma" tends to imply for me that the
diatonic semitone is the smaller of the two semitones, as in
"positive tunings" where the fifth is larger than 700 cents. The large
chromatic semitone is then called an apotome -- in Pythagorean,
2187:2048 (~113.69 cents).

What may be happening with the "large limma" term for 27:25 is that,
by analogy, it's the "large diatonic semitone" of a 5-limit system,
where the diatonic semitones are _larger_ than the chromatic ones.

Specifically, if we take C-D as a large 5-limit tone at 9:8 (the same
size as all regular tones in Pythagorean), and the chromatic semitone
F-F# as 25:24 (the difference between A-C at 6:5 and A-C# at 5:4),
then the "remainder" C#-D will be 27:25, or about 133.24 cents.

Thus in this case, C#-D is indeed the "remainder" or "difference"
between the fourth A-D and the 5:4 major third A-C# -- except that, if
not adjusted, the pitches A-D here don't form a perfect fourth at 4:3,
but rather an interval a syntonic comma of 81:80 wider (~21.51 cents),
or 27:20.

What I'd guess is that limma is taken here as meaning "diatonic
semitone," a category which does C#-D -- the larger such semitone, in
contrast to the 16:15 semitone which does define the "remainder"
between a 4:3 fourth and 5:4 major third.

By the way, a semitone of around 133 cents can be very pleasing in a
neo-Gothic kind of setting also, where some regular temperaments have
chromatic semitones near this size. It's also a type of semitone
available in certain schemes with two chains of fifths a given
distance apart.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

/tuning/topicId_31270.html#31270

> Hello, there, Joe Pehrson, and maybe I can give some reply to two
> questions you've raised in recent threads.
>
> First, from reading Mark Lindley, it's my understanding that viols
are typically fretted more or less for 12 equal semitones to the
octave,but with a good deal of leeway for adjusting each note -- a
kind of
> "adaptive tuning."
>
> Lindley remarks that any Renaissance viol ensemble or the like with
a fair degree of skill can achieve some near-just sonorities, and
> suggests that in the 17th century, a skillful violist (for example
in France) could accord well with a meantone keyboard.
>

Thank you very much, Margo, for helping me with this.... Yes, of
course now I remember discussions about the viols being tuned
similarly to 12-tET... not meantone, but apparently the just
sonorities can be achieved in performance... that's certainly what it
sounds like, anyway...

> A limma, most properly, means to me a diatonic semitone in
> Pythagorean, 256:243 or ~90.22 cents. It comes from the Greek for
> "remainder" -- what remains in a diatonic tetrachord when you take
4:3 fourth (~498.04 cents) and subtract the two 9:8 whole-tone steps
of
> ~203.91 cents each, together adding up to a ditone or major third of
> 81:64 (~407.82 cents). That is, it's the difference between a fourth
> and a major third.
>
> More generally, the term "limma" tends to imply for me that the
> diatonic semitone is the smaller of the two semitones, as in
> "positive tunings" where the fifth is larger than 700 cents. The
large
> chromatic semitone is then called an apotome -- in Pythagorean,
> 2187:2048 (~113.69 cents).
>

Why, of course, I understand there can be different sizes of "whole
tones" and "semitones" dependent on the tuning system... relating it
to 12-tET was momentary confusion.

Thanks so much again!

Joe