Re: No apologies necessary: consonance and beatlessness

In a recent post (Tuning Digest 227), David J. Finnamore wrote:

> Do you still think I don't have anything to apologize for? ;-) I'll
> never post anything on 3 hours of sleep again!

Hello, there, and I'd like indeed to say that no apologies are called
for in expressing one's opinions on tuning in this forum, and raising
some very interesting issues and distinctions for further discussion.

As for posting in the circumstances you describe, while the results
might be a bit unpredictable <grin>, I would hardly say that drawing
an equation between "maximal consonance" and "beatless" that one would
later want to quality is anything meriting an apology.

In fact, as I see it, maybe part of the issue is to _which intervals_
one might want to apply an ideal of maximal "consonance" or
"beatlessness" in a given tuning system or style.

Your post is a helpful reminder that these terms may not always be
synonymous (some musicians report finding 400 cents more pleasing for
a major third than 5:4 in styles where this is a prime stable
consonance), but nevertheless I find it easy to treat them as
synonymous in many circumstances.

Maybe Pythagorean (3-limit just intonation or JI) and 1/4-comma
meantone are classic cases of "consonance maximalization" within a
certain stylistic framework.

Whatever we may conclude about "sensuousness" or "humanism" in
medieval music (and I agree we should be rather careful to be sure
that such a discussion stays focused mainly on tuning), 3-limit JI
_does_ maximize the beatlessness of certain intervals: obviously the
prime concords of 3:2 and 4:3, and also 9:8 and 16:9. At the same
time, other intervals may become more "beatful" or "tense":
e.g. 81:64 (vs. 5:4), 243:128 (vs. 15:8), etc.

In 1/4-comma meantone, it is the major third (5:4) which becomes
ideally beatless -- after all, it's the prime concord in 16th-century
music -- at a cost of slight "beatfulness" of the fifth (~5.38 cents
narrower than just). This raises an interesting question: is the
choice of 1/4-comma rather than pure 5-limit merely an accommodation
to conventional keyboards and their players, or does the more active
fifth somehow give a "Renaissancy" flavor to keyboard music of this
era which pure 5-limit might not?

If we define "maximal consonance" in one sense as "beatlessness for
the prime concords in a given system or style," this leaves open the
possibility either of seeking beatlessless (or least beatful ratios)
for _all_ intervals in a system, or of applying the ideal more
selectively.

There are various directions we might go in from here, but I wonder if
I would be writing this without the help both of your earlier posts,
and of your "apology."

> David

Most appreciatively,

Margo Schulter
mschulter@value.net

Margo Schulter wrote,

>3-limit JI
>_does_ maximize the beatlessness of certain intervals: obviously the
>prime concords of 3:2 and 4:3, and also 9:8 and 16:9. At the same
>time, other intervals may become more "beatful" or "tense":
>e.g. 81:64 (vs. 5:4), 243:128 (vs. 15:8), etc.

To evaluate this statement, one would need some criterion for deciding how
much mistuning would be allowed before an "interval" changes to another
interval. Clearly the syntonic comma is not enough in your opinion. Is the
septimal comma enough? If not, then I would have to disagree that 9:8 and
16:9 are the most consonant tunings for their respective intervals. 8:7 and
7:4 are more consonant if the intervals are played alone and not accompanied
by other notes.

>In 1/4-comma meantone, it is the major third (5:4) which becomes
>ideally beatless -- after all, it's the prime concord in 16th-century
>music -- at a cost of slight "beatfulness" of the fifth (~5.38 cents
>narrower than just).

I think this is an oversimplification. I would say that tuning the major
third, rather than one of the other equally important 5-limit concords,
close to just was common because it leads to a smaller error in the fifth
than tuning the minor third just, and it leads to a smaller error in the
minor third than tuning the fifth just. Using various mathematical criteria,
we recently derived various "optimal" meantone tunings (by considering all
three 5-limit concords), ranging from 5/18-comma to 3/14-comma meantone --
all close enough to 1/4-comma meantone to explain the latter's historical
position of favor.

>This raises an interesting question: is the
>choice of 1/4-comma rather than pure 5-limit merely an accommodation
>to conventional keyboards and their players, or does the more active
>fifth somehow give a "Renaissancy" flavor to keyboard music of this
>era which pure 5-limit might not?

I think the former may have started out as the case, but the keyboard music
created in the Renaissance uses the subtle balances of meantone in such a
way that a non-meantone tuning would destroy the meaning of the music.
Sweelinck's music, for example, seems particularly to cry out for meantone
(you wouldn't know this if you hadn't heard it in meantone).