<< >Usual Pythagorean
> traits such as active thirds and sixths inviting efficient resolutions
> to stable 3-limit concords, and narrow diatonic semitones for
> expressive melody and incisive cadential action, are heightened in a
> form of intonational mannerism.
<<"
What is a 3-limit concord.?>>
3-limit refers to the use of ratios that use no more than a 3. Thus, a
3 limit tuning is created by use of intervals 2:1 and 3:2. You will not find
a 5:4 interval in this tuning.
Earlier, I had written Margo the following, and I will include her reply:
: Hmm, the meantone tunings are accomplished by tempering the first 4
: fifths between C and E. Then notes such as G# are tuned as Just M3 with E,
: the
: D# is tuned Just with B, which was tuned Just with G, etc. This makes
notes
: such as that note between F and G a F# , not a Gb. The result of this
tuning
: is that you create a wolf interval between the F#and Bb, that ain't a
third.
Regards,
Ed Foote
>From Margo:
Hello, there, and I would just add that this is a very nice description of
"meantone" in its 1/4-comma version, where each fifth is tempered by 1/4
syntonic comma so that four such fifths make a pure major third of 5:4.
Some people would say that this is the only "true" meantone, since each
pure 5:4 major third is divided into two "mean-tones" each equal precisely
to the average of the unequal 9:8 and 10:9 whole-tones of tertian just
intonation.
However, the term "meantone" is often used more broadly to describe a
range of tunings described in the 16th-18th centuries with _equal_
whole-tones -- not necessarily equal specifically to the "mean" of 9:8 and
10:9, as in the special (and paradigm) case of 1/4-comma.
Thus in the 16th century we find descriptions not only of 1/4-comma (often
attributed to Aron, although as I recall Lindley may argue that this
involves some interpretation), but also of 2/7-comma (Zarlino) and
1/3-comma (Salinas). Around 1700, a rival to the new well-temperaments was
Silbermann's 1/6-comma.
As you suggest, what all of these characteristic meantones have in common
is that two pure or near-pure major thirds, e.g. c-e, e-g#, leave a third
interval to complete the pure octave which is much wider than a 5:4 or
even a Pythagorean 81:64 major third, a diminished fourth (here g#-c').
Vicentino describes it as a "proximate major third," and suggests that it
might be tolerable as a quick interval, but not very satisfactory
harmonically. In a European context of these eras, it is indeed a "Wolf"
of first order!
For 14th-century or 20th-century music, such a "supermajor third" can have
its uses -- but that's outside the scope of the musical languages we're
considering for characteristic meantone.
Incidentally, some 18th-century theorists describe what might be called
"well-tempered meantones." If we reduce the temperament of the fifth to
about 1/7-1/8 comma, then both our odd 12th fifth and the diminished
fourths become "playable." These tunings maybe have a quality somewhat
between unequal well-temperaments (with greater contrasts in color) and
12-tet with its uniformity.
Margo Schulter
mschulter@value.net
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