<< >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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