source file: m1570.txt Date: Sat, 31 Oct 1998 21:49:36 -0800 (PST) Subject: Re: Consonance: self-correction and further discussion From: "M. Schulter" Hello, there, and this is a post first to ask pardon for my public display of innumeracy in Tuning List Digest #1569 in an article on the "Spiral of Fifths/Fourths," and secondly to respond to some very interesting remarks on consonance concepts, JI, and 12-tet. First, my blooper duly confessed and corrected: > Yet a doubly diminished fourth (fifth #16, e.g. d#-gb) of about 271 > cents is only about 3.8 cents from a pure 7:6, which I suspect many > people on this list would regard as more concordant than a comma > (fifth #12) or even a major seventh (fifth #5)... In fact, the first quoted line should have said "fourth #15," which does yield an interval very close to 7:6. My twofold error was think in terms of 16 _notes_ (actually 15 tuning intervals), and to write "fifth" when "fourth" is correct (i.e. 15 fourths up or fifths down). Now we move to a much more delicate question of definition and taste raised in a post on "Mathematical explanation of consonance" from monz@juno.com: > How does one explain the fact that so many musicians who are less > "well tuned" than us will find 12-equal "3rd"s to be consonant, for > instance? Most of us on this List wouldn't say that, but lots of > other listeners would. Personally, I might rephrase this a bit: "How does one explain the fact that so many musicians ... apparently assume that 12-tet 3rds are the 'ideal' and 'natural' versions of these intervals to use for all periods and styles of music, as if no other tunings were known?" This whole question could lead into lots of areas, and here I'll just try to address a few. First of all, as someone working mainly with just (3-based or Pythagorean) and "semi-just" (1/4-comma meantone) tunings, and also with 17-tet, I'd say that 12-tet 3rds can and do serve as full consonances in triadic or post-triadic music, and have done so at least since the middle of the 16th century, when 12-tet was already being described as a standard tuning for the lute. This doesn't mean that 16th-century musicians, who were likely _not_ generally accustomed to 12-tet but rather to 5-based just intonation for voices and meantone approximations of it on keyboard, found a 12-tet third _ideally_ "concordant." Vincenzo Galilei, in his _Fronimo_, speaks as both an accomplished lutenist and a tuning theorist when he recognizes that major thirds with the standard 12-tet fretting _are_ larger than the Renaissance ideal of 5:4, but nevertheless finds them acceptably concordant. Interestingly, he explains this acceptability by pointing to the quality of the lute's strings, which tend to "soften" the sharpness of these thirds. I don't know if this means a new trend with people wearing t-shirts saying: _Vincenzo Galilei was a proto-Setharian_, but he did distinguish between the timbre of a lute and a harpsichord (customarily tuned in meantone), the latter being more intonationally sensitive. At the same time, he pokes a bit of fun at people who add _tastini_ or "little frets" to the lute in order to get more just thirds. His argument is basically: (1) These people often in practice wind up playing intolerably out-of-tune fifths when the system gets too complicated; and (2) A true virtuoso (like Galilei himself) doesn't need such extra gadgets to impress an audience. What we're dealing with, then, is what has been called "categorical perception": in music, as in natural language, a vowel, consonant, or harmonic consonance for that matter may vary somewhat in its pronunciation or size and still be recognizable to the listener. Differences between dialects of various languages might be analogous to variations in tuning a "major 3rd," for example: it might be 386 cents in 5-based JI or 1/4-comma meantone, 379 cents in 1/3-comma meantone or 19-tet (proposed in the 16th-century), 400 cents in 12-tet, or shift depending on the key between values ranging anywhere from 386 cents to around 408 cents in the various well-temperaments standard for harpsichords and soon also pianos during the era of about 1680-1880. For a major third that the listener is expected to recognize as fully concordant or stable in a Renaissance-Romantic setting, there do seem to be outer tolerances. One line of demarcation is that an interval much larger than the regular Pythagorean M3 (81:64 or 408 cents) may be heard as a quite "dissonant" or "Wolvish" interval obviously "out of tune." Another line, proposed by Easley Blackwood, is around 406 cents, the point where he feels that major thirds are just subdued enough to form stable triads. By his standard, 12-tet is clearly within this range. However, "acceptable" and "ideal" are two different questions. For 16th-century music, I'd agree with Mark Lindley and _lots_ of attuned listeners that 12-tet on an organ-like or harpsichord-like keyboard would be a very serious compromise; which doesn't mean that one couldn't enjoy the music, only that it would be an arrangement of necessity rather than ideal beauty. In fact, I'd tend to stick pretty much with 1/4-comma meantone (or 1/3-comma/19-tet for at least one chanson by Costeley where it is explicitly indicated, and maybe Zarlino's 2/7-comma for certain kinds of pieces), going to 13 notes per octave or 15 or even 17 or 19 where required. Such are the advantages of microtunable synthesizers. Also, as has been pointed out by others in this thread, consonance is a complex matter of stylistic expectations and "acclimatization" as well as tuning ratios. In 13th-14th century European music, major thirds are stylistically treated as "partial concords," and Pythagorean tuning does a nice job of putting a bit of an accent on the _partial_ in that statement. However, even if I played Perotin in 1/4-comma meantone with pure 5:4 major thirds, these intervals would sound "_relatively_ concordant but unstable" to me, because this is the musical definition of the language around 1200. While I'd agree with 16th-century theorists that 5:4 and 6:5 are ideal ratios for thirds in a style based on concordant triads, maybe a view fitting my preference for Renaissance and Manneristic music as my favorite triadic styles, lots of other approaches are possible. For example, as Ed Foote has discussed, well-temperaments with their "key color" exploit a subtle scale of microtonal consonance and dissonance, and one I might add which seems based both on categorical recognition (even a triad in one of the most remote keys is still recognizable as a triad) and on tangible distinctions. Having become acquainted with the experiments of Gary Morrison with 88-cet, for example, where even his "supermajor" thirds of 440 cents (roughly 9:7) may successfully be treated as "consonant," I'd say that there is indeed a xenharmonic continuum, and that 400 cents is just one point on that continuum, neither especially bad nor necessarily the best. There are circumstances where 400 cents might be an ideal choice, specifically late Romantic pieces which seem to assume a close equivalence of all transpositions on the circle of fifths, and also 12-tone serialist works (yes, 11-tet serialism is a viable alternative, too). Maybe this just goes to prove what I'd call Darreg's Law: _all_ n-tet's offer a basis for beautiful music. Anyway, I'd say that musical grammar, categorical perception, and custom can explain a lot about why perceptions of consonance are fluid and flexible, as well as often a matter of some divergence in views, among n-limit JI and n-tet and "non-just, non-equal" enthusiasts on this list as well as among other people. Most respectfully, Margo Schulter mschulter@value.net ------------------------------ End of TUNING Digest 1570 *************************