Meantone Explanation

Somebody asked me about the exact meaning of fifth-comma meantone.
Those of you who are new to unusual tunings might find this useful
information.


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The usual definition of an nth-comma meantone would be something like "a
tuning based upon a circle of identical-sized fifths tuned flat of the 3:2
by 1/n commas, wrapping within an octave".

Assuming that you use the syntonic comma in that definition, as opposed
to the pythagorean comma, then quarter-comma meantone is devised to create
a just 5:4 at the sacrifice of the accuracy of the 3:2. That is because a
stack of four fifths lands you a major third above where you started: C G
D A E. If those fifths were exactly just, then you'd arrive at an 81:64
ratio, which is a syntonic comma (81:80) sharp of the idea 5:4. So the
quarter-comma meantone solution to that problem is to compromise the P5
down by 1/4 of a comma so that when you stack up 4 of them you end up 4/4
of a comma flat of 81:64, which is of course 5:4.

Similarly, third-comma meantone is devised to temper the 27:16 major
sixth down to the cleaner-sounding 5:3.

As you've probably heard by now, 31TET is virtually identical with
quarter-comma meantone, and 19TET, for all realistic purposes, *IS*
third-comma meantone. If you stack up 31 quarter-comma flat fifths you
land only 6 cents off from where you started, and if you stack up 19
third-comma flat fifths you land only about 1 cent off.

So what about fifth-comma meantone? As with sixth-comma meantone, that
is probably best characterized as well temperament. That particular size
of fifth was probably chosen more to generally equalize tuning errors
within a 12-toned framework, rather than to make any particular interval
sound accurate. That in the sense that whether a 243:128 major seventh is
really all that much more harmonically unpleasant than 15:8 major seventh
(a comma flatter) or than any major seventh for that matter, is pretty
marginal.

Or relative to the tonic anyway; 15:8 is of course is mediant of the
dominant, so perhaps one could argue it that way. Perhaps one could argue
that fifth comma meantone was devised to make the third of V-chord just in
the same sense that third-comma meantone makes the IV-chord just. That
combined with the fact that the total error between the third of I (the
mediant) and the third of V (the leading tone) is smaller that way. In the
quarter-comma meantone case, the mediant's error is 0 and the leading
tone's is 1/4 comma, whereas in fifth-comma meantone a total error of 1/4
comma, the mediant's is 1/5 comma, and the leading tone's is 0, a total of
1/5 comma.

But I think that when you add to that formulation the error in the third
of the IV-chord (the submediant), and also consider the effects upon minor
keys, I suspect that the well-temperament explanation may be more
realistic. Well anyway, fifth- and sixth-comma meantone definitely
straddle the boundary between well temperaments and classic meantones. By
the pure definition, they're certainly meantone tunings, but their
motivation seems to match more closely that of well temperaments.

Oh, by the way, I think that Switched-On Bach 2000 has some fifth-comma
meantone on it.

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