Paul Erlich's meantone

> In 1/4-comma meantone temperament, the major thirds are tuned
> pure. The Ensoniq VFX-SD, despite its cents-based tuning tables, has
> a true internal tuning resolution of 512 notes per octave
> (unbeknowest to most). The best approximation to the major third is
> 165 steps of the tuning, or 386.7188, as compared with the just
> value of 386.3137 cents. A major third played in the middle
> register with the 165/512 oct approximation would beat once every
> two seconds, too slow to be noticeable in most music.

Hello, there. In fact, doing a calculation for f'-a' (easy using
a'=440), I got a beat once every 2.43 seconds or so. This is much
closer to just, of course, than the schisma thirds described by
Helmholtz -- and, as we now know, used in lots of early 15th century
keyboard music where they contrast with regular Pythagorean thirds.

> The major third is constructed from four fifths, which would have to
> be approximated by an average of (165+512+512)/4, or 297.25
> units. So tuning every fourth fifth to 298 units (698.4375 cents)
> and the rest to 297 units (696.0938 cents) makes all 8 major thirds
> in the tuning virtually just. This is my 1/4-comma meantone setting
> on my Ensoniq.

Interestingly, 512-step and 1024-step synthesizers share this best
approximation for 1/4-comma meantone. With a 1024-step device like the
Yahama TX-802, tuning two fifths of 594/1024 oct (696.0938 cents, the
same as 297/512 oct) and two of 595/1024 oct (697.2656 cents). That
averages out as 594.5 units in 1024-tet -- the same as your 297.25 in
512-tet -- likewise yielding a major third of 386.7188 (330 steps,
equivalent to 165 steps out of 512).

Curiously, 768-tet (e.g. Yahama TX81Z) does a bit better than either
512 or 1024, although on the narrow rather than wide side. Three
fifths of 446/768 oct (696.8750 cents) and one of 445/768 oct
(695.3125 cents) -- an average fifth of 445.75 steps -- yield a major
third of 385.9375 cents (247 steps), or 0.3762 cents narrow. With 512
or 1024 steps, our best approximation of 386.7188 cents is 0.4051
cents or so wide.

> Any tuner during the period when meantone was predominant would be able
> to tune the four augmented triads (Eb G B, Bb D F#, F A C#, C E G#)
> almost perfectly by ear by eliminating beats in the constituent major
> thirds, but tuning them to one another using approximately identically
> tempered fifths was an imprecise endeavor. So in practice the tuning
> could rarely have been much better than the approximation I have
> described.

Thank you for explaining this process, and especially in focusing on
this major source of imprecision, which I hadn't considered.

Incidentally, I've heard that Zarlino's 2/7-comma meantone tuning fits
lots of 16th-century music very nicely (especially pieces having lots
of emphasis on minor thirds), but must have been hard to tune.

Also, congratulations to you and John Chalmers and all the other
contributors on _Xenharmonikon_ 17! Your discussion on the
stylistic criteria for a scale in the context of a given harmonic
system are very intriguing, and I hope to be asking some questions
here.

Most appreciatively,

Margo Schulter
mschulter@value.net