RE: Nano-temperament

>The method of nano-temperament seems useful in a range of musical
>contexts, for example in approximating various shades of meantone.
>Since a meantone major third is derived from four fifths, it is
>possible by varying the size of _some_ of these fifths by one tuning
>unit to achieve shades of _average_ temperament only 1/4 tuning unit
>apart. The major thirds will vary by an amount four times this size: a
>single tuning unit.

The principle Margo delineates for realizing Pythagorean tuning is one I
recently used to realize 1/4-comma meantone, the most important tuning
of 1500-1700, on my Ensoniq synthesizer. Though I recently posted the
basic idea in this, I will repeat myself for the benefit of this
discussion.

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.

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.
Just as in Margo's Pythagorean nano-temperament, the minor thirds here
vary in size.

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.

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End of TUNING Digest 1486
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