Re: More N-out-of-M (What is microtempering?)

Hello, there, Dave Keenan and everyone.

An example of what I suspect may fit our similar concepts of
"microtempering" occurs to me, with a nod in the direction of Graham
Breed: tempering the fifths of a regular Pythagorean tuning very slightly
wider than pure in order to obtain purer ratios involving factors of 7
(e.g. 9:7, 7:4, 7:6).

For example, as I recall, Graham favors 135-tET, where the fifth is at
702.222... cents, or about 0.27 cents wider than a pure 3:2, producing
almost exactly pure ratios of 8:7 or 7:4.

Here the "microtempering" is so small that the fifth would meet your test
of "just," that is, within about 0.5 cents of the theoretically pure
ratio.

By the way, taking this standpoint, I might urge by contrast that an
historical meantone is _not_ "microtempered," because the very tangible
impurity of the fifths is noted as a leading feature in some of the
earliest practical guides for tuning this type of temperament.

As has been mentioned, 1/4-comma meantone might be considered "semi-just"
in a different sense because the 5:4 major thirds _are_ pure, and in some
keyboard compositions the closing sonority has a major tenth above the
bass without any fifth, for example a just 5:10:12:20 (e.g. D3-D4-F#4-D5,
with C4 as middle C). Mark Lindley refers to the "coupling" of partials
for the 5:4 as a feature of 1/4-comma contrasting with Zarlino's
2/7-comma, where all the intervals have a gentle beating which can be very
pleasant in some 16th-century textures.

In a very pragmatic discussion of organ tuning, for example, Arnold
Schlick (1511) expresses the idea of the syntonic comma in vivid albeit
nonmathematical terms: the fifths must be made to "suffer" in order that
the thirds may not be "altogether too high" for a Renaissance style.
This has a sound almost of John deLaubenfels and his "pain."

The main point is that in historical meantones, the fifth is quite
noticeably tempered; in a "microtemperament" such as Graham's 135-tET, the
temperament is little if at all noticeable.

Of course, we can agree in distinguishing 0.27 cents from 5.38 cents while
still leaving open the question of how much "microtemperament" borders on
"macrotemperament."

One possible rough border zone: might something like a 3-5 schisma (~1.95
cents), or the amount by which the fifth is tempered in 12n-tET, serve as
one example of a rather large "microtemperament" in a "JI-like" setting.
For example, Helmholtz and Ellis seem to take the view that this is a
rather small deviation from just, and actually implement a 1/8-schisma
tuning (a 3-5 schisma, for a ~5:4 from eight fifths down) with some fifths
tempered a full schisma narrow, or 16384:10935, virtually identical to
12n-tET.

Here my purpose isn't to propose an exhaustive definition, only to give
some examples and invite your comments on whether we are in general
agreement that "microtemperament" means tuning intervals slightly _away_
from just ratios (or the most beatless or "locking-in" ones, if you
prefer), in order to make other consonances more accurate or to permit
more such consonances, etc.

Most appreciatively,

Margo Schulter
mschulter@value.net

Margo,

I totally agree. My current thinking is that it can still be called a
microtempereament if it has up to about a 2.7c error, provided that
the error in ratios of 11 or higher primes is smaller than that.

So yes, 12-tET is microtempered at the 3-limit only. Meantone is not
microtempered anywhere.