Re: Meantone 31 (1/4-comma, 5:4 major thirds) circulates

Hello, everyone, and after reading some recent discussions about
1/4-comma meantone (pure 5:4 major thirds) vs. 31-tone equal
temperament (31-tet), I just wanted to clarify the point (in case
there was any doubt) that these are both very nice circulating
temperaments when carried to a full set of 31 notes per octave.

Thus if people prefer the pure major thirds of 1/4-comma, there's no
need to adopt 31-tet instead for the sake of circulation. Similarly, a
pure Pythagorean tuning carried to 53 notes is a fine circulating
tuning -- although in neither case is there _absolute_ symmetry or
_precise_ mathematical closure as with 31-tet or 53-tet.

A circulating tuning need not be an _equal_ temperament, and 31-note
meantone with pure major thirds or 53-note Pythagorean provides quite
satisfactory _musical_ closure, although not exact mathematical
closure.

The slight "inaccuracy" in the closure of Pythagorean 53 (~3.62 cents)
or 1/4-comma meantone 31 (~6.07 cents) is in either case a much
smaller factor of variation than that tolerated in 18th-century
well-temperaments, for example.

In the case of a circulating 31-note meantone tuning, we have two
lines of historical and recent tradition: the Vicentino-Colonna line
calling for a "common practice meantone" tuning of the first 12 notes
(Eb-G#) or 19 notes (Gb-B#), very likely meaning 1/4-comma with pure
major thirds; and the Rossi-Huyghens-Fokker line calling for precise
31-tet with major thirds very slightly larger than pure.

In either case, we have 6.07 cents or so of "imperfect closure" to
deal with -- the amount by which 31 fifths tempered by 1/4-comma fall
short of 18 pure octaves. In 31-tet, this discrepancy gets distributed
nice and evenly among _all_ the fifths. In a 31-note version of
1/4-comma meantone, it all gets distributed to _one_ odd fifth.

Let's see if there are any terrible consequences from the viewpoint of
a circulating temperament of dumping this 6.07 cents on a single
fifth:

1. The affected fifth, which would regularly be ~5.38 cents
narrow, is instead almost exactly a just 3:2! --
actually, ~0.69 cents wide (~702.65 cents). Maybe this
phenomena provides the best response to the argument
against this kind of "semi-JI" meantone with pure 5:4's:
"All the fifths are too narrow for my taste."

2. Four major thirds including this odd near-pure fifth
will be ~6.07 cents wider than just, somewhat as in
2/11-comma meantone, about the limit in tastes for
regular 16th-century meantones (~392.38 cents).
These major thirds are built from 27 fifths down
rather than the usual 4 fifths up; when playing
them, we might recall Mark Lindley's comment that
1/5-comma meantone can be "sprightly" for some
English keyboard music around 1600.

3. Three minor thirds will be ~6.07 narrower than
their regular meantone size (already ~5.38 cents
smaller than 6:5) -- in all, ~11.45 cents from
just, or ~304.19 cents. These minor thirds are
built from 28 fourths down rather than 3 fourths
up. Maybe we could promote these three thirds as
leaning a bit toward 19:16.

4. Intervals built from longer chains of fifths,
such as the "proximate minor thirds" or "neutral
thirds" a diesis wider than minor (~11:9), and
the "minimal seventh" a diesis narrower than
minor (~7:4), will have two slightly different
"flavors" differing by ~6.07 cents.

In short, people seeking a 31-note circulating meantone temperament
who like pure major thirds can happily stick to the Vicentino-Colonna
tradition; people who like precise mathematical symmetry or
intervallic consistency can go with the Rossi-Huyghens-Fokker
tradition of 31-tet.

Similarly, the choice between Pythagorean 53 and 53-tet might be
influenced by various factors, but either choice circulates.

Please let me add a disclaimer: my own involvement is mainly with
noncirculating versions of Pythagorean and meantone with 12-24 notes.

Most respectfully,

Margo Schulter
mschulter@value.net

Margo!
I agree and feel this is an important point! Let me say that even
though my very first instrument was in 31 ET I found that when i went to a
JI tuning, I wished to preserve the features of a 31 tone scale. Even if it
was different that the fine ones you suggest merely shows its versatility!

"M. Schulter" wrote:

> From: "M. Schulter" <MSCHULTER@VALUE.NET>
>
> Hello, everyone, and after reading some recent discussions about
> 1/4-comma meantone (pure 5:4 major thirds) vs. 31-tone equal
> temperament (31-tet), I just wanted to clarify the point (in case
> there was any doubt) that these are both very nice circulating
> temperaments when carried to a full set of 31 notes per octave.
>
> Thus if people prefer the pure major thirds of 1/4-comma, there's no
> need to adopt 31-tet instead for the sake of circulation. Similarly, a
> pure Pythagorean tuning carried to 53 notes is a fine circulating
> tuning -- although in neither case is there _absolute_ symmetry or
> _precise_ mathematical closure as with 31-tet or 53-tet.
>
> Most respectfully,
>
> Margo Schulter
> mschulter@value.net

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com