Re: 53-tet, meantones, etc. -- for Monz, Paul, and Graham

Dear Monz, Paul, Graham, and everyone.

Synchronicity of ideas and discoveries can be a fascinating thing: for
example, the simultaneous formulation of the calculus by Newton and
Leibnitz. This morning, preparing to put the final touches on a paper
I should soon by posting here (probably in two parts because of
length), I was fascinated to see on reading Tuning Digest 700 that all
of you, and especially Graham, were touching on some central themes of
this paper. Of course, this is in part a reminder that these themes
in turn owe much to the Tuning List, and it should be interesting to
see how conventional or unconventional people regard my viewpoint when
I post this article.

Anyway, here are a few responses as part of the current dialogue on
53-tet, meantones, Xeno-Gothic and congenial n-tet's, etc.

Note that I've attempted quickly to revise the opening of the first part
for Monz and Paul after reading a new Tuning Digest (701) that just came
in; my apologies for any possible errors or infelicities.

-------------------------------------------------------------
1. For Monz and Paul: 53-tet and meantones -- yes, and 22-tet
-------------------------------------------------------------

First of all, Monz, apart from the whole "meantone" issue on which you and
Paul now seem in agreement, I agree, of course, that if we are looking for
the closest possible emulation of Pythagorean (for whatever purpose), then
53-tet (or some applicable subset, e.g. 24 of 53 for a usual Xeno-Gothic
mapping on two keyboard) is the obvious choice. It's further correct, of
course, that 53-tet has "schisma thirds" of a near-5-limit size almost
identical to the Pythagorean variety used in the early 15th century, as
well as "septimal schisma" intervals varying a "comma" (1/53 octave) from
their regular Pythagorean flavors in the opposite direction.

At the same time, Paul (and Monz), I agree that if we take "meantone" in
its traditional historical sense to imply "a scale where fifths are
tempered on the narrow side so that four of them form a pure or near-pure
5-limit major third," then 53-tet is _not_ a 5-limit meantone
approximation.

A very simple way of demonstrating this is to note that in 53-tet
treated as a 5-limit approximation, a ~5:4 major third will be 17
steps. Now 17 is an odd number, and cannot be divided into two
"mean-tones" of equal integer size which together produce this
third.

Instead we have what you have nicely described as an emulation of
classic 16th-century just intonation (JI) based on Ptolemy's syntonic
diatonic, with large whole-steps at ~9:8 (9 steps) and small ones at
~10:9 (8 steps), a la Fogliano and Zarlino. Together, these two
unequal whole-tones do form a near-pure major third, just as a 9:8 and
a 10:9 together produce a 5:4.

If we want a 53-tet-like tuning that _does_ support meantone in the
usual sense, then maybe 106-tet would be an ideal solution. Here our
near-5:4 major third would be 34 steps, so that a mean-tone of 17
steps (the mean of 9/53 octave and 8/53 octave) would indeed answer to
your requirements. This mean-tone is ~192.45 cents, very slightly
smaller than the whole-tone in 1/4-comma meantone (~193.16 cents)
because our 53-tet or 106-tet "schisma third," like its Pythagorean
counterpart, is slightly narrower than 5:4.

Of course, if we take "meantone" in the broadest generic sense as a
regular tuning (all regular fifths equal), and consider Pythagorean as
"zero-comma meantone," then 53-tet is an excellent meantone. We have a
whole-tone of 9 steps, and two make a major third (or ditone) of 18
steps, an almost exact emulation of the Pythagorean 81:64.

This is the framework in which I tend to look at 53-tet, in which
those intervals of 17 steps are "schisma thirds." In fact, I remark in
a footnote in my paper that from a Pythagorean point of view, people
who use 53-tet as a 5-limit JI system are redefining the schisma
thirds as regular thirds.

Curiously, while writing my paper, I realized that there is another
n-tet which has a close analogy with 1/4-comma meantone, with a
whole-tone almost at the mean between two integer ratios, but forming
a different flavor of major third. Which scale? -- yes, it's no other
than 22-tet, taken in a neo-Gothic manner.

In this interpretation, we have whole-tones of 4 steps (~218.18
cents), just slightly larger than the mean between 9:8 and 8:7; and
two of these "mean-tones" make a regular 22-tet major third just a tad
larger than a pure 9:7 (8 steps, ~436.36 cents). When I recognized
this yesterday, I saw the analogy with 1/4-comma and its mean-tone
between a 9:8 and a 10:9 so that two make a 5:4.

This example also shows how in n-tet's, the number of steps in
whole-tones and semitones can depend on your musical and stylistic
viewpoint. To me, from a neo-Gothic viewpoint where major thirds in
the region of 81:64 to 9:7 expanding to fifths are a prime attraction,
22-tet is a regular tuning, indeed a close approximation of an exact
"meantone" based on the 9:7 (as 31-tet is very close to 1/4-comma).
From a decatonic viewpoint, or some other viewpoint than a
3-stability-limit (neo-Gothic) one, the step counting for intervals as
well as musical style would be quite different, as your article in
_Xenharmonikon_ 17 shows.

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2. For Graham -- n-tet's and Xeno-Gothic
----------------------------------------

Thank you for some very perspicacious and intriguing comments which
very much share a major focus of my forthcoming paper: the use of
"neo-Gothic tunings and temperaments" including n-tet's where the
fifths are either pure (or minutely smaller in 53-tet) or wider than
pure.

Thank you for the very important insight that lots of Xeno-Gothic
progressions (conceived in Pythagorean 24, but indeed "portable" to
many n-tet's, with fascinating shades of variation) don't involve
"near-5-limit" ratios at all.

Near-5-limit schisma thirds (Pythagorean diminished fourths and
augmented seconds) are important in early 15th-century music, and so
are part of the Xeno-Gothic mix, but I would say that thirds modified
in the opposite direction (even more active and dynamic than
Pythagorean) may be at least equally important and a better clue to
what Xeno-Gothic or neo-Gothic is really about.

In fact, I'm much attracted to temperaments (equal and otherwise)
which have fifths reasonably close to 3:2 on the wide size, but no
close approximations of simple 5-limit intervals, at least within the
subset that I'm likely regularly to use. Here I say "simple" 5-limit
intervals, because the diatonic semitone of 17-tet (1 step, ~70.59
cents) could be viewed as an excellent ~25:24, the _chromatic_
semitone of 5-limit <grin>.

While your latest post and my almost-completed paper focus on some of
the same tunings (17-tet, 29-tet, 41-tet, 53-tet), you discuss a point
I really don't get into, and which is definitely worth lots of
attention in musical practice and theory: comma size, especially in
relation to commas as melodic intervals.

As you point out, as fifths get larger than pure, the comma
(difference between diatonic and chromatic semitones) rapidly gets
larger also. It's interesting that you find 41-tet, where this comma
is around 29 cents, as a point where it becomes more of a clear
"melodic elaboration." I wonder if there could be a connection here
with the question of when people starting hearing an interval as a
very small minor second rather an imprecise unison.

In 29-tet, the comma is around 41 cents, interestingly very close to
the diesis of 1/4-comma meantone which Vicentino uses in his
"enharmonic" music. This tuning, and others in the general vicinity
with slightly larger fifths, thus somewhat resemble Renaissance
meantones in having a dramatic contrast between the size of the two
semitones, say around 40-60 cents (with the _diatonic_ semitone
smaller in neo-Gothic and the chromatic one in usual meantone, of
course). Making the most of this feature in a neo-Gothic setting, as
Renaissance keyboard composers did with their meantone chromaticism
(and Vicentino's "enharmonicism" using the diesis as a melodic
interval), is a fascinating theme for new music as well as theory.

Your mention of 135-tet is a good alert to the limitations of only
considering n-tet's through 53, an arbitrary limitation of my paper,
and also raises very interesting questions about the ways different
people view n-tet's in general -- topics for lots more dialogue.

Most appreciatively,

Margo Schulter
mschulter@value.net

In-Reply-To: <Pine.BSF.4.20.0007061350010.59696-100000@value.net>
Margo Schulter wrote:

> Thank you for the very important insight that lots of Xeno-Gothic
> progressions (conceived in Pythagorean 24, but indeed "portable" to
> many n-tet's, with fascinating shades of variation) don't involve
> "near-5-limit" ratios at all.

Well, you'll know about the "lots", I only know the one you showed us
before.

> As you point out, as fifths get larger than pure, the comma
> (difference between diatonic and chromatic semitones) rapidly gets
> larger also. It's interesting that you find 41-tet, where this comma
> is around 29 cents, as a point where it becomes more of a clear
> "melodic elaboration." I wonder if there could be a connection here
> with the question of when people starting hearing an interval as a
> very small minor second rather an imprecise unison.

Ah, what I said is that 41 and 53-equal have characteristically different
commas. That is, the cutoff is between a 41st and 53rd part of an octave.
It happens to be a lot closer to 53. I find Pythagorean, or at least the
perfect 702 cent fifth, tuning to have an adequate comma. It takes a bit
of getting used to, but doesn't have the troubling quality of 53=. That
means the exact point must be somewhere between 22.6 and 24.0 cents (or my
synthesizer's approximations thereof).

These are conclusions I came to a while ago, and follow from my first
impressions after setting up the schismic keyboard tuning. I've mostly
avoided the flat-fifth tunings since.

> Your mention of 135-tet is a good alert to the limitations of only
> considering n-tet's through 53, an arbitrary limitation of my paper,
> and also raises very interesting questions about the ways different
> people view n-tet's in general -- topics for lots more dialogue.

Oh, I think 53 is already on the high side for an ET. 135-equal is purely
a convenience for calculation. A just 7:4 leads to a 1/44.9 oct comma, so
setting it to 1/45 oct is an obvious simplification. That happens to
define 135= where the semitones are 7, 10 and 13 steps. Which means the
whole tuning can be described in terms of integers, keeping the arithmetic
nice and easy.

A just 7:4 means the 9-limit is accurate to within 5 cents, and the 3.3.7
subset to within 0.6 cents.

Graham