Yahoo Tuning Groups Ultimate Backup tuning What size is G-B? (for Joseph Pehrson)

Hello, there, Joe Pehrson, and thank you for an opportunity to
appreciate how maybe my outlook is "early music" in a way you discuss:
I am generally quite happy with "sloppy notation" that simply
indicates which notes on a keyboard are played, and maybe includes a
remark about the tuning, or general type of tuning, intended.

Please let me caution that I am not addressing the specific issues of
interval naming or notation in Blackjack, but rather my experience
with the tunings I typically use in terms of questions like "How large
is a major third like G-B, and what size should 'G-B' indicate?"

Here I'd go with some medieval scholastic opinion which asserted that
although Aristotle had proposed that the Earth is the "proper place"
at which falling objects seek their rest, an object dropped near the
surface of the Moon would seek the "center" of the Moon, and so
forth.

In other words, each tuning system or style has its own sense of
"normalcy," and to me it makes sense to let "G-B" represent,
typically, whatever the regular "major third" is in that tuning.

From my perspective, since a regular major at G-B often has a size of
around 408-429 cents, I find either 5:4 or 400 cents a rather curious
"standard." However, I'd gladly use 5:4 in 1/4-comma meantone, or 400
cents in 24-EDO or 36-EDO -- where this is indeed the size of G-B.

Why don't we move through a quick survey of tunings, including one
suggested, Dave Keenan, by your scenario with a meantone where the
major third is equal to 23/72 octave or 383-1/3 cents?

Joe and Dave, actually I sometimes use a tuning very close to that
specification, which also happens to be the first regular temperament
in Western Europe documented with mathematical precision: Zarlino's
2/7-comma meantone of 1558. In this tuning, the major third is 1/7 of
a syntonic comma narrower than just (and the minor third wider by the
same amount), or about 383.24 cents, within a tenth of a cent of the
Blackjack interval you were discussing.

In this meantone, G-B is this size -- the "major third" of this
particular intonational planet. The regular thirds aren't the only
just thirds, by the way -- the diminished fourth (e.g. E3-Ab3) is
around 433.52 cents, actually a bit more accurate than 36-EDO,
although the fifth and minor third are definitely a bit less so!
Nevertheless, we get a very nice 6:7:9 or 7:9:12, and in a wonderful
16th-century tuning.

What would I call the near 9:7 third in Zarlino's tuning? Maybe a
diminished fourth (in a more or less usual Renaissance style), or a
supermajor third if I'm using it in a neo-Gothic fashion. Anyway, I
would call it something other than a simple "major third," which in
this tuning is about 383 cents -- G-B, the regular third from a chain
of four fifths (G-D-A-E-B).

By the way, speaking of historical authenticity, following Zarlino
himself we could call that near-9:7 third a "semidiatessaron," meaning
a "less-than-fourth" -- in other words, a diminished fourth. We could
also follow Vicentino and call it a "proximate major third," rather
like the "supermajor third" concept. My notation confirms that
whatever we call it, this third is something other than regular. For
example, on a 24-note keyboard with two manuals a diesis apart (about
50.28 cents), here are two ways of writing a cadence from a
7:9:12-like sonority to a near-2:3:4 in neo-medieval style, with the
first also available in a basic 12-note tuning of Eb-G#:

Eb4 E4 C#*5 D5
Bb3 B3 G#*4 A4
F#3 E3 E4 D4

In the second example, the * shows a note on the upper keyboard,
raised by a diesis. I find this quite descriptive, and it's how I
write down a progression I find in a tuning like this.

Now suppose we shift to 36-EDO, where, like 72-EDO, a regular major
third is 400 cents. To me this isn't a general "standard," because I
consider the symmetry of 12n-EDO a rather specialized choice, but
certainly when I'm using it, I'm ready to recognize a 400-cent
interval such as G-B as the "usual" third.

In 36-EDO, a near-just third of special interest to me is the near-9:7
at 433-1/3 cents (13/36 octave or 26/72 octave), but I recognize again
that this is something different than the _regular_ major third.

Of course, one could remark, as you have, Dave: "Suppose that you were
playing a 22-EDO keyboard with the notes generated from a usual chain
of fifths, e.g. a 12-note tuning of Eb-G#. Now G-B would be about
436.36 cents, almost the same size as this 36-EDO interval."

This is quite correct, and in each tuning G-B in my notation stands
for the regular third formed by four fifths. It happens that either
the diminished fourth of Zarlino's tuning, or the third a sixthtone
larger than the regular 400-cent third of 36-EDO, is very close to the
regular major third of 22-EDO. However, each system is distinct.

Thus when I write "G-B" in meantone, I mean something at or near 5:4;
in 36-EDO, 400 cents; and in 22-EDO, something very close to 9:7. All
of these are regular "major thirds," and each tuning has its own
"standard."

Now let's move to a tuning where the regular major third, G-B for
example, has a size very close to another 72-EDO interval. In the
"Noble Fifth" tuning mentioned in the Scale Tree of Erv Wilson, and
described here during the summer of 2000 by Keenan Pepper, the fifth
is about 704.10 cents, giving a major third of about 416.38 cents,
almost identical to the 72-EDO third at 25/72 octave or 416-2/3
cents. Either interval is an excellent approximation of 14:11 (~417.51
cents).

The Wilson/Pepper tuning also has a large major or supermajor third at
around 446.76 cents, very close to 22:17 (~446.36 cents), which I
would notate, for example, as G*3-C4 in a 24-note tuning -- a fourth
less the diesis of about 49.15 cents.

It also, curiously, has a few rather exotic thirds in a 24-note tuning
at about 386.01 cents, a virtually just 5:4 (~386.31 cents), which I
would notate, for example, Eb4-F#*4. These thirds are formed from an
augmented second plus diesis, and as the notation may suggest, are
special "altered" intervals in this context, formed from 21 fifths up,
although in meantone they (or intervals of much the same size) would
define the usual "major third."

So far, we've been discussing tunings where all regular fifths, and
thus also regular major thirds, have a single size -- be it 383 cents,
400 cents, 416 cents, or 436 cents.

In some well-temperaments and related just systems, however, regular
major thirds may have a range of sizes. Thus in George Secor's 17-note
well-temperament, they vary from a just 14:11 to around 428.88 cents,
with the largest thirds (closest to the simple ratio of 9:7) in the
nearest keys or transpositions, and G-B as an example.

In a 17-note circulating temperament I came up with before learning of
that one, these thirds range from 14:11 to 9:7; in a 17-note just
tuning I've recently developed with much inspiration from Kathleen
Schlesinger and others, they range from the Pythagorean 81:64 (~407.82
cents) to about 440.24 cents. As it happens, G-B is a just 14:11 in
the first tuning, and very slightly narrower in the second.

Thus I would say that the notation should follow the system and the
style: a 418-cent major third feels "right" in one setting, and a
383-cent major third in another.

Of course, especially if one is writing a piece that might be
performed in a range of different systems, or by performers with
different expectations, then something like George Secor's notation
could be very helpful.

However, I must admit that I tend to take the regular chain of fifths
in a tuning as a "standard," which generally leads to a "regular major
third" (e.g. G-B) fitting the main stylistic purpose at hand. In
another words, letting G-B equal 383 cents for a 16th-century
meantone, but 418 cents for a neo-Gothic tuning, not only is
conceptually convenient but happens to fit what I'm doing musically.

By the way, there are circumstances where a "usual major third" could
be close to either a 5:4 or an 81:64, specifically in historical
tunings of the early 15th century or the late 17th-18th century era in
Europe.

It's well-known that in an 18th-century well-temperament, the major
thirds can range from around a just 5:4 to around Pythagorean, and a
similar situation occurs in the early 15th-century Pythagorean tunings
of the Dufay era that we've sometimes discussed.

In a 12-note tuning of Gb-B, for example, from a performer's or
composer's viewpoint, either G-B (81:64, ~408 cents) or A-C# (actually
A-Db, 8192:6561, ~384 cents) is a "usual major third." That means a
mixed "standard," one of the special charms of the epoch.

In a Pythagorean tuning of the Eb-G# variety, I'd say that 81:64 is
the "standard major third," while something like C#4-F4 is a
"diminished fourth," which is how Johannes Boen describes it in 1357.
As it happens, this altered interval at about 384.36 cents is very
close to the small major third in Blackjack at ~383.33 cents, or to
Zarlino's regular third in 2/7-comma meantone at ~383.24 cents.

Moving from 1357 to 1427, however, we find that _either_ size of third
is standard; and by 1487 or so, at least judging from what Ramos
(1482) and Gaffurius (1496) tell us, some kind of meantone temperament
has made a regular major third maybe quite close to 5:4 the new
keyboard "standard."

In my view, this _variability_ of intervals such as "regular major
thirds" in different tuning systems and stylistic contexts is a basic
element of xenharmonicism or "microtonality."

For me, to "get people to take microtonality seriously" means to let
people know about the range of available choices out there, with
beautiful music (historical or new) a very powerful form of advocacy.

Most appreciatively,

Margo Schulter
mschulter@value.net

What size is G-B? (for Joseph Pehrson)

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗jpehrson2 <jpehrson@rcn.com>