Re: tuning question

In rec.music.early Dale <[EMAIL PROTECTED]> wrote:

> Given a musician with an ear trained to meantone tunings (say circa
> 1630), and a properly tuned modern piano in even-temper, could you
> derive a well-temper tuning from the two?

Hello, there, and my response might be that indeed one _could_ "derive" an 
unequal 12-note well-temperament as a kind of "compromise" between a 
characteristic 16th-17th century meantone temperament as 12-note equal 
temperament -- but that this would be a very laborious way of going about 
things if one simply wants some kind of well-temperament. There are other, 
more likely, ways to go about it -- some of them described in 16th-century 
sources, and others described or inferred from certain 17th-century 
sources.

First of all, 12-note equal temperament or a close approximation _was_ 
known and common at least from around 1545 or 1550 on for lutes, although 
some people preferred other tuning systems for this instrument, and one 
might tune a harpsichord, for example, with such an instrument as a guide. 
Vincenzo Galilei reports trying this, or something like it, in his 
treatise of 1581, but found the tuning, although "perfect" on the lute, to 
be less supportable on the harpsichord with its more "vehement" sound 
production and different material for strings. He preferred 2/7-comma 
meantone, the favorite tuning of his former teacher Zarlino, with whom he 
differed on many other points.

Without the help of such an instrument, one question is just how 
accurately one was likely to tune 12-note equal temperament in the 1630's 
by ear; even in the late 19th century, when this tuning was the new 
theoretical ideal, Victorian pianos often tended toward "shades of equal 
temperament" with some degree of diverse if subtly different colors for 
different transpositions or keys. It would be must easier to tune such a 
"gently shaded" well-temperament than something closely equivalent to 
12-equal.

My main reaction is that to get a well-temperament around the 1630's, 
starting with meantone or the like, there are two easier kinds of methods 
-- depending on just how one defines a "well-temperament."

The simplest approach, which could actually have started as a kind of 
"mistake" in reading the instruments for a regular meantone, is to tune 
eight or nine of the fifths in a usual meantone, and then tune the 
remaining fifths about equally _wide_ of pure. For 1/4-comma or 2/7-comma, 
typical 16th-century meantones, we could tune eight fifths or nine notes 
as usual (e.g. F-C#), and then make the fifths Bb-F, Eb-Bb, C#-G# -- and 
also the "odd" fifth G#-Eb, a "Wolf" diminished sixth quite different from 
a usual concordant fifth in regular meantone! -- about equally wide.

Whether we choose to call this a "well-temperament" depends on how wide 
we're willing to have the largest major thirds. In this kind of scheme, 
which I call a _temperament extraordinaire_ (a variation on the French 
term _temperament ordinaire_ used for various 17th-18th century schemes, 
some rather like what I describe), the widest major third or diminished 
fourth (C#/Db-F) is the same size as a diminished fourth in the 
regular meantone version of the basic temperament -- 32:25 (~427 cents) in 
1/4-comma, and around 434 cents in the Zarlino 2/7-comma scheme (very 
close to a pure 9:7, ~435 cents). This is fine where the music calls for a 
diminished fourth, and not so optimal if a regular major third like Db-F 
is desired! Such a scheme does _not_ fit the conventional 18th-century 
definition of "well-temperament" as a system where all major thirds are 
no larger than Pythagorean (81:64, ~408 cents), and all fifths either pure 
or narrow.

Personally I love this kind of temperament extraordinaire because I find 
ratios such 32:25 or 9:7 very useful for introducing 14th-century style 
progressions: thus I tend to use remote transpositions for a 
"neo-medieval" style interspersed with more conventional Renaissance or 
Manneristic (16th-early 17th century) progressions in the nearer 
transpositions often identical or close to a standard meantone.

For something closer to a 17th-century temperament ordinaire, we might 
choose a basic meantone temperament around 1/5-comma or 1/6-comma, and 
temper _nine_ fifths (or ten notes) regularly, for example F-G#. Then the 
fifths Bb-F, Eb-Bb, and also G#/Ab-D#/Eb, are made equally wide; this is 
the kind of thing a tuner might have done by "creative mistake" in 
following standard meantone tuning instructions for the fifths involving 
only naturals or sharps, but then tuning fifths involving flats wide 
rather than narrow.

In this kind of scheme, the tempering on all the fifths is milder (narrow 
or wide), and the widest major thirds or diminished fourths not so 
much larger than Pythagorean, although still somewhat larger. For someone 
acquainted with standard meantone tuning in the 1630's, and especially 
someone leaning toward a milder degree of temperament (and more impure 
regular thirds), this could be the shortest and easiest road to something 
like a "well-temperament," or more precisely a temperament ordinaire of 
the kind which Mark Lindley suggests Couperin might have used. Here the 
widest major thirds are definitely "special effects" intervals.

An approach actually described by Arnold Schlick around 1511 uses more 
than two sizes of fifths: he advises tuning tempering the white-note 
fifths (F-B) someone more heavily than the narrowed fifths involving 
accidentals (Bb-F, Eb-Bb, B-F#, F#-C#), and with all regular major thirds 
somewhat wider than pure. Then the fifth Ab-Eb is made _wide_ so that Ab-C 
can serve as a reasonably good third, and E-G# as a passable third for 
ornamented cadences on A (calling for the semitone G#-A). Depending on how 
one interprets this scheme, the odd fifth C#-Ab/G# might be around 8-10 
cents wide -- more impure than in any regular meantone tuning of the era, 
but not necessarily an outright "Wolf."

This scheme, from the 16th century, has the main difference from a classic 
"well-temperament" of the 18th century that some of the major thirds or 
diminished fourths will, again, be considerably wider than Pythagorean. 
One could take this either as "unacceptably dissonant," or as "strikingly 
colorful."

Going to a classic 18th-century well-temperament has the main drawback 
that meantone color in some prominent locations often gets compromised 
considerably more than in these schemes. The mathematics require that if 
you have eight or more fifths in a characteristic meantone, then some or 
all of the other fifths will have to be wide. Conversely, if you want all 
fifths pure or narrow (to keep all major thirds or diminished fourths no 
wider than Pythagorean), then you'll have to use less narrow temperament 
within the range of the more common transpositions than in a 
characteristic meantone.

When doing a classic well-temperament of the kind described by
Werckmeister in 1681, it's convenient to measure temperament in fractions
of a Pythagorean comma (531441:524288, ~23.46 cents), since this is the
amount of narrow temperament to be balanced out in the tuning circle
without tuning any fifths wider than pure.

For example, we might tune the four fifths C-G-D-A-E narrow by 1/4-comma 
(here a Pythagorean comma), and tune all of the other fifths pure. In 
practice, there might be slight variations, but the object would be to get 
the tempered fifths about equally impure, and the others pure, with all 
fifths and thirds reasonably "playable."

Another approach is to temper five of the fifths by 1/5-comma each, or six
by 1/6-comma each (again Pythagorean), and keep the others pure.

How accurately these schemes might have been tuned in the 1630, if someone 
were so inclined -- or were tuned in the 18th century, when they were 
standard for stringed keyboards and sometimes also used on organs (where 
meantone was still a popular standard) -- are open questions.

However, there's no need for 12-equal as a "standard" in approaching any 
of these temperaments. True, there are reports in the 16th and early 17th 
century of a few people using 12-equal on keyboard: Vincenzo Galilei 
(father of the astronomer Galileo), as I mentioned, liked the idea in 
theory but found it less pleasing in practice; in the early 17th century, 
there are reports of people in France and Italy doing it. One reaction is 
that it requires lots of skill, and another is that it might sound better 
"if we were more accustomed to it."

The advantage of an unequal temperament extraordinaire or ordinaire, or of 
a classic well-temperament of the kind that Werckmeister described 
starting in 1681, is that you can follow a simple or familiar procedure 
for many of the fifths: tune most of them in meantone (temperament 
extraordinaire or ordinaire), or pure (classic well-temperament), with the 
rest "modified."

Mathematically, of course, 12-equal might be called an "isotropic" tuning
for 12 notes forming a circulating system: each fifth is tempered by
precisely 1/12-Pythagorean comma (very close to 1/11-comma meantone, by
the way, since meantones are generally measured by the smaller syntonic
comma at 81:80 at ~21.51 cents describing the difference, for example,
between an 81:64 Pythagorean major third and a 5:4 major third at ~386.31
cents). Thus you _can_ describe any other 12-note tuning as having
intervals which deviate from the "absolute symmetry" of 12-equal -- but 
this isn't necessarily the simplest, or historically likeliest, approach 
to deriving a temperament ordinaire or well-temperament or the like.

We can discuss this more, from the viewpoint of a science fiction 
scenario: how, for example, might listeners or theorists of the 1630's 
react to a 12-equal piano. Here we'd want to take into account not only 
the tuning itself, but also, as Vincenzo Galilei reminds us, the question 
of timbre: a piano, like a lute, has less prominent fifth partials than a 
harpsichord, so that wider major thirds at 400 cents might not seem so 
tense as on the "brighter" instrument.

With the benefit of some actual observations from the period of interest 
by people such as Mersenne and Doni, we can at least consider some 
"historically informed" science fiction scenarios.

Most appreciatively,

Margo Schulter
[EMAIL PROTECTED]