Re: Salinas and 19-note tunings -- response to Bill Alves

Hello, there, and this is a quick response to an inquiry posted by
Bill Alves regarding a 19-tone-per-octave tuning scheme of Salinas and
its connection to a chromatic keyboard with this number of notes per
octave.

First, please let me confess that I know of the Salinas scheme only in
general and second-hand terms, and am unfamiliar with the specific
instrument described. However, I can possibly offer a "semi-educated
guess" as to how such a keyboard _might_ be arranged.

As I understand it, Salinas described a form of meantone with
1/3-comma of temperament -- that is, the narrowing of each regular
fifth by 1/3 of an 81:80 (~21.51 cents) syntonic comma, producing pure
6:5 minor thirds. In this tuning, regular major thirds as well as
fifths are narrower than a just 5:4 and 3:2 respectively by this
factor of 1/3-comma or ~7.17 cents.

I've read that Salinas himself expressed mixed feelings about this
drastic a temperament in his treatise describing it (1571 as I
recall), and indeed 1/3-comma seems about the historical limit of
meantone in the narrow direction.

However, this 1/3-comma tuning is almost identical to 19-tone equal
temperament (19-tet), and around the same epoch, French composer and
theorist Guillaume Costeley advocated 19-tet and used it in a
spiritual chanson.

While 5-limit just intonation (with alternating 9:8 and 10:9
whole-tones) was often taken as the "natural" tuning for voices in
this period, I've not aware of any proposal to implement such a scheme
on keyboard. Rather, as Paul Erlich has rightly emphasized, meantone
was the assumed standard, and the basis for various keyboards (in
practice and theory) with more than 12 notes per octave.

Thus as Erlich himself aptly adds to underscore this point, Zarlino
himself favored a 2/7-comma meantone scheme for keyboards, which in
modern terms might be said to "distribute the error" rather equitably
between major thirds (pure at 1/4-comma) and minor thirds (pure at 1/3
comma). Mark Lindley has suggested that Zarlino's tuning might give a
special character to pieces with prominent minor thirds.

A 19-tone keyboard might use a wide range of meantone temperaments,
with the special advantage of 19-tet being that it makes possible a
neatly closed system. How important such a "closure" property is for
even much "wildly experimental" Renaissance music seems unclear to me.

Maybe a diagram might make this "closure" property clearer. In any
characteristic meantone tuning, the diatonic semitone is the _large_
semitone (e.g. c#-d), and the chromatic semitone the _small_ semitone
(e.g. c-c#). In 19-tet, more specifically, each diatonic semitone is
2/3-tone, and each chromatic semitone 1/3-tone:

fb cb'
c# db d# eb e# f# gb g# ab a# bb b#
c d e f g a b c'

Since the effect of a sharp is to raise a note by 1/3-tone (a
chromatic semitone), and the effect of a flat is to lower it by the
same amount, either e# or fb precisely bisects the 2/3-tone interval
e-f, and likewise cb'/b# bisects b-c'. In each case, these pairs of
identical tones can be represented by a single key, and the circle
closes on itself.

It would also be very possible to map the same 19-tone keyboard in
Nicola Vicentino's 31-tet (almost identical to 1/4-comma tuning with
pure 5:4 major thirds), and indeed the first 19 notes of Vicentino's
_archicembalo_ or "superharpsichord" are tuned in just this way. Note
that this tuning has diatonic semitones of 3/5-tone, and chromatic
semitones of 2/5 tone, so that the pairs E#/Fb and B#/Cb remain
distinct notes, with room left for the other basic 12 notes of the
31-tet scheme (a @ shows a note raised by Vicentino's "diesis" or
1/5-tone):

db@ eb@ gb@ ab@ bb@
c@ d@ e@ f@ g@ a@ b@
---------------------------------------------------------------------
c# db d# eb e# f# gb g# ab a# bb b#
c d e f g a b c'

Here e@ (1/5-tone up from e) might also be written as fb (2/5-tone
down from f), and likewise b@ as cb.

(If I'm correct, a curious property of 31-tet is that e# is _higher_
than fb, and b# higher than cb, contrary to the usual meantone pattern
of the first 19 notes that c# is _lower_ than db, etc.)

It may be noted that neither in a "Salinas/Costeley 19" keyboard
(19-tet), nor in a "Vicentino 19 out of 31" keyboard, do we have
intervals smaller than the diesis of 1/3-tone or 1/5-tone
respectively. This diesis represents the difference between c# and db,
for example, and between a regular major third (e.g. d-f#) and a
diminished fourth (e.g. d-gb).

However, Vicentino _does_ propose a 31-tet keyboard with 36 or 38
actual notes per octave in order to include a few "comma keys"
permitting alternative intonations for certain frequently used
sonorities. While the term "comma" can apparently mean various kinds
of small intervals in this treatise, one attractive interpretation is
that these keys might actually be raised by about _1/4 syntonic comma_
(~5.38 cents) in order to permit sonorities like d-f#-a with just 3:2
fifths. Or, if one presumes precise 31-tet, they would be raised by
~5.18 cents, with major thirds actually a tidge wider than 5:4,
etc. -- but "virtually" just intonation, in any case.

Yet more interestingly, in an Aristoxenian (or Erlichian?) fashion,
Vicentino proposes that his 31-tet keyboard be taken as a standard for
singers in judging and perfecting intervals. Maybe the recent Tuning
List topic of "Adaptive JI" might be as good a name as any for this
kind of approach.

In short, either a Salinas/Costeley 1/3-comma or 19-tet scheme, or a
Vicentino 1/4-comma or 31-tet scheme, might be implemented on a
19-tone instrument. In the former case, we have a closed tuning; in
the latter, we have in effect "19 out of 31."

To borrow Vicentino's terms, a 19-tone keyboard under either scheme
implements the diatonic and chromatic genera. With 19-tet, this is a
complete and closed system; with 31-tet, it remains open, leaving room
for the additional 12 notes of Vicentino's enharmonic genus.

Another way of stating this is that with 19-tet, each whole-tone of
the diatonic scale is divided into three parts, and each diatonic
semitone (2/3-tone) into two parts, for (5*3 + 2*2) or 19 steps. In
31-tet, each diatonic whole-tone is divided into five parts, and each
diatonic semitone (3/5-tone) into three parts, for (5*5 + 2*3) or 31
steps.

With a keyboard limited to 19 steps, the 19-tet scheme has the
advantage of closure, but also has more highly tempered fifths, and
yields pure or virtually pure minor thirds while somewhat compromising
the often more important major third.

In comparison, the "19 out of 31" scheme yields less tempered fifths,
plus pure major thirds. As long as the music remains within the range
of gb-b#, this tuning provides all the notes we actually need.
Gesualdo's very daring madrigal repertory, for example, includes
according to Glenn Watkins only a single note -- an instance of cb --
outside this range.

To conclude, I would guess that a 19-tone keyboard from around the
late 16th or early 17th century would likely be tuned in some variety
of meantone in a range conveniently expressed as gb-b# (with eb-g#
being the usual subset of this range found on a 12-note keyboard).

While a closed 19-tet or open "19 out of 31" scheme are two obvious
possibilities actually described in the period for such keyboards,
one might also choose, for example, Zarlino's 2/7-comma tuning. Also,
Mark Lindley has suggested that some musicians of the time may have
favored meantones of a bit less than 1/4-comma, with smoother fifths,
a 1/5-comma tuning being one possibility.

Most respectfully,

Margo Schulter
mschulter@value.net