Re: Lindley, Gb-B Pythagorean tuning (for Paul Erlich)

Hello, there, and I'd like briefly (I hope) to respond to a perceptive
question from Paul Erlich regarding Mark Lindley's attractive
hypothesis that a popular 12-note keyboard tuning of the early 15th
century used Pythagorean (3-limit) intonation with sharps tuned on the
flat side of the chain (Gb-B):

> A key piece of evidence for this hypothesis would be an avoidance,
> in these earlier 15th-century compositions, of the fifth B-F#
> (actually B-Gb), except in passing, as this fifth would be off by
> 24� and so quite rough. Does Lindley (or can you) present any such
> evidence?

The quick answer would be that Lindley indeed focuses on this
criterion, as likely would anyone playing these pieces and seeking the
best tuning, since a Pythagorean diminished sixth or "Wolf" fifth like
B-Gb is a classic illustration of a nonequivalent for a 3:2 fifth, to
put it mildly.

Thus in his germinal article "Pythagorean Intonation and the Rise of
the Triad," _Royal Musical Association Research Chronicle_ 16:4-61
(1980), ISSN 0080-4460, Lindley shows that for the Robertsbridge Codex
(c. 1335?) an F#-B (or actually Gb-B) tuning "is out of the question"
(p. 33) because of the prominent use of the fifth B-F#. He concludes
that a typical 14th-century tuning of Eb-G# would very nicely fit this
collection, a conclusion which might fit most 14th-century pieces.

In contrast, in suggesting that the Gb-B tuning may have influenced
the vocal compositions of certain composers in the epoch of around
1400, Lindley remarks (p. 43):

"Anyone familiar with the secular composition of
Matteo of Perugia (active 1402-16) will readily
observe that most them avoid the harmonic interval
B-F#, especially between the lower voices, but at
the same time contain prominent written-in sharps
involving those triads that would have a special
sound in the F# x B tuning."

On p. 44, he offers a suggested list of "other such works" from the
late 14th and early 15th centuries. By the way, rather than using
Lindley's term "triad"[1], I would prefer to use such terms as _quinta
fissa_ ("split fifth"), for example, to describe an outer fifth
"split" by a middle voice into a major and minor third (e.g. D3-F#3-A3
-- or, in this tuning, D3-Gb3-A3 with near-pure schisma thirds D3-Gb3
and Gb3-A3). Here C4 is middle C, and higher numbers show higher
octaves.

In practice, of course, the question gets much more complicated,
because while the presence of prominent sonorities involving the fifth
B-F# tends to exclude a 12-note F#-B (i.e. Gb-B) tuning, there are
several other tuning possibilities to consider.

Also, some of Lindley's examples suggest to me that different people
can reach different conclusions on how "prominent" a Wolf fifth should
be to motivate one to choose some other tuning.

First, let's consider some of the tuning options during the late
medieval and early Renaissance eras:

A. Traditional Gothic (1300-1420, sometimes later)
1. A "classic" 12-note Pythagorean tuning, e.g. Eb-G#;
B. Transitional Pythagorean (1380?-1450?)
2. A 12-note Pythagorean tuning of F#-B (Gb-B);
3. A 13-note Pythagorean tuning of Gb-B plus a true F# (Gb-F#);
4. A 15-17 note Pythagorean tuning, e.g. Eb-G# plus Gb-B (Gb-G#);
C. Renaissance meantone (starting around 1450?)
5. A 12-note meantone tuning (e.g. Eb-G# or Ab-C#);
6. An extended meantone tuning like Eb-G# plus Ab and D# (Ab-D#).

In deciding between these alternatives, period and general style -- as
well as the preferences of the interpreter -- are important factors.

For example, in his article, Lindley gives an excerpt from a keyboard
composition which he regards as appropriate for the Gb-B tuning:
Buxheimer Organ Book #180, _Ob lieb din Lieb_ (Lindley, p. 48,
ex. 25). However, this excerpt itself includes a fifth B-F# with the
duration in transcription of a semiminim (crotchet) or quarter-note,
and at least at my tempo of playing this piece, I definitely prefer to
avoid such a Wolf fifth, although like Lindley I find a Pythagorean
tuning fitting, and agree with him that some prolonged noncadential
sonorities could serve as ideal illustrations of the schisma third
effect (D3-A3-F#4 as D3-A3-Gb4; E3-A3-C#4-E4 as E3-A3-Db4-E4).

Interestingly, Lindley elsewhere suggests one quite viable alternative
for this piece and others like it: a 13-note tuning with Gb-B plus an
extra key providing a true F#, and thus a concordant fifth B-F#.
Having earlier cited some 15th-century sources mentioning or endorsing
the idea of _two_ accidental keys between F and G, Lindley remarks (at
p. 45):

"Were evidence found that keyboard instruments with
thirteen notes per octave were built prior to the
late fifteenth century (that is, including two
forms of F#, as hinted by Ramos's friend Tristan
de Silva), then another group of compositions
would become particularly significant: those
which exploit the fifth B-F# as well as the
other intervals containing sharps."

Thus while Lindley might consider the fifth B-F# in _Ob lieb din Lieb_
to be transient enough that a Wolf intonation is tolerable in G-Bb,
those of us who prefer to avoid this dissonance might choose a 13-note
Pythagorean tuning. This tuning provides all the schisma thirds and
sixths of Gb-B plus a pure fifth B-F#.

My own choice, however, is to go with a 15-note Pythagorean tuning of
Gb-G# -- basically a combination of the "classic" 14th-century tuning
Eb-G# with the usual flats and sharps, and the early 15th-century Gb-B
for the schisma thirds. In addition to solving the B-F# Wolf problem,
this kind of tuning gives the option of using near-pure schisma thirds
and sixths for "coloristic" sonorities like E3-A3-C#3/Db4-E4, but
regular Pythagorean thirds and sixths for direction cadential
sonorities like E3-C#4 leading to D3-D4 (M6-8).

Two theorists of the early 15th century, Prosdocimus of Beldemandis
and Ugolino of Orvieto, express their preference for "full perfected"
major thirds and sixths in M3-5 and M6-8 progressions -- that is, the
regular wide and active Pythagorean flavors of these intervals,
resolving by narrow diatonic semitones. Both these authors describe
and endorse a 17-note Pythagorean tuning (Gb-A#), which Ugolino notes
would permit the "intelligent organist" to perfect such thirds and
sixths with discretion.

As Lindley (p. 45) remarks, 15th-century musicians may likely have
found the wider cadential semitones entailed by a 12-note Gb-B tuning
as something of a "necessary evil" for the sake of the beguiling
schisma intervals. With a keyboard of 15 or more notes, however, we
can choose freely between "smoothed" sonorities with schisma intervals
(using Gb, Db, Ab for written F#, C#, G#) and dynamic cadential
sonorities with regular Pythagorean thirds and sixths (with true F#,
C#, G#).

As we reach the mid-15th century, another alternative arises: a
meantone tuning, which solves the B-F# problem while providing smooth
thirds and sixths for all such intervals with regular spellings.

A complication in all this is that a piece might be conceived in an
era when one kind of tuning was likely in vogue, but also give very
pleasant results for performers and listeners of a somewhat later
epoch when realized in a newer tuning.

Lindley himself mentions this kind of situation when he notes that
certain 14th-century compositions would take on "an unforeseen
acoustical aura" when performed around 1400 in a Gb-B or similar
tuning transforming thirds and sixths involving sharps into near-pure
schisma intervals (p. 44).

Another way of putting this is that some compositions of Machaut, for
example, might be satisfyingly realized either in an Eb-G# tuning
(c. 1370?) or a Gb-B tuning (a "modern" interpretation of c. 1410?).
Lindley, commendably, lists some compositions of Machaut as lending
themselves to this treatment while leaving open the question of the
composer's original intent.

His suggestion that by around 1400-1410, and a bit earlier in some
Italian centers, the Gb-B may have been influencing such composers as
Matteo de Perugia, Ciconia, and likewise the young Dufay in the
1420's, seems quite credible and attractive.

As we enter the epoch around 1450, similarly, we face the possibility
that many pieces originally conceived for Pythagorean keyboards may
have been played on early meantone instruments.

Anyway, for the late Pythagorean repertory, we have in effect three
possible criteria in choosing between the various tunings:

1. Wolf avoidance, specifically B-F# in Gb-B tuning;
2. Schisma thirds/sixths with Gb, Db, Ab;
3. Regular cadential thirds/sixths with F#, C#, G#.

A classic 14th-century tuning of Eb-G# gives us (1) and (3); a 12-note
tuning of Gb-B gives us (2); a 13-note tuning of Gb-B plus a true F#
gives us (1) and (2); a 15-note or larger tuning with F#/Gb, C#/Db,
and G#/Ab gives us all three advantages.

Maybe my main point in all this is that whatever criteria we decide
on, people applying them to actual music may reach different
conclusions -- is that B-F# sonority "merely passing," or prominent
enough to make a Wolf fifth in the Gb-B tuning a "contraindication"?

Anyway, Paul, thank you for a question which not only invites more
discussion of Lindley's very important article, but shows how taste as
well as logic can play a role in tuning decisions.

----
Note
----

1. Note, by the way, that I'd prefer to use some term other than
"triad" to describe unstable sonorities around 1400 such as an outer
fifth plus lower and upper third (e.g. D3-F#3-A3) or an outer sixth
plus lower third plus upper fourth, or vice versa (e.g. E3-G#3-C#4 or
G3-C4-E4). Here I use C4 for middle C, and higher numbers for higher
octaves.

The problem I have with "triad" is that not only is it anachronistic,
it may for many readers imply stability, and distract from the
directed cadential implications of a sonority such as E3-G#3-C#4,
often resolving to a stable D3-A3-D4 (M6-8 + M3-5 by stepwise contrary
motion).

Most appreciatively,

Margo Schulter
mschulter@value.net