Lucca / Frescobaldi / Zarlino [was FAQ -- What is microtonality? What is paucitonality?]

Dear List members,

> Date: Sun, 4 Mar 2001 22:18:33 -0800 (PST)
> From: "M. Schulter" <MSCHULTER@VALUE.NET>
> Subject: Re: FAQ -- What is microtonality? What is paucitonality?

...
> After considering the arguments and opting for Ab, Ramos added that
> some people prefer to satisfy both sides of the question by designing
> a keyboard with both accidentals -- typically by splitting the key for
> an accidental so that pressing the front portion would sound G#, for
> example, while the back portion would sound Ab.

Does Ramos specify exact whether the front key is/should be g# or ab?

> This approach was
> followed, for example, in the organ at Lucca in Italy with such split
> keys for G#/Ab and Eb/D#, providing 14 notes in each octave, and
> enjoyed widespread favor in 16th-17th century Europe.

Is that better known now, than what I have in my database and if I'd be
grateful to get the source:

Lucca, Italy,
Cathedral S. Martino,
1480, Maestro Domenico di Maestro Lorenzo

Subsemitones:
2 pairs of split keys (exact distribution unknown):
eb/d#, g#/ab or
eb/d#, bb/a#
[note that the order of the keys doesn't state which is the resp. front key!]

Remarks, present state and history:
Dupont writes, probably erroneusly, that the organ was commissioned in
1450 and finished 1484. Whether the date of completion of the organ
might be correct or not, 1450 might be doubted.
Stembridge reports Luigi F. Tagliavini suggested that a bb/a# could be
an alternative. This might be compared with the concept of eb/d# and
bb/a# in the Hagerbeer-organs like Alkmaar 1643-1645/6 or Den Haag 1641.

Source:
Dupont 1935, p. 45.
Meister 1991 (after Jeppesen 1960), p. 45.
Ratte 1991, pp. 189 and 359.
Stembridge 1994, p. 162

It is interesting however that a much later instrument in Lucca is know
to have included bb/a# as well

Lucca, Italy
Accademia di Tomaso Raffaelli
before 1609, Andrea Lucchese

Subsemitones:
eb/d#, g#/ab, bb/a#
or
g#/ab, bb/a#, eb1/d#1

Source:
Stembridge 1994, p. 167.

-----------------------

> In the 1630's, the great composer Girolamo Frescobaldi reportedly
> advocated that a new organ be tuned in 12-tET -- and was roundly
> ridiculed by one theorist of the time for allegedly being ignorant of
> the difference between a large and small semitone. Others remarked
> that 12-tET might be more palatable if it were less unfamiliar -- a
> comment sometimes offered concerning "microtonal" music in more recent
> times.

On this topic I wrote last year to the harpsichord list:
--------
This is based on a anecdote by Doni who wrote 1638 to Mersenne, that
Frescobaldi didn't even know what a minor or major semitone was and
hardly ever would play on the black keys. 1647 he "added" that an old
ragged sicilian man managed to convince Frescobaldi to praise equal
temperament against his ear - by spending him many free drinks.

Does that sound reasonable to anyone?
Doni seems to be what is called "Rufm�rder" in German ("reputation murderer").

> Anyone ha ve comment on the veracity of these remarks?

Frescobaldi was a pupil of Luzzaschi, known as a virtuoso on enharmonic
instruments. Frescobaldi had certainly instruments with split keys
around and available (the Blasi-organ of San Giovanni in Laterano, Roma,
has been restored!). His music boasts of typical meantone fetaures ...

Until it is without doubt proven that Frescobaldi used and promoted
equal temperament (12-ET), he must be regarded as innocent (meantone).
Has anyone produced such a proof until now.

----------

And I'd like to know about your view of Zarlino

From a previous post to the list

> (Zarlino does define his 2/7-comma system rigorously in 1558, but
> other meantones such as 1/4-comma only in 1571).

What would you regard as the reason, that Zarlino would describe the
quarter-comma-meantone in the later publication as "new temperament", if
it was already around frequently since ca. 100 years. Or is he
referring to something else which he thinks is new in it?

This question seems especially challenging to me, because otherwise I'd
interpret the sources I know or have read about, in the same way like
you.

Many thanks

kind regards

Ibo Ortgies

Hello, there, Ibo Ortgies, and everyone.

Thank you for your questions both concerning the Lucca organ of 1480,
and the possible views of Ramos concerning the placement of split
keys, specifically G#/Ab (G# in front) or Ab/G# (Ab in front).

With Ramos, the quick answer is that, as I recall, he doesn't discuss
the design of split keys, but simply notes the opinion that a keyboard
should have both Ab and G#, raises the objection that this would
introduce an interval foreign to the diatonic order, and later says
that this proposal nevertheless has merits.

As I'll discuss in my FAQ item on Pythagorean tuning schemes and
possible keyboards with 13-17 notes (1370-1482), Ramos also mentions
the opinion of his friend Tristan de Silva that there should be a
another key added between F and G -- very likely a Gb/F# proposal,
which as Mark Lindley notes would very nicely fit the modified
Pythagorean tuning of Gb-B, popular around 1400-1450. Ramos finds such
a scheme inclined toward discord rather than harmony -- which might be
taken as another bit of evidence that he's discussing some kind of
meantone keyboard rather the popular 15th-century Pythagorean scheme
with a Wolf at B-Gb (written B-F#).

Incidentally, while your FAQ focuses on actual known instruments, mine
will focus more on theory and possible musical applications, since
various 15th-century theorists discuss 13-17 note tunings or keyboards
in some form of Pythagorean tuning, but at least as of Mark Lindley's
article of 1980, no examples of actually constructed instruments were
known.

As for the Lucca organ, please let me admit that I cannot match your
scholarship and knowledge of sources in this area; my information is
based only on discussions by Mark Lindley. However, the source which
he quotes, if accurate, does offer a fairly clear answer to the
question of which accidentals were included, and gives at least a hint
as to a likely arrangement of the split keys. He I quote Lindley's
text and translation[1], adding two notes:

inoltre li tasti soprascritto In addition to the above-mentioned keys,
la tersa del b quadro the third above B-natural{1}
et la tersa del fa delle f and the minor third above F{2}
et insieme cum li loro octavi together with their octaves
uve saranno necessarij where necessary.

1. Here Lindley translates _b quadro_ in the English manner as
"B-natural"; this is "square-B" or B-mi, a symbol providing the
origins of both the natural and sharp signs, and also the Germanic H
still used to represent English "B-natural," with Germanic B showing
Bb (the "round-B" of medieval notation, and the origin of the flat
sign). On these signs and their origins in medieval hexachord theory,
see, for example, http://www.medieval.org/emfaq/harmony/hex.html.

2. Here Lindley translates _tersa del fa_ as minor third; we might
also say "flat third," the third above F notated with a fa-sign or
flat, i.e. Ab.

As Lindley explains, this is a 14-note instrument with D# and Ab as
the extra notes.

To me, at least, the viewing of these notes as the extra ones might
suggest an arrangement of Eb/D# and G#/Ab, the one that becomes
standard in the 16th century (along with Eb-G# as the usual 12-note
tuning).

My crude musical information, not grounded I would emphasize in your
kind of knowledge of these instruments and their distributions of
notes in various countries and eras, would be that Bb/A# might be a
less likely choice in this early meantone era, as opposed to the
mid-16th century.

Please forgive me for what may be a bit of a digression on the
possible earlier 15th-century Pythagorean role of A#; as Ramos
suggests, the issues in early meantone would have been somewhat
different, and in fact Lindley focuses on some musical differences in
persuasively (at least for me) demonstrating that Ramos was discussing
a keyboard in meantone.

In the early 15th-century schemes of Prosdocimus of Beldemandis for a
17-note monochord (1413), and of Ugolino of Orvieto (2nd quarter of
15th century) for a 17-note organ, the range of Gb-A# does include
both notes, but A# seems maybe the least important, since one of
Ugolino's 12-note monochords has four sharps (F#, C#, G#, D#) but Bb.

Here a distinction may be that Bb is a _musica recta_ note, part of
the regular medieval gamut, while the other accidentals are _musica
ficta_ or "contrived notes" outside this gamut. However, both
Prosdocimus and Ugolino do include A# as well in their full 17-note
tunings.

This is not a "scientific" observation, nor evidence on how
15th-century musicians or instrument builders may have viewed things,
but in performing early 15th-century music on a Pythagorean keyboard I
have found occasion to use every note on a Gb-D# instrument, but at
least to me, A# seems of mostly theoretical interest. If I were
improvising or composing my own music, of course, I could use A# for
more remote transpositions of cadences, and also for more cadential
choices: e.g. F#3-A#3-D#4 in a cadence with ascending semitones to
E3-B3-E4; or a cadence from E3-G#4-C#4 to D#3-A#3-D#4 (with descending
semitone in lowest voice) rather than to the usual D3-A3-D4 (with
ascending semitones).

Lindley does present evidence in another article that one German organ
collection, the Breslau manuscript (I.F. 687) dating around 1430, may
have been written for a 12-note instrument with five sharps, that is
an F-A# tuning, with A#-D (written Bb-D) as a schisma third; but he
comments that this proposal would not seem very popular, because Bb is
a very common _musica recta_ note (Bb/B or B/H, Bfa/Bmi, both being
integral forms of this step in the standard gamut).

Having a Wolf at the frequently used fifth Bb-F (here tuned A#-F)
would cause problems with a very large proportion of the usual
repertory, although Lindley notes that the collection he describes
cleverly avoids this problem by often using the written third Bb-D
(played A#-D, the schisma third of 8192:6561 or ~384.36 cents, only
~1.95 cents narrower than a pure 5:4) without a fifth.[2]

If this five-sharps scheme had become more popular in the early 15th
century, then A#/Bb might have become the most common Pythagorean
split key scheme with 13 notes.

Returning to the meantone era, my impression is that A# may come into
vogue by around 1550: Orlando di Lasso uses it often in his
_Prophetiae Sibyllum_, and for the Prologue of this collection (which
I might assign to the Eighth Mode, or G Hypomixolydian), I have used a
14-note scheme of Eb-A#.

However, since I am not familiar with the earlier 16th-century
experiments in remote accidentalism, I would want to speak with some
caution here.

Anyway, if Lindley's Italian source is correct, the Lucca organ of
1480 had Eb-G# plus D# and Ab, with a hint of Eb/D# and G#/Ab if we
assume that the notes considered part of the "usual" 12-note set would
likely be placed in front.

-----
Notes
-----

1. Mark Lindley, "Fifteenth-Century Evidence for Meantone
Temperament," _Proceedings of the Royal Musical Association_ 102
(1976), 37-51 at 48-49 and n. 29, citing as the source of this text
from the contract for the rebuilding of the Lucca organ: Luigi Nerici,
_Storia della Musica in Lucca_ (Lucca, 1879), pp. 141-143.

2. Lindley, Mark, "Pythagorean Intonation and the Rise of the Triad,"
_Royal Musical Association Research Chronicle_ 16:4-61 (1980), ISSN
0080-4460, at pp. 33, 37, and Examples 14(a)-14(c).

Most appreciatively,

Margo Schulter
mschulter@value.net

Hello, there, Ibo Ortgies and everyone.

You have raised a very important question in the history of meantone,
which I'd like to discuss here and seek further comments before
writing my FAQ draft about meantone in the era 1450-1640: what
conclusions might we draw from Zarlino's remark in 1571 that he wishes
to describe a "new" temperament, 1/4-comma meantone?

Certainly we agree that Zarlino deserves the highest degree of
attentive respect, especially in a discussion of intonation, an area
in which he is most justly famed (pun intended) for his contributions.
One obvious conclusion, suggested by Mark Lindley, is that in 1571 the
1/4-comma temperament was not the self-evident "universal standard" of
common practice, contrary to a widely held assumption about
16th-century meantone.

Here I would like to argue that while more or less regular meantone
temperaments with more or less pure major thirds were very likely in
use at various times and places in Europe during the era 1450-1571,
Zarlino's tuning was indeed a "new" _formulation_ specifying and
_defining_ the entity called "1/4-comma meantone" for the first time.

Similarly, when we speak of Newton's laws of gravitation as "new," we
do not mean that gravitation was itself introduced as a natural
phenomenon around 1689, only that Newton's "discovery," explanation,
and precise mathematical formulae defining its operation were new.

To appreciate the importance of Zarlino's "new temperament" of 1571,
while leaving room for various interpretations of earlier statements
and tuning practices, we might turn to the first definitive
theoretical evidence for meantone, given by Gafurius in 1496.

Describing the practices of organ tuners, Gafurius notes that they use
_participatio_, or "temperament" as we should say, a technique in
which the fifths are narrowed by "a small and hidden amount."

This mathematically imprecise language may suggest a possible
limitation of the theory of the time: an assumption that the comma was
the smallest audible or measurable interval. Since meantone tunings
involve the narrowing of the fifths by a fraction of the comma, a more
precise formulation than "a small and hidden amount" might require new
developments in theory reflecting the new practice.

With Aaron's _Toscanello_ (1523), and with Vicentino's descriptions of
his archicembalo (1555) and arciorgano (1561), theory in this specific
aspect still seems quite imprecise. While even the cautious Mark
Lindley appears to imply that the _Toscanello_ instructions do set up
the first five notes as the pure major third C-E divided into four
equally tempered fifths (C-G-D-A-E), these instructions say nothing
about commas or ratios.

In his treatise of 1577, Francisco Salinas tells us that during his
younger years in Rome (around 1538), he was known as the inventor of
1/4-comma meantone. Possibly Salinas did develop a precise
mathematical formulation during his youth, and one might be tempted to
wonder whether the just intonation system of Lodovico Fogliano (1529)
with its discussion of the syntonic comma might have stimulated such a
quantitative approach to temperament. However, he did not _publish_ at
that time, and in 1577 he acknowledges and agrees with Zarlino's
formulation of 1571.

In his _Ancient Music Adapted to Modern Practice_ (1555), Vicentino
describes his archicembalo temperament as a division of the octave
into 31 parts, and of the tone into five such parts, each called a
minor diesis. He describes the minor (chromatic) and major (diatonic)
semitones as consisting of 2/5-tone and 3/5-tone respectively. The
tuning is circular: all intervals are available from all notes.

In his advertisement of 1561 for his arciorgano, Vicentino likewise
notes that one may begin a piece of music on any step of the
instrument; he further describes what appears to be the same as his
alternative "pure fifths" tuning of 1555 for the archicembalo, a
tuning combining "perfect fifths" with "perfect thirds," in other
words, adaptive just intonation.

If we take Zarlino's announcement of his "new" 1/4-comma temperament
as an assertion that "nothing like this has previously been tuned,"
and especially if we take such a reading of Zarlino as a conclusive
statement of fact, then the interpretation of Vicentino's scheme
raises some perplexing problems.

If his archicembalo divides the octave into a circulating system of 31
dieses, then the fifths must have at least an _average_ size quite
close to 10/31 octave, which in turn is a measurement very close to
that of a fifth tempered by 1/4 syntonic comma.

Similarly, if his arciorgano offers "perfect fifths" and "perfect
thirds" in an adaptive JI tuning, then some of those intervals very
likely are derived from the usual tuning of the first manual, which
Vicentino has described in his treatise of 1555 as the usual common
practice tuning with the fifths slightly "blunted" or "foreshortened"
(i.e. meantone). Since this same first manual can also be the basis of
a 31-note circulating scheme (rather than one of 19 notes), pure major
thirds rather than pure minor thirds on this manual seem implied.

To reason that since Zarlino declared his 1/4-comma temperament was
"new" in 1571, therefore Vicentino could not have tuned something
substantially identical in 1555 or 1561, might well require setting
aside the basic laws of acoustics and mathematics -- or proposing that
Vicentino's scheme was actually realized as a kind of appreciably
unequal 31-note well-temperament.

However, Vicentino's writings may also illustrate precisely how
Zarlino's 1/4-comma temperament of 1571 was "new": we might consider
not only what Vicentino says and implies about his keyboard tunings,
but what he does _not_ say.

Nowhere does Vicentino define anything like "1/4-comma meantone," nor
does he say anything about measuring meantone temperaments in terms of
fractional commas. Like Gaffurius 61 years earlier, he describes the
"foreshortening" of the fifths; for Vicentino, this _participatio_ or
tempering is indeed one of the two distinguishing characteristics of
modern common practice music, together with what he views as a
"mixing" of the diatonic, chromatic, and enharmonic genera (_musica
participata & mista_).

However, while Vicentino espouses temperament, discusses it, and
describes some of the resulting features of his keyboard tunings (a
circulating 31-note fifthtone system, or an adaptive JI system with
sonorities featuring pure fifths and thirds), he does not _define_ the
amount of temperament he uses, or divide the comma into fractional
parts.

Rather than splitting the syntonic comma in the manner of Zarlino and
Salinas, Vicentino "splits" his definition of the term "comma" itself:
its meanings can include either the syntonic comma of 81:80, or half
of his "minor diesis" or fifthtone, roughly the same size[1]; or the
amount by which a fifth is tempered.

These multiple uses of the term "comma" have caused considerable
confusion for modern scholars such as Barbour and Kaufmann: we
apparently must take the term in the sense of "the amount by which the
fifth is tempered" to understand his adaptive JI system with "perfect
fifths" -- that is, fifths a "comma" wider than the usual tempered
ones. Here a size of around 702 cents, rather than 716 cents, is
evidently intended.

The laws of "musical geometry" suggest this "comma" of Vicentino was
very close to the "1/4-comma" of Zarlino, but in the writings of
Vicentino himself it is, to paraphrase Gafurius, a small and
mathematically undefined amount.

The first mathematical definition, at least in print, was Zarlino's;
in 1558 he had been the first to split the comma and specify his
2/7-comma temperament in this elegant and precise manner, and in 1571
he applied this method to the "new" formulation of 1/4-comma with its
pure major thirds.

It remains a moot question whether by his announcement of this
temperament as "new" Zarlino meant to imply that the _sound_ of a
keyboard instrument in a regular tuning with pure or virtually pure
major thirds was itself new (at least to him), or whether he simply
meant such a _formulated_ tempering scheme, recognized and specified,
was being presented for the first time.

However we interpret Zarlino's intended meaning, his philosopical view
of music as a science which should have consistent laws and results
might well serve as the basis for an argument that "1/4-comma
meantone" had not really been "discovered" until it was _defined_.

On the theme of consistency, for example, Zarlino asserted that a
sixth expanding to an octave should _always_ be major, apart from some
narrow exceptions, because this was its nature, and that to permit it
to be either major or minor would be to imitate the mistaken approach
to medicine of those who treat every disease with the same remedy. He
also asserts that even "peasants" in their singing use the major sixth
before an octave, thus demonstrating the "natural" basis of this law.

Looking at some actual early to middle 16th-century tablatures of
instrumental pieces indicating precise semitones, as Robert Toft has
done[2], will show that some of Zarlino's contemporaries felt free
either to alter a minor sixth to major before an octave in line with
the usual rule of "closest approach," or at times to leave it
unaltered, the latter choice being a frequent "custom" in some German
practices.

If Zarlino's "natural" laws of counterpoint in 1558 did not
necessarily reflect practices in all parts of Western Europe -- and
given the diversity of musical styles and "dialects," we should hardly
ask this of any theorist -- then his announcement in 1571 of 1/4-comma
meantone as a "new" temperament, however read, need not necessarily be
taken to imply the absence of a tuning with a substantially similar or
identical _sound_ in 1523, 1538, or 1555.

In fact, if asked to guess when a meantone featuring pure or virtually
pure major thirds may first have been tuned, I would lean more toward
1471 than 1571 -- also noting, as does Lindley, that such temperaments
may have varied considerably in their degree of regularity, a feature
of Zarlino's mathematical model but not necessarily of the models and
methods used by tuners during these many decades.

However, to say that schemes with pure or virtually pure major thirds
were _tuned_, and that Vicentino's 31-note cycle logically implies or
even entails such a scheme (at least if we assume the degree of
regularity his description seems to suggest), is not to say that
"1/4-comma temperament" was widely known or defined before 1571,
although Salinas suggests that it was known in about these terms in
some Roman circles during his residence there as a young musician.

Here Zarlino's rightful place in the history of temperament may be
like that of Galileo's in the history of the physics of acceleration:
both formulated elegant and mathematically precise models which
earlier precedents do not render less "new" and significant.

Around 1545, with some earlier precedents in the scholastic physics
of 14th-century Oxford and Paris, Domingo de Soto stated the "mean
speed" theory that falling objects accelerate at a uniform rate.
However, it was left for Galileo to test this model by experiment with
his famous inclined planes, and to formulate it in an elegant
mathematical fashion in his dialogues of 1632 and 1638.

Similarly, in 1523, Aaron's _Toscanello_ seems in persuasive and
sometimes poetic language to demonstrate how a pure 5:4 major third as
"sonorous and just" as possible can be divided into four equal fifths,
each tempered by the same amount in the narrow direction. However, it
mentions neither the ratio of 5:4, nor the syntonic comma, nor the
quantity by which each fifth is tempered (Zarlino's "1/4-comma").

Zarlino, like Galileo, presents a mathematical formulation in which
the relevant parameters are both recognized and defined -- first in
his account of 2/7-comma temperament in 1558, then in his application
of the same approach to 1/4-comma and 1/3-comma in 1571.

Is this "discovery" a mere theoretical nicety, or something with vital
practical implications also?

As already suggested, I would not read Zarlino's announcement of 1571
as evidence that keyboard tunings with pure or virtually pure major
thirds had previously been unknown in Europe. However, I would take
his statement and formulation -- as well as what is _not_ formulated
in earlier instructions or descriptions by Aaron and Vicentino -- as
evidence for a "new" kind of tuning worldview.

In envisioning the tuning practices of European musicians from Conrad
von Paumann (c. 1450) to Vicentino and his contemporaries, we may find
it all too easy to imagine tuners setting up regular temperaments in
terms of fractions of a comma. After all, this is the "meantone
model," a model it seems natural to associate with this category of
tuning.

Such a view of history may capture one facet of the first century or a
bit more of meantone: the use of various shades of temperament. It may
not, however, capture another facet: the impressionistic and not
necessarily so regular nature of the tuning process, a facet which the
"small and hidden amount" of temperament described by Gafurius may put
into words better than the fractional comma models of Zarlino.

This is likely a world of blurred, or more positively stated,
"flexible" distinctions and boundaries; for example, a scheme such as
Arnold Schlick's of 1511 might be described as either a modified
meantone tuning or a kind of marginal "semi-well-temperament."

While I am inclined to conclude that a natural reading of the
_Toscanello_ instructions of 1523 would at least _invite_ a tuning
with pure or near-pure major thirds, and that Vicentino's parameters
of 1555 or 1561 come close to entailing such a result, I am also now
inclined to say that describing such a result as "1/4-comma meantone"
may be a conceptual anachronism, although a close muslcal equivalence.

Here is the kind of formuation toward which I would now lean:

"Meantone tunings with pure or virtually pure thirds could have been
and likely were used in various times and places between the advent of
meantone as a common practice around 1450-1480 and Zarlino's
mathematical definition of such a tuning in 1571.

"However, we should not assume that such a shade of meantone was a
'standard,' much less that tuners during these many decades conceived
of the tuning process in terms of fractions of a comma. Nor, as Mark
Lindley has wisely cautioned, should we assume that absolute
regularity was a general ideal of the time in theory or practice.

"Zarlino's precise definitions of 2/7-comma meantone in 1558, and of
1/4-comma and 1/3-comma temperaments in 1571, for the first time (at
least in print) did establish systematic models of regular
temperaments. Earlier theorists had described the tempering process
(e.g. Aaron's _Toscanello of 1523), or had described results implying
a certain approximate shade of temperament (e.g. Vicentino's 31-note
archicembalo cycle of 1555, suggesting pure or virtually pure major
thirds), but Zarlino _specified_ and _defined_ it.

"For these reasons, if we wish to capture something of the empirical
and flexible temperament practices of meantone's first century or a
bit more, we might do well to speak in terms of general shades of
temperament (e.g. `major thirds pure or close to pure,' or `major
thirds somewhat wider than pure') rather than in terms of Zarlino's
fractional commas.

"It is perhaps a special tribute to Zarlino that his fractional comma
measurements seem so natural that they have become, in effect, an
integral part of the conception of `meantone.'"

-----
Notes
-----

1. The syntonic comma of 81:80 is about 21.51 cents, and Vicentino's
discussion of JI ratios may be based on that of Fogliano; Vicentino's
comma equal to half of a minor diesis or fifthtone, or about
1/10-tone, would have a size of around 20.53 cents if we take the
diesis to approximate the 128:125 ratio (~41.06 cents) found in JI or
Zarlino's 1/4-comma meantone, or around 19.35 cents if we take it
approximate 1/31 octave (~38.70 cents) as defined in 17th-century
formulations by Lemme Rossi and Christiaan Huygens of 31-tone equal
temperament (31-tET).

2. Robert Toft, _Aural Images of Lost Traditions: Sharps and Flats in
the Sixteenth Century_ (University of Toronto Press, 1992), especially
Chapter 3, pp. 95-102.

Most respectfully,

Margo Schulter
mschulter@value.net

On Tue, 6 Mar 2001, M. Schulter wrote:

> If his archicembalo divides the octave into a circulating system of 31
> dieses, then the fifths must have at least an _average_ size quite
> close to 10/31 octave, which in turn is a measurement very close to
> that of a fifth tempered by 1/4 syntonic comma.

Hello, there, and this should, of course, read "quite close to 18/31
octave" for the fifth, with the _major thirds_ having an average size
close to the stated 10/31 octave (very slightly wider than a pure 5:4).

Most respectfully,

Margo Schulter
mschulter@value.net