Yahoo Tuning Groups Ultimate Backup tuning Re: Harmonic evolution

Hello, there, everyone, and I want to wish everyone a Happy New Year
while responding to a few questions and comments about "harmonic
evolution" and the development of rhythm in Western European music.

First, as far as I am aware, rhythmic divisions of both 5 and 7 are
first indicated in the notation of Petrus de Cruce and his followers
around the late 13th century, maybe 1280-1300. This is accomplished
with a _punctus divisionis_ or "dot of division" which shows that a
group of notes together have the value of a breve, literally the
"short" unit, and divide it as might be said in modern theory into
n-tuplets.

Around 1325, Jacobus of Liege points to this ability to divide the
breve into 5 or 7 (or in theory 9) equal parts as an advantage of the
Ars Veterum of "Art of the Ancients" (i.e. the generation of his
youth) in contrast to the Ars Nova of the early 14th century, which
directly recognizes only divisions of 2 or 3.

There are a number of means used in late medieval notation to indicate
polymeters or irregular grouping of various kinds, for example:

(1) The Petronian _punctus divisionis_, which
can divide the breve into an equal group of
3, 4, 5, 6, or 7 notes, with the divisions
of 5 and 7 occurring in actual late 13th-century
pieces;

(2) The regular mensurations of the 14th century and
later -- 9/8, 3/4, 6/8, and 2/4 in modern notation --
which can be alternated in one voice or mixed in
different voices;

(3) Syncopation, termed by Willi Apel "insertion
syncopation" or "displacement syncopation,"
in which for example one might take the notes
for the first 5 units of a 6/8 measure, then
then continue for some 6/8 measures from that
point, eventually supplying the odd eighth-note
unit to get this voice "back in synch" with the
others. A related technique would permit odd
groups of 3/8 or 5/8, etc., to alternate freely
in a given voice -- this may not be a very
precise explanation, but it should give some
general idea;

(4) Proportions, sometimes with the same name as
those defining interval ratios, such as
_sesquialtera_ (3:2), permitting such things
as singing 9 notes in one voice against 4 in
another.

These techniques seem to reach their height in the _Ars subtilior_ or
"14th-century Manneristic period" in the decades of about 1370-1410,
although there are some vivid echoes in the early music of Guillaume
Dufay (1397?-1474), often viewed as a key transitional figure between
Gothic and Renaissance eras.

There are also some later Renaissance techniques recalling these: the
"mensuration canons" of the era around 1500 where a single melody
might be sung by several voices, each with its own mensuration,
forming a concordant whole; and also occasional divisions by odd
factors such as 5 or 7. Juan del Encina around 1500 has a piece, a
"jazzy" one I would say, with rhythms of 5; and as late as 1597,
Thomas Morley includes in his _Plain and Easy Introduction to
Practical Music_ a piece including proportional divisions of 5 and 7.

In response to the "Harmonic Evolution" discussion, I might caution
that prime limits are only one aspect of a more complicated process,
in 1200 or 1500 or 1900 or 2002.

As a medievalist, I would not surprisingly emphasize that simply
looking at the highest prime (or odd) factor in the most complex
stable sonority gives a very partial view of musical technique.
Having expressed this opinion, I would like to suggest several factors
to be considered in considering changes in vertical or harmonic and
intonational style:

(1) The development of progressions contrasting unstable
and stable sonorities, with considerations such as
smooth melodic motion as well as vertical variety;

(2) The tendency of urgent cadential sonorities or
"discords" in one era to become "relatively
concordant" or "floating" sonorities in the next,
complete stable sonorities in the next after that,
and "incomplete" forms of some more complex stable
sonorities in some following era;

(3) The role of cadential efficiency as well as vertical
"euphony" (as regarded in a given stylistic context)
in possibly favoring certain intonational changes;

(4) The possible factor of categorical perception in
the treatment of new interval types as "concords"
or "discords" of whatever degree; and

(5) The harmonic and melodic potential of intervals
arising in a given tuning system fortuitously, as
it were.

Knowing that discussions of this kind can get a bit abstract, and
wanting to follow what I might call the Szanto Axiom of keeping this
relevant to my own musical experience as well as that of others, I'll
try to develop some of these themes from a certain first-person
viewpoint, drawing also on the first-person accounts of other
musicians.

First of all, the question is often not whether one can theoretically
describe a given ratio, but whether at a given moment it "fits in"
with one's musical style -- or maybe is just there in a tuning and
is hard to resist using in the context of one's stylistic inclination.

For example, around 1325, Jacobus of Liege discusses ratios of 5 and
7, notes that instruments could be built using such intervals, and
demonstrates both the 81:80 and 64:63 commas. However, he concludes
that such intervals are not in practical use because they cannot be
generated from the proper steps of Pythagorean intonation.

In 1998, I had a specific reason to use ratios of 7 such as 9:7, 7:6,
12:7, 7:4 and 8:7 in a neo-medieval style: they would permit more
efficient cadential resolutions to stable fifths, fourths and
octaves. It is not that the usual Pythagorean intervals are less than
excellent for this task, only that a set of even more efficient
cadences using new ratios very much appealed to me.

One response I got in Tuning List discussions was a certain puzzlement
that I would use these ratios of 7 as unstable, like their Pythagorean
counterparts, rather than regard them as stable, since they are small
integer ratios and _could_ be treated as stable.

Here it may be a matter of taste rather than acoustical imperative:
similarly I tend to treat 0-277-738-1015 cents or 0-462-738-923 cents
in 13-tET as an unstable sonority resolution to 0-738 cents or
0-738-1200 cents, although there is no reason why the first two
sonorities might not be treated as conclusive.

Returning to the theme of cadential efficiency, one might ask whether
the popularity of the 4:5:6:7 intonation of the 18th-19th century
dominant seventh chord could reflect not only a tendency toward
greater "euphony" or "concord" (as it does for Euler in 1764), but
also a more efficient two-voice resolution from the smaller 5:7
diminished fifth to a major third at or near 4:5, with a compact 20:21
semitone.

However, my own musical experience makes me regard the appeal of
certain ratios as an open question: I find that I like a "tritonic"
interval at or near either 5:7 (e.g. 29-tET) or 8:11 (e.g. 22-tET) in
progressions where this interval moves by parallel motion to a stable
fifth (or sometimes fourth) in a medieval fashion.

Shifts toward the use of a given interval category in freer or more
"floating" treatments may or not be accompanied by intonational
shifts, and trying out different permutations can be a fine approach
for creating new music.

For example, I find that early 15th-century fauxbourdon, the use of
parallel sixth sonorities with a lower third and upper fourth, can be
very pleasing in a standard 14th-century Pythagorean tuning, or in a
modified early 15th-century scheme where thirds involving sharps are
often realized as near-5 "schisma thirds" (or "skhismic thirds" as
Helmholtz/Ellis would spell the term), or using ratios of 3 and 7.

In addition to urging this kind of experimentation, where tastes may
certainly differ, I would look to the accounts of xenharmonicists who
describe their own experiences with new tunings, for example Nicola
Vicentino (1555) and Fabio Colonna (1618) in exploring their 31-note
divisions of the octave into equal or near-equal fifthtones.

Categorical perception may play a role in Vicentino's remarks, for
example, that a near-11:9 third is rather concordant, but an interval
we would describe as near 7:6 or 7:4 more dissonant. From a
16th-century viewpoint, a "seventh" generally is a dissonance, and a
small seventh at 7:4 fits this pattern -- as I also noted when
listening to this interval in a 16th-century kind of context. In a
14th-century kind of setting I might find it more "mellow" or
"relatively concordant," and indeed Guillaume de Machaut uses minor
sevenths more freely than in typical 15th-16th century practice.

Thus "harmonic evolution" need not follow set patterns: Vicentino can
find 11:9 more concordant than 7:6 or 7:4. The first interval
apparently better fits his categorical concept of a "concordant third"
of some kind. In contrast, the 7:6 tends toward a dissonant second,
while the 7:4 suggests a kind of more-or-less dissonant seventh.

For me, sometimes, I suspect that musical context can influence
categorical perception: "If it _acts_ like a minor third, it is to a
certain extent a `minor third,' or `quasi-third.'" This happens in
20-tET when I resolve a 240-cent interval to a unison by contrary
motion.

Sometimes changes in concepts of stability or "concord/discord" may
occur without any obvious changes in intonation. For example, in
12-tET, a sonority of 0-400-700-900 cents or "added-sixth chord" is
inherently unstable in a Classic-Romantic style, but often conclusive
in certain 20th-century styles.

My own inclination is to focus on the role of complex as well as
simple intervals and ratios, and to seek a subtle continuum of
concord/discord.

In my view, much of the creative process involves not so much "moving
up the harmonic series" as discovering the wealth of materials, often
unexpected ones, in a given type of tuning, for example when carried
to more notes.

Thus while meantone tunings were developed, possibly around the
mid-15th century, to provide simple diatonic scales and modes with
approximate ratios of 3 and 5, by 1555 two fortuitous resources (among
many others) had been noted by Vicentino in his 31-note cycle: the
availability of new kinds of intervals such as "proximate minor
thirds" near 11:9; and the melodic step of the enharmonic diesis
(e.g. G#-Ab) or fifthtone.

As has often been remarked here, standard Western European keyboard
tunings tend to have a note of "regularity," from medieval Pythagorean
to Renaissance/Manneristic meantone to the _mildly_ irregular
well-temperaments and the isotropic geometry of 12-tET.

However, dramatic asymmetry (as measured against the "geometry" of
one's own usual expectations) can be a very stimulating variant. For
example, 13-tET has excellent minor thirds and major sixths at 277
cents and 923 cents which could be the pride of any neo-medieval
tuning system -- but no usual major thirds or minor sixths. It has
fine submajor thirds (369 cents) and supraminor or "Phi" sixths (931
cents), but no supraminor thirds or submajor sixths like those found
in regular temperaments in the general neighbhorhood of 704 cents.

There is also the charm of ambiguity, as with the 5-step interval of
13-tET, which nicely serves in appropriate timbres as the equivalent
of a medieval fourth and also of a large cadential major third
expanding to a fifth.

Intonational structure makes possible, but does not necessarily
compel, certain stylistic choices. For example, in a 31-note meantone
structure like that of Vicentino following or approximating 31-tET,
one might follow a conventional style based on 16th-century ratios of
3 and 5; or a 20th-century technique (anticipated, for example, by
Euler) of seeking combined ratios of 3, 5, and 7; or a kind of
"Neo-Gothic/Xeno-Renaissance" style favoring now ratios of 3 and 5,
and now ratios of 3 and 7, with some Vicentino/Colonna enharmonic
idioms featuring fifthtone motions or shifts.

There is another kind of question: Can concepts based on "n-limit"
harmony apply to a range of tempered systems, or might it be better to
speak in categorical interval terms: "Minor sevenths and major seconds
are often treated as relatively or even conclusively concordant in a
range of 20th-century sonorities."

We can also imagine alternate histories where certain stylistic
changes _might_ have been accompanied by intonational ones. For
example, suppose that the Debussyian exploration of more complex
"floating" harmonies around 1900 had been followed in the next decades
of the 20th century by the widespread adoption of Busoni's 36-tET,
with a focus on the near-pure ratios of 3 and 7 making possible such
sonorities as 0-267-700-967 cents (very close to a just 12:14:18:21).

Such alternate histories should be not only an opportunity for
reflection, but an instigation to action.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗genewardsmith <genewardsmith@juno.com>

🔗paulerlich <paul@stretch-music.com>

🔗graham@microtonal.co.uk

🔗paulerlich <paul@stretch-music.com>

🔗graham@microtonal.co.uk

Me:
> > Fourths, on the other hand, were treated as dissonances in
> > certain contexts. One reason for writing minor thirds explicitly
> in the
> > score was so that singers didn't bend the written major third the
> wrong
> > way, and end up with a fourth. If they didn't bend far enough, and
> ended
> > up with a neutral third, is that so bad? Don Nicola seems to think
> not.

Paul Erlich:
> Is this argument his, or your interpolation?

I think he says something about implicit accidentals, but haven't been
able to find it. On neutral and supermajor thirds, here's an excerpt from
Book V Chapter 8, pp. 336-8 of the Maniates translation

"""
There are also two other kinds of thirds that can be used, even though
they do not have the just measurement of the others. Nonetheless, they
can be employed more readily in playing than in singing, because the
minute difference between the third we normally use and those we shall now
adopt is not audible if players do not linger on them. It may be argued,
moreover, that if totally dissonant seconds and sevenths are used, the
proximates of the minor and major third are much more serviceable, since
they seem consonant when newly composed on the archicembalo.

If a player fails to pay attention to the proximate and most proximate
consonances, he will be deceived by them, for they are so proximate to
imperfect consonances that they seem identical to them. Thus, when
playing the archicembalo, you may use the third larger than the minor
third, that is, the proximate third that is one minor diesis larger than
the minor third. This step resembles the major third without being a
major third, and the minor third without being a minor third. The minor
third we use below the low A la mi re [1A] is the second G sol re ut
[2F#]. Its proximate is on the third F fa ut on the fourth rank [4F+],
and it seems better than the minor third because it is not as weak as the
minor third in comparison to the major third. Still, the proximate is
somewhat weaker then the major third because it is smaller by one
enharmonic diesis. thus, the proximate or most proximate to the minor
third sounds acceptable and can be played. I believe that some people
sing proximate and most proximate thirds as they sharpen these minor and
major consonances when performing compositions, and they do not create
discords despite the fact that the former are not the same size as the
latter.

Moreover, the same thing happens with the proximate to the major third,
which seems to be both a major third and a fourth without being either.
This proximate third is less tolerable to the ear than the proximate of
the minor third. The reason is that the minor third itself moves toward
the major third so that its proximate tends toward this good thirds,
whereas the proximate of the major third moves toward the fourth as if
tending toward a dissonance. Thus, the proximate of the minor third below
A la mi re [1A] is the third F fa ut in the fourth rank [4F+], and the
proximate of its major third is the second F fa ut in the third rank
[3E#]. Its fifth is D sol re [1D].

In addition, there are two other new consonances that occur in the same
way: the proximate and most proximate of the major and minor sixth. Just
as the minor sixth below A la mi re [1A] is located on the second D sol re
[2C#] and its proximate on the third C fa ut in the fourth rank [4C+], so
the major sixth below A la mi re [1A] is the first C fa ut [1C], and the
third C fa ut in the fourth rank [4C+] has its proximate below A la mi re
[1a[ which is the second C fa ut in the third rank [3B#]. The latter
proximate is harsh, for it tends toward the seventh. But it is salvaged
by its sixth.
"""

I use + in place of the superscript dot for raising by a minor diesis.
"Proximate" means enlarge by such a diesis, and "most proximate" by a
comma, whatever that might be. The most proximate major third is probably
the Pythagorean major third of 81:64. Perfect fourths were perfect
consonances in Book II.

Me:
> > For them to be passably smooth try a medium register, not too loud
> and a
> > simple, harmonic timbre (don't get carried away with adding 11th
> > partials).

Paul:
> With a piano sound they sound jarringly dissonant to me.

I've played neutral thirds with my Korg's piano settings, and they sound
okay. But there are a lot if different piano sounds, especially if you
remove the restriction of them being produced by a piano.

Graham

🔗paulerlich <paul@stretch-music.com>

🔗graham@microtonal.co.uk

Me:
> > I use + in place of the superscript dot for raising by a minor
> diesis.
> > "Proximate" means enlarge by such a diesis, and "most proximate" by
> a
> > comma, whatever that might be.

Paul Erlich:
> Vicentino calls the 1/4-comma that separates the two manuals in his
> second tuning, a "comma". However, he's talking about his first
> tuning here, right?

Vicentino gives the word "comma" at least three contradictory meanings, so
it isn't always obvious which one he intended. I don't think even he knew
some of the time. I can't find it used with reference to the second
tuning. The most likely definition in this context is 1/62 of an octave,
or something like that.

Here we go. Book V, Chapter 59, pp. 432-3 "... every diesis can be
partitioned into two commas, the whole tone having ten commas, the minor
semitone having four commas, and the major semitone having six commas."

The archicembalo has 5 keys more than you need for 31-equal. They get
left out of the first set of tuning instructions. But these are what the
"comma" symbols refer to. They were probably tuned as in the second
tuning, giving just fifths with the lower manual. There are references to
"just fifths" in the catalog, which all involve these commas. Although he
makes numerous mistakes, I don't see what else he could have meant.

> > The most proximate major third is probably
> > the Pythagorean major third of 81:64. Perfect fourths were perfect
> > consonances in Book II.
>
> The way I read it, he simply meant the 11/31-oct. interval. No?

Oh no, not that for most proximate. But not 81:64 either. Probably a
quarter comma larger than just. It's this passage from Book V, Chapter 62
pp436-7 that confussed me

"Next on my instrument, we have the major third plus a comma. This
consonance is considered to be not good because it is not blunted like the
one we use. It contains two sesquioctaval ratios similar to Boethius'
ditone. Finally, the major third plus one enharmonic diesis, although not
good, is passable in running passages."

It can't be the same as the Pythagorean ditone, although Vicentino may not
realize this. That's also another reference for the consonance of the
neutral third.

In Book V, Chapter 6 (p.334) on the second tuning, he does actually say
that Pythagorean thirds are "more justly tuned than the ones we use".
That's probably an example of him getting confused. He was no
mathematician and I suspect he had little experience of the second tuning,
or those 5 extra keys. I mention this to show that you can't really tell
what Vicentino was about from isolated quotes. You need to get hold of
the treatise, and exercise your judgement in context.

> I guess so. Korgs tend to have pretty realistic piano sounds, I've
> found . . . I guess it's an aesthetic difference we have.

This is an X5D, so not in the same league as their digital pianos. I also
use it with the built-in filter, which may over-soften the sound. But I
suspect it's a matter of taste. I don't much like pure 5-limit
consonances with these sounds, so neutral thirds are no problem.

Graham

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., graham@m... wrote:
> Me:
> > > I use + in place of the superscript dot for raising by a minor
> > diesis.
> > > "Proximate" means enlarge by such a diesis, and "most
proximate" by
> > a
> > > comma, whatever that might be.
>
> Paul Erlich:
> > Vicentino calls the 1/4-comma that separates the two manuals in
his
> > second tuning, a "comma". However, he's talking about his first
> > tuning here, right?
>
> Vicentino gives the word "comma" at least three contradictory
meanings, so
> it isn't always obvious which one he intended. I don't think even
he knew
> some of the time. I can't find it used with reference to the
second
> tuning. The most likely definition in this context is 1/62 of an
octave,
> or something like that.

1/62 of an octave? Are you serious? What tuning system is this???

> Here we go. Book V, Chapter 59, pp. 432-3 "... every diesis can
be
> partitioned into two commas, the whole tone having ten commas, the
minor
> semitone having four commas, and the major semitone having six
commas."

Oh dear. Margo, any insight?

> The archicembalo has 5 keys more than you need for 31-equal. They
get
> left out of the first set of tuning instructions. But these are
what the
> "comma" symbols refer to. They were probably tuned as in the
second
> tuning, giving just fifths with the lower manual. There are
references to
> "just fifths" in the catalog, which all involve these commas.
Although he
> makes numerous mistakes, I don't see what else he could have meant.

So it's about 1/4 of a syntonic comma, yes?

>
> > > The most proximate major third is probably
> > > the Pythagorean major third of 81:64. Perfect fourths were
perfect
> > > consonances in Book II.
> >
> > The way I read it, he simply meant the 11/31-oct. interval. No?
>
> Oh no, not that for most proximate. But not 81:64 either.
Probably a
> quarter comma larger than just.

Gotcha.

> I mention this to show that you can't really tell
> what Vicentino was about from isolated quotes. You need to get
hold of
> the treatise, and exercise your judgement in context.

I guess his music would tend be illuminate more, yes?

> > I guess so. Korgs tend to have pretty realistic piano sounds,
I've
> > found . . . I guess it's an aesthetic difference we have.
>
> This is an X5D, so not in the same league as their digital pianos.
I also
> use it with the built-in filter, which may over-soften the sound.
But I
> suspect it's a matter of taste. I don't much like pure 5-limit
> consonances with these sounds,

Well then, things like Blackjack are not likely to be of much
interest to you . . .

> so neutral thirds are no problem.

. . . unless you use them for their neutral thirds!

🔗graham@microtonal.co.uk

🔗M. Schulter <MSCHULTER@VALUE.NET>

Hello, everyone, and maybe I can comment about some of the questions
raised concerning Vicentino and his archicembalo tunings, as discussed
in _Ancient Music Adapted to Modern Practice_ (1555), and also,
briefly, in his circular advertising his similar arciorgano in 1561.

Thank you, both Graham and Paul, for raising these questions and
giving me the opportunity to clarify some of the inferences drawn here
-- and the fact that one must indeed look to the context and seek a
"likely" musical interpretation.

First, Graham, I strongly agree that "comma" can mean a number of
things in Vicentino, among them his "half-diesis" or "tenth-tone" of
about 19 cents (if one takes the dieses or fifthtones as near-equal,
and thus as averaging at about 1/31-octave), and the amount by which
the fifth is tempered or "blunted" (1/4 syntonic comma, or
thereabouts).

Also, I would agree that some of his comments can be ambiguous or even
confusing. As I view it, the question of the usual "proximate thirds"
is the easier one to address, since his descriptions seem nicely to
fit either a 31-note cycle of 1/4-comma meantone or 31-tET, and these
intervals do not involve the more problematic "comma keys."

The question of the comma keys and the second tuning might seem a bit
more disputable, since I'm not aware of any of his music where he
expressly indicates "comma" as opposed to diesis alterations. One
might guess that he optionally left such refinements to a skilled
player -- as one would lead adaptive tuning or JI in a usual
16th-century vocal piece to the performers of flexible-pitch
instruments, writing in a notation which could also show the steps on
a fixed-pitch meantone instrument.

As you observe, the question of the "most proximate" intervals
involving commas also involves this question of interpretation, and I
would be inclined to read them as varying by a "half-diesis" from the
regular ones -- an interval close to a syntonic comma.

At the same time, Paul, I would say that the familiar interpretations
of Vicentino's first and second tunings still seem very attractive to
me. The most likely reservation I might have here is that in the first
tuning, the comma keys _might_ have been tuned in accordance with
either of the leading interpretations of a "comma" -- either the
amount by which the fifth is tempered (a 31-note meantone cycle plus a
few notes in adaptive JI), or the "half-diesis" reading (resulting,
for example, in a few near-Pythagorean intervals).

Addressing first the usual "proximate" intervals, I might also say
something more about the way that Vicentino and Zarlino, although
strongly disagreeing on some basic issues regarding the chromatic and
enharmonic genera in polyphonic music (an understatement, one might
say), seem to take similar approaches to the fourth. Since Vicentino
ties the "proximate major third" to the fourth, this point may be
relevant.

Both Vicentino and Zarlino take the fourth from one point of view as a
"perfect consonance" grouped with but after the fifth -- an approach
standard around the 13th century, when 3:2 and 4:3 are often ranked
together as "intermediate concords" (a term used by Zarlino also).
Interestingly, both of them apply the Renaissance rule against
parallel "perfect concords" to fourths as well as octaves and fifths,
although Vicentino characteristically recognizes an exception in
theory, and it would interesting to see how strictly he follows his
general recommendation against parallel fourths in practice.

At the same time, both Vicentino and Zarlino recognize that as a
simple interval or dyad -- or interval directly above the bass -- the
fourth is generally treated rather cautiously in practice, somewhat in
the manner of a discord. Between upper voices, it is a concord in
practice as well as theory. Thus the interval has an ambivalent
status.

Let us know consider the usual "proximate" thirds of Vicentino, the
ones involving diesis but not comma modifications, as I interpret them
in 1/4-comma meantone or 31-tET:

7/5-tone 8/5-tone 9/5-tone 10/5-tone 11/5-tone

minimal 3rd minor 3rd proximate major 3rd proximate
minor 3rd major 3rd
[~7:6] [~6:5] "~5-1/2:4-1/2" [~5:4] "~4-1/2:3-1/2"
(~11:9) (~9:7)

min3 less min3 plus maj3 plus
diesis diesis diesis

From Vicentino's perspective, the minimal third is too small to have a
concordant "thirdlike" flavor, tending toward the dissonance of the
major second as judged by 16th-century standards. The minor third is
the smallest third to give satisfying concord. The "proximate minor
third," whose ratio he gives as approximately 5-1/2:4-1/2 (i.e. 11:9)
is rather concordant, and "tends towards" to the ideal concord of the
major third, which he like Zarlino finds more joyful as a simultaneous
consonance.

He finds that the proximate major third near "4-1/2:3-1/2" or 9:7
tends rather toward dissonance, or toward the ambiguous status of the
fourth. Actually, it is in some ways more "discordant" than the
fourth, which is routinely used as a full concord between the _upper_
voices of a sonority while the proximate major third remains more of a
special effects interval either in conventional practice (where the
diminished fourth is sometimes boldly used) or in Vicentino's style.

These categories nicely fit either 1/4-comma meantone or the very
similar 31-tET. Also, if one reasonably concludes that Vicentino's
melodic enharmonicism dividing a chromatic semitone into two dieses
(e.g. C4-C*4-C#4) calls for an equal or near-equal division, then
these tunings (41.06-34.99 or 38.7-38.7 cents) would fit this
requirement.

Here it's only fair to note, on the diesis question, that
unconventional views (proposing something other than 1/4-comma
meantone or 31-tET) can be found in the literature. Thus Mark Lindley,
in one article from the 1980's, suggested a kind of 31-note
well-temperament, based in part on Vicentino's remarks about the
_second_ tuning.

However, let's assume that we take an equal or near-equal 31-note
division as the basis for the first tuning -- apart from the question
of what to do with the five remaining "comma keys" on his 36-note per
octave keyboard.

As you observe, Graham, Vicentino does make an observation about a
kind of major third equivalent to the Pythagorean, which would not be
"blunted." Is it possible that here he is suggesting that from a
melodic viewpoint, the full 81:64 ditone (or the close approximation
of a major third plus his "half-diesis" comma, say 5:4 plus 20.53
cents or 406.84 cents if we calculate a diesis at 128:125) is ideal
for melody, although the 5:4 is required for stable vertical concord?

Anyway, however he might have disposed the five "comma keys" in his
first tuning -- and Maniates and Palisca in fact treat them as
"half-diesis" keys 1/62-octave higher than their unmodified
counterparts (e.g. C-E'3 near 81:64) -- his description of his second
or "just fifth" tuning in his 1555 and 1561 treatises suggests to me
as well as to other interpreters that his "comma" equal to the
temperament of the fifth (~1/4-comma) is the one likely intended.

Vicentino says that this tuning features "perfect fifths" and "perfect
thirds" sounding together (1561), and that suggests to me the pure
ratios based on 5 rather than the Pythagorean ratios (which to
Vicentino and Zarlino cannot sustain harmonic concord).

Also, if one takes "perfect thirds" to mean 5:4 and 6:5, then this
could be taken as implying a regular temperament specifically of
1/4-comma meantone for each manual (ideally both having 19 notes in
Gb-B# a "comma" apart, and on his actual instruments having 19 and 17
notes respectively). Then each manual would have regular "perfect" 5:4
thirds, with the 3:2 fifths and 6:5 minor thirds supplied by the
"comma" (i.e. 1/4-comma) adjustments between the manuals.

Canvassing the literature, I have found that Vicentino is ambiguous
enough to have suggested a range of hypotheses -- for example, one
creative author connected Vicentino's "just fifth" with the near-pure
fifth which in fact results if we tune 31 notes in 1/4-comma meantone,
where the "odd" fifth will be about 6.07 cents wider than the others
(which are tempered by about 5.38 cents in the narrow direction).

With the first tuning, I find an arrangement of the first 31 notes in
1/4-comma or 31-tET to fit his theory and music, including the matter
of the usual "proximate intervals." Deciding how to interpret the
"comma keys" could go in two directions.

Like Bill Alves and some others, I find the "comma as 1/4-comma"
reading very attractive: it would provide some optional pure concords
for a few comma diatonic sonorities often appearing in prolonged
durations at the end of a composition or section where pure intonation
might be especially striking. Here I mean "pure intonation" in
16th-century terms, with 4:5:6 or 10:12:15 sonorities.

The "half-diesis" division maybe seems to me a bit more "antiquarian."
While it might nicely fit a kind of mixed "Gothic-Renaissance" style
of the kind I enjoy cultivating myself, I don't see any avowed
tendencies of this kind in Vicentino. Maybe it's a bit like Ugolino of
Orvieto's suggestion that to his 17-note Pythagorean keyboard -- an
MOS keyboard, in Wilsonian terms (Gb-A#) -- one might add two notes
dividing E-F and B-C into equal dieses in the ancient Greek manner,
although he remarks that this is not used in modern (i.e. early
15th-century) practice.

In deciding which interpretation _might_ be more musically likely --
and I agree, Graham, that certain remarks of Vicentino can raise a
real possibility of more than one "solution" -- I would consider his
remarks defining modern music as _musica mista e participata_, that is
"mixed and tempered."

Both what he describes as a "mixture" of genera (i.e. the use of steps
from the diatonic, chromatic, and enharmonic genera), and the
tempering of the fifth in order to get more concordant thirds and
sixths, are basic traits for him of common practice as well as his own
fifthtone music.

Possibly, as you suggest, Graham, Vicentino himself may not always
have had a single interpretation for those "comma keys" -- maybe he
wanted to show how they _could_ be used to emulate Pythagorean thirds,
a feature one noted ancient system of tuning (the half-diesis
reading). Whether he would have tuned them this way in his first
tuning, some or all of the time, remains an open question.

Graham and Paul, I would say that the use in Vicentino's notated music
of the diesis but not the comma, as far as I am aware, makes the
latter concept (and its implementation in his tunings) somewhat more
problematic. The 1/4-comma hypothesis might fit his 1561 treatise, and
also my own musical intuition -- but we are indeed dealing with
probabilities and interpretations, much as when performing
16th-century music and deciding which accidental inflections to add.

Here scholarly humility is well called for: as astute a commentator
and musician as Bill Alves has set an example by remarking that a
cycle in 1/4-comma meantone seems a "probable" interpretation, and
that the "comma as 1/4-comma" is one reading for Vicentino's "comma
keys."

Most appreciatively,

Margo Schulter
mschulter@value.net