Yahoo Tuning Groups Ultimate Backup tuning post from SMT list

Here is an interesting article from the Society for Music Theory list. Note
that the author is not subscribed to this list.

>Date: Wed, 28 Mar 2001 04:25:51 -0800 (PST)
>Reply-To: smt-list@boethius.music.ucsb.edu
>Originator: smt-list@boethius.music.ucsb.edu
>Sender: smt-list@boethius.music.ucsb.edu
>Precedence: bulk
>From: Jeffrey Dean <dean@okeghem.demon.co.uk>
>To: alves@orion.ac.hmc.edu
>Subject: Re: equal temperament; Lassus Prophetiae
>
>There's been more on the list lately than I can find the time to
>reply to, but there are a couple of points I do want to address
>briefly.
>
>Equal temperament was first described rigorously at the beginning of
>the 17th century by the Dutch mathematician Simon Stevin in his
>unpublished _Vande Spiegheling der Singconst_, written before 1608;
>he used several equivalent notations for the 12th root of 2, but
>Napier's logarithms were not published until a few years later. They
>did make it much easier for mathematicians to calculate
>interval-sizes, but (partly because Stevin's treatise was
>unpublished) no-one did so before the 1660s. There is an excellent
>account of Stevin's work in Floris Cohen, _Quantifying music: the
>science of music at the first stage of the Scientific Revolution,
>1580-1650_ (Dordrecht: D. Reidel, 1984); he makes it clear that
>Stevin didn't consider what he was describing as a temperament, but
>rather as the true mathematical foundation of the musical intervals
>-- that is, he didn't believe in the small-integer ratios of
>Pythagorean and just intonation at all. For the uptake in the later
>17th century, see Penelope Gouk, _Music, science and natural magic in
>seventeenth-century England_ (Yale U.P., 1999).
>
>One of Gouk's arguments is that the natural philosophers of the Royal
>Society (etc.) believed in equal temperament because they tended to
>be gentleman-amateur musicians who typically played the viol. There
>is evidence that fretted instruments had been tuned (to a close
>approximation) to equal temperament from early in the 16th century;
>Mark Lindley has written a great deal about this and other practical
>tuning and temperament systems, and I won't single out a particular
>essay of his. It *may* be pertinent that in 1482 Bartolomeus Ramis de
>Pareia argued that the tritone is identical to the diminished 5th;
>this is strictly true only in equal temperament, but he may simply
>have been counting keys (he often cited the keyboard, sometimes
>specifically the organ, to illustrate his points).
>
>
>Lassus's _Prophetiae Sibyllarum_ is a knotty problem, and I don't
>pretend to have a solution to it. I was exposed very early to William
>Mitchell's Schenkerian analysis of the prologue in _The Music Forum_
>(1970); I concur on the whole with Peter Bergquist's analysis
>recently posted. I want to comment specifically on the issue of
>"chromaticism". This meant something entirely different in Lassus's
>time from what we normally understand today. When we talk about
>chromaticism we mean the 12-note semitonal octave. When Lassus's
>contemporaries (we don't have any verbal testimony from Lassus
>himself, only the music) referred to "chromaticism" they meant the
>chromatic genus of ancient Greek music, in which the octave is
>constituted from two chromatic tetrachords
>(semitone-semitone-semiditone) separated by a tone [1132113]. When
>Claude Le Jeune wrote "Quelle eau" (1585), "Qu'est devenu ce bel
>oeil" (1608), and "Hellas, mon Dieu" (1612) he exemplified
>chromaticism *not* by saturating his pitch space with semitones but
>by expressing the chromatic tetrachord specifically. For Nicolo
>Vicentino, the minor 3rd was as much a diagnostic interval for
>chromaticism as the semitone, and this was the thrust of his famous
>(losing) debate with Vicente Lusitano in 1551 over whether the music
>of the time was diatonic or not.
>
>What Lassus was doing in the _Prophetiae_ (not just in the prologue)
>is *not* the same as what Le Jeune did. There are no expressions of
>the chromatic tetrachord, and precious few melodic minor 3rds;
>instead there are many chromatic semitones -- as distinct from
>diatonic semitones, though these are also present. The tuning of
>semitones in the 15th and 16th centuries is another difficult
>problem, which has been very little discussed.* In a nutshell,
>whether one is following Pythagorean or just intonation or any
>unequal temperament (such as one of the meantone alternatives), the
>diatonic semitone between B and C or between C-sharp and D is a
>different size to the chromatic semitone between C and C-sharp. In
>Pythagorean or just intonation the diatonic semitone is smaller than
>the chromatic; in meantone temperament it is larger. Lusitano's
>winning argument against Vicentino was that in normal music the
>chromatic semitone is never used, therefore music is diatonic.
>Composers immediately (if they hadn't already) proved him wrong, as
>they usually do with overgeneral theoretical pronouncements. Cipriano
>de Rore in particular wrote in 1555 a brilliant setting of Horace's
>_Calami sonum ferentes_ (about the invention of music from the sound
>of the wind in the reeds; I may have the wrong poet here, but it's
>one of the classic authors), which is based on an ascending semitone
>scale. There is some good stuff about late-16th- and 17th-century
>chromaticism in Eric Chafe, _Monteverdi's tonal language_ (NY:
>Schirmer, 1992).
>
>I think the inequality of sung semitones (which must have been very
>demanding on the singers, though experience shows it is not
>unattainable; Vicentino called for microtones in some of his works)
>is significant to our understanding, not only of the sound of this
>music, but of the composers' purposes. I have not undertaken a close
>analysis of these works myself, but if I did I should want to be
>extremely alert to the differing significance of diatonic and
>chromatic semitones in the voice-leading. I don't disagree with the
>harmonically-based analyses that have been posted before this, but I
>think they ought to be supplemented by attention to what the
>composers thought their "chromaticism" meant.
>
>*The real problem alluded to here is that there is evidence that
>singers (and players of instruments with flexible tuning) did not
>hold strictly to the basic tuning they were following. (The evidence
>that singers used an essentially just tuning rather than the
>Pythagorean goes back to Ramis in 1482 again, and is probably valid
>for about a generation earlier.) But there are other witnesses from
>the beginning of the 14th century (Marchettus of Padua) well into the
>16th (the preface to a French chanson publication of Lassus's time;
>I'm afraid I've mislaid the precise reference) that the supposedly
>diatonic semitone between a sharped note and the note above was
>narrower than the proper semitone (solmized fa-mi) between a flatted
>note and the note below, or between F and E or C and B. The extent of
>this narrowing is expressed differently by the different witnesses,
>but I conclude that singers were doing something essentially
>equivalent to the well-known phenomenon in which string players
>narrow their leading tones. (I refer again to my article on
>"Okeghem's attitude towards modality" cited in a recent posting.)
>--
>--------------------------------------------------------------------
> Jeffrey Dean Internet: dean@okeghem.demon.co.uk
> 4 Chandos Road Phone: +44 (0)161 861 7542
> Chorlton-cum-Hardy Fax: +44 (0)161 861 7543
> Manchester M21 0ST
> England 'Harmonia est discordia concors.'
>====================================================================
>

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

🔗Graham Breed <graham@microtonal.co.uk>

Bill Alves wrote:

> Here is an interesting article from the Society for Music Theory
list. Note
> that the author is not subscribed to this list.

You'll have to act as go-between then.

> >Equal temperament was first described rigorously at the beginning
of
> >the 17th century by the Dutch mathematician Simon Stevin in his
> >unpublished _Vande Spiegheling der Singconst_, written before 1608;

Didn't Partch declare this a tie between Europe and China?

> >It *may* be pertinent that in 1482 Bartolomeus Ramis de
> >Pareia argued that the tritone is identical to the diminished 5th;
> >this is strictly true only in equal temperament, but he may simply
> >have been counting keys (he often cited the keyboard, sometimes
> >specifically the organ, to illustrate his points).

Or comparing 45/32 and 1024/729

> >Lassus's _Prophetiae Sibyllarum_ is a knotty problem, and I don't
> >pretend to have a solution to it. I was exposed very early to
William
> >Mitchell's Schenkerian analysis of the prologue in _The Music
Forum_
> >(1970); I concur on the whole with Peter Bergquist's analysis
> >recently posted.

All sounds interesting. Bill, can you forward me this analysis?

> >Claude Le Jeune wrote "Quelle eau" (1585), "Qu'est devenu ce bel
> >oeil" (1608), and "Hellas, mon Dieu" (1612) he exemplified
> >chromaticism *not* by saturating his pitch space with semitones but
> >by expressing the chromatic tetrachord specifically.

A new name to me. Is anybody familiar with this music?

> > The tuning of
> >semitones in the 15th and 16th centuries is another difficult
> >problem, which has been very little discussed.* In a nutshell,
> >whether one is following Pythagorean or just intonation or any
> >unequal temperament (such as one of the meantone alternatives), the
> >diatonic semitone between B and C or between C-sharp and D is a
> >different size to the chromatic semitone between C and C-sharp. In
> >Pythagorean or just intonation the diatonic semitone is smaller
than
> >the chromatic; in meantone temperament it is larger.

The diatonic semitone usually larger in JI. Not always, because it
may arise from a string of fifths.

> >I think the inequality of sung semitones (which must have been very
> >demanding on the singers, though experience shows it is not
> >unattainable; Vicentino called for microtones in some of his works)
> >is significant to our understanding, not only of the sound of this
> >music, but of the composers' purposes. I have not undertaken a
close
> >analysis of these works myself, but if I did I should want to be
> >extremely alert to the differing significance of diatonic and
> >chromatic semitones in the voice-leading. I don't disagree with the
> >harmonically-based analyses that have been posted before this, but
I
> >think they ought to be supplemented by attention to what the
> >composers thought their "chromaticism" meant.

In many cases, the distinction between semitones arises naturally from
the harmony, and a meantone continuo would merely support this. My
impression of Gesualdo is that he did treat the two semitones
differently, but never left a "smoking gun" where the "wrong" semitone
was written in the context of the rules he seems to follow.

As there's a big gap between Vicentino's independently published
music, and that to illustrate his theories, it indicates to me that
singers were having trouble with the enharmonicism, and so he avoided
it.

> > there are other witnesses from
> >the beginning of the 14th century (Marchettus of Padua) well into
the
> >16th (the preface to a French chanson publication of Lassus's time;
> >I'm afraid I've mislaid the precise reference) that the supposedly
> >diatonic semitone between a sharped note and the note above was
> >narrower than the proper semitone (solmized fa-mi) between a
flatted
> >note and the note below, or between F and E or C and B. The extent
of
> >this narrowing is expressed differently by the different witnesses,
> >but I conclude that singers were doing something essentially
> >equivalent to the well-known phenomenon in which string players
> >narrow their leading tones. (I refer again to my article on
> >"Okeghem's attitude towards modality" cited in a recent posting.)

That's another interesting detail. The use of split-key instuments
for accompaniment may count against it for the late C16th.

Graham

🔗jpehrson@rcn.com

🔗Alison Monteith <alison.monteith3@which.net>

🔗M. Schulter <MSCHULTER@VALUE.NET>

Hello, there, and thanks to Bill Alves for sharing a post originally
made to the Society for Music Theory (SMT) list by Jeffrey Dean, on
which I would have a number of comments, inviting a repost of what
follows or any portion thereof to the SMT list if Bill or another
subscriber finds it appropriate and helpful.

> There is evidence that fretted instruments had been tuned (to a
> close approximation) to equal temperament from early in the 16th
> century; Mark Lindley has written a great deal about this and other
> practical tuning and temperament systems, and I won't single out a
> particular essay of his.

Here the main qualification might be that while 12-tone equal
temperament (12-tET) with equal semitones does appear to become the
"standard" tuning for lute by around the middle of the century,
Lindley notes that some lute pieces by composers such as Milan around
the 1530's seem ideally to fit a meantone tuning.[1]

Also, as Lindley notes, it happens that because of the physics of lute
playing with the factor of pressure applied to the frets in stopping
them, Vincenzo Galilei's "18 rule" of frets spaced by a repeated
rational factor of 18:17 (~98.95 cents) in practice give a better
approximation of 12-tET than the "correct" logarithmic spacing.

> It *may* be pertinent that in 1482 Bartolomeus Ramis de Pareia
> argued that the tritone is identical to the diminished 5th; this is
> strictly true only in equal temperament, but he may simply have been
> counting keys (he often cited the keyboard, sometimes specifically
> the organ, to illustrate his points).

While the views of Ramis (or Ramos) on the tritone are of great
interest, I would emphasize that whatever his views on its size in
relation to the diminished fifth, his treatse of 1482 clearly points
to some tuning for keyboards using _unequal_ semitones, which Lindley
persuasively argues is likely some shade of meantone temperament.[2]

In his treatise, in discussing how to find "good" and "bad" thirds and
other intervals on a keyboard instrument, Ramos gets into a discussion
of the relative merits of having Ab or G# in a 12-note tuning -- that
is, Ab-C# vs. Eb-G# as the chain of fifths. In a 12-tET tuning where
G# and Ab are acoustically equivalent, this debate (in which Ramos
finds Ab the more "provident" choice) would seem rather beside the
point.

In the course of this discussion, Ramos argues that the utility of G#
in an Eb-G# tuning is limited because there is no "good" fourth D#-G#,
stating that Eb-G# does not form a proper fourth, as it would, of
course, in 12-tET.

Further, he observes that some people like to satisfy both sides of
the question by providing a keyboard instrument with keys for both Ab
and G# -- the "split-key accidentals" of a kind advocated by certain
early 15th-century theorists in a Pythagorean setting, and used in the
Lucca organ of around 1480 (G#/Ab, and either Eb/D# or Bb/A# depending
on which sources or interpretations one favors).

Like Lindley, I find his remarks very much in line with the hypothesis
of the widespread use of meantone temperament by 1482, a practice
expressly described by Gafurius in 1496, although not precisely
quantified in terms of fractions of a syntonic comma until Zarlino
(1558 and later). At any rate, he is evidently describing some tuning
other than 12-tET.

A quick aside: the fine distinction between tritone and diminished
fifth might or might not be made at various eras. Thus a typical
13th-century reckoning based on Pythagorean intonation recognized a
diatonic set of 13 simple intervals ranging from unison to octave,
with a single category of tritone (_tritonus_). Around 1325, however,
Jacobus of Liege pointed out that there were actually 14 such
intervals, since the _semitritonus_ or diminished fifth had a
different and smaller size of 1024:729 (~588.27 cents) than the
_tritonus_ or augmented fourth at 729:512 (~611.73 cents).

> I want to comment specifically on the issue of "chromaticism". This
> meant something entirely different in Lassus's time from what we
> normally understand today. When we talk about chromaticism we mean
> the 12-note semitonal octave. When Lassus's contemporaries (we don't
> have any verbal testimony from Lassus himself, only the music)
> referred to "chromaticism" they meant the chromatic genus of ancient
> Greek music, in which the octave is constituted from two chromatic
> tetrachords (semitone-semitone-semiditone) separated by a tone
> [1132113].

Certainly the Greek chromatic genus was one defining concept in the 16th
century, but not necessarily the only one. As Thomas Morley observes in
1597, musicians such as organists often regarded a "chromatic" figure as
one alternating diatonic and chromatic semitones in stepwise motion,
giving the example in semibreves or whole-notes of E-F-F#-G-G#-A. Since
the chromatic semitone might be considered the distinctive interval
defining the chromatic genus, this looser sense has some connection to the
stricter one. Morley himself remarks that a theme such as that he states
"is not right Chromatica, but ... patched up of half Chromatic, and half
Diatonic."[3]

> When Claude Le Jeune wrote "Quelle eau" (1585), "Qu'est devenu ce
> bel oeil" (1608), and "Hellas, mon Dieu" (1612) he exemplified
> chromaticism *not* by saturating his pitch space with semitones but
> by expressing the chromatic tetrachord specifically. For Nicolo
> Vicentino, the minor 3rd was as much a diagnostic interval for
> chromaticism as the semitone, and this was the thrust of his famous
> (losing) debate with Vicente Lusitano in 1551 over whether the music
> of the time was diatonic or not.

Yes, Vicentino's thesis was that whenever a melodic minor third
appeared, it marked a use of the chromatic genus, causing him to
describe music of the common practice as _participata & mista_, or
"tempered and mixed," featuring both the slight narrowing or
"blunting" of the fifth characteristic of meantone, and the mixing of
the genera. Zarlino (1558) condemned this opinion of the
"chromaticists," asserting that the chromatic semitone and enharmonic
diesis rather than the minor and major third were the defining
intervals of these two genera, with steps of a major or minor third
belonging equally to the diatonic.

Vicentino also maintained that a strictly "diatonic" composition could
not use _any_ accidental inflections such as sharps applied by
composer or performer to obtain major sixths before octaves, etc. In
contrast, Zarlino took for granted the use of "justifiable" and indeed
necessary accidentals as a feature of normal diatonic style.

> In Pythagorean or just intonation the diatonic semitone is smaller
> than the chromatic; in meantone temperament it is larger.

Indeed in Pythagorean intonation, the diatonic semitone or limma
(256:243, ~90.22 cents) is smaller than the chromatic semitone or
apotome (2187:2048, ~113.69 cents), while in meantone, e.g. 1/4-comma
with pure 5:4 major thirds, the diatonic semitone (~117.11 cents) is
larger than the chromatic (~76.05 cents).

However, for "just intonation" taken in the sense of a system based on
pure ratios of 3 and 5 (as in the systems of Fogliano in 1529 and
Zarlino in 1558), as for meantone, the diatonic semitone is larger,
most typically 16:15 (~111.73 cents), than the chromatic semitone,
most typically 25:24 (~70.67 cents). These semitone sizes vary,
because of the use of differently sized whole-tones at 9:8 and 10:9,
but with the diatonic semitone characteristically larger.

> Lusitano's winning argument against Vicentino was that in normal
> music the chromatic semitone is never used, therefore music is
> diatonic. Composers immediately (if they hadn't already) proved him
> wrong, as they usually do with overgeneral theoretical
> pronouncements.

Interestingly, the first bold use of what is sometimes termed "direct
chromaticism" -- the use of a chromatic semitone as a melodic step --
in Western European polyphonic music may be the practice of Marchettus
of Padua and some of his colleagues during the early 14th century,
described and advocated in his _Lucidarium_ of 1318, and also
documented in some compositions from this epoch cited by Jan
Herlinger.[4]

However, according to much conventional theory, the apotome or
chromatic semitone was "unsingable," and this view is expressed as
late as 1565 by Tomas de Santa Maria in a treatise on four-voice
keyboard textures and the art of _fantasia_ or improvisation, where he
further cautions that "what cannot be sung cannot be played." As you
rightly note, the experimental compositions of Lasso, Rore, and others
of this same era might serve as counterexamples.

As far as the Vicentino-Lusitano debate goes, I might award the formal
and narrow issue of the disputation to Lusitano (a melodic leap of a
minor third as tone-plus-diatonic-semitone, or of a major third as a
"ditone" or tone-plus-tone, seems routinely diatonic to me).

However, I would emphatically join Vicentino in celebrating the
artistic merits and possibilities of polyphonic music using the steps
of the chromatic semitone and enharmonic diesis -- the latter
fortuitously and felicitously realized by the usual meantone diesis of
128:125 (~41.06 cents), about 1/5-tone. Based on my own experience
with a 24-note archicembalo (or the synthesizer equivalent) in
1/4-comma meantone as a subset of Vicentino's full instrument with 36
or 38 notes per octave, I would say that his germinal contributions
leave immense scope for further exploration.

Maybe the Vicentino-Lusitano affair is an example of how great
aesthetic controversies can focus on issues not always so happily
defined.

> I don't disagree with the harmonically-based analyses that have been
> posted before this, but I think they ought to be supplemented by
> attention to what the composers thought their "chromaticism" meant.

As someone looking at 16th-century music from something of a
medievalist perspective (which means maybe that my 14th-century biases
and other people's 18th-century biases may average out to a balanced
analysis <grin>), I would focus on such things as the theme of
"closest approach" progressions by contrary motion such as m3-1, M3-5,
M6-8 as they guide motion between the pervasive tertian sonorities of
the newer style.

Scholars such as Richard Crocker and Carl Dahlhaus have emphasized
this point, and I might describe the 13th-16th century approach as one
of "combinative verticality," with two-voice intervals as the
elementary particles, but with a three-note unit of complete
sonority.

In Gothic music, this unit is Johannes de Grocheio's _trina harmoniae
perfectio_ or "threefold perfection of harmony" expressed by the
combination of outer octave, lower fifth, and upper fourth
(e.g. d-a-d' or D3-A3-D4 in MIDI notation), which another theorist
writing in the same era around 1300 derives from the series of ratios
2-3-4. Interestingly, this "natural" series with 2:3 fifth below 4:3
fourth coincides with the modern frequency ratio of 2:3:4; the
medieval string ratio is 12:8:6 or 6:4:3.

By the second quarter of the 16th century, it is clearly what Zarlino
terms _harmonia perfetta_, "the third plus fifth or sixth," ideally
outer fifth, lower major third, and upper minor third, with a string
ratio of 15:12:10 and a frequency ratio of 4:5:6.

In my view, an analysis of 16th-century music, chromatic or otherwise,
should approach the vertical dimension in terms of directed two-voice
progressions (some borrowed from earlier Gothic practice and theory,
others new) and in terms of the different combinations and
arrangements of _harmonia perfetta_, with Zarlino's remarks on these
points providing one valuable outlook. Also, Vicentino and Santa Maria
bring their own vital perspectives to this question, and I've found
that the latter's four-voice patterns and progressions can be relevant
for analyzing a range of 16th-century pieces, especially those
involving note-against-note textures.

> *The real problem alluded to here is that there is evidence that
> singers (and players of instruments with flexible tuning) did not
> hold strictly to the basic tuning they were following. (The evidence
> that singers used an essentially just tuning rather than the
> Pythagorean goes back to Ramis in 1482 again, and is probably valid
> for about a generation earlier.)

As Bill Alves has aptly emphasized on the Tuning list, flexible
intonation is indeed flexible, so that a tuning for fixed-pitch
instruments is a model rather than a definition of what an ensemble
may actually do.

Here I would agree that "about a generation" before the treatise of
Ramos, or around 1450, is a very reasonable date for a shift from
Pythagorean to meantone tuning for keyboards, and to some
approximation of just intonation based on ratios of 3 and 5 (i.e. 5:4
and 6:5 thirds) for singers and players of flexible-pitch
instruments.

The problem of intonation in the era of the Gothic-Renaissance
transition, arguably making up most of the 15th century, is an
intriguing one. For the era of the young Dufay (say 1420-1450), I
would be inclined with Mark Lindley to posit some kind of modified
Pythagorean tuning with smoother "schisma thirds" (diminished fourths
and augmented seconds with ratios very close to 5:4 and 6:5) used for
certain sonorities, especially prolonged noncadential ones.

Experimenting with a 15-16 note Pythagorean tuning on two synthesizer
manuals (one in the likely typical Eb-G# of the 14th century, or
Bb-D#, the other in the popular Gb-B tuning of the early 15th
century), I've found myself agreeing with Prosdocimus de Beldemandis
(1413) and Ugolino of Orvieto (c. 1425-1440?) in favoring regular and
active Pythagorean thirds and sixths for cadences in the "classic"
14th-century manner combining M6-8 and M3-5 (e.g. E-G#-C# to D-A-D).
At the same time, noncadential schisma third sonorities like E-Ab-B
(written E-G#-B) can have a beguiling effect which to my ears very
much evokes the fresh new aura of the Dufay era.

Lindley suggests 1450 as an approximate date for the transition to
meantone, with the compositions of Conrad Paumann for keyboard and the
vocal works of the later Dufay and Ockeghem favoring such a system
with a consistent use of pure or near-pure thirds.

However, some scholars lean toward the view that singers may have
still often favored a somewhat "Pythagorean" approach in the middle to
late 15th century, so that the kind of theoretical disarray discussed
by Lindley may reflect the transitional nature of practice as well as
conceptual models.

For example, singers may have at times leaned toward the compact
diatonic semitones of the Pythagorean tradition, and at other times
toward smoother vertical thirds and sixths, for example in resolving
the suspension dissonances playing a central role in the new style.

> But there are other witnesses from the beginning of the 14th century
> (Marchettus of Padua) well into the 16th (the preface to a French
> chanson publication of Lassus's time; I'm afraid I've mislaid the
> precise reference) that the supposedly diatonic semitone between a
> sharped note and the note above was narrower than the proper
> semitone (solmized fa-mi) between a flatted note and the note below,
> or between F and E or C and B.

The expression of such views by Renaissance theorists is very
interesting, and I'd love to learn more about this.

What you describe is a very apt summary of Marchettus, who indeed
advocates the use of an extra-narrow cadential semitone or "diesis" in
certain directed vertical progressions from an unstable to a stable
interval by stepwise contrary motion (specifically M3-5, M6-8, or
M10-12).

In his fivefold division of the tone -- with at least one passage
suggesting to both Jan Herlinger and me that he may have meant a
division into five _equal_ dieses -- a usual mi-fa semitone such as
E-F or B-C is "two parts" of the tone, possibly 2/5-tone, and the
usual apotome (e.g. Bb-B, or German B-H) "three parts," possibly
3/5-tone.

In contrast, a cadential diesis such as G#-A is only "one part" of the
tone, possibly 1/5-tone, or about half the size of a usual semitone
step. The complementary "chroma" or "chromatic semitone," e.g. G-G#,
is thus equal to the remaining "four parts," possibly 4/5-tone.

Note that Marchettus is addressing vocalists, who could made small
adjustments to achieve these different step sizes while maintaining or
closely approximating the pure Pythagorean ratios which Marchettus
makes part of his system (2:1 octave, 3:2 fifth, 4:3 fourth, 9:8
whole-tone).

In a keyboard tuning system, one must make some compromises one way or
another -- this is not a theoretical flaw, only a limitation of
modelling vocal intonation on a fixed-pitch instrument.

One moot question may be whether Marchettus, in advocating his
extra-narrow diesis specifically for progressions with _ascending_
semitones involving _musica ficta_ (mi-signs or sharps outside the
Guidonian gamut of the diatonic notes plus Bb), may in part be
reflecting notational concerns.

He advocates a distinction between the usual "square-B" sign (like a
modern natural sign) showing a note below a normal semitone step such
as B-C, and a special diesis sign (somewhat like a modern sharp)
showing at once a semitone alteration and his special division of the
tone into _chroma_ and diesis (e.g. G-G#-A).

In usual practice, both types of signs are often used interchangeably
as "mi-signs" showing a note below a semitone. Marchettus proposes
that the diesis sign be reserved specifically to show his special
chroma/diesis division.

Thus his system permits a notated distinction between regular
semitones and special cadential ones using signs already familiar. To
show a similar chroma/diesis division for _descending_ semitones, he
would have had to invent a new kind of "fa-sign" in addition to the
established "round-B" sign (the source of the modern flat). Of course,
one could reply, "If he wanted his special semitones in cadences with
descending semitonal inflections (e.g. Eb-D), or in such progressions
within the regular or _musica recta_ gamut, he could have invented
appropriate signs, or at least described such a practice."

The Berkeley Manuscript or Paris Anonymous of around 1375 -- part of
it dated to that year -- includes one treatise advocating that singers
intone a usual mi-fa semitone as 2/3-tone, but a cadential semitone as
1/3-tone. Oliver Ellsworth has interpreted this statement, directly
addressing only the finding of semitones for singers, not polyphony or
vertical intervals, as suggesting 19-tone equal temperament (19-tET),
the system used and described about two centuries later by Costeley
(1570).[5]

However one interprets these 14th-century statements and systems --
another topic -- they do suggest a distinction for some musicians
between cadential and other semitones. Here I should emphasize that I
use "cadential" (from _cadentia_) in the medieval sense of Jacobus of
Liege: a progression from a more tense or unstable interval to a
stable one, not necessarily limited to the end of a phrase or the
like.

While accentuated cadential progressions with narrower-than-usual
semitones seem to me a not-unlikely practice in various eras and
places, I'm not sure about any consistent tendency to make ascending
cadential semitones narrower than descending ones. For theorists such
as Prosdocimus and Ugolino in the earlier 15th century, these
semitones should be the same size in either direction, and I have
heard reports that performers may tend to narrow certain semitones in
_either_ direction (e.g. either C#-D or Eb-D).

-----
Notes
-----

1. Mark Lindley, _Lute, viols, and temperaments_ (Cambridge
University Press, Cambridge, 1984), for example pp. 52-55 on Milan.

2. Mark Lindley, "Fifteenth-Century Evidence for Meantone
Temperament," _Proceedings of the Royal Musical Association_ 102
(1976), 37-51.

3. Thomas Morley, _A Plain & Easy Introduction to Practical Music_,
Alec Harman, ed., 2nd ed. (W. W. Norton, 1973), p. 103.

4. See Marchettus of Padua, _The Lucidarium of Marchetto of Padua_,
Jan W. Herlinger, ed. (University of Chicago Press, 1985); and Jan
W. Herlinger, "Marchetto's Division of the Whole Tone," _Journal of
the American Musicological Society_ 34 (1981), pp. 193-216.

5. Oliver B. Ellsworth, "A Fourteenth-Century Proposal for Equal
Temperament," _Viator_ 5 (1974), pp. 445-453. For Costeley's tuning,
see Kenneth J. Levy, "Costeley's Chromatic Chanson," _Annales
Musicologues: Moyen-Age et Renaissance_, Tome III (1955), pp. 213-261.

Most respectfully,

Margo Schulter
mschulter@value.net

🔗jpehrson@rcn.com

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:

/tuning/topicId_20497.html#20744

I very much enjoyed Margo Schulter's repost to the Society of Music
Theory post. As you will see in the following, the SMT will NOT
accept "forwarded" posts. They are a little "stuffy" about what they
will include in their posts, in my view. ALL of my short posts have
been rejected so far.

I believe the rejection has mostly to do with the fact that my posts
were simply short comments and not scholarly articles or commentary.
Frankly, I believe this is a shortcoming of that particular list.
Although I very much enjoy reading thorough and scholarly articles, I
also like, sometimes, isolated ideas that can illuminate the larger
posts. These can also be quite valuable.

This is one reason that I feel the Tuning List is such a "healthy"
format as opposed, say, to SMT. However, I should say that I have
been enjoying reading the SMT posts. At the moment there is a lively
discussion of counterpoint, much like the one we had on THIS list,
minus the emphasis on tuning. Additionally, there is now another
active discussion of that old chestnut -- tonality vs. atonality. On
the overall, I have not found the SMT posts to be "overly academic"
at all. It is quite enjoyable, and has a couple of "offbeat
characters" who are not involved in ANY institutions and who post
regularly. Additionally, the list "owner" Bob Kosovsky, is
affiliated with the New York Public Library, not a University, per
se.... not that it matters, except for the fact that the "initiation
rites" of the SMT seem rather threatening.... I will post them below
for the curious!

In order to JOIN the SMT, one must follow the following procedure:

"2. SUBSCRIBING AND UNSUBSCRIBING

NOTE: The SMT home page now has links to forms for subscribing
and unsubscribing, at the URL http://smt.ucsb.edu/
smt-list/smthome.html, in the section SMT Subscriber Services.

If you have not yet made a request to join the E-conference and
would like to, send a message to the following listproc address
(the "site" address):

listproc@smt.ucsb.edu

Leave the "Subject:" line blank, if possible, and include the
following single line as the content of the message:

subscribe smt-list YourFirstName YourLastName

Do *not* include your email address anywhere in the message.
Replace "YourFirstName" and "YourLastName" with (obviously) your
first and last names. If your email utility is configured to
include a personal letterhead or signature, disable those features.
To cancel a subscription, send the following single line to the
site address given above:

unsubscribe smt-list

Do *not* include your name in a cancellation request.

Subscription to the E-conference is not automatic. The list manager
will make contact by email to finalize the subscription. Once a
subscription has been confirmed, subscribers may send messages to the
smt-list at the following address (the "list" address):

smt-list@smt.ucsb.edu

Note that the list address is different from the site address. Do
not confuse the two. Messages meant for distribution
to all subscribers are sent to the list address, *not* to the
site address. The site address is used for subscribing/unsubscribing,
and for configuring, among other things, subscribers' personal
preferences for receiving email from the smt-list.

Authors *must* include their full name, institutional affiliation, and
email address(es) at the end of all messages sent to smt-list.
Messages without this information will be returned to their authors.
Messages sent to smt-list by non-subscribers will not be distributed
to list members. Instead, the messages will be forwarded to the list
manager, who will contact the author about joining the smt-list."

NOW, to the fun, "threatening" part. Well, maybe this isn't as bad
as it sounds, initially, but after you first send a subscription
request, one must be, ahem, "approved." This is the policy,
according
to the SMT:

"***Society for Music Theory***

Thanks for your interest in the SMT Email Conference. The
Conference is a private list for professional and pre-professional
music theorists, musicologists, composers, performers, and others
with the background for participating in discussion on music-
theoretical, analytical, cognitive, and related subjects at an
advanced level. Typically, subscribers to the SMT conference are
professors and graduate students in the aforementioned areas of
study.
If you have the background for the conference, please fill in the
form below and send it to me. Thanks in advance for your cooperation
and patience.

Sincerely,

Bob Kosovsky
smt-editor@boethius.music.ucsb.edu

Cut Here

Name:

Email Address:

Academic Affiliation:

Societal Affiliation (SMT, AMS, SEM, CMS, or other society):
Discipline (theory, musicology, composition, performance, or other
academic field such as psychology, acoustics, computer programming,
etc.):

Primary Areas of Research/Study:

Where did you hear of the smt-list: "

So why am I a little bit "opposed" to such a structure for such a
list....?? Well, in my view, it's a little like George Bush making
everybody "dress up" in the Oval Office....

Essentially, people who are not "suitable" for such a list, as in the
case of the Tuning List, will stop posting of their own accord...
It's a "self-selecting" process. Additionally, the lack of formal
restrictions leads to a wider range of views. Admittedly, the Tuning
List can sometimes sound like a "chat room..." but, on the overall,
the contrast between the shorter posts and comments with the more
scholarly articles and posts makes the Tuning List a more interesting
read.... at least that's my PERSONAL take on things...

In any case, let me hasten to add that I am finding the SMT list very
interesting to read at present. I would recommend it, if one is
willing to go through the "initiation rites." (I wore socks, though,
when I did the hot coals, however...)

Right now, it is a particularly pleasant and informative read. Just
don't try to post anything, unless it's REALLY detailed...

______ _____ _____ _____
Joseph Pehrson

post from SMT list

🔗Bill Alves <ALVES@ORION.AC.HMC.EDU>

🔗Graham Breed <graham@microtonal.co.uk>

🔗jpehrson@rcn.com

🔗Alison Monteith <alison.monteith3@which.net>

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗jpehrson@rcn.com