Re: Medieval intonations -- reply to Joe Monzo

Hello, there, and I'd like to respond to some comments by Joe Monzo on
issues regarding the use of Pythagorean tuning in medieval European
music, both plainsong and polyphony.

Please let me state at the outset that the following remarks focus on
written music which has come down to us, especially liturgical chant and
various types of polyphonic compositions. Traditional musics of Europe in
the medieval era, as later, may have used various intonational systems.

> And it *is* possible to perform plainchant in a JI scale with
> *every* note consonant with those already being heard - it just
> wouldn't be a type of scale that we usually recognize as a 'basic'
> scale.

Here I might have a couple of different responses.

First, just about any chant will have melodic semitones, regarded when
they occur as simultaneous intervals as dissonances in just about any
system of Western European composition before around 1900. Many chants
also have prominent tritone relations, e.g. between F and B-natural in
G Mixolydian (Modes 7 and 8, or _tetrardus_). A minor second has been
regarded as a strong dissonance whether tuned at 256:243 (~90 cents),
16:15 (~112 cents), 1/4-comma meantone (~117 cents), etc. These
tunings may vary the degree of dissonance, of course.

Secondly, it might be relevant to point out that degrees of "concord"
are a product of musical context as much as acoustics. Even if tuned
in meantone or 5-limit JI, thirds and sixths would be unstable in a
medieval setting rather than "fully concordant" simply because the
musical language treats them thus, and attuned listeners respond in
accordance with the language's expectations. Similarly, if one
attempted to make Bach's or Beethoven's triadic harmonies "richer" by
adding pure 7:4's where stable triads are expected, this would be
"ungrammatical," however attractive from the viewpoint of 7-limit JI
as a goal in itself.

> This really doesn't have much bearing on the question anyway,
> because by the time period under consideration, unaccompanied
> monodic plainchant had mostly disappeared from the church rituals.

Here I must respectfully disagree. While polyphony indeed became more
and more common during the later medieval era, plainsong remained a
vital basis of the liturgy. Indeed, the many hymn settings by
composers of the 15th century -- the period of transition from 3-limit
to 5-limit on the Continent -- featuring the alternation of plainsong
and polyphony nicely symbolize the interaction of these two
forms. Dufay (1397?-1474) comes especially to mind.

Further, this interaction is suggested by the fact that, for example,
a polyphonic Mass setting of the Gloria opens with a plainsong
intonation of _Gloria in excelsis Deo_, the polyphony beginning
on the words _Et in terra pax_.

While it might be argued that the Renaissance practice of chant had
become "corrupted" -- and in the late 16th century, Palestrina was
commissioned to reform this practice -- plainsong continued alongside
polyphony in the liturgy.

> Since the Ars Nova movement in Paris in the 1200s, most church music
> thruout Europe had become quite emphatically polyphonic.

The term "Ars Nova" (New Art) refers to a movement dating to around
the beginning of the 14th century in France (e.g. Philippe de Vitry,
Johannes de Muris) and Italy (e.g. Marchettus de Padua). The late 13th
century, in contrast, was now known as the "Ars Antiqua" (Old Art) or
"Ars Veterum" ("Art of the Ancients").

Actually, both liturgical and secular music played a role in both
movements -- especially if we take Ars Antiqua to include the whole
13th century, and also the end of the 12th century, including the
great liturgical works of Perotin and his colleagues for three and
four voices.

By the late 13th century, the main focus was on the motet, a form
which could have liturgical applications, and was certainly popular in
clerical circles (see Johannes de Grocheio, c. 1300), but often had a
secular focus.

Interestingly, one trend of the 14th century was a growing practice of
polyphonic settings for the Ordinary of the Mass, Machaut's Mass being
one famous example. However, while these settings indeed made
polyphony a more and more prominent element in the liturgy, they did
not make plainsong obsolete, as can be heard, for example, in some
modern performances of Machaut's Mass including also plainsong items
to accompany the composer's settings of the Ordinary.

> Of course, Marchetto of Padua in 1318 discussed the use of intervals
> smaller than the 'standard' Pythagorean semitones. But because of
> his obscure numerical descriptions, exactly what he meant is not
> entirely clear. See [plug alert]:

> http://www.ixpres.com/interval/monzo/marchet/marchet.htm

From the view of a 3-limit vs. 5-limit debate, my take on Marchettus
seems similar to that of Mark Lindley and Karol Berger: his approach,
however unorthodox, fits in nicely with a general preference in the
14th-century for some approximation of Pythagorean intonation with
wide major thirds and sixths, and narrow diatonic semitones.

> There is also evidence suggesting that the shift from Pythagorean to
> Just was already in process in English popular music as early as the
> 1200s (see Walter Odington and Theinred of Dover).

Here we are in agreement, and I might add that actual music from
certain English sources in this period indeed shows a pervasive use of
thirds nicely corroborating the remarks of these theorists (_Sumer is
icumen in_ being the most famous example, and some religious settings
in Latin also concluding on sonorities with thirds). Around 1275(?),
Coussemaker's Anonymous IV remarks that in the "Westcountry" of
England, thirds are esteemed as the "best concords," a remark again
fitting the musical evidence. I might add that there is considerable
variation in 13th-century English styles, some being more similar to
those on the Continent, where Pythgorean tuning would seem ideal in
theory and practice.

By the way, personally I would prefer to regard Pythagorean (3-limit)
as one of the many forms of Just intonation -- but I realize that
"Just" often implies 5-limit or higher. Maybe arguing such fine points
of usage, where the meaning is clear enough in either usage, is a bit
like arguing about English spelling <grin>.

> So based on my historical knowledge, it appears that the use of
> polyphony, the tuning shift from 3- to 5-limit in 'serious' music,
> and the building of Gothic cathedrals, are all intertwined, taking
> place largely during the 1200s and 1300s.

For polyphonic music on the Continent in the Gothic and Renaissance
and Manneristic eras, I would suggest this rough timetable:

Pythagorean, usual flats and sharps 1200-1400
Pythagorean, sharps often tuned as flats 1400-1450
5-limit JI or meantone for keyboards 1450-1650

In this chronology, regular Pythagorean tuning is the norm for the
13th and 14th centuries. In the late Ars Nova, however -- maybe as
early as 1370-1380 in certain places such as Florence -- there was
apparently a movement to tune Pythagorean sharps as flats, converting
the usual wide major thirds (very effective in cadences to stable
intervals) into diminished fourths or "schisma thirds" (8192:6561)
only ~1.95 cents from a pure 5:4.

Mark Lindley documents this movement as it manifests in early
15th-century keyboard music, and also possibly some vocal music around
the epoch of the young Dufay. As a result both of internal
developments in style on the Continent, and the English influence
possibly linked to the occupation of France after the battle of
Agincourt (1415), both the more pervasive use of thirds and sixths,
and a certain fascination with the altered Pythagorean forms of these
intervals, followed.

By around 1450, as Lindley argues, keyboard compositions such as those
of Conrad Paumann seem to invite a meantone tuning where all the
regular thirds -- not just four in each octave -- can be pure or
near-pure.

In 1482, Ramos appears to describe a meantone keyboard tuning as well
as his famous 5-limit just intonation scheme.

As Joe Monzo very aptly notes, theory during the late 15th and early
16th century was in a state of flux. Writers such as Gaffurius took
Pythagorean tuning as their theoretical model, although noting that in
practice other schemes prevailed -- for example, the narrowing of
fifths on keyboards.

My own view on this tension between theory and practice around 1500
might be similar to that of Carl Dahlhaus and Mark Lindley. In
medieval practice, the melodic ideal of narrow diatonic semitones and
the harmonic ideal of active thirds and sixths "striving" toward
stable 3-limit intervals were in equilibrium. With the 15th-century
shift toward thirds and sixths as primary concords, however, the new
harmonic style required wider leading-tones -- and it took theorists a
bit of time to accept that the _major_ semitone was now the normal
diatonic interval.

Also, I would agree with Joe Monzo that Zarlino (1558) and Lippius
(1610, 1612) were two leading advocates of the "syntonic diatonic"
tuning of Ptolemy with its pure thirds and sixths. It's interesting
that these two theorists were also champions of _harmonia perfetta_
(Zarlino) or the _trias harmonica_ (Lippius) -- that is, the complete
5-limit harmony of third plus fifth or sixth, to which Lippius gave
the name "triad" -- as a central element of good composition.

Here Bill Alves might welcome my adding that Nicola Vicentino (1555)
also championed the syntonic diatonic, while pragmatically advocating
a system of 31-tone equal temperament as the standard for its
realization in an "Aristoxenian" manner. This theorist's modern
editors Maria Rika Maniates and Claude V. Palisca, mention Fogliano as
a likely source for some of Vicentino's teachings regarding pure
thirds and sixths. Also, Vicentino formulated the rule that
a composition for three or more voices should wherever possible have
complete 5-limit sonorities -- three years before the publication of
Zarlino's more extensive treatment of this point.

Most respectfully,

Margo Schulter
mschulter@value.net

Thanks to Margo Schulter for providing detail for what I only
skimmed over.

I would like to emphasize that her dating of the shift
from Pythagorean to 5-limit tuning in European music follows
the orthodox views more closely than mine, thus explaining
some of our differences of opinion.

I am firmly convinced that the singing of 5-limit intervals
either never went out of use in Europe once it was described by
the ancient Greeks, or, if it did go out of use, it was only
during a few centuries immediately after the German invasions
of the late 400s AD.

At present, the only references I can cite to back this claim
are a few postings I've made to this forum, and my own book.

But hopefully in the next few months, I'll have a webpage
finished which explains a diatonic system described by Boethius
as Pythagorean, but the Greek-letter notation of which (provided
by Boethius himself) seems to indicate a differentiation between
two tones separated by a syntonic comma [~21.5 cents] but an
equation of two tones separated by a skhisma [~1.95 cents].
This describes what amounts to a skhismatic-tempered version
of a 5-limit JI system.

As far as the issue of polyphony, Margo is correct in that
I may have argued too strongly for an 'early death of plainchant'
in the cathedrals of Europe.

The earliest indisputable description of polyphony is in the
_musica enchiriadis_, which has been dated to approximately 900 AD.

In some churches at the present time, monophonic plainchant can
still be found in use during the Mass or service.

Some years ago, I wrote a paper comparing Schutz's and Bach's
settings of the _St John Passion_, at the beginning and end,
respectively, of the Baroque period in music. I find it a
telling commentary on plainchant's role in European music
*in general* (that is, outside of the church in particular)
that Schutz had sections of his Passion set to monophonic
plainchant, whereas Bach did not.

-monz

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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Margo Schulter's reply to me [TD 204.10] was so rich in
informative detail that I found much more to which I wanted
to respond.

> Please let me state at the outset that the following remarks
> focus on written music which has come down to us, especially
> liturgical chant and various types of polyphonic compositions.
> Traditional musics of Europe in the medieval era, as later, may
> have used various intonational systems.

A very good point. I should have made a similar disclaimer.

>> And it *is* possible to perform plainchant in a JI scale with
>> *every* note consonant with those already being heard - it just
>> wouldn't be a type of scale that we usually recognize as a
>> 'basic' scale.
>
> Here I might have a couple of different responses.
>
> First, just about any chant will have melodic semitones, regarded
> when they occur as simultaneous intervals as dissonances in just
> about any system of Western European composition before around
> 1900.

You're entirely correct about that one, and I let it slip by.

> Many chants also have prominent tritone relations, e.g.
> between F and B-natural in G Mixolydian (Modes 7 and 8, or
> _tetrardus_).

Well... of course we can't be certain about what happened
in practice, but indications are that the B-natural in these
modes was frequently, or perhaps virtually always, altered thru
_musica ficta_ to a Bb.

Of course, that still leaves the A-Bb semitone you discussed above.

>> This really doesn't have much bearing on the question anyway,
>> because by the time period under consideration, unaccompanied
>> monodic plainchant had mostly disappeared from the church >> rituals.
>
> Here I must respectfully disagree. While polyphony indeed became
> more and more common during the later medieval era, plainsong
> remained a vital basis of the liturgy. <...etc.>

Again, you are entirely correct about this.

I suppose what I was really thinking about here was that
'significant composed music' was based on liturgical forms
which were largely polyphonic by about 1350, until about 1600,
when they gave way to opera, which was fully homophonic.
The point was that unaccompanied plainchant was not a *major*
factor in European 'art' music after around 1300.

>> Since the Ars Nova movement in Paris in the 1200s, most church
>> music thruout Europe had become quite emphatically polyphonic.
>
> The term "Ars Nova" (New Art) refers to a movement dating to
> around the beginning of the 14th century in France (e.g. Philippe
> de Vitry, Johannes de Muris) and Italy (e.g. Marchettus de Padua).
> The late 13th century, in contrast, was now known as the "Ars
> Antiqua" (Old Art) or "Ars Veterum" ("Art of the Ancients").

I stand corrected on my terminology as well.

>> Of course, Marchetto of Padua in 1318 discussed the use of
>> intervals smaller than the 'standard' Pythagorean semitones.
>> But because of his obscure numerical descriptions, exactly what
>> he meant is not entirely clear. See [plug alert]:
>
>> http://www.ixpres.com/interval/monzo/marchet/marchet.htm
>
> From the view of a 3-limit vs. 5-limit debate, my take on
> Marchettus seems similar to that of Mark Lindley and Karol
> Berger: his approach, however unorthodox, fits in nicely with
> a general preference in the 14th-century for some approximation
> of Pythagorean intonation with wide major thirds and sixths, and
> narrow diatonic semitones.

The point I was emphasizing about Marchetto was precisely that
his numerical description is unclear, and as I've noted on
my webpage about him, he may have been indicating either
a Pythagorean *or* a 5-limit tuning; even after a close
investigation, I'm still not sure which is correct, and indeed,
I'm not sure that either one of them *must* be picked as correct;
on the contrary, it's beginning to look to me like Marchetto
was intentionally being fuzzy and allowing multiple interpretations
of his 'fifth-tones'.

As for the rest of Margo's reply to me, while I can agree with
all her other comments and the rest of her timetable, I must
'respectfully disagree', as she likes to say, on the preponderance
of Pythagorean tuning in vocal music up to about 1370.

Certainly, up until about 1370 (the beginning of the Florentine
swapping of sharps and flats Margo cited), or 1318 if Marchetto
is admitted, theorists considered the '3rds' and '6ths' to
be tuned as Pythagorean ratios, which made them dissonances.

And certainly, composers were following the theorists's
descriptions by treating '3rds' and '6ths' as dissonant
intervals which needed to resolve to '5ths' and 'octaves'.

But if the singers were unaccompanied, it is highly doubtful
to me that they sang these 'dissonances' as anything other than
their 5-limit ratios, at any time in history.

If they were sharpening these intervals in accordance with their
understanding of the compositional treatment, as Margo suggests,
they were probably singing other less-dissonant 5-limit intervals
that made the intervals even wider than the Pythagorean ratios;
or perhaps the were *really* attuned to the compositional treatment
and sang them as the Pythagorean, or even more dissonant, ratios.

Altho I would consent that the above is possible, I still think
it very unlikely that a group of unaccompanied singers will
*not* sing simultaneous 'major 3rds' as 5:4, 'minor 3rds' as
6:5, 'minor 6ths' as 8:5, or 'major 6ths' as 5:3.

According to the extant manuscripts, simultaneous harmonies
were first used in music around 800 or 900, and I would argue
that from that time onward, unless they were accompanied by
a Pythagorean-tuned organ, the choir sang 5-limit JI.

If this is correct, then there is a lag of *at least* 400 years
until Marchetto caught up with practice in his theoretical
description, and that's only if I'm correct in thinking that
Marchetto was describing 5-limit JI. Even if this is the
*minimum* delay in theorist's recognition of actual intonational
practice, four centuries is a long time.

This shows more than anything the stranglehold that received
theoretical doctrine can have on even 'progressive' music
theory. I think this historical 'lock on Pythagoreanism'
was far worse than the current 'lock on 12-eq' that we're
dealing with. At any rate, it lasted for a much longer time.

No intrinsic evaluation of Pythagorean tuning in itself
is implied by my use of the word 'worse' - it's the situation of
an *unconscious mental limitation* that I'm describing as 'worse'.
:)

I speak on this based on my own first-hand experience. My ears
told my brain all along that the 12-eq scale was inadequate
for describing what I was hearing when I listened to music,
but my conservatory training (strictly 12-eq) and book learning,
until encountering Partch's _Genesis_, never gave me a clue
that this was the case. That's why I used to have so much
trouble trying to play the blues correctly: I was thinking of
it entirely in terms of the 12-eq scale, which just doesn't
work for the real low-down blues.

Of course, learning about microtones made it possible for me
to give an excruciatingly accurate analysis of a blues song
recorded 60 years ago by a musician who had no 'formal' training
in music-theory at all.

IMO, this is an exact analogy to the situation in the 1300s,
when European theorists re-discovered the ancient Greek treatises
filled with non-Pythagorean ratios, and applied the concepts and
measurements to the music that was already being made all around
them.

-monz

PS - Please give the citation for Mark Linley's study of
14th-century keyboard music. I'm fascinated.

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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Margo Schulter wrote,

> Secondly, it might be relevant to point out that degrees of "concord"
> are a product of musical context as much as acoustics. Even if tuned
> in meantone or 5-limit JI, thirds and sixths would be unstable in a
> medieval setting rather than "fully concordant" simply because the
> musical language treats them thus, and attuned listeners respond in
> accordance with the language's expectations. Similarly, if one
> attempted to make Bach's or Beethoven's triadic harmonies "richer" by
> adding pure 7:4's where stable triads are expected, this would be
> "ungrammatical," however attractive from the viewpoint of 7-limit JI
> as a goal in itself.

I agree, I agree, I wholeheartedly agree.

Margo Schulter wrote,

>> Many chants also have prominent tritone relations, e.g.
>> between F and B-natural in G Mixolydian (Modes 7 and 8, or
>> _tetrardus_).

Joe Monzo wrote,

>Well... of course we can't be certain about what happened
>in practice, but indications are that the B-natural in these
>modes was frequently, or perhaps virtually always, altered thru
>_musica ficta_ to a Bb.

That would create a prominent tritone relation between E and Bb.

>But if the singers were unaccompanied, it is highly doubtful
>to me that they sang these 'dissonances' as anything other than
>their 5-limit ratios, at any time in history.

I totally disagree. It is highly doubtful to me that 5-limit ratios had any
importance in unaccompanied singing, at least outside of huge reverberant
caves or Gothic cathedrals.

>If they were sharpening these intervals in accordance with their
>understanding of the compositional treatment, as Margo suggests,
>they were probably singing other less-dissonant 5-limit intervals
>that made the intervals even wider than the Pythagorean ratios;

Huh? That makes sense for a minor third (6:5 is wider than 32:27), but for a
major third, what less-dissonant 5-limit tuning is even wider than
Pythagorean?

>Altho I would consent that the above is possible, I still think
>it very unlikely that a group of unaccompanied singers will
>*not* sing simultaneous 'major 3rds' as 5:4, 'minor 3rds' as
>6:5, 'minor 6ths' as 8:5, or 'major 6ths' as 5:3.

>According to the extant manuscripts, simultaneous harmonies
>were first used in music around 800 or 900, and I would argue
>that from that time onward, unless they were accompanied by
>a Pythagorean-tuned organ, the choir sang 5-limit JI.

>If this is correct, then there is a lag of *at least* 400 years
>until Marchetto caught up with practice in his theoretical
>description, and that's only if I'm correct in thinking that
>Marchetto was describing 5-limit JI. Even if this is the
>*minimum* delay in theorist's recognition of actual intonational
>practice, four centuries is a long time.

>This shows more than anything the stranglehold that received
>theoretical doctrine can have on even 'progressive' music
>theory. I think this historical 'lock on Pythagoreanism'
>was far worse than the current 'lock on 12-eq' that we're
>dealing with. At any rate, it lasted for a much longer time.

I totally disagree with all of this but have little to add to Margo's
arguments.

>I speak on this based on my own first-hand experience. My ears
>told my brain all along that the 12-eq scale was inadequate
>for describing what I was hearing when I listened to music,

I hate to say it but music of 1000 years ago was totally different from, and
probably tuned totally differently from, music of today. There was a
"paradigm shift" at one point when thirds and sixths moved from dissonant to
consonant, and this is clear in the grammar of the musical examples that
have survived. IMO, 5-limit ratios would have sounded as wrong 1000 years
ago as 7- and 9-limit intervals sound to most musicians today.