Chant, resonanant spaces, medieval polyphony etc.

It is probably safe to say that the original intonation of Gregorian chant
is not known. The development of what at present appears to be a solid and
unified repertoire took place over such a long period and with sources
drawn from a wide geo-cultural region. This suggests a diversity of
intonational resources rather than a single scheme.

Once established in notation, a monophonic repertoire is certainly pliable
to variations in intonation and local performance traditions almost
certainly developed taking full advantage of this pliability. To this day,
the chant sung in moveable do countries is probably going to be closer to
triadic just intonation than that in fixed do countries where "perfect
pitch" -- meaning, in this case, memory of 12tet -- is valued and rehearsal
is conducted with a reference keyboard nearby. Similarly, there is a lot of
evidence that singers rehearsed to a monochord in the late middle ages and
a pythagorean division of the instrument was ubiquitous.

The idea that the intonation is influenced or determined by the reverberant
effects of resonant spaces is an attractive one (works by Stuart Dempster
and Alvin Lucier are very suggestive modern responses to this), but
difficult to pin down historically. (Possibly of more interest would be to
trace the emergence of accellated harmonic rhythm in the Baroque era in the
increasingly dry acoustics of the Baroque architecture.) In any case, the
chant repertoire was composed very much prior to the beginning of the
period of large church construction.

But what effects might have the larger churches had on subsequent
performance practice? My own experiences in playing in Gothic and
Romanesque churches have been radically different acoustical situations,
but the fact that these rooms resonate very low frequencies so well
contributes to my impresssion that a sustained 9:8 major second in the
treble range with its implied fundamental 3 octaves below is not an order
of magnitude less consonant than a 5:4 major third with an implied
fundamental 2 octaves below. My very tentative conclusion is that it is
both possible and hospitable to sing in pythagorean intonation in large
churches.

I find the recordings by Anonymous 4 to be wildly inconsistant in
intonation. They perform in a register that is most likely to be higher
than that of the original pitch level, so any conclusions about the
historical intonation are suspect when made on the basis of these
performances.

More interested would be an analysis of the earlier (i.e. while Paul
Hillier was still a member) recordings of the Hilliard Ensemble. In
addition to a more propbably pitch level, it was their stated intention to
sing consistantly in pythagorean intonation. (This actually created some
difficulties when they commissioned a new work, largely triadically tonal,
from Gavin Bryars.)

Daniel Wolf
Frankfurt-am-Main

Daniel Wolf wrote:

> It is probably safe to say that the original intonation of Gregorian chant
> is not known. The development of what at present appears to be a solid and
> unified repertoire took place over such a long period and with sources
> drawn from a wide geo-cultural region. This suggests a diversity of
> intonational resources rather than a single scheme.

An excellent and thought-provoking observation!

> To this day,
> the chant sung in moveable do countries is probably going to be closer to
> triadic just intonation than that in fixed do countries where "perfect
> pitch" -- meaning, in this case, memory of 12tet -- is valued and rehearsal
> is conducted with a reference keyboard nearby.

Hear, hear! No, I mean really, hear, hear! :-)

I used to be in an cow punk band called The Moveable Do
Countries. Just kidding. Took me a minute to figure out
what that meant. I thought there was a letter missing
somewhere or something. :-)

This may relate to my perception that American singers tune
more tempered than European ones. I don't know
anything about European music training, but I suppose it
must
rely more on harmonic theory and orchestral instruments, and
less on
pianos, than American music training (with which I'm all too
familiar). Or maybe there's just more of a continuation of
tradition over there.

> Similarly, there is a lot of
> evidence that singers rehearsed to a monochord in the late middle ages and
> a pythagorean division of the instrument was ubiquitous.

I have thought some about the influence of habit on
intonation. I think it's clear that someone who practices
everyday tuning his voice to a piano or a Pythagorean
monochord is not going to walk into an echoy environment and
instantly start tuning his thirds to 5/4. But if he stopped
practicing singing with the cursED PIANO FOR A WHILE
([breathing hard and calming down] sorry!) and sang often in
reverberant spaces, he might well begin gravitating that
way.

> My own experiences in playing in Gothic and
> Romanesque churches have been radically different acoustical situations,
> but the fact that these rooms resonate very low frequencies so well
> contributes to my impresssion that a sustained 9:8 major second in the
> treble range with its implied fundamental 3 octaves below is not an order
> of magnitude less consonant than a 5:4 major third with an implied
> fundamental 2 octaves below. My very tentative conclusion is that it is
> both possible and hospitable to sing in pythagorean intonation in large
> churches.

Wouldn't it be more fitting to compare major thirds with
major thirds? The 81/64 implied fundamental, at 6 octaves
below, would normally be sub-sonic for male vocals (Tiny Tim
notably excepted). There's maybe a good order of magnitude
between that and 5/4 (and between Tiny Tim and most other
male vocalists). Even medieval theory holds the ditone
somewhat unstable. It might also be pointed out that 10/9
is only slightly more dissonant than 9/8 by this criterion,
and 7/4 is less so.

> I find the recordings by Anonymous 4 to be wildly inconsistant in
> intonation.

Good ears. My spectrographic analysis agrees. (Strangely,
this hasn't reduced the severity of my crush on them at
all.) Of course, they're American. Actually, they have an
Irish lass now, but she's only on the most recent release "A
Lamas Ladymass" which I, regrettably, don't yet own.

Come to think of it, when I listen to European-made madrigal
recordings, the female voices are the ones that most often
noticeably miss (NPI) Just Intonation. It's not uncommon to
hear wonderful consonance when the soprano is on the root of
the chord and not when she's got the third. This could be
because melodic concerns are stronger than vertical ones for
the soprano part in motets with some homophonic character.
Or could it be that women have a lesser bent toward JI than
men? (Ohhh, now I've gone and done it...)

> They perform in a register that is most likely to be higher
> than that of the original pitch level, so any conclusions about the
> historical intonation are suspect when made on the basis of these
> performances.

But you said that movable dos increase the likelihood of JI
intonation.

Plus, why should register matter to the supposition whether:

a) unaccompanied singers of diatonic melody tend toward JI,
b) and if so, whether the "virtual polyphony" created by
long and loud reverberation, as found in Gothic
architecture, promotes tuning to higher primes.

Actually, I guess the original supposition was that
intonation of chant, while
previously in tunings unknowable, began to shape towards
5-limit JI when it began to be sung in the great
cathedrals. It should be noted that while historical
questions lay at the root of this discussion, the pursuit
and analysis of data from a cappella singing has taken on a
life of its own. (I hope it doesn't come back in a rage and
burn down our laboratory!)

> More interested would be an analysis of the earlier (i.e. while Paul
> Hillier was still a member) recordings of the Hilliard Ensemble. In
> addition to a more propbably pitch level, it was their stated intention to
> sing consistantly in pythagorean intonation.

Good to know. I don't have any of those at hand. Dante?
Anyone else? Actually I might have them on a cassette of a
Harmonia show or something. I'll see what I can dig up.

David

>From: "David J. Finnamore" <dfin@bellsouth.net>
>

>> More interested would be an analysis of the earlier (i.e. while Paul
>> Hillier was still a member) recordings of the Hilliard Ensemble. In
>> addition to a more propbably pitch level, it was their stated intention
to
>> sing consistantly in pythagorean intonation.
>
>Good to know. I don't have any of those at hand. Dante?
>Anyone else? Actually I might have them on a cassette of a
>Harmonia show or something. I'll see what I can dig up.
>

I have the cd of the Hilliard doing Machauts mass (w/P. Hillier) and I have
just started to look at some spectrograms. Im going to the UK for two weeks
tomorrow so it will have to wait till I get back.

In the meantime, I put a link to David's Anon4 analysis in the
"Proceedings", added a look at a 1908 recording of Emma Calve (speaking of
crushes on singers), and a jpg of a cool looking spectrogram of an old
Csound piece that I dug up and have begun revising.
http://www.users.interport.net/~dante/hartheory2.html

see y'all when I get back
peace

dante

Daniel Wolf wrote,

>But what effects might have the larger churches had on subsequent
>performance practice? My own experiences in playing in Gothic and
>Romanesque churches have been radically different acoustical situations,
>but the fact that these rooms resonate very low frequencies so well
>contributes to my impresssion that a sustained 9:8 major second in the
>treble range with its implied fundamental 3 octaves below is not an order
>of magnitude less consonant than a 5:4 major third with an implied
>fundamental 2 octaves below. My very tentative conclusion is that it is
>both possible and hospitable to sing in pythagorean intonation in large
>churches.

I don't know what you're trying to say about the very low frequencies, but
that fact is that combination tones exist not in the room, but only in the
ear and brain. Helmholtz was wrong about this, but it has since been
demonstrated that air does not demonstrate significant acoustical
nonlinearities until about 160dB.

Daniel Wolf wrote,

>> My own experiences in playing in Gothic and
>> Romanesque churches have been radically different acoustical situations,
>> but the fact that these rooms resonate very low frequencies so well
>> contributes to my impresssion that a sustained 9:8 major second in the
>> treble range with its implied fundamental 3 octaves below is not an order
>> of magnitude less consonant than a 5:4 major third with an implied
>> fundamental 2 octaves below. My very tentative conclusion is that it is
>> both possible and hospitable to sing in pythagorean intonation in large
>> churches.

David Finnamore wrote,

Wouldn't it be more fitting to compare major thirds with
major thirds? The 81/64 implied fundamental, at 6 octaves
below, would normally be sub-sonic for male vocals (Tiny Tim
notably excepted).

a) there is no way for the human auditory system to acheive the accuracy
needed to extract the correct implied fundamental from a heard 81:64
interval. There are far too many ratios of equal or lesser complexity within
1% of the 81:64 (I listed these some time ago).

b) I already made the point that combination tones between singers' voices
exist not in the room but only in the ears. Well, implied fundamentals exist
neither in the room nor in the ears, but only in the brain (as dichoic
experiments have proved).

Daniel Wolf wrote:

> More interested would be an analysis of the earlier (i.e. while Paul
> Hillier was still a member) recordings of the Hilliard Ensemble. In
> addition to a more propbably pitch level, it was their stated intention to
> sing consistantly in pythagorean intonation.

About 10 years ago, I heard the Hilliard Ensemble sing an all DuFay
concert in Buffalo, NY. After the concert, I cornered Rogers
Covey-Crump to ask why the intonation was *not* Pythagorean. He
answered (after recovery from the shock that anyone could ask such a
question in a bush-league town like Buffalo) that Dunstable's influence
in introducing the "English countenance" (ie, 5-limit tuning) to the
Continent could validly be considered in a HIP of DuFay's music.

- Fred

> Daniel Wolf wrote,
>
>>> My own experiences in playing in Gothic and
>>> Romanesque churches have been radically different acoustical situations,
>>> but the fact that these rooms resonate very low frequencies so well
>>> contributes to my impresssion that a sustained 9:8 major second in the
>>> treble range with its implied fundamental 3 octaves below is not an order
>>> of magnitude less consonant than a 5:4 major third with an implied
>>> fundamental 2 octaves below. My very tentative conclusion is that it is
>>> both possible and hospitable to sing in pythagorean intonation in large
>>> churches.
>
> David Finnamore wrote,
>
> Wouldn't it be more fitting to compare major thirds with
> major thirds? The 81/64 implied fundamental, at 6 octaves
> below, would normally be sub-sonic for male vocals (Tiny Tim
> notably excepted).

Paul E. helpfully chimed in:

> a) there is no way for the human auditory system to acheive the accuracy
> needed to extract the correct implied fundamental from a heard 81:64
> interval. There are far too many ratios of equal or lesser complexity within
> 1% of the 81:64 (I listed these some time ago).
>
> b) I already made the point that combination tones between singers' voices
> exist not in the room but only in the ears. Well, implied fundamentals exist
> neither in the room nor in the ears, but only in the brain (as dichoic
> experiments have proved).

Excellent! All the more reason, then, to expect that
singers of chant in Gothic cathedrals should have
gravitated, over time, towards tuning thirds to 5/4 and 6/5
instead of the 81/64 and 32/27 that their vocal coaches (or
the monkish equivalent) probably tried to make them
practice. This is the strongest argument yet for the
original supposition, I think.

I was purposefully extending Dan's own example beyond where
I thought it could hold up, to coax him to clarify.
Nevertheless, you found my implied fundamental point, even
though my reasoning system hadn't yet extracted it. :-)
Thanks for straightening us out, Paul.

Does this mean that the 81/64 doesn't really "exist" in the
auditory part of the brain, but only in the air, ear, and
theory, even when seemingly sung accurately? What does it
say about the likelihood that 81/64 would be sung accurately
(a cappella), even in compositions that used the interval in
theory?

David

Message text written by INTERNET:tuning@onelist.com
>> David Finnamore wrote,
>
> Wouldn't it be more fitting to compare major thirds with
> major thirds? The 81/64 implied fundamental, at 6 octaves
> below, would normally be sub-sonic for male vocals (Tiny Tim
> notably excepted).
<

I made no such claim for the 81/64, nor did I claim that difference tones
were produced anywhere but in the listener's head. My claim was for the
consonance of the 9/8 in an environment with long reverberation. If I sing
c' then d' a 9/8 above in a room with sufficient reverb, I have no trouble
hearing the

THAT SAID, in both the chant and Gothic repertoire in question a melodic
Major third is _relatively_ infrequent in comparison with seconds, minor
thirds, fourths or fifths. But 81/64s outlined by two successive melodic
9/8s are very common. And easy to sing.

The English repertoire does include singing in parallel just thirds and
already, with Dufay it is hard to escape the conclusion that the triadic
character demands 5 limit intervals. But this line of discussion began with
a claim of Joe Monzo's about the performance of chant, and I wanted to
counter that a pythagorean performance was equally or even more plausible
in a cathedral setting.

Message text written by INTERNET:tuning@onelist.com
>From: Daniel Wolf <DJWOLF_MATERIAL@compuserve.com>

Message text written by INTERNET:tuning@onelist.com
>> David Finnamore wrote,
>
> Wouldn't it be more fitting to compare major thirds with
> major thirds? The 81/64 implied fundamental, at 6 octaves
> below, would normally be sub-sonic for male vocals (Tiny Tim
> notably excepted).
<

I made no such claim for the 81/64, nor did I claim that difference tones
were produced anywhere but in the listener's head. My claim was for the
consonance of the 9/8 in an environment with long reverberation. If I sing
c' then d' a 9/8 above in a room with sufficient reverb, I have no trouble
hearing the difference tone.

THAT SAID, in both the chant and Gothic repertoire in question a melodic
Major third is _relatively_ infrequent in comparison with seconds, minor
thirds, fourths or fifths. But 81/64s outlined by two successive melodic
9/8s are very common. And easy to sing.

The English repertoire does include singing in parallel just thirds and
already, with Dufay it is hard to escape the conclusion that the triadic
character demands 5 limit intervals. But this line of discussion began with
a claim of Joe Monzo's about the performance of chant, and I wanted to
counter that a pythagorean performance was equally or even more plausible
in a cathedral setting. <

I wrote,

>> a) there is no way for the human auditory system to acheive the accuracy
>> needed to extract the correct implied fundamental from a heard 81:64
>> interval. There are far too many ratios of equal or lesser complexity
within
>> 1% of the 81:64 (I listed these some time ago).
>
>> b) I already made the point that combination tones between singers'
voices
>> exist not in the room but only in the ears. Well, implied fundamentals
exist
>> neither in the room nor in the ears, but only in the brain (as dichoic
>> experiments have proved).

David J. Finnamore wrote,

>Excellent! All the more reason, then, to expect that
>singers of chant in Gothic cathedrals should have
>gravitated, over time, towards tuning thirds to 5/4 and 6/5
>instead of the 81/64 and 32/27 that their vocal coaches (or
>the monkish equivalent) probably tried to make them
>practice. This is the strongest argument yet for the
>original supposition, I think.

I don't agree with your conclusion at all.

>I was purposefully extending Dan's own example beyond where
>I thought it could hold up, to coax him to clarify.
>Nevertheless, you found my implied fundamental point, even
>though my reasoning system hadn't yet extracted it. :-)
>Thanks for straightening us out, Paul.

>Does this mean that the 81/64 doesn't really "exist" in the
>auditory part of the brain, but only in the air, ear, and
>theory, even when seemingly sung accurately? What does it
>say about the likelihood that 81/64 would be sung accurately
>(a cappella), even in compositions that used the interval in
>theory?

81/64 would probably occur with a rather large error. I wouldn't be
surprised if it occasionally came out as small as 5/4 or as large as 9/7 in
medieval practice. Harmonic thirds were not stable sonorities so they went
by fairly quickly in the music, too quickly for much accuracy to be
attained. In principle though, the logic of the scale was Pythagorean, so
the "target" for the major third would be 81/64. The implied fundamental, or
lack thereof, has absolutely nothing to do with it.

[Daniel Wolf, TD 220.13]
>
> But this line of discussion began with a claim of Joe Monzo's
> about the performance of chant, and I wanted to counter that a
> pythagorean performance was equally or even more plausible
> in a cathedral setting.

OK, what you say sounds reasonable about chant.

Note that my theories about perceiving 5-limit overtones in
a cathedral, which I would defend, are specifically in connection
with polyphonic singing.

In reference to monophonic chant, I was just speculating.

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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[David J. Finnamore, TD 220.10]

> Does this mean that the 81/64 doesn't really "exist" in the
> auditory part of the brain, but only in the air, ear, and
> theory, even when seemingly sung accurately? What does it
> say about the likelihood that 81/64 would be sung accurately
> (a cappella), even in compositions that used the interval in
> theory?

Paul's Harmonic Entropy (or 'cognitive dissonance', his
preferred name for it) theory simply says that we are much
more likely to *perceive* any particular interval as the
smallest-number ratio which falls within the probablility
range for that intevallic _gestalt_.

The probability of perceiving any particular ratio depends
upon two things: the size of its numbers, and exactly where
it lies under the probability bell curves surrounding the
small-number ratios lying near it.

By his theory, 81:64 is a ratio which simply has such large
numbers that it's unlikely that we'll perceive it as such.
This is exactly what I was saying in this thread about a
week ago, when I suggested that other simpler ratios would
probably be sung for the wider-than-5:4 'major 3rd'.

However, at the same time, given my prediliction for prime
factoring, I would say that 3^4 (= 81:64) is not such a
complicated interval. We've already had lots of arguments
here about this point of view; no need to repeat it now.

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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Paul H. Erlich wrote:

> 81/64 would probably occur with a rather large error. I wouldn't be
> surprised if it occasionally came out as small as 5/4 or as large as 9/7 in
> medieval practice. Harmonic thirds were not stable sonorities so they went
> by fairly quickly in the music, too quickly for much accuracy to be
> attained.

In polyphony, yes. In plain chant, "harmonic," "stable,"
and "quickly" are less relevant concepts to the abstract
composition, though possibly not to the performance in a
highly resonant environment. It's not at all uncommon in
plain chant to have a phrase like:

C C D E E D E F E D E D

which would cause the Es to be sung over the reverb of the
Cs. In such as case, the simple 5-limit tetrachord

1/1 9/8 5/4 4/3

would produce maximum consonance for the singers' ears, not
Pythagorean, and certainly not a 9/7 between the C and E.
It would surprise me less in such a case to hear a 10/9
between the C and D (and 9/8 above) than to hear a stack of
9/8s. Now if you get a short enough reverb that the Cs are
almost inaudible by the time the Es arrive, then 5-limit
probably has as little chance as Pythagorean.

Paul, you've been very helpful in clearing up
misunderstandings and factual errors, and you've stated that
you disagree with my conclusion, but you haven't said why.
Or maybe I just can't read what's between your lines? I'm
not firmly decided on the issue yet, BTW. Just exploring
the possible validity of the supposition, and leaning
somewhat toward it for plain chant.

> In principle though, the logic of the scale was Pythagorean, so
> the "target" for the major third would be 81/64. The implied fundamental, or
> lack thereof, has absolutely nothing to do with it.

I agree with that. Neither Dan nor I was trying to say that
it does. My purpose in bringing it up was to show that it
doesn't. Dan had seemingly compared 5/4 to 9/8, and I felt
that was not a valid, since they represent two different
intervals. Later he explained more thoroughly what he
meant, and it was not what I had taken it to mean, so my
"argument" served only to prompt a clearing up of what was
being said, as in fact it was designed to do.

David

Monz wrote:

> By [Paul Erlich's] theory, 81:64 is a ratio which simply has such large
> numbers that it's unlikely that we'll perceive it as such.
> This is exactly what I was saying in this thread about a
> week ago, when I suggested that other simpler ratios would
> probably be sung for the wider-than-5:4 'major 3rd'.

I think that's looking at the situation backwards. A singer
isn't waiting to perceive an interval one way or the other.
S/he knows when s/he is singing a third against another tone
and will try to tune it based on:

1) a practiced tuning
2) ear (which is influenced by #1)
3) a theory about how it "should" be tuned

The stronger the role played by #2, the more likely, IMHO,
that a third will be tuned 5-limit. I said to Paul earlier
that he was right about Pythagorean tuning for polyphonic
song, which goes against what you said:

> Note that my theories about perceiving 5-limit overtones in
> a cathedral, which I would defend, are specifically in connection
> with polyphonic singing.

Now that I think about it again, I'm not so sure one way or
the other. Even with the thirds being regarded as unstable,
and seemingly going by quickly, if the environment is
causing virtually the whole scale (plus ficta) to sound at
once, 5-limit JI should reduce the amount of perceived
dissonance as compared with Pythagorean tuning, making for a
more physically sensuous experience. But were they
concerned about making music a sensuous experience? It
might be argued that our current obsession with experience
results from the relatively recent widespread acceptance of
humanism, which is antithetical to medieval philosophy.

The musical form of the question is whether singers of such
compositions would have more likely relied on their ears or
on their knowledge of the theory to tune thirds. How
concerned were they about maximizing consonance? Judging by
the compositions themselves, the composers didn't lose a lot
of sleep over it. The theorists evidently saw it in terms
of a continuum to be traversed. But the performers? Could
be a different story. It's seeming less likely to me now
that we'll ever be able to make anything more than a guess.

David

[David J. Finnamore, TD 222.5]

> It might be argued that our current obsession with experience
> results from the relatively recent widespread acceptance of
> humanism, which is antithetical to medieval philosophy.

A very good observation.

> It's seeming less likely to me now that we'll ever be able to
> make anything more than a guess.

Well, basically, that's along the lines of what I was saying
last week.

I wouldn't be so negative as to say we'll *never* be able to
figure it out, but it's a much more complicated situation than
anyone's ever really claimed.

Other factors, such as geography, ethnicity, religious
and philosophical attitudes, and architectural acoustics,
have to be taken into account, in addition to the manuscripts,
theoretical treatises, and notational chronology that have
already been considered.

-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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