Yahoo Tuning Groups Ultimate Backup tuning Dialogue: 16th-century European music and meantone

Dear Marcel,

Here are some replies to specific points you raised.

> Also worth mentioning is perhaps that Ramis, who is accredited with
> making the 5/4, and therefore ultimately meantone temperaments,
> popular, based this 5/4 major third on his observations on how
> singers naturally sing the major third. Which also suggests that
> (at least some) practice of 5/4 major thirds probably pre-dated the
> meantone era.

While the question of whether Ramis himself may have used
meantone is an interesting one, I should begin by agreeing that
indeed the use of 5/4 before the meantone era is supported not
only by some 13th-14th century English music, but by the
schismatic Pythagorean tunings of the era around 1400 as
documented by Mark Lindley.

In these tunings, for example Gb-B on a 12-note keyboard with all
accidentals tuned as Pythagorean flats, diatonic thirds remain
Pythagorean, but major thirds involving sharps at 8192:6561 or
384 cents are very close to 5/4. In the Gb-B tuning, we have
written D-F#-A, E-G#-B, and A-C#-E at a near-just 4:5:6. Lindley
reasonably proposes that by around 1450, the desire to obtain as
many major thirds as possible at or around 5/4 led to meantone
temperament.

In his treatise of 1482, Ramis or Ramos describes not only his
theoretical 5-limit monochord, but a practical 12-note keyboard
for which he advocates a tuning of Ab-C#, noting that some people
prefer Eb-G#. As Lindley argues, Ramis describes these tunings in
a way that strongly suggests meantone rather than Pythagorean.

> However, what I do not see is a strong link between
> differences in meantone and Pythagorean leading to different
> music.

This is precisely what I do see, hear, and experience, and I
speak as someone for whom medieval, Renaissance, and Manneristic
music, together with Near Eastern styles, are my everyday musical
fare.

> The music is all fifth based, and all music from the middle
> ages till now functions perfectly well in Pythagorean. And in
> my opinion would have just as well been written had the
> popular tuning of the 16^th century till the 20^th century
> been Pythagorean instead of meantone.

Assuming harmonic timbres -- a very big assumption, but a not
unlikely one, with timbre recognized as an important factor by
Vincenzo Galilei in 1581! -- I must dissent from this
conclusion. If William Sethares and his technologies had been
available around 1400 rather than 2000, then it becomes a much
more open question!

In a 13th-14th century style, 2:3:4 is the ideal concord, the
point of harmonic saturation, with anything more complex seeking
its sense of sonorous stability and repose. Seconds, thirds,
sixths, and sevenths all serve as unstable and contrasting
intervals introducing various degrees of creative tension,
sometimes in very prominent and prolonged sonorities. This can be
heard in Perotin and Machaut, and likewise in the rather
different style of a composer such as Landini.

This 13th-century improvisation may illustrate these points:

<http://www.bestII.com/~mschulter/MixolydianDiversionTE1.mp3>

In a 16th-century style, the ideal is 4:5:6, or more specifically
a voicing such as 2:3:4:5 or 2:4:5:6, etc. The smooth concord of
5/4 is just as essential as that of 3/2 in forming this new
standard of stable saturation.

<http://www.bestII.com/~mschulter/LassoPrologue.mid>

Both the explicit rules and the implicit aesthetics of
13th-century and 16th-century polyphony are thus radically
different, although equally meritorious and beautiful!

In a 13th-century approach, we have a continual alternation and
contrast between stable sonorities (ideally the complete 2:3:4,
or often the simple 3:2 fifth) and unstable ones built from
various interval combinations, many of which (e.g. 4:6:9) would
be prohibited or drastically restricted by 16th-century rules.

In a 16th-century approach, we often have a smooth and seamless
flow of saturated 4:5:6 or 10:12:15 sonorities, decorated by
rather inconspicuous passing or ornamental tones, and sometimes
punctuated by the telling but muted tension of suspension
dissonances. That major and minor thirds, now points not of
creative tension but of restful concord, be intonationally smooth
and fully blending, is of the very essence!

In a style based on major/minor tonality, as in the late
17th-19th centures, the situation is again different, because the
level of vertical tension and contrast is again much higher than
in the 16th century, although thirds and sixths remain the most
complex stable intervals. Here various intonational tastes and
compromises develop, including the well-temperaments used
throughout this era from Werckmeister on, and styles of
"expressive intonation" as favored by some string players from at
least the 19th century on, and today by George Secor, where, for
example, the major third of a dominant triad might well be
Pythagorean or even larger.

[On Pythagorean (or wide-fifth) tunings as "unstylish"
specifically in a usual 16th-century context]

> Yes agreed on "unstylish". Not in an "objective" musical
> context (as if meantone is implied by the music), but in a
> historically correct context meantone is of course known to be
> correct.

I would say that, assuming a harmonic timbre, meantone or
something like it (e.g. classic or adaptive 5-limit JI) is
implied by most 16th-century European music.

This situation is in contrast to that of the era around 1400, for
example, where a classic Pythagorean tuning (e.g. Eb-G#), or a
schismatic tuning (e.g. Gb-B), or meantone or 5-limit JI, or even
a modern wide-fifth temperament, might all be reasonable musical
choices, although history suggests some kind of Pythagorean
tuning (maybe schismatic or not) as the likeliest period
practice. Here's a fine rendition of such a piece in a wide-fifth
temperament by Aaron Johnson:

<http://www.akjmusic.com/audio/greygnour.ogg>

> I think you may have heard them before. 2 short piano pieces,
> one by Mozart and one by Mussorgsky, tuned to 12tet,
> Pythagorean, 1/4-comma meantone and 17tet (in that order).

Please let me emphasize that my argument for meantone focuses
specifically on a 16th-century or early 17th-century style, and
these pieces are from the 18th and 19th centuries. From Lasso in
the later 16th century to Mozart in the later 18th century is as
long a period as from Machaut in the mid-14th century to Lasso.

In my post opening this new thread, I give some examples of music
from the 16th and early 17th centuries, or composed in similar
styles, where meantone seems to me the appropriate choice.

> I don't have them online in a different format. But you can
> download the audio for the videos using an online converter
> like [43]http://www.listentoyoutube.com

Thank you for calling attention to what should be for many people
a very useful site!

When I tried this and some other similar sites, what happened is
that I got the "Loading..." message -- but nothing happened even
when I waited a couple of minutes for the conversion to mp3. My
strong suspicion is that these sites don't support my text-based
browser, or vice versa, although they do support the vast
majority of current personal computers and software.

> Well I don't really see it the same as you do.
> I hear Pythagorean (or 12tet even) as fitting in all music.

Interestingly, in 1581, Vincenzo Galilei as a strong advocate of
12-tet, the standard tuning for the lute, sought to use it on a
harpsichord as well. Although he very much wanted it to succeed,
holding that it was a "perfect" temperament, in contrast to the
"defects" of meantone with its augmented and diminished
intervals, he found that in practice 2/7-comma meantone was more
supportable on keyboards. He connected this with the different
mechanisms of the lute and harpsichord -- or, in modern terms,
with the brighter fifth partials on the latter instrument.

> The music itself does not determine a preference for a
> different temperament to me.

For me, it is a matter of style and tuning/timbre. As John
Chalmers pointed out to me sometime around 1998, Pythagorean
tuning has many attractions even for music with stable major and
minor thirds, but there is in many such styles the problem of the
fifth partial. Adjusting the timbre can address this problem.
However, here I believe that we are assuming usual harmonic
timbres.

> The functioning of western music is fifth based. I see a major
> third as de-constructible into 4 fifths (like a circle
> progression does for instance) and so does western music.

That 4 fifths up form a major third is indeed a feature of fixed
European tunings from Pythagorean to meantone to unequal
circulating systems to 12-tet. And it applies to modern
wide-fifth temperaments also.

However, patterns of consonance/dissonance and directed or
functional progressions can change radically through the
centuries. As someone who loves both 13th-14th century and
16th-century cadences, this is an everyday given. In medieval
music, stepwise motion in all parts is generally the most
powerful. In the 16th century, there's a balance between bass
motion by step, by thirds, or by fourths or fifths. In
major-minor tonality of the later 17th-19th centuries, motion by
fifths or fourths is most important.

> I don't see any functional change by using 5/4 major thirds,
> nor does this change the music. It's simply a different
> "timbral color" to me.

> So why would some music fit meantone better than Pythagorean?

In the kinds of mostly harmonic timbres used around 1450-1600,
the degree of acoustical tension in a major third becomes more
than merely a fine nuance of intonation. It becomes of
grammatical significance. True, 5/4 or 81/64, or for that matter
14/11, is recognizable as a major third. But the tuning of that
third can either reinforce or rather comically contradict the
basic grammar of the counterpoint and aesthetic of the music.

The rules of classic 16th-century counterpoint privilege thirds
and sixths, and impose restrictions on fifths and fourths as well
as seconds and sevenths, precisely because thirds and sixths now
define the very standard of "rich" or "complete" harmony. They
are no longer points of tension, but points of rest and ideal
euphony.

Whether to tune a major third at 379 cents (1/3-comma or 19-tet)
or 383 cents (2/7-comma) or a pure 5/4 (1/4-comma, 5-limit JI) or
1/5-comma (390 cents) are fine questions of shading which do not
affect the basic grammar of the music.

Whether to tune a major third at 5/4 or at 81/64 is a distinction
which makes a basic musical difference: between a restful and
concordant third and an active one admirably fitting the
different musical grammar of Perotin or Machaut, but
contradicting that of Lasso or Palestrina -- or Gesualdo, for
that matter.

> But necessary, I don't see it. Meatone does give harsher sounds
> a more pleasant timbre than Pythagorean.

One point I would make as an advocate of Pythagorean and indeed
wide-fifth tunings for much medieval music: active thirds are not
so much "harsh" as dynamic and exciting in a musical context
where they are expected to present moments of creative tension
and motion, or pauses rather than more final conclusions. The
problem in a 16th-century setting is that we are getting
harshness in the sense of tension where smooth repose is
expected.

> Can't agree with counterpoint rules though.

Just to clarify for those who may not have read our earlier
discussion: you are, of course, disagreeing not with the rules
themselves in different eras, but with my conclusions drawn from
them :).

> To me they imply the exact opposite.

> The thirds and sixths are not perfect consonances in
> counterpoint, but specifically imperfect consonances.

True, and this terminology can be found from the 13th century at
least on, but the musical implications of these terms do change.

> And the rules for imperfect consonances are very different from
> the perfect consonances.

They are -- but in a different way during the 13th century than
during the 16th century!

In the 13th-14th centuries, "imperfect" means "partially" or
"relatively" concordant, with a considerable level of tension or
instability. Thus thirds must sooner or later resolve to stable
concords, with a complete 2:3:4 as the ideal destination in
multi-voice writing.

By the mid-16th century, Vicentino's "richness of harmony" or
Zarlino's "perfect harmony" not only permits but mandates using
the new complete sonority, 4:5:6, as often as possible, typically
even in closing. Thomas Morley (1597), who shows an interest in
historical questions, brings up the point that it is now common
to close on "imperfect" concords, although they are still called
imperfect.

> For instance: no parallel fifths or octaves allowed, but
> parallel thirds and sixths are fine (within reason).

But parallel fifths, and even octaves, are freely allowed in
13th-century music, always with the caution that contrary motion
is generally preferred, as for serious counterpoint in any era.
And in the 14th century, although "modern" theory prohibits
parallel fifths or octaves in simple two-voice writing, they
remain allowed even in theory in writing for three or more
voices, with practice yet freer. Banning parallel fifths and
octaves would rule out many of the most common 13th-14th century
cadences. To take an example from Machaut with both fifths and
octaves:

C# A
C# D
G# A
E D

This is one of the most compelling cadences, at least as I
perceive it, in the history of European music! The musical
grammar depends on the fact that thirds and sixths are
dynamically unstable, while 2:3:4, or here 2:3:4:6, is ideally
harmonious and restful. Pythagorean tuning, a temperament around
704 or 705 cents, 17-EDO, George Secor's 17-WT, or a 2-3-7 JI
tuning of the first sonority at 7:9:12:18 can all realize this
music grammar and syntax very effectively!

If we move through two centuries to the later 16th century, then
we find an often radically changed musical language, although one
with certain connections to that of Machaut. Consider, for
example, this beautiful progression with the outer voices
expanding from octave to major tenth to twelfth:

E F# G
B D E
G# A C
E D C

Here the motion from 4:5:6:8 to 2:3:4:5 to 2:4:5:6 is very
smooth, and involves not a contrast between instability and
rich stability as in Machaut, but rather a flow from one richly
stable sonority to another. A common thread of the Machaut
example and this is the two-voice progression in which a major
third or tenth expands to a fifth or twelfth -- a progression
standard in the 14th century, and still noted, for example, by
Zarlino in the 16th century. But the musical meaning is quite
different, calling for a difference in intonation. Meantone
brings out the fully concordant quality of the thirds.

May I conclude by adding that just as the rules of counterpoint
indeed change from medieval to Renaissance times, so they again
change by the 18th century. The routine use of bold sonorities
involving seconds and sevenths in ways which would be foreign to
a conventional 16th-century style produces, as in a 13th-14th
century setting, dramatic contrasts of stability/instability or
consonance/dissonance which might influence intonation, together
with the frequent desire for a 12-note circulating system.

Most appreciatively,

Margo

🔗Marcel de Velde <marcel@...>

Dear Margo,

First of all, thank you for your in depth reply!

Ill reply without quoting as my reply will become large enough already
without them Im guessing ;)

Our main point of disagreement seems to be that you think 5/4 or near 5/4
major thirds are implied in certain music.

And if I understand you correctly, this is in a way irrespective of the
period of composition.

So were not just talking about historical correctness here but also about
(near) 5/4 major thirds having a restfulness and that this is implied by
the music itself.
This I see very differently, and I will try to explain why.

First of all. I think tuning is first and foremost about relative pitch
height (intervals) of notes.

This is often described by chords, but melody is nothing different. We judge
the pitches of a melody against one another.

The difference is that in a monophonic melody the pitches do not sound at
the same time. But this does not matter in my opinion.

Ill illustrate:

Play the chords C-E-G, A-C-E, D-F-A, G-B-D, C-E-G-C
And now play the melody C, E, G, A, C, E, D, F, A, G, B, D, C, E, G, C

As you can hear, they express the exact same thing. To say the chord version
implies something different, in tuning or otherwise, is absurd in my
opinion. (not that I think you said so, just to be clear)

Or as another example, out of a polyphonic composition, you can pick out and
play by itself any of the melodic lines. Doing so of course does not change
the melody or its tuning, and vice versa.

I happen to have somewhat of an example of the latter online in Pythagorean
https://soundcloud.com/justintonation/tuning-demo-elgar-enigma-theme

(part of Chris Vaisvils Enigma assignment:
http://soonlabel.com/xenharmonic/archives/884 )

Here you can hear one monophonic melodic line first and then on each
repetition another melody comes in and together they form the chords.

Now of course it would make absolutely no musical sense at all to change the
tuning of the melody when other melodies are introduced in counterpoint.

What I do agree with is that when one plays a chord in isolation, say C-E-G.
That when one compares C-E as 5/4 and C-E as 81/64, that the 5/4 version
sounds more locked, and has a timbral quality that one could describe as
more restful. And this indeed most clear with harmonic timbres.

But, is this timbral lock the definition of in tune? I used to think so
after the first time I played a 5/4 major third on my keyboard. And many
other people will have thought the same after they first heard a 5/4 major
third.

Since that experience I started listening for this timbral lock in music and
used it as my identifier of whether that music was in tune or not.

But after many years of research, experimenting and comparing tunings etc, I
have come to very different ideas about what makes music in tune.

Ill refer back to the melodies vs chords example I wrote about above. I
find that the 5/4 major third sounds too low to me when I play for instance
1/1 9/8 5/4 4/3 melodically. Yet 1/1 9/8 81/64 4/3 sounds of
perfect pitch height to me.

Furthermore, there is ample evidence that western music functions according
to fifths and octaves, and that the major third has included in its
functionality that of being 4 fifths from its root.

If one breaks this functionality one gets horrible sounding comma shifts and
or wolves and or drifting. And not only do these sound horrible, they also
make no sense in proper musical logic I think (I wont go deeper into this
or this email may take on the length of a book).

And if we look at higher harmonics and their timbral qualities, we find that
the 7th harmonic for instance makes a very nice dominant seventh chord of
1/1 5/4 3/2 7/4, which sounds great / locked etc in isolation, but also
even more clearly out of tune in the musical context of normal music,
and it has all the same problems as 5-limit JI, comma shifts, wolfs and
drifting, only even more so.

So again without going further into this, Ill from now on talk about the
5/4 in the context of ¼ comma meantone and fifth based thinking.

Just like with the 5/4 major third, and the 7/4 minor seventh, we like the
timbral lock of 2/1 and 3/2.

The octave at 2/1 is well supported to coincide to how we perceive music
(for instance octave equivalence), and theres not much sense in making the
octave anything other than 2/1 other than small timbre specific
modifications (like slightly stretched tunings for pianos for instance).

The 3/2 we like too to be pure, it sounds best. But does our ear / brain
expect it to be pure? Or 12tet, or meantone?
If found that it does indeed work best when used pure as 3/2. Diatonic music
sounds most clear and in tune to me with 3/2 fifths. And furthermore I find
that more remote augmented and diminished intervals still sound perfect to
me when they are tuned according to many pure fifths.

Here an example:

Play C-C-E-G -> F-C-F-Ab -> E-B-E-G#, A-C-E-A in Pythagorean and in ¼
comma meantone.

In Pythagorean this progression sounds perfectly natural and good to me, the
comma shift of Ab going up to G# sounds clear and right.

In ¼ comma meantone however, the comma shift goes downwards instead, and it
sounds bad and unnatural to me. The errors of the flat fifths of meantone
add up for more remote intervals and this error become more and more
audible.

For diatonic music, a comparison between different fifth sizes Ive uploaded
the demonstration I referred to earlier to soundcloud:

https://soundcloud.com/justintonation/tuning-demo-mussorgsky

https://soundcloud.com/justintonation/tuning-demo-mozart-alla-turca

After uploading these, I realized you may not be able to play these either
as soundcloud uses flash as well ;) so Ive uploaded them here as well:
https://mega.co.nz/#!TRdmTJbD!RZQz1VOpAohZBQP7TKYxtCQu6hg0drRUH4J8DjF6yME
https://mega.co.nz/#!zFMhzaBS!NSQX3TeErk92uGCxzukSNQP3ORRLWyk4eya7FcOBIWc

Hope this works for you.

The order of tuning is 12tet, Pythagorean, ¼ comma meantone, 17tet for both
files.

To my ears, both the 12tet version and Pythagorean version sound correct,
and upon closer listen the Pythagorean version wins over the 12tet version
as it is slightly more clear and expressive and right sounding to me.

The ¼ comma meantone and 17tet versions both sound out of tune to me.

Many people on this list may prefer the ¼ comma meantone version though. And
thats fine with me. The music comes across in that version as well. They
may be listening to the timbral effect of those 5/4 major thirds, just like
I used to do, and use this to determine what is in tune to them, or what
they prefer.

However, the majority of the general public has not learned to listen for
the timbral effects of 5/4, and even though the timbral effect of 5/4 is in
ways objectively pleasant, Ive found that normal people (without
exception so far) heard the ¼ meantone version and 17tet version as out of
tune. ¼ meantone is not general public friendly anymore. It may have been
when they didnt know much else, or when there is no direct comparison
available, but now that they know 12tet it is not an improvement over it for
them (and not to me either).

Now to get back to the original point. You say that some music implies
meantone to give it rest in certain chords.

I say this is in the ears of the beholder. For me it is clearly not so. All
music from the meantone period sounds best to me in Pythagorean, not
meantone. And the Pythagorean or 12tet major chords just as well express
rest etc.

It may sound best to you in meantone, and thats fine with me. But it is not
linked to the music in an objective way I think.

I see music in its basis as separate from the composer and period.

Music is like a book and notes are like words. One can put on a different
cover, include a picture and small bio from the author, use a different
font, etc. But as long as the font does not make the words unreadable, all
these things may add something for some readers extra context that may make
it more enjoyable to some, or more authentic, etc, but the words say the
same thing.

To try to base meantone or 5-limit or higher limit JI perception and link
this to music on objective qualities is not something that is defendable I
think, and I hope that you can agree with my arguments for this. (and I have
many more in reserve ;) ).

Btw Ive listened to your audio examples.

And I say this with all respect, but they do not sound good to me, sorry.

Especially the comma shifting ones sound very bad to me. Also could not see
anything like that become general public friendly.

The Gesualdo song sounded most acceptable to me (in a meantone tuning?).

But if these things do work for your ears, then of course Im not objecting
to that :)

As for the historical context. As Ive explained above I think it is not
relevant / tied to the music in an objective sense.

It is of course relevant in order to play it historically correct.

I do question however the link between popular tunings of the time and type
of compositions made.

Ill try to write back to you on this later.

But even if meantone was influential in using major and minor chords as for
instance final chords (again I do not think this is the case), then still
this does not mean that the music itself implies meantone.

Kind regards,

Marcel

🔗Margo Schulter <mschulter@...>

> Dear Margo,

> First of all, thank you for your in depth reply!

You're welcome, and your reply and pieces give me a chance to
clarify some of my views.

> I'll reply without quoting as my reply will become large enough
> already without them I'm guessing ;)

What I'll do is to quote here selectively, focusing especially on
questions you raise where clearer answers may help, whether to
reveal areas of agreement or to make disagreements more
understandable to each of us and to other readers.

Also, I should say something about scholarly and artistic
humility, since here I am somewhat in the role of defending the
conventional wisdom, which might or might not be right.

A famous or infamous example of less than infallible judgment
occurred a bit more than 400 years, when Artusi wrote of the many
"imperfections" in the music of Monteverdi. Both Artusi's
knowledge of conventional counterpoint, and his ears, told him
that Monteverdi was using dissonances and also certain melodic
intervals (e.g. diminished fourths) in a way which violated not
only the familiar rules but good taste. The verdict of history
has generally been quite different as to the value of his music.

I am equally capable of fallible, not to say highly quirky,
judgment. All that I write should be read with that important
caution.

> Our main point of disagreement seems to be that you think 5/4
> or near 5/4 major thirds are implied in certain music.

Yes, in certain styles of music, especially assuming harmonic
timbres.

> And if I understand you correctly, this is in a way
> irrespective of the period of composition.

Actually I'd say it's specific to certain periods or styles, with
16th-century European music as the focus for my statement about
5/4 or something close to it being appropriate in harmonic
timbres for this specific style. A fine point as to "period of
composition" is that whether a piece is composed in 1555 or 2005,
if it follows this style (fairly well defined by the usual rules
of 16th-century counterpoint and harmony), then my argument would
apply.

Yet more specifically, I have in mind a style by around 1520-1530
where major thirds are increasingly preferred in closing
sonorities, and more generally at points of repose (if a third is
used at all); and three or more voices move at moderate speed in
note-against-note writing. In fast, two-part counterpoint,
intonation might not be so sensitive; and around 1500,
Pythagorean as well as 5-limit or meantone ideals may have shaped
vocal performances, for example.

To give a minority view: Roger Wibberley has suggested that
Pythagorean may have remained common in vocal practice through
the epoch of Willaert (c. 1540), and raises the issue of comma
drift in 5-limit. So there are various sides to this question.

> So we're not just talking about historical correctness here
> but also about (near) 5/4 major thirds having a "restfulness"
> and that this is implied by the music itself.

For the kind of 16th-century style I'm discussing, I would say it
is in a sense implied by the music itself -- at least in a strong
harmonic timbre. On the lute, 12-tET was standard, but a heavier
meantone temperament on keyboards -- then, for example, the organ
and harpsichord, rather than the later piano with its somewhat
more subdued fifth partials (analogous to those of the lute).

With some other kinds of music, "historical correctness" may be
more easily separated from the nature of the music itself. Your
caution to keep this distinction in mind is itself very wise! For
example, Bach's music seems to fit many irregular systems as well
as 12-tet, thus the many debates about the "correct" tuning(s).

> This I see very differently, and I will try to explain why.

Your arguments help in identifying the issues.

> This is often described by chords, but melody is nothing
> different. We judge the pitches of a melody against one
> another.

> The difference is that in a monophonic melody the pitches do
> not sound at the same time. But this does not matter in my
> opinion.

In my view, and that of George Secor, the melodic and harmonic or
vertical dimensions of music may sometimes lead intonation in the
same direction, and sometimes be in tension. See

<http://anaphoria.com/Secor17puzzle.pdf>

> I happen to have somewhat of an example of the latter online
> in Pythagorean
> [42]https://soundcloud.com/justintonation/tuning-demo-elgar-enigma-theme

> (part of Chris Vaisvil's Enigma assignment:
> <http://soonlabel.com/xenharmonic/archives/884>

> Here you can hear one monophonic melodic line first and then
> on each repetition another melody comes in and together they
> form the chords.

This piece I like! In fact, there have been studies suggesting
the performers of classic European music at least sometimes tend
to approximate Pythagorean rather than 5-limit JI or 12-tET.

Again, as I also emphasized to Jake, my statements were focusing
specifically on a 16th-century style.

> But, is this timbral lock the definition of "in tune"? I used
> to think so after the first time I played a 5/4 major third on
> my keyboard. And many other people will have thought the same
> after they first heard a 5/4 major third.

Since I often tune major thirds at around 414-418 cents, and have
done so for many years, clearly I would not regard 5/4 as the
definition of "in tune." But it can be the definition of
"stylistically in tune" for most 16th-century music. Different
writers may draw different lines. Thus Vincenzo Galilei reports
that 81/64 is "dissonant," and 12-tET ideal both in theory and
practice on the lute, but 2/7-comma better on the harpsichord
(although this in part, he suggests, might be matter of habit),

> Play C'-C-E-G -> F''-C-F-Ab -> E'-B'-E-G#, A''-C-E-A in
> Pythagorean and in 1/4 comma meantone.

> In Pythagorean this progression sounds perfectly natural and
> good to me, the comma shift of Ab going up to G# sounds clear
> and "right".

> In 1/4 comma meantone however, the comma shift goes downwards
> instead, and it sounds bad and unnatural to me. The errors of
> the flat fifths of meantone add up for more remote intervals
> and this error become more and more audible.

In 1/4-comma meantone, I find this diesis shift of 128/125 or 41 cents, about 1/5 tone, as striking and in keeping with the
system. For me, in a Renaissance setting, Ab is already a bit of
a "remote accidental" rather than part of the usual Eb-G# system,
so I'm ready for interesting things to happen. Your example is a
great illustration of how the diesis moves downward in meantone!

The Pythagorean is interesting also, and acceptable for me, if
the stylistic context fits these third sizes.

With meantone, however, if one wanted to avoid the downward
diesis shift, there is, for me, a neat alternative solution:

G G#*/Ab G#* A
E F E* E
C C B* C
C F E* A

Here I'm using a *, like Vicentino's dot, to show a note raised
by an enharmonic diesis or fifthtone. So the first two sonorities
are as in your example -- and then we treat the highest note of
the second sonority, G#* or Ab, as a common tone for the
following E*-B*-E*-G#*, which resolves back to A. The cadence
with the last two sonorities is the kind of thing Vicentino often
writes in this enharmonic style with the fifthtone steps. I'm
aware that you don't care for diesis shifts, as noted below, but
while I didn't have trouble with your original progression, this
is what I might well use, since the cadence is so idiomatic!

> For diatonic music, a comparison between different fifth sizes
> I've uploaded the demonstration I referred to earlier to
> soundcloud:

> [44]https://soundcloud.com/justintonation/tuning-demo-mussorgsky
> [45]https://soundcloud.com/justintonation/tuning-demo-mozart-alla-turca

These links were easy for me to download from: thank you!

With the Mussorgsky, a piece from the Romantic era I really like,
all four versions sounded acceptable to me, including the 17-tET!

With the Mozart, the 17-tET sounded a bit curious, but everything
else fine.

> I say this is in the ears of the beholder.

That is clearly true!

> All music from the meantone period sounds best to me in
> Pythagorean, not meantone. And the Pythagorean or 12tet major
> chords just as well express "rest" etc.

Of course, my perceptions reflect my own musical experience,
which could be anything but typical of the "general public."
We are dealing with interactions of tuning/timbre/style.
Also, since I spend so much of time with music where thirds are
unstable, I may tend to want to put music where they're stable in
a special category, by using meantone (or 5-limit JI). In either
the 16th or the 21st century, most of the "general public" might
not have this focus on 13th-14th century European music!

Please let me add that I focused on Pythagorean or larger thirds
in harmonic timbres. Obviously people find major triads quite
stable in 12-tET, and on a piano, I've routinely played
16th-century music without any problem (as they did back then on
lute). Easley Blackwood has suggested he perceives a limit for
convincing classic major/minor or the like at around 405 cents,
although he wasn't, as I recall, addressing a specific style or
timbre, each of which, in my view, could make a big difference.
My own view is that 12-tET can represent either Pythagorean or
meantone, but in a timbre with a bright fifth partial might tend
more to the former.

> Btw I've listened to your audio examples.

> And I say this with all respect, but they do not sound good to
> me, sorry.

Fair enough, and thank you for your honest opinion, which is all
that I could ask.

> Especially the comma shifting ones sound very bad to me. Also
> could not see anything like that become general public
> friendly.

Well, Vincenzo Galilei said rather similar things about Vicentino
in 1581, focusing a lot on the general public's lack of demand
for such music. Then, and now, such music is xenharmonic, and
stands apart from any relevant "common practice"; and for my
Invocation, Vicentino was certainly one model! The Coimbra
Manuscript piece, maybe from around 1600, also uses his
enharmonic steps.

A small technical point, as also in your example of a progression
in 1/4-comma meantone with descending Ab-G#, is that I would call
this an enharmonic diesis shift, 128/125, the difference between
12 meantone fifths and 7 octaves at 2/1, and also between three
pure 5/4 thirds and a 2/1 (the "lesser diesis" of 5-limit JI).
However, since this 12-note meantone diesis is equivalent to the
Pythagorean comma, your "comma shift" could also be correct.

> The Gesualdo song sounded most acceptable to me (in a meantone
> tuning?).

It's a variation on 1/4-comma described by Vicentino, where we
have two chains of fifths at 1/4-comma apart, making possible
pure 5-limit intervals with only very small shifts of 5.38 cents
rather than the comma shifts of classical 5-limit JI. There
weren't any diesis shifts.

> But if these things do work for your ears, then of course I'm
> not objecting to that.

I understand. And finding the Mussorgsky pleasant in 17-EDO was
really interesting. As I note above, all your examples in
Pythagorean sounded fine to me. From my point of view, apart from
the fact that these are not in 16th-century style, and more
specifically in a slower note-against-style, possible factors
could include timbre, tempo, rules of counterpoint, etc.

Best,

Margo

🔗Marcel de Velde <marcel@...>

Dear Margo,

I forgot to write about the different models for consonance, so I'll add
it here.

On these tuning lists the models for consonance are often harmonics based.
For instance something like consonance in this order: 1/1, 2/1, 3/1,
3/2, 4/1, 4/3, 5/1, 5/2, 5/3, 5/4, 6/5, 7/1, 7/2, 7/3, 7/4, 7/5, 7/6,
8/1, 8/3, 8/5, 8/7, etc becoming progressively more dissonant.
And the harmonic entropy model popular here is based on identification
of harmonics as well.
These models do not agree with Pythagorean very well (or at all is
perhaps more like it).
But there is an in my opinion much simpler model for consonance which
does fit Pythagorean like a glove and also fits actual music and
existing music theory much better than the harmonics based models.
It is the well known and explored model of "auditory" roughness (first
described by Helmholtz I belief) combined with simplicity of ratio in
Pythagorean (number of fifths spanning the interval).
I have searched a bit and have not found this model described anywhere,
so there's a small chance I may have invented it myself, though this is
probably not very likely as I can't imagine someone else hasn't done
this before me based on 12tet fifths instead of Pythagorean fifths
(which gives about the same results except for remote intervals).
The model goes like this. We simply take the Pythagorean intervals, and
say that by definition the simpler interval is in a way the most consonant.
So intervals go like this from simple to more complex: 2/1, 3/2, 4/3,
9/8, 16/9, 27/16, 32/27, 81/64, 128/81, 243/128, 256/243, 729/512,
1024/729, etc..
Now we put on top of this the roughness model. Intervals that are closer
together have more roughness (unless they're so close together that the
2 tones are perceived as one tone, but we're not dealing with that here).
And we also put on top of this octave equivalence and harmonic timbres.
And root of chord as I'll explain below.
The harmonics 2/1 and 3/1 are most often the strongest of course.
So we see that 2/1 and 3/2 which are the simplest Pythagorean ratios,
are the most consonant tones.
Now for the next simplest ratio, 4/3, we see something different. While
it is consonant when played with the octave, it gives roughness due to
the proximity of the 2/1 overtone of the F 4/3 note, and the 3/1
overtone of the C 1/1 note.
I am in favor of using the model of root of chord being determined by a
perfect fifth above it. (due to harmonic timbres having 2/1 and 3/1
harmonics so strongly, and even in inharmonic timbres these overtones
may be generated in the ear to some degree)
So 4/3 is very consonant if it is seen as an inversion of a root 3/2
chord, but it produces roughness with the 3/2 if it is seen as a 4/3
above the root of the chord.
I'd go as far as saying that by in my definition a root position chord
implies a 3/2 and 2/1 above the root.
So when we explore the other simple ratios, 9/8 and 16/9, they produce
roughness again with either the 1/1 or 2/1.
27/16 produces roughness with 3/2 (but if we combine it with 4/3 instead
of 3/2, and see it as an inversion then there is minimal roughness).
32/27 produces the least amount of roughness with 1/1, 3/2 and 2/1.
81/64 also produces the least amount of roughness with 1/1, 3/2 and 2/1.
Both 32/27 and 81/64 are the most consonant tones possible when combined
with 3/2 and 2/1. In other words, the major and minor triads.
Then we get intervals which are more complex in Pythagorean but also all
produce more roughness. 128/81 is rough with 3/2, 243/128 is rough with
2/1, 256/243 is rough with 1/1, 729/512 is again rough with 3/2, and the
intervals are getting more dissonant as well simply by their increasing
complexity / number of fifths.
So in the model of 4 tone chords inside the octave (including the
octave) from root position the most consonant chords are 1/1 81/64 3/2
2/1, and 1/1 32/27 3/2 2/1.
If we extend this to 5 tones including octave and within the octave and
in root position we get something like this: 1/1 32/27 3/2 16/9 2/1 is
simplest, perhaps together with 1/1 81/64 3/2 27/16 2/1 as most
consonant, followed by 1/1 81/64 3/2 16/9 2/1 and 1/1 32/27 3/2 27/16
2/1, and then perhaps 1/1 81/64 3/2 243/128 2/1 and 1/1 32/27 3/2 128/81 2/1? Though not sure on the order of consonance of these chords as it
become a balance between interval complexity and roughness weighing of
multiple tones together etc.
But in any case, this model seems to explain a lot of things we see in
actual music, and fits normal music theory very well.
It is also flexible enough I think to not impose these consonance
ratings as absolute rules, but allows for personal / cultural preferences.
(One small note: in normal music theory a perfect fifth above root is
not always implied, but disagree with this definition and it is
inconsistent.
Take for instance the B-D-F chord, it is often described as having a
root of B due to the peculiarities of the diatonic naming system the F
is a diminished fifth. I think the actual root will most often be G for
this chord. And if we were to add a note to this chord making B-D-F-Ab
then all of a sudden normal music theory agrees with me and also says
that the true root of this chord is G (most simple interpretation), so I
think it's better and more consistent theory to say the true root always
implies a perfect fifth above it.)

Furthermore, often Pythagorean is discredited by the many stacked fifths
which does not make sense to some people who favor a more "simple" model
based on harmonics.
This is still relevant to the meantone discussion I think as somewhere
underneath meantone thinking I still see some form of harmonic thinking.
For instance the harmonic entropy model is based on the perception and
identification of harmonics.
What most people don't realize is that the model of interval perception
based on harmonics is actually much more complex and much larger than
the Pythagorean model of interval identification!
If we take for instance the first 16 harmonics and reduce them to the
octave so we get a small scale of a mere 8 tones per octave like this:
1/1 9/8 5/4 11/8 3/2 13/8 7/4 15/8 2/1
This small scale already has 48 unique intervals contained within the
octave. And it's not even very musical in its possibilities yet (one
can't even play proper pentatonic scales in it, let alone the major or
minor scales).
Now in contrast, here an 8 tone Pythagorean scale:
1/1 9/8 32/27 81/64 4/3 3/2 27/16 16/9 2/1
It has a mere 14 unique intervals per octave (34 less than the harmonic
scale!), and it is already much broader in musical application that the
harmonic scale above (it has the major and minor scales + an extra tone)
And here a 12 tone Pythagorean scale, 4 tones larger than the harmonic
scale above, and one can already play most western music in it if one
transposes the key to the key of the piece.
1/1 256/243 9/8 32/27 81/64 4/3 729/512 3/2 128/81 27/16 16/9 243/128 2/1
It has a mere 22 unique intervals per octave. Still much less than the 8
tone harmonic scale. And every one of these 22 unique intervals which we
know to have a specific place and function in music, from the octave all
the way to the diminished sixth.
So while the Pythagorean model may seem more complex due to the larger
ratios, it is the exact opposite. The Pythagorean model is actually the
most simple and the smallest model possible for unique identifiable
intervals.

Also, I already mentioned in my previous email how 1/4 comma meantone
starts sounding particularly bad to me when going further than the
diatonic system (gave the Ab to G# example).
What also happens is that 1/4 comma meantone doesn't really go anywhere
useful I think when it is extended.
Pythagorean on the other hand is. It gives things like the 22-tone
Indian shruti and Arabic/Turkish maqam intervals.

So I see in total also a much stronger theoretical argument for
Pythagorean for any music than I see for meantones or other systems that
try to exploit harmonics overtones other than 3/2.
And I see only historically correctness and personal preferences as
valid arguments for meantone tuning for music from the meantone period.

Kind regards,

Marcel de Velde

> Dear Margo,
>
> First of all, thank you for your in depth reply!
>
> I'll reply without quoting as my reply will become large enough
> already without them I'm guessing ;)
>
> Our main point of disagreement seems to be that you think 5/4 or near
> 5/4 major thirds are implied in certain music.
>
> And if I understand you correctly, this is in a way irrespective of
> the period of composition.
>
> So we're not just talking about historical correctness here but also
> about (near) 5/4 major thirds having a "restfulness" and that this is
> implied by the music itself.
> This I see very differently, and I will try to explain why.
>
> First of all. I think tuning is first and foremost about relative
> pitch height (intervals) of notes.
>
> This is often described by chords, but melody is nothing different. We
> judge the pitches of a melody against one another.
>
> The difference is that in a monophonic melody the pitches do not sound
> at the same time. But this does not matter in my opinion.
>
> I'll illustrate:
>
> Play the chords C-E-G, A-C-E, D-F-A, G-B-D, C-E-G-C
> And now play the melody C, E, G, A, C, E, D, F, A, G, B, D, C, E, G, C
>
> As you can hear, they express the exact same thing. To say the chord
> version implies something different, in tuning or otherwise, is absurd
> in my opinion. (not that I think you said so, just to be clear)
>
> Or as another example, out of a polyphonic composition, you can pick
> out and play by itself any of the melodic lines. Doing so of course
> does not change the melody or it's tuning, and vice versa.
>
> I happen to have somewhat of an example of the latter online in
> Pythagorean
> https://soundcloud.com/justintonation/tuning-demo-elgar-enigma-theme
>
> (part of Chris Vaisvil's Enigma assignment:
> http://soonlabel.com/xenharmonic/archives/884 )
>
> Here you can hear one monophonic melodic line first and then on each
> repetition another melody comes in and together they form the chords.
>
> Now of course it would make absolutely no musical sense at all to
> change the tuning of the melody when other melodies are introduced in
> counterpoint.
>
> What I do agree with is that when one plays a chord in isolation, say
> C-E-G. That when one compares C-E as 5/4 and C-E as 81/64, that the
> 5/4 version sounds more "locked", and has a timbral quality that one
> could describe as more "restful". And this indeed most clear with
> harmonic timbres.
>
> But, is this timbral lock the definition of "in tune"? I used to think
> so after the first time I played a 5/4 major third on my keyboard. And
> many other people will have thought the same after they first heard a
> 5/4 major third.
>
> Since that experience I started listening for this timbral lock in
> music and used it as my identifier of whether that music was "in tune"
> or not.
>
> But after many years of research, experimenting and comparing tunings
> etc, I have come to very different ideas about what makes music "in tune".
>
> I'll refer back to the melodies vs chords example I wrote about above.
> I find that the 5/4 major third sounds too low to me when I play for
> instance 1/1 -- 9/8 -- 5/4 -- 4/3 melodically. Yet 1/1 -- 9/8 -- 81/64
> -- 4/3 sounds of perfect pitch height to me.
>
> Furthermore, there is ample evidence that western music functions
> according to fifths and octaves, and that the major third has included
> in its functionality that of being 4 fifths from its root.
>
> If one breaks this functionality one gets horrible sounding comma
> shifts and or wolves and or drifting. And not only do these sound
> horrible, they also make no sense in proper musical logic I think (I
> won't go deeper into this or this email may take on the length of a book).
>
> And if we look at higher harmonics and their timbral qualities, we
> find that the 7^th harmonic for instance makes a very nice dominant
> seventh chord of 1/1 5/4 3/2 7/4, which sounds "great" / "locked" etc
> in isolation, but also even more clearly "out of tune" in the musical
> context of "normal" music, and it has all the same problems as 5-limit
> JI, comma shifts, wolfs and drifting, only even more so.
>
> So again without going further into this, I'll from now on talk about
> the 5/4 in the context of ¼ comma meantone and fifth based thinking.
>
> Just like with the 5/4 major third, and the 7/4 minor seventh, we like
> the "timbral lock" of 2/1 and 3/2.
>
> The octave at 2/1 is well supported to coincide to how we perceive
> music (for instance octave equivalence), and there's not much sense in
> making the octave anything other than 2/1 other than small timbre
> specific modifications (like slightly stretched tunings for pianos for
> instance).
>
> The 3/2 we like too to be pure, it sounds best. But does our ear /
> brain expect it to be pure? Or 12tet, or meantone?
> If found that it does indeed work best when used pure as 3/2. Diatonic
> music sounds most clear and in tune to me with 3/2 fifths. And
> furthermore I find that more remote augmented and diminished intervals
> still sound perfect to me when they are tuned according to many pure
> fifths.
>
> Here an example:
>
> Play C'-C-E-G -> F''-C-F-Ab -> E'-B'-E-G#, A''-C-E-A in Pythagorean
> and in ¼ comma meantone.
>
> In Pythagorean this progression sounds perfectly natural and good to
> me, the comma shift of Ab going up to G# sounds clear and "right".
>
> In ¼ comma meantone however, the comma shift goes downwards instead,
> and it sounds bad and unnatural to me. The errors of the flat fifths
> of meantone add up for more remote intervals and this error become
> more and more audible.
>
> For diatonic music, a comparison between different fifth sizes I've
> uploaded the demonstration I referred to earlier to soundcloud:
>
> https://soundcloud.com/justintonation/tuning-demo-mussorgsky
>
> https://soundcloud.com/justintonation/tuning-demo-mozart-alla-turca
>
> After uploading these, I realized you may not be able to play these
> either as soundcloud uses flash as well ;) so I've uploaded them here
> as well:
> https://mega.co.nz/#!TRdmTJbD!RZQz1VOpAohZBQP7TKYxtCQu6hg0drRUH4J8DjF6yME
> <https://mega.co.nz/#%21TRdmTJbD%21RZQz1VOpAohZBQP7TKYxtCQu6hg0drRUH4J8DjF6yME>
> https://mega.co.nz/#!zFMhzaBS!NSQX3TeErk92uGCxzukSNQP3ORRLWyk4eya7FcOBIWc
> <https://mega.co.nz/#%21zFMhzaBS%21NSQX3TeErk92uGCxzukSNQP3ORRLWyk4eya7FcOBIWc>
>
> Hope this works for you.
>
> The order of tuning is 12tet, Pythagorean, ¼ comma meantone, 17tet for
> both files.
>
> To my ears, both the 12tet version and Pythagorean version sound
> correct, and upon closer listen the Pythagorean version wins over the
> 12tet version as it is slightly more clear and expressive and "right"
> sounding to me.
>
> The ¼ comma meantone and 17tet versions both sound "out of tune" to me.
>
> Many people on this list may prefer the ¼ comma meantone version
> though. And that's fine with me. The music comes across in that
> version as well. They may be listening to the timbral effect of those
> 5/4 major thirds, just like I used to do, and use this to determine
> what is "in tune" to them, or what they prefer.
>
> However, the majority of the general public has not learned to listen
> for the timbral effects of 5/4, and even though the timbral effect of
> 5/4 is in ways objectively pleasant, I've found that "normal" people
> (without exception so far) heard the ¼ meantone version and 17tet
> version as "out of tune". ¼ meantone is not general public friendly
> anymore. It may have been when they didn't know much else, or when
> there is no direct comparison available, but now that they know 12tet
> it is not an improvement over it for them (and not to me either).
>
> Now to get back to the original point. You say that some music implies
> meantone to give it "rest" in certain chords.
>
> I say this is in the ears of the beholder. For me it is clearly not
> so. All music from the meantone period sounds best to me in
> Pythagorean, not meantone. And the Pythagorean or 12tet major chords
> just as well express "rest" etc.
>
> It may sound best to you in meantone, and that's fine with me. But it
> is not linked to the music in an objective way I think.
>
> I see music in its basis as separate from the composer and period.
>
> Music is like a book and notes are like words. One can put on a
> different cover, include a picture and small bio from the author, use
> a different font, etc. But as long as the font does not make the words
> unreadable, all these things may add something for some readers extra
> context that may make it more enjoyable to some, or more authentic,
> etc, but the words say the same thing.
>
> To try to base meantone or 5-limit or higher limit JI perception and
> link this to music on objective qualities is not something that is
> defendable I think, and I hope that you can agree with my arguments
> for this. (and I have many more in reserve ;) ).
>
> Btw I've listened to your audio examples.
>
> And I say this with all respect, but they do not sound good to me, sorry.
>
> Especially the comma shifting ones sound very bad to me. Also could
> not see anything like that become general public friendly.
>
> The Gesualdo song sounded most acceptable to me (in a meantone tuning?).
>
> But if these things do work for your ears, then of course I'm not
> objecting to that J
>
> As for the historical context. As I've explained above I think it is
> not relevant / tied to the music in an objective sense.
>
> It is of course relevant in order to play it historically "correct".
>
> I do question however the link between popular tunings of the time and
> type of compositions made.
>
> I'll try to write back to you on this later.
>
> But even if meantone was influential in using major and minor chords
> as for instance final chords (again I do not think this is the case),
> then still this does not mean that the music itself implies meantone.
>
> Kind regards,
>
> Marcel
>

🔗Marcel de Velde <marcel@...>

Dear Margo,

I just sent my other email and only now read yours!

> You're welcome, and your reply and pieces give me a chance to
> clarify some of my views.
>
Glad to hear that :) And this works for me too.
I may not be replying in depth to your historical arguments. But they are very much appreciated and I'm learning a great deal from it.
The reason I'm not replying in depth to them is simply because your knowledge on this goes much further than mine! And I don't have much to add or contest.

> A famous or infamous example of less than infallible judgment
> occurred a bit more than 400 years, when Artusi wrote of the many
> "imperfections" in the music of Monteverdi. Both Artusi's
> knowledge of conventional counterpoint, and his ears, told him
> that Monteverdi was using dissonances and also certain melodic
> intervals (e.g. diminished fourths) in a way which violated not
> only the familiar rules but good taste. The verdict of history
> has generally been quite different as to the value of his music.
>
Ah it sounds like I may agree with Artusi there. I have not seen the music of Menteverdi that he's referring to, but if it is the case that Monteverdi may have used diminished fourths in placeswhere a major third makes sense, then I may well also see it as a case where a major third is interpreted by our brain even though a diminished fourth is written. In which case writing a diminished fourth would make no musical sense. I've argued for correct enharmonic spelling in such cases before.
Though again I have not seen this music by Monteverdi, so perhaps he used the diminished fourths in a way I do agree with.

> I am equally capable of fallible, not to say highly quirky,
> judgment. All that I write should be read with that important
> caution.
>
I can appreciate you saying this! But it goes for everybody I think. Myself very much included ;)
I have spent years spewing the greatest nonsense on these tuning lists while trying to make 5-limit JI work haha.

> > Our main point of disagreement seems to be that you think 5/4
> > or near 5/4 major thirds are implied in certain music.
>
> Yes, in certain styles of music, especially assuming harmonic
> timbres.
>
> > And if I understand you correctly, this is in a way
> > irrespective of the period of composition.
>
> Actually I'd say it's specific to certain periods or styles, with
> 16th-century European music as the focus for my statement about
> 5/4 or something close to it being appropriate in harmonic
> timbres for this specific style. A fine point as to "period of
> composition" is that whether a piece is composed in 1555 or 2005,
> if it follows this style (fairly well defined by the usual rules
> of 16th-century counterpoint and harmony), then my argument would
> apply.
>
Well.. If I'm allowed to put "style" under "historically correct", then we actually have no disagreement!

> > I say this is in the ears of the beholder.
>
> That is clearly true!
>
Again, this seems to indicate we have no disagreement! :)

> From my point of view, apart from
> the fact that these are not in 16th-century style, and more
> specifically in a slower note-against-style, possible factors
> could include timbre, tempo, rules of counterpoint, etc.
>

Well I thought we had theoretical disagreement on the use of Pythagorean vs meantone in an objective way, as if Pythagorean would express something different in a musical way than meantone.
Timbral differences and historical style differences I agree with fully (as long as I can put this part of "style" under historical authenticity, and not as an objective quality implied by the very notes of the music).
I therefore defended Pythagorean strongly, something which I've come to expect to have to do in tuning circles. However I see now that this may not have been needed :)
If there is still some question if Pythagorean can play 16th-century style music and express it convincingly, I'm more than willing to tune a piece to meantone and Pythagorean to give a good comparison. If so, would you happen to know of a piece in proper 16th century style of which the MIDI is available somewhere online, and preferably one which does not use more than 12 enharmonics per octave? (so it's easily put in both Pythagorean and meantone without a lot of manual labour).

Kind regards,
Marcel