Yahoo Tuning Groups Ultimate Backup tuning Werckmeister Continuo Temperament Spreadsheet

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

>
> I'm a little confused about the mention of wide fifth(s) in
> the spreadsheet, given that I thought I read old W.M. to say
> that all fifths should be flat.

Nope.

"Da denn das dis' von
dem gis ein klein wenig über sich schweben kan . . .
Auf dieses dis kan nun wider die Quinta b gestimmet werden / welches
auch ein gar wenig über sich schweben kan . . .
Zu dem b kan die Quinta f' gezogen
werden / wieder ein wenig über sich schwebend / oder gar rein . . ."

Literally "hovering a little bit above itself", i.e. tempered wider
than pure.

> And the temperament that it defaults to has thirds of 410
> cents. Yuck!

I take it you mean those few "bad" thirds. Well, as Werckmeister said,
he's trying to keep the good keys better at the expense of the less
used keys. The Lord giveth and the Lord taketh away, as they say.
There ain't nuthin' we can do 'bout that as long as we only have 12
tones. It's the same as all the Froggie modified meantone temperaments.

Ciao,

Paul

🔗Carl Lumma <clumma@yahoo.com>

🔗Paul Poletti <paul@polettipiano.com>

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

>
> Sharp fifths in WTs lead to "harmonic waste" -- you can get the
> thirds just as good without it.

What does a term invented by a modern American piano tuner have to do
with how German keyboard players tempered their instrument in the
early 18th century, as reported by a man who was a practicing
organist, a certified organ inspector, and highly-regarded temperament
theorist? Talk about the blind leading the blind. . .

> Think about it: the ideal fifth
> from the third's point of view is a 1/4-comma flat. The ideal
> fifth from a fifth's point of view is not flat or sharp at all.
> So why would one use sharp fifths?

I begin to think you (a) have not read Werckmeister's instructions and
(b) don't grasp the concept of cumulative overcompensation. If you
had, and did, you would not pose such questions.

As we used to say to the customers years ago when I worked as a stereo
tech, "When all else fails, read the instructions."

Ciao,

Paul

🔗Carl Lumma <clumma@yahoo.com>

🔗Paul Poletti <paul@polettipiano.com>

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > > Sharp fifths in WTs lead to "harmonic waste" -- you can get the
> > > thirds just as good without it.
> >
> > What does a term invented by a modern American piano tuner have
> > to do with how German keyboard players tempered their instrument
> > in the early 18th century, as reported by a man who was a
> > practicing organist, a certified organ inspector, and highly-
> > regarded temperament theorist? Talk about the blind leading the
> > blind. . .
>
> Depends on what you see as historically accurate.

Generally, in the business I am professionally involved in, it means
"doing what they did", to the best of our ability to understand what
it is that they did. It does NOT mean making something up and
proclaiming it to be historic, or worse yet, "better" - though some
have great success with this approach, especially in the field of
"copying" historical instruments.

> Do you
> follow in the footsteps of great men, or seek what they sought?

That's a statement painted with a broad brush that could cover a
multitude of sins. But I don't find much difference between what
Werckmeister DID here and what he claimed he was seeking; the solution
satisfies the criteria quite well.

> We have lots of tools and understanding that Werckmeister
> didn't,

This is an interesting hypothesis. I know large number of people in
the organ building and early music world who would dismiss you as an
arrogant modernist who knows nothing of the acoustical and
mathematical knowledge of ancient organ builders (which was
extensive), but I'll give you the benefit of the doubt. Just what are
these invaluable advances and tools which will help us with the rather
simplistic problem of balancing the problem of the confilciting
distributions of the syntonic and Pythagorean commas?

> and there's a good chance we can come up with a better
> tuning.

Better according to whose defintion? Yours, Owen Jorgenson's, or
Werckmeister's?

> And this was, after all, a treatise for beginners.

The basic approach is more or less the same as a large number of other
surviving temperament instructions, not all designed for amateurs.
They all include wide fifths. Plus if that is going to be a defining
characteristic, what aspect of using large fifths would make it easier
for amateurs?
>
> > (b) don't grasp the concept of cumulative overcompensation.
>
> Why not explain it?

Since the two commas are more or less the same size (the difference
being only about 1/12th P comma), you can for the purpose of
discussion discount the difference and just talk about "commas" (as
many did). The problem is that in order to make pure thirds the comma
must be elminated over four fifths and in ordr to close the cirlce it
must be eliminated over twelve fifths. If you fix C-E you've used up
your whole comma and the circle will already close (except for the
schisma); you've got nothing left to fix the remaining thirds. Thus,
in purely circulating terms (elminating P. comma), any further use of
small fifths beyond the first four is "overcempensation". That's why
1/4 meantone ends up with the wolf fifth (and the wolf thirds, as
Preatorius said); all the cumulative overcompensation, employed to
produce a large number of good thirds, ends up leaving a hole in the
circle large enough to drive a truck through. Modified meantones of
any stripe (Froggie or German) are designed to bridge this hole,
either by canceling the cumulative overcompensation with a small
number of large and/or pure fifths, or, as in the case of Werckmeister
and Schlick, by using wide and pure fifths in combination with fifths
wider than 1/4 S. comma but still narrow enough to accumulate some
overcompensation.
>
> Do you grasp the concept of adaptive tuning?

Yep. I've read Tuning Timbre Scale Spectrum. I suspect it is more or
less what folks did in the old days, though there are no easy answers
to the problem of how to account for the role of the
keyboard/lute/gamba, unless we follow Rameau's advice and just ignore
them, except for tonic/dominant.

Ball's in your court. I really wanna know how we are smarter today
than Werckmeister was, and what tools allow us to make "better'
temperaments, and how we define "better" better than W did. Poor old
Werckmeister.

Ciao,

Paul

🔗Aaron Krister Johnson <aaron@dividebypi.com>

I'm confused. Are you the same Paul Poletti who critiqued WerckIII in
a paper, calling it 'economy class 1/4 pyth.; favored Neidhardt and
Sorge, etc., and now you are here defending Werckmeister?

Irony of ironies!

I don't see the problem of using modern language and tools to solve
old problems. It happens in mathematics all the time. The ancient
greeks knew of problem they approximated by brute force that modern
calculus does in a jiffy....so I guess unless one is in a really
uptight 'authenticity game', I'm with Carl on this one.

-A.

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> >
> > > > Sharp fifths in WTs lead to "harmonic waste" -- you can get the
> > > > thirds just as good without it.
> > >
> > > What does a term invented by a modern American piano tuner have
> > > to do with how German keyboard players tempered their instrument
> > > in the early 18th century, as reported by a man who was a
> > > practicing organist, a certified organ inspector, and highly-
> > > regarded temperament theorist? Talk about the blind leading the
> > > blind. . .
> >
> > Depends on what you see as historically accurate.
>
> Generally, in the business I am professionally involved in, it means
> "doing what they did", to the best of our ability to understand what
> it is that they did. It does NOT mean making something up and
> proclaiming it to be historic, or worse yet, "better" - though some
> have great success with this approach, especially in the field of
> "copying" historical instruments.
>
> > Do you
> > follow in the footsteps of great men, or seek what they sought?
>
> That's a statement painted with a broad brush that could cover a
> multitude of sins. But I don't find much difference between what
> Werckmeister DID here and what he claimed he was seeking; the solution
> satisfies the criteria quite well.
>
> > We have lots of tools and understanding that Werckmeister
> > didn't,
>
> This is an interesting hypothesis. I know large number of people in
> the organ building and early music world who would dismiss you as an
> arrogant modernist who knows nothing of the acoustical and
> mathematical knowledge of ancient organ builders (which was
> extensive), but I'll give you the benefit of the doubt. Just what are
> these invaluable advances and tools which will help us with the rather
> simplistic problem of balancing the problem of the confilciting
> distributions of the syntonic and Pythagorean commas?
>
> > and there's a good chance we can come up with a better
> > tuning.
>
> Better according to whose defintion? Yours, Owen Jorgenson's, or
> Werckmeister's?
>
> > And this was, after all, a treatise for beginners.
>
> The basic approach is more or less the same as a large number of other
> surviving temperament instructions, not all designed for amateurs.
> They all include wide fifths. Plus if that is going to be a defining
> characteristic, what aspect of using large fifths would make it easier
> for amateurs?
> >
> > > (b) don't grasp the concept of cumulative overcompensation.
> >
> > Why not explain it?
>
> Since the two commas are more or less the same size (the difference
> being only about 1/12th P comma), you can for the purpose of
> discussion discount the difference and just talk about "commas" (as
> many did). The problem is that in order to make pure thirds the comma
> must be elminated over four fifths and in ordr to close the cirlce it
> must be eliminated over twelve fifths. If you fix C-E you've used up
> your whole comma and the circle will already close (except for the
> schisma); you've got nothing left to fix the remaining thirds. Thus,
> in purely circulating terms (elminating P. comma), any further use of
> small fifths beyond the first four is "overcempensation". That's why
> 1/4 meantone ends up with the wolf fifth (and the wolf thirds, as
> Preatorius said); all the cumulative overcompensation, employed to
> produce a large number of good thirds, ends up leaving a hole in the
> circle large enough to drive a truck through. Modified meantones of
> any stripe (Froggie or German) are designed to bridge this hole,
> either by canceling the cumulative overcompensation with a small
> number of large and/or pure fifths, or, as in the case of Werckmeister
> and Schlick, by using wide and pure fifths in combination with fifths
> wider than 1/4 S. comma but still narrow enough to accumulate some
> overcompensation.
> >
> > Do you grasp the concept of adaptive tuning?
>
> Yep. I've read Tuning Timbre Scale Spectrum. I suspect it is more or
> less what folks did in the old days, though there are no easy answers
> to the problem of how to account for the role of the
> keyboard/lute/gamba, unless we follow Rameau's advice and just ignore
> them, except for tonic/dominant.
>
> Ball's in your court. I really wanna know how we are smarter today
> than Werckmeister was, and what tools allow us to make "better'
> temperaments, and how we define "better" better than W did. Poor old
> Werckmeister.
>
> Ciao,
>
> Paul
>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

🔗Paul Poletti <paul@polettipiano.com>

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:
>
>
> I'm confused. Are you the same Paul Poletti who critiqued WerckIII in
> a paper, calling it 'economy class 1/4 pyth.; favored Neidhardt and
> Sorge, etc., and now you are here defending Werckmeister?
>
> Irony of ironies!

You have it somewhat right, but you are confusing/forgetting a couple
things. Actually, in the very paper to which you refer, I also highly
recommeded Werckmeister's Continuo temperament, exactly as I do here.

I have criticized the ubiquitous use of Werkmeister III in the modern
"historical performance" world, because it is being applied broadly in
many contexts when Werckmeister offered it as a solution to a specific
problem. He himself said that it was an economic solution to retuning
a 1/4 S. comma meantone organ in a passable circulating (what some
call "Well") temperament. He specifically pointed out that because
several pipes could remain untouched, it would require less atention
from the organ builder. Retempering an organ in those days was a vast
and hugely expensive endeavor, since many pipes had to be physically
shortened or extended.

I do indeed recommend Neidhardt and Sorge solutions for the music of
the 18th century which does not work well in some sort of modified
meantone, and certainly for Classical music.

The temperament I am talking about is specifically recommended by
Werckmeister for continuo playing, and it is very similar to a large
number of temperaments from other sources.

So there is no irony here. Perhaps you suffer from the misconception
(as do many) that Werckmeister offered III as a general purpose
temperament, and that he never recommended anything else. This could
indeed lead you to think that there was some sort of a contradiction
in my position.

> I don't see the problem of using modern language and tools to solve
> old problems. It happens in mathematics all the time. The ancient
> greeks knew of problem they approximated by brute force that modern
> calculus does in a jiffy....so I guess unless one is in a really
> uptight 'authenticity game', I'm with Carl on this one.
>
What I take issue with is the judgement of Werckmeister's own criteria
based upon the ideas of Owen Jorgenson. It is not that "harmonic
waste" is a modern term, but only that the CONCEPT to which it refers
represents a hypothetical disadvantage which many 16th, 17th, and 18th
century musicians seemed quite happy to accept, from Schlick to
D'Alembert. Thus using it to criticize the structure of Werckmeister's
Continuo temperament, as did Carl, has no validity. By doing so, you
are not "solving" any "old problem", you are using your viewpoint to
redefine the problem in a way that comes up with a solution that you
are more comfortable with. There is nothing historical about this
endeavor, neither vaguely historical nor "uptight" historical.

Ciao,

Paul

🔗Carl Lumma <clumma@yahoo.com>

> > Depends on what you see as historically accurate.
>
> Generally, in the business I am professionally involved in, it means
> "doing what they did", to the best of our ability to understand what
> it is that they did. It does NOT mean making something up and
> proclaiming it to be historic, or worse yet, "better" - though some
> have great success with this approach, especially in the field of
> "copying" historical instruments.

If you think copying instruments is a forward activity, we
have little in common. That's not to say a lot of good ideas
in, for example piano actions, weren't lost along the way to
20th-century mass-produced hegemony...

> > Do you
> > follow in the footsteps of great men, or seek what they sought?
>
> That's a statement painted with a broad brush that could cover a
> multitude of sins.

One thing the great instrument builders and theoreticians
of antiquity all seemed to share is a relentless practice
of experimentatation. They were constantly trying to improve
their instruments and their understanding. There's little
evidence they were looking back to the medieval period and
copying instruments. For all the people who insist Bach
must be played on a harpsichord, there's at least one
account of Bach proclaiming pianos to be the greatest thing
since sliced bread.

> But I don't find much difference between what Werckmeister
> DID here and what he claimed he was seeking; the solution
> satisfies the criteria quite well.

There's a clear discussion of the goals of temperament in
Werkmeister's article. The more perfect consonances admit
to less tempering -- a principle embraced by much of the work
presented on this list over the last decade (though some of
it concludes that "absolutely none" for octaves isn't ideal).
1/4-comma fifths are too flat -- Werckmeister seems to
overtstate this, and it is more of an opinion, which his
contemporaries may or may not have shared. Remember, no
matter how historically accurate you try to be, you're still
limited by the lens of History. If this is indeed the best
document we have on 17th-century temperament in Germany,
then we're not in very good shape.

The actual tuning presented is for beginners, and we presume
is meant to be easy to tune by ear. "Werckmeister III", on
the other hand, does not have "harmonic waste". . .

> > We have lots of tools and understanding that Werckmeister
> > didn't,
>
> This is an interesting hypothesis. I know large number of people
> in the organ building and early music world who would dismiss you
> as an arrogant modernist who knows nothing of the acoustical and
> mathematical knowledge of ancient organ builders (which was
> extensive),

I take it they don't use things like spectral analysis and
physical models for studying pipe acoustics?

> but I'll give you the benefit of the doubt. Just what are
> these invaluable advances and tools which will help us with
> the rather simplistic problem of balancing the problem of the
> confilciting distributions of the syntonic and Pythagorean
> commas?

Logarithms are a big one, though by the end of the 17th
century they could very well have made it to organ tuners
in Germany. But did they? Temperaments still seem to be
described by bearing plans. Whereas we use cents.

Computers can be helpful. So today we can do things like
precisely balance the tempering of the consonances according
to their degree of "perfection". We can arrage for triads
to be proportional-beating. We can search the space of
possible temperaments and find ones that meet certain
criteria.

The same thing can be done for bearing plans. We can take
a temperament and produce any number of bearing plans,
and find the one that would be easiest to tune by counting
beats, or by counting beat ratios, or...

> > > (b) don't grasp the concept of cumulative overcompensation.
> >
> > Why not explain it?
>
> Since the two commas are more or less the same size (the
> difference being only about 1/12th P comma), you can for the
> purpose of discussion discount the difference and just talk
> about "commas" (as many did). The problem is that in order to
> make pure thirds the comma must be elminated over four fifths
> and in ordr to close the cirlce it must be eliminated over
> twelve fifths. If you fix C-E you've used up your whole comma
> and the circle will already close (except for the schisma);
> you've got nothing left to fix the remaining thirds. Thus, in
> purely circulating terms (elminating P. comma), any further use
> of small fifths beyond the first four is "overcempensation".
> That's why 1/4 meantone ends up with the wolf fifth (and the
> wolf thirds, as Preatorius said); all the cumulative
> overcompensation, employed to produce a large number of good
> thirds, ends up leaving a hole in the circle large enough to
> drive a truck through. Modified meantones of any stripe
> (Froggie or German) are designed to bridge this hole, either
> by canceling the cumulative overcompensation with a small
> number of large and/or pure fifths, or, as in the case of
> Werckmeister and Schlick, by using wide and pure fifths in
> combination with fifths wider than 1/4 S. comma but still
> narrow enough to accumulate some overcompensation.

I think I'd choose the pure fifths.

> > Do you grasp the concept of adaptive tuning?
>
> Yep. I've read Tuning Timbre Scale Spectrum.

That's a different kind of adaptive tuning. I should have
said "adaptive JI". Sorry. So let me repeat. Do you
consider the following scheme a temperament?: melodic
intervals in 31-ET, and harmonic intervals in 5-limit JI.

http://en.wikipedia.org/wiki/Archicembalo

(see Tuning # 2)

-Carl

🔗Paul Poletti <paul@polettipiano.com>

Carl Lumma wrote:

> If you think copying instruments is a forward activity, we
> have little in common.

Well I'm not sure what is meant by a "forward activity". I serve a large group of musicians and listeners who enjoy hearing the sounds as people of the past heard them. One man's meat is another man's poison. Whether or not this has any "value" is not a debatable point, because it is subjective. Let's not even start with this waste of time.

> > > > Do you
> > > follow in the footsteps of great men, or seek what they sought?
> >
> > That's a statement painted with a broad brush that could cover a
> > multitude of sins.
> > One thing the great instrument builders and theoreticians
> of antiquity all seemed to share is a relentless practice
> of experimentatation. They were constantly trying to improve
> their instruments and their understanding.

Wrong. There are many periods of stability within which musicians/instrument makers/composers were quite happy to just carry on doing what had been done.

> For all the people who insist Bach
> must be played on a harpsichord, there's at least one
> account of Bach proclaiming pianos to be the greatest thing
> since sliced bread.

I am not aware of this account. Could you find it for us, please?

> > There's a clear discussion of the goals of temperament in
> Werkmeister's article. The more perfect consonances admit
> to less tempering -- a principle embraced by much of the work
> presented on this list over the last decade (though some of
> it concludes that "absolutely none" for octaves isn't ideal).
> 1/4-comma fifths are too flat -- Werckmeister seems to
> overtstate this, and it is more of an opinion, which his
> contemporaries may or may not have shared.

I keep reading this paragraph looking for some substance. You make some statements but they don't make any point. W. said fifths should be less tempered than thirds, true. His temperament manifests just that. Regarding contemporary preferences, as I keep saying over and over again, a point which you seem to keep refusing to acknowledge, is that there are many instructions from many different sources which are very similar. Can you deall with that, Carl?

> Remember, no
> matter how historically accurate you try to be, you're still
> limited by the lens of History. If this is indeed the best
> document we have on 17th-century temperament in Germany,
> then we're not in very good shape.

Why not? Is it because you don't like what it tells us?
> > The actual tuning presented is for beginners, and we presume
> is meant to be easy to tune by ear. "Werckmeister III", on
> the other hand, does not have "harmonic waste". . .

You still have yet to demonstrate WHY a non-harmonic wasted temperament is more challenging to tune. W III is one of the easiest things in the world to set, which is one of the reasons why it is so popular. W. Continuo is much more difficult to set, requiring far more tempering and some judgements about relative qualities of thirds. So once againn, yo keep raising an issue which doesn't make a point.

> > I take it they don't use things like spectral analysis and
> physical models for studying pipe acoustics?

How does this aide our ability to construct temperaments? Please try to stay on point, Carl, and address the issues within the context.
> > Logarithms are a big one, though by the end of the 17th
> century they could very well have made it to organ tuners
> in Germany. But did they? Temperaments still seem to be
> described by bearing plans. Whereas we use cents.

Werckmeisters monochord engravings are highly sophisticated mathematically. There's no way we can claim his ability to address the problem with numbers has anything to do with his methodology and conclusions.
> > Computers can be helpful. So today we can do things like
> precisely balance the tempering of the consonances according
> to their degree of "perfection".

Is this precise "balancing" an aim of Werckmeister? I don't find such language in his writings.

> We can arrage for triads
> to be proportional-beating.

Is this arrangement an aim of Werckmeister? I don't find such language in his writings.

> We can search the space of
> possible temperaments and find ones that meet certain
> criteria.

Your criteria, not Werckmeister's.
> > The same thing can be done for bearing plans. We can take
> a temperament and produce any number of bearing plans,
> and find the one that would be easiest to tune by counting
> beats, or by counting beat ratios, or...

Again, YOUR criteria, not Werckmeister's.
> > I think I'd choose the pure fifths.

What you would choose is not what W. would choose.
> > > > Do you grasp the concept of adaptive tuning?
> >
> > Yep. I've read Tuning Timbre Scale Spectrum.
> > That's a different kind of adaptive tuning. I should have
> said "adaptive JI". Sorry. So let me repeat. Do you
> consider the following scheme a temperament?: melodic
> intervals in 31-ET, and harmonic intervals in 5-limit JI.

By definition, a "temperament" is ANY system which substitutes impure intervals for pure intervals. A JI of any stripe is a "tuning". What's the point?

Ciao,

🔗Carl Lumma <clumma@yahoo.com>

> > If you think copying instruments is a forward activity, we
> > have little in common.
>
> Well I'm not sure what is meant by a "forward activity".

You said something about people being daring by doing
reproductions (instead of restorations?). Maybe I misread
you.

> > One thing the great instrument builders and theoreticians
> > of antiquity all seemed to share is a relentless practice
> > of experimentatation. They were constantly trying to improve
> > their instruments and their understanding.
>
> Wrong. There are many periods of stability within which
> musicians/instrument makers/composers were quite happy to
> just carry on doing what had been done.

What were they, precisely? Not including 1900 to the
present day, of course.

> > For all the people who insist Bach
> > must be played on a harpsichord, there's at least one
> > account of Bach proclaiming pianos to be the greatest thing
> > since sliced bread.
>
> I am not aware of this account. Could you find it for us, please?

I forget the biographer, but it's the famous one of his
visit to Frederick the Great, who was collecting Silberman
pianos. I understand some scholars aren't terribly
convinced this visit happenned ... I'm not up on the latest
about that.

> > I take it they don't use things like spectral analysis and
> > physical models for studying pipe acoustics?
>
> How does this aide our ability to construct temperaments?
> Please try to stay on point, Carl, and address the issues
> within the context.

If you've read Sethares' book, you should be able to guess.

> > Logarithms are a big one, though by the end of the 17th
> > century they could very well have made it to organ tuners
> > in Germany. But did they? Temperaments still seem to be
> > described by bearing plans. Whereas we use cents.
>
> Werckmeisters monochord engravings are highly sophisticated
> mathematically. There's no way we can claim his ability to
> address the problem with numbers has anything to do with his
> methodology and conclusions.

Was he using logarithms or not? Because without them, you
can't make the useful abstraction between tunings and
bearing plans.

> > Computers can be helpful. So today we can do things like
> > precisely balance the tempering of the consonances according
> > to their degree of "perfection".
>
> Is this precise "balancing" an aim of Werckmeister?
> I don't find such language in his writings.

He says octaves the least, fifths next, thirds next. But
it's clear you'll find what you like in his writing, while
I'll do the same.

> > We can arrage for triads
> > to be proportional-beating.
>
> Is this arrangement an aim of Werckmeister?

Is it?

> > We can search the space of
> > possible temperaments and find ones that meet certain
> > criteria.
>
> Your criteria, not Werckmeister's.

So?

> > The same thing can be done for bearing plans. We can take
> > a temperament and produce any number of bearing plans,
> > and find the one that would be easiest to tune by counting
> > beats, or by counting beat ratios, or...
>
> Again, YOUR criteria, not Werckmeister's.

I think I'm sensing a theme here.

> > I think I'd choose the pure fifths.
>
> What you would choose is not what W. would choose.

Wait, you lost me on this one.

> > > > Do you grasp the concept of adaptive tuning?
> > >
> > > Yep. I've read Tuning Timbre Scale Spectrum.
> >
> > That's a different kind of adaptive tuning. I should have
> > said "adaptive JI". Sorry. So let me repeat. Do you
> > consider the following scheme a temperament?: melodic
> > intervals in 31-ET, and harmonic intervals in 5-limit JI.
>
> By definition, a "temperament" is ANY system which substitutes
> impure intervals for pure intervals. A JI of any stripe is
> a "tuning". What's the point?

The point is, it's a model for the flexible intonation you
keep foaming on about.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

🔗Paul Poletti <paul@polettipiano.com>

Carl Lumma wrote:

> > Let me try to sum up my view without being crass.
> It's a series of questions:
> > - How much stock should we put in one document
> as a representation of Werckmeister's goals and
> conclusions?
> > - How much stock should we put in Werckmeister as
> a representative of the baroque tuning ideal?
> > - How much stock should we put in baroque methods
> toward the fulfillment of the baroque ideal?
> > Carl, you forget that this whole discussion began with regard to interpreting ONE particular temperament. You seem to be expanding this to some overreaching "Baroque Ideal", which is equally silly.

The Baroque was a complex time for tuning/temperament. The biggest problem was not musical, but purely practical: the need to combine instruments made at different pitch levels in performance. Thus organists/harpsichordists where often required to transpose by 1/2 step, a whole step, or a minor third in order to accommodate the four basic pitch levels, from about 390 to 465, each separated by a half step. The lucky ones had sliding transposing keyboards/pedal boards, but all were required to be able to do it at sight. Of this there is no doubt, as there is an overwhelming amount of organological and documentary evidence. The fact the temperament theorists were struggling with it is also proved by documentary evidence. In regards to this aspect ONLY, we would have to say the "Baroque Ideal" was ET, the conclusion reached by Werckmeister and Neidhardt.

Problem is, organs sound terrible in ET, especially organs from then, which often had tierce mixtures. So, despite some brave attempts to use equal or circulating temperaments, the overwhelming number of organs remained tuned in 1/4 S. comma strict meantone, as the new book coming out by Ibo Ortgies based on church organ maintenance contracts will demonstrate. So in this limited sense we could say the "Baroque Ideal" was strict meantone.

Between these two extremes their is a world of possibilities, all matching one or another degree of the "Baroque Ideal". Werckmeister presented several "ideals" for different contexts. Neidhardt specified 4 "ideals" based upon the social setting. This he did twice, altering the ideal degree of "the ideal" slightly, though in both cases, ET was "the ideal Ideal" for the most cosmopolitan setting: the court. Why? We can only speculate, though I doubt it is because sophisticated listeners preferred the sound of equal. There are many comments to the contrary, that the best players inflected sharps and flats differently. No, it was most likely because the court was the setting most likely to be visited by traveling musicians bearing their own instruments made at different pitch levels, requiring transpostion by 1/2, 1, or 1 1/2 steps by the court organist/harpsichordist.

And what are we to make of Rameau? At first he recommended a modified meantone similar to Werckmeister's, though more conservative in that it used 1/4 S comma fifths for the regularly-tempered fifths. But he later jumped clear to the other end of the spectrum, recommending ET. What was his "ideal"? How much does he reflect the French "Baroque ideal"? If everybody thought like him, then why did D'Alembert recommend yet another modified meantone in the 1750's, twenty years after Rameau's ET conversion?

Nachersberg, writing at some time near the end of the 18th century (the most wide-spread publication of his work is the 1805 'Gall' version) said just this; there is 1/4 comma meantone (he did not use those terms but he described in such a way that there is no doubt), and there is equal, which most people find wanting because of the sour thirds. Between them there is a cornucopia of possibilities

So, if anybody is being silly here, I'd say it is someone who coins a phrase like "Baroque Ideal". What an utterly silly notion!

Now, about Werckmeister specifically, this particular document is one of his last, so we probably ought to give it more weight than either of the earlier two in which the infamous WIII is found. His final words on the topic, published posthumously, were that he liked equal thought he preferred a system which kept the more commonly used keys somewhat purer sounding. This 1698 temperament, when tweaked as far toward equal as the criteria will bear, satisfies that description fairly well.

But again, I think we make a mistake in assuming that Werckmeister would say their is one ideal for all situations. So I while I think this particular document may well be Werckmeister's final word on the type of temperament (NOTE: not THE temperament) best suited for continuo work circa 1700, I think it is rather silly to talk about "Werckmeister goals and conclusion" in any broader sense. This whole discussion began when you doubted the use of wide fifths in THIS particular temperament. That these fifths help to achieve Werckmeister's goals, whatever they may have been in this particular instance, is undeniable. He put them there, after all; the man wasn't stupid!

> You seem to answer all of these questions with
> "100%", which to me seems rather silly.

Were on earth did you get this silly conclusion?

Ciao,

Paul

🔗Carl Lumma <clumma@yahoo.com>

🔗Paul Poletti <paul@polettipiano.com>

Carl Lumma wrote:
> > > > Rameau ... recommending ET.
> > Where does he do this?
> I'm traveling at the moment so do not have access to my archives. This is the best I can do for you at the moment:

***

In his Gï¿½nï¿½ration harmonique (1737) Rameau endorsed equal temperament and, by way of retracting his own views of 11 years before, introduced a new argument in its favour:

"He who believes that the different impressions which he receives from the differences caused in each transposed mode by the temperament [now] in use heighten its character and draw greater variety from it, will permit me to tell him that he is mistaken. The sense of variety arises from the intertwining of the keys [lï¿½entrelacement des Modes] and not at all from the alteration of the intervals, which can only displease the ear and consequently distract it from its functions."

Distracting the musical ear from its proper functions is an unpardonable fault in a tuning. Rameauï¿½s argument might well have applied more palpably in France than in Germany, if French unequal tunings were, as they generally appear to have been, less subtle than their German counterparts. Rameauï¿½s authority as a musician was such that the 1749 register of the Paris Acadï¿½mie Royale des Sciences could state, "M. Rameau assures us that experience is not opposed to the temperament that he proposes; and in this regard he has earned the right to be taken at his word". Equal temperament continued to be identified with his name throughout the 18th century in France and occasionally in Italy as well.

Source: Grove's Dictionary of Music and Musicians, current on-line edition, article on "Temperament" written by mark Lindley, which is substantially identical to the last paper edition.

***

I thought this was more or less general knowledge.

Ciao,

Paul

🔗Paul Poletti <paul@polettipiano.com>

Carl Lumma wrote:

> > > > One thing the great instrument builders and theoreticians
> > > of antiquity all seemed to share is a relentless practice
> > > of experimentatation. They were constantly trying to improve
> > > their instruments and their understanding.
> >
> > Wrong. There are many periods of stability within which
> > musicians/instrument makers/composers were quite happy to
> > just carry on doing what had been done.
> > What were they, precisely? Not including 1900 to the
> present day, of course.

Well for one, the modern piano has changed hardly at all since about 1870. Ya, Steinway claims the have constantly improved, but it's all in really minor areas. Other periods of stability are most the Renaissance and Baroque. There were a few bursts of changes in harpsichord construction, but generally speaking, instrument makers built the same thing most of their lives. The only period of really intense continuous keyboard development is the piano from about 1770 to about 1870.

> > > > I take it they don't use things like spectral analysis and
> > > physical models for studying pipe acoustics?
> >
> > How does this aide our ability to construct temperaments?
> > Please try to stay on point, Carl, and address the issues
> > within the context.
> > If you've read Sethares' book, you should be able to guess.

Sethares has brought together the obvious in a convincing package, essentially regurgitating Helmholtz with thte trappings of modern tehcnology. This is all very good, Helmholtz needs to be repeated now and then. I think TTSS is a great book EXCEPT for his chapter on western temperament history which is completely out to lunch. His stuff on Gamelon is very good, and the idea of constructing scales based on timbre for nonharmonic tones is of course spot on.

That said, organ pipes, wind instruments, bowed strings, and the keyboard instruments used in the Baroque all produce harmonic tones. Taken as a body, they generally produce all of the first 16 or so harmonics in roughly natural proportions (saw wave). Since our harmony/scale system is also derived from the same set of harmonics, our ability to do spectral analysis of these instruments in no way offers us any improved ability to design temperaments, nor will a temperament designed by taking the spectrum of such instruments into account be radically different from anything the historical record has left us, unless perhaps it is based on a purely acoustic analysis without regard to the requirements of harmonic syntax of western music.

So I think the whole argument that we somehow have greater analytical powers is a red herring.

> >
> > Werckmeisters monochord engravings are highly sophisticated
> > mathematically. There's no way we can claim his ability to
> > address the problem with numbers has anything to do with his
> > methodology and conclusions.
> > Was he using logarithms or not? Because without them, you
> can't make the useful abstraction between tunings and
> bearing plans.

I don't understand the connection. If I want to calculate bearing plans, I do it using cents as the basic calculator, but only because it is easier to set up a spreadsheet that way. I could also do it with fractions. As the text under discussion clearly demonstrates, Werckmeister was quite aware of the most important thing for devising bearing plans: intervals tempered by the same amount increase or decrease in their beat speeds in proportion to the fraction of the interval of transposition, i.e. if C-G is tempered the same as G-D, G-D beats 3 for every 2 of C-G. This is where Jorgenson is completely off the mark, claiming nobody knew about this until "modern science" opened our ears. This is one of the main reasons why those who have actually gone and read old treatises in German think Jorgenson, although well-meaning, is so uniformed that his work can only be taken as the speculative ramblings of an amateur armchair musicologist.

So I don't see where your hypothetical limitation lies. Could you elaborate?

> > >
> > > That's a different kind of adaptive tuning. I should have
> > > said "adaptive JI". Sorry. So let me repeat. Do you
> > > consider the following scheme a temperament?: melodic
> > > intervals in 31-ET, and harmonic intervals in 5-limit JI.
> >
> > By definition, a "temperament" is ANY system which substitutes
> > impure intervals for pure intervals. A JI of any stripe is
> > a "tuning". What's the point?
> > The point is, it's a model for the flexible intonation you
> keep foaming on about.

It's not my model. My model is the musicians decide who is the law, usually the Basso Continuo. The rest play in tune to the law. If the law has a bad fifth or third, he leaves it out and lets the perfect instruments intone properly. I would not attempt to quantify this process in any way. Just LISTEN, man, and play what you need to play. Who needs a system? For those who are so deaf they cannot hear, perhaps.

Ciao,

Paul

🔗Carl Lumma <clumma@yahoo.com>

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, Paul Poletti <paul@...> wrote:
>
> Carl Lumma wrote:
> >
> >
> > > Rameau ... recommending ET.
> >
> > Where does he do this?
> >
>
>
> In his Génération harmonique (1737)

The Rameau text is available online at the following address:

http://www.chmtl.indiana.edu/tfm/18th/RAMGEN_TEXT.html

"... comme notre Tempéramment se réduit à une proportion Géométrique,
les curieux pourront l'examiner dans la formule suivante, qui consiste
à trouver onze moïennes proportionnelles entre les deux termes de
l'Octave 1. 2."

"Par ce moïen tous les Demi-tons sont égaux, ce qu'il suffit de
reconnoître dans notre Tempéramment, pour juger de sa proportion."

It was a notorious fact among (French, at least) music theorists after
that that Rameau had claimed to have invented ET.

See also the following summary for details of the circulating
temperaments with wide fifths associated with Rameau's earlier writing
and other Frenchies:

http://harpsichords.pbwiki.com/Temperament_Ordinaire

The bottom line on 'harmonic waste' is that if your intention is to
have some thirds which are *further* from the 'pure'
5:4 or 6:5 than are 32:27 and 81:64, one must obviously use wide fifths.

If some thirds happen to be used very rarely, or only in extreme
musical contexts, it makes sense to have them rather 'impure', to the
benefit of those more frequently used. For example between F-A and A-C
# and C#-F (enh.E#), one may judge it desirable to have C#-F wider
than Pythagorean in order to let both other thirds be nearly pure.

Alternatively, if one wishes to have a large difference in intonation
between adjacent thirds (round the circle of fifths), it may be
necessary to introduce one or more wide fifths.

That is what Werckmeister seems to have been doing with his wide
fifths, both in 1691 with his 'IV' and 'V' and 'VI' monochords, and in
1698 with his continuo. The distribution of thirds in the 1698
instruction is quite similar to the 'IV' and 'V' tunings, but the
tempering of the fifths is much more regular.

~~~T~~~

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > In his Generation harmonique (1737) Rameau endorsed equal
> > temperament and, by way of retracting his own views of
> > 11 years before, introduced a new argument in its favour:
> >
> > (...)
>
> Interesting. I wonder if he had a suggestion of how to set
> it. Apparently equal-beating fifths was the closest they
> could do to ET.
>
> -Carl
>

Same text, http://www.chmtl.indiana.edu/tfm/18th/RAMGEN_TEXT.html

"Prenez telle Touche du Clavecin qu'il vous plaira, accordez-en
d'abord la Quinte juste, puis diminuez-la si peu que rien, procédez
ainsi d'une Quinte à l'autre, toujours en montant, c'est-à-dire, du
grave à l'aigu, jusqu'à la derniere, dont le Son aigu aura été le
grave de la premiere, vous pouvez être certain que le Clavecin sera
bien d'accord.

"Comme le jugement de l'Oreille n'est pas si exact sur les Sons trop
graves, ou trop aigus, que sur les moïens, il faut commencer sa
Partition au milieu du Clavier, [-101-] et quand on se trouve un peu
trop haut, on accorde très-juste l'Octave au-dessous du Son aigu de la
Quinte qu'on vient d'accorder, puis l'on continue à l'ordinaire.

"La preuve de la parfaite Partition consiste à ce que la derniere
Quinte se trouve d'accord d'elle-même,..."

In English: Go round the cycle of fifths somewhere about the middle of
the keyboard flattening each one 'as little as nothing', the check is
the last fifth, which should be in tune automatically.

Nothing about beats, equal or otherwise. It cannot have produced
*precise* results, but it probably shared many musical qualities with
'modern' ET.

~~~T~~~

🔗Carl Lumma <clumma@yahoo.com>

> > > > One thing the great instrument builders and theoreticians
> > > > of antiquity all seemed to share is a relentless practice
> > > > of experimentatation. They were constantly trying to improve
> > > > their instruments and their understanding.
> > >
> > > Wrong. There are many periods of stability within which
> > > musicians/instrument makers/composers were quite happy to
> > > just carry on doing what had been done.
> >
> > What were they, precisely? Not including 1900 to the
> > present day, of course.
>
> Well for one, the modern piano has changed hardly at all since
> about 1870.

Ok, you say 1870, I say 1900. It's this period of unbelievable
stagnation, augmented with nothing more than museum reproductions
of baroque instruments, that I'm railing against. There's
nothing even remotely like it in antiquity. I studied
harpsichord and fortepiano design with a student of John Challis,
if that gives you any idea where I'm coming from.

> Other periods of stability are most the Renaissance
> and Baroque. There were a few bursts of changes in harpsichord
> construction, but generally speaking, instrument makers built
> the same thing most of their lives.

The baroque produced the piano. It was so rife with
experimentation it's hard to overstate.

> The only period of really intense continuous
> keyboard development is the piano from about 1770 to about 1870.

The harpsichord had to evolve at some point. When
do you think that was?

I can tell.

-Carl

🔗Paul Poletti <paul@polettipiano.com>

Carl Lumma wrote:

> > Interesting. I wonder if he had a suggestion of how to set
> it. Apparently equal-beating fifths was the closest they
> could do to ET.

Carl, this is Jorgenson nonsense. The sooner you forget it, the faster you'll get to a real understanding of historical tempering.

Werckmeister's text shows beyond all doubt that people were aware that identically-tempered intervals beat at different rates depending on their position on the keyboard. There is nothing mysterious about this, and if you are paying attention, this lesson is automatic even if you don't temper. Set Pythagorean and then check the beat rates of stacked sets of major thirds. As long as all the thirds are among the wide ones, any third will beat 5 times for every four of the third below it. Now, everyone knew that these thirds are all too wide by exactly the same amount - 1 Syntonic comma - and yet they all beat at different rates. This simple fact completely destroys all of Jorgenson's rambling on this aspect, especially his laughable assertion that tuning true regular meantone was impossible because nobody knew that identically-tempered intervals beat at different rates. What a load o' crap!

This is part of the reason why I refuse to translate "schweben" as "beating", because it ain't, even though any modern German will tell you it is. This mistranslation has caused a whole bunch of people to think that "Gleichschweben" means "Equal beating", and they all merrily remove one shoe and run off waving the Holy Gourd. Jorgenson is their True Messiah.

So, free your mind and your ears will follow. . .

Regarding how they set it, since I'm traveling I don't have all my files, but as I recall, in an article on 18th century German tuning recipes published back in the '80's in Early Music, I am pretty sure there were a couple that suggest the 3-thirds method. Since everybody knows how to identically-temper the four fifths that make up a major third, the first step of doing meantone of any flavor, you simply divide the octave into 3 identically-tempered major thirds and then divide them each into 4 identically-tempered fifths. So you tune C-E-G#-C so that each successive third beats 5 times for every 4 of the third below it. Or you simple set them so that each beats "a little faster than the other". If the progression of increase in beat speeds is more or less even, you'll get damn close even if you don't actually count the beats. A friend of mine who learned modern organ tuning in the late '60's in Germany tells me this is the method he was taught. This method is self explanatory if you have a reasonable amount of tempering experience with meantone in various flavors and modified meantones.

So, take that big red book and toss it. You'll be much better off.

Ciao,

Paul

🔗Paul Poletti <paul@polettipiano.com>

Carl Lumma wrote:
> > > It's this period of unbelievable
> stagnation, augmented with nothing more than museum reproductions
> of baroque instruments, that I'm railing against. There's
> nothing even remotely like it in antiquity. I studied
> harpsichord and fortepiano design with a student of John Challis,
> if that gives you any idea where I'm coming from.

I won't argue this point. Either you want to know how things sounded in the past or you don't. If you do, the idea of "progress" has no more validity than if you want to know how the Egyptians built the pyramids, or how Drake sailed the globe.
> > > Other periods of stability are most the Renaissance
> > and Baroque. There were a few bursts of changes in harpsichord
> > construction, but generally speaking, instrument makers built
> > the same thing most of their lives.
> > The baroque produced the piano. It was so rife with
> experimentation it's hard to overstate.

The Baroque produced the piano, and it largely lay dormant until the Classical. Granted, there was lots of experimentation in the early 18th century, but it was largely to make new instruments, not to "improve" existing ones.
> > > The only period of really intense continuous
> > keyboard development is the piano from about 1770 to about 1870.
> > The harpsichord had to evolve at some point. When
> do you think that was?
> Go read Hubbard.

Ciao,

Paul

🔗Carl Lumma <clumma@yahoo.com>

> The Rameau text is available online at the following address:
>
> http://www.chmtl.indiana.edu/tfm/18th/RAMGEN_TEXT.html

My alma mater. Just one catch: it's in French.

> "... comme notre Tempéramment se réduit à une proportion
> Géométrique, les curieux pourront l'examiner dans la formule
> suivante, qui consiste à trouver onze moïennes
> proportionnelles entre les deux termes de l'Octave 1. 2."
>
> "Par ce moïen tous les Demi-tons sont égaux, ce qu'il suffit de
> reconnoître dans notre Tempéramment, pour juger de sa proportion."

Why didn't you say so! Seriously, does anyone here read
French?

> If some thirds happen to be used very rarely, or only in extreme
> musical contexts, it makes sense to have them rather 'impure',
> to the benefit of those more frequently used.

For continuo this may make sense, since keys like C# simply
weren't seen in the ensemble repertoire. For a keyboard tuning
meant for all 24 keys, it makes no sense that I can see.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

--- In tuning@yahoogroups.com, Paul Poletti <paul@...> wrote:
>
> Carl Lumma wrote:
>
> >
> > Interesting. I wonder if he had a suggestion of how to set
> > it. Apparently equal-beating fifths was the closest they
> > could do to ET.
>
> Carl, this is Jorgenson nonsense.

I've never read Jorgenson, but apparently it was repeated
here.

> This is part of the reason why I refuse to translate "schweben"
> as "beating", because it ain't, even though any modern German
> will tell you it is. This mistranslation has caused a whole
> bunch of people to think that "Gleichschweben" means "Equal
> beating", and they all merrily remove one shoe and run off
> waving the Holy Gourd. Jorgenson is their True Messiah.

Yes, I'm aware of this criticism. Still, is there even a
single bearing plan extent describing checks of major thirds?
Equal-beating fifths does, in fact, get pretty close to ET.

> 18th century German tuning
> recipes published back in the '80's in Early Music, I am pretty
> sure there were a couple that suggest the 3-thirds method.

I'd like to know about that.

> So, take that big red book and toss it.

I've never even seen a copy.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

🔗Paul Poletti <paul@polettipiano.com>

🔗Carl Lumma <clumma@yahoo.com>

> > > > That's a different kind of adaptive tuning. I should have
> > > > said "adaptive JI". Sorry. So let me repeat. Do you
> > > > consider the following scheme a temperament?: melodic
> > > > intervals in 31-ET, and harmonic intervals in 5-limit JI.
> > >
> > > By definition, a "temperament" is ANY system which substitutes
> > > impure intervals for pure intervals. A JI of any stripe is
> > > a "tuning". What's the point?
> >
> > The point is, it's a model for the flexible intonation you
> > keep foaming on about.
>
> It's not my model. My model is the musicians decide who is the
> law, usually the Basso Continuo. The rest play in tune to the
> law. If the law has a bad fifth or third, he leaves it out and
> lets the perfect instruments intone properly. I would not
> attempt to quantify this process in any way. Just LISTEN, man,
> and play what you need to play. Who needs a system? For those
> who are so deaf they cannot hear, perhaps.

Your model is pretty vague. And it only applies to baroque
music.

A bass pitch is really all baroque ensembles take from the
continuo, 95% of the time. That's it's musical function, and
the rapid decay of the harpsichord and the style used in
continuo playing suit that function well. The reason the 3rd
is omitted is the same reason it is omitted in jazz 'comping --
it muddies the sound (by putting too many partials too close
together... which is the role of the solo instruments :) .

As for ludditism, as I say, I'm glad people are working to
preserve the great discoveries of the past. If that's ALL
you're interested in, I can't help but find this a bit
regrettable, but hey, if that's you're niche, more power
to you. But various statements of yours seem to advocate
that others not push forward either, or worse, that there is
nowhere to push. And that's not cool.

-Carl

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > The Rameau text is available online at the following address:
> >
> > http://www.chmtl.indiana.edu/tfm/18th/RAMGEN_TEXT.html
>
> My alma mater. Just one catch: it's in French.
>
> > "... comme notre Tempéramment se réduit à une proportion
> > Géométrique, les curieux pourront l'examiner dans la formule
> > suivante, qui consiste à trouver onze moïennes
> > proportionnelles entre les deux termes de l'Octave 1. 2."
> >

Since our temperament can be reduced to a geometric series, curious
people can examine it in the following formula, which consists of
finding eleven mean proportionals between the two terms of the Octave
1 and 2.

> > "Par ce moïen tous les Demi-tons sont égaux, ce qu'il suffit de
> > reconnoître dans notre Tempéramment, pour juger de sa proportion."

By this means all the semitones are equal, which is all that is
necessary to be known in our temperament in order to judge of its
proportion.

> > If some thirds happen to be used very rarely, or only in extreme
> > musical contexts, it makes sense to have them rather 'impure',
> > to the benefit of those more frequently used.
>
> For continuo this may make sense, since keys like C# simply
> weren't seen in the ensemble repertoire. For a keyboard tuning
> meant for all 24 keys, it makes no sense that I can see.

Before 1750 I only know of one composer who wrote interesting,
listenable pieces in 'all 24 keys' ... you know his name already. The
biggest name in Baroque keyboard music was the biggest exception in
the use of tonalities. The rest of the world probably had little or no
reason to desire a 'tuning meant for all 24 keys'.

Not to speak of any possible desire for obvious contrasts in
intonation. See for example Scarlatti K.518: beginning in F major, in
the middle he has a sequence going from A maj to B maj to C# maj then
breaking off - the same music repeated in each key. What would be the
point of such a sequence, if each successive key sounded about the
same? (As it does in most of these supposed Bach-tunings, except
possibly for Lindley's...) Whereas in one of these tunings with wide
fifths it sounds amazingly audacious.

Curiously enough, even Rameau didn't suddenly start writing pieces in
F# major or G# minor after he took up ET. Perhaps he didn't think his
music-buying public would follow suit...

~~~T~~~

Werckmeister Continuo Temperament Spreadsheet

🔗Paul Poletti <paul@polettipiano.com>

🔗Carl Lumma <clumma@yahoo.com>

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🔗Carl Lumma <clumma@yahoo.com>

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🔗Carl Lumma <clumma@yahoo.com>

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🔗Carl Lumma <clumma@yahoo.com>

🔗Tom Dent <stringph@gmail.com>

🔗Tom Dent <stringph@gmail.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗Paul Poletti <paul@polettipiano.com>

🔗Paul Poletti <paul@polettipiano.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗Paul Poletti <paul@polettipiano.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗Tom Dent <stringph@gmail.com>

🔗George D. Secor <gdsecor@yahoo.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗George D. Secor <gdsecor@yahoo.com>