Yahoo Tuning Groups Ultimate Backup tuning Re: Re Meantone, was Metastable intervals, re Paul

Paul: I did, but I didn't see anything on the first few pages that came up.
If you have some URL, why not just post it instead of sending people
on a wild goose chase?

Johnny: It may not be up any longer. I found my print out, beginning with a
quote from Beethoven:

"Also please kindly send me the Kirnberger, to add to mine. I am teaching
someone counterpoint and I cannot find my own manuscript under my pile of
papers." In 1817 (probably March) Beethoven asked Tobias haslinger, his friend,
publisher and music dealer in Vienna, for one of the standard works of music
theory. The "someone", whom Beethoven does not wish to mention, is without a
doubt the Archduke Rudolph. He was the only person to whom Beethoven taught
counterpoint, music theory and composition. uiton had already begun in
1809. For this purpose Beethoven compiled extra cts from the most important
contemporary music books, mainly no couterpoint and the theory of the fugue.
[Deethoven-Haus Bonn Digital archives] Sammlung H.C. Bodmer, HCB Mh 46i

There is much more, with Beethoven copying many (I believe there were 9
pages of different Kirnberger-Beethoven related documents.

Paul: Kinrnberger's KRS is a big work covering lots of stuff. I don't doubt
Beethoven could have used some aspects of it. It's the temperament
that I think is amateurish and of little or no practical value for
music of >1780. Most especially Beethoven. And whenever I mention
Kirnberger and temperament in the same breath, I mean his original
temperament, the one with 1/2 comma fifths.

Johnny: You are somewhat unfair. You lambaste the fellow, and then begin
hedging. Having present Beethoven in live performance in Kirnberger (II and
III), I think you are full of ... nonsense. You really just don't know.
Kirnberger III is only a bow towards just, and this serves Beethoven's piano
music well. As an example, listen to the Moonlight Sonata in the radio archive
of the AFMM web site _www.afmm.org_ (http://www.afmm.org)
played by Joshua Pierce. You have to be tone deaf not to listen to that
performance and then to say the tuning is "amateurish and of little or no
practical value."

> JR: Yes, it is true, you do not get any of this. :)
> There are a number of reasons why there was a difference between
church
> organs and keyboards tuned for wind instruments like oboes and
bassoons.
> Firstly, woods sound better when they can be elongated a bit (note
the early music
> fixation on 415 herz).

Paul: Woodwinds were built at many different pitches at many different
times. They sound different, I hesitate to say "better".

Johnny: That's because you are not a woodwind player. Even a half step
flatter, and I'm sure Mr. Haynes would agree, makes for a warmer sound in the
instrument. In this regard, Frederick the Great had been pushing for the
lowest pitches so he would sound best. Warmer is better.

Paul: Depends on
what you are after; a mellow, warm sound, or a bright piquant sound.
It just so happens that bassons and clarinets were largely of French
origins, and French pitch was low

Johnny: Actually, there is a different between the German "fagot" and the
French "basson" in that they are completely different instruments to play with
the German instrument played dark due to a heavy spine running down the
reed, and the French emphasizing the higher register with a reedier or "buzzier"
sound (an little if any spine). I'm sure the oboes that were longer sounded
better than otherwise and were associated with lower kammer pitch.

. Early Dulcians and cornetti, on the
other hand, were high.

> Secondly, organ parts are cheaper when the pipes are smaller.

This idea was floating around the Netherlands 15 years ago or so, but
I don't know anybody who believes it.

Johnny: The people who wrote about it, like Werckmeister, believed it.

> Thirdly,
> higher pitch is symbolically closer to God, angel like, for an
instrument
> designed to get God's attention.

Paul: That's very creative, but it doesn't wash. Organs were originally
pitched lower, in the old Chorton, when Cammerton was the higher of
the two because of the cornetti. As cornetti began to be used more in
churches, the organ were jacked up to agree with them.

Johnny: Are you saying that the wooden cornetti couldn't have their pitched
changed more easily than organs? Are you saying they were only able to be
constructed smaller rather than larger? This seem really odd.

Paul: Only later did
the two switch place.

Johnny: Yes, I know about the switches. Praetorius had a part to play in
recognizing this phenomenon. And Kuhnau got some credit for the swithc, I
remember from somewhere, for making a change in Leipzig.

PauL: Did 16th musicians somehow feel less of a need
to be close to God?

Johnny: This is not a polemic, only that a different generation has
different emphases. There is no reason to deride a congregation adding a new aspect
to their devotion. Incidentally, this was not an original idea, but one I
retrieved from "Tuning of Music" by R. Murray Schaefer, a work based mainly on
different amplitudes of sound around the world.

Paul: Plus I don't think there is much stock in your
suggestion that intent was "to get God's attention". When we are
dealing with an ostensibly omnipotent and omnipresent being, by
definition you've already got his attention. No, I suspect it was more
so to put the FEAR of God into the church goers, and for that it
merely needs to be loud.

Johnny: whatever

> Fourthly, there was no pitch standard, with
> pitches set at different rates in different churches in the same
city. And there
> was no standard of pitch between Chor and Kammer pitches. A whole
tone apart
> does not tell anyone what kind of whole tone is intended.

Paul: That is more or less true but only when you want to pin it down to
something as small as the difference between various believable whole
sizes. This assertion is generally disproved by the number of
instruments built with transposing keyboards (like the early 18th c
Frankfurt cathedral organ, that could slide manuals and pedal by a
minor third, making all half step stops along the way), and the truly
vast number of sources talking about the organist needing to transpose
up or down a whole tone in order to play with winds pitched
differently.

Johnny: Um, don't you realize that this is true _only_ in meantone? Man,
there is another part to music making that you seem to be missing. Slide
manuals were made by Silbermann for use with his meantone keyboards. In
contradistinction, irregular tunings (where each key is distinctive sounding) can not
make transpositions without changing character, even sentiment. That's
another valuable Kirnberger lesson; modulation outside of a close key is harmful
to the meaning of the music. It would be much like Tom has warned, changing
the meaning of the music by changing one meantone variant for another.

Paul: Or works like the Krebs Fantasy for organ and oboe, with
the manuscript organ part written in f minor (!) and the oboe in g
minor (could be a case of subsemitones).

Johnny: Um, with all respect, you are missing out on a different but
concurrent musical aesthetic. It's like the lute, or the bagpipes, something
outside the conservative status quo. The Krebs situation was likely in
Werckmeister III. If so, then the oboe would be a whole step flat to the organ. For
them to match, but only as close as possible to non-church pitch interactions,
the oboe reads g to the organist playing in f. The oboist still has to
match the actual pitch of g minor no matter what notation is on the paper.

written
f
0 90 198 294 390 498 594 702 792 894 996 1092

sounding
g
0 96 192 300 396 504 594 696 798 894 1002 1092

Accordingly, the oboist has a large difference in fifth. Other intervals
are six cents apart. The composer Krebs must have wanted the key of g minor,
transposing the oboe as best as possible to allow them to play together.
Mattheson wrote about how funny this process normally sounds. But it didn't stop
Bach's St. Matthew's Passion, perhaps Bach's greatest chromatic achievement,
still unheard outside of ET. If it was truly in ET, it would never have
needed to have retained both church and chamber pitch.

Have you read Haynes/Story of A?

Ciao,

Johnny: I know of Haynes's writing, but not Story of A. I tried to contact
him in Canada, but I must have had an old address.

may we both have good future adventures that further enlighten, Johnny

************************************** See what's new at http://www.aol.com

🔗Paul Poletti <paul@polettipiano.com>

--- In tuning@yahoogroups.com, Afmmjr@... wrote:

>
> Johnny: It may not be up any longer. I found my print out,
beginning with a
> quote from Beethoven:
>
> "Also please kindly send me the Kirnberger, to add to mine. I am
teaching
> someone counterpoint and I cannot find my own manuscript under my
pile of
> papers."

[snip]
>
> There is much more, with Beethoven copying many (I believe there
were 9
> pages of different Kirnberger-Beethoven related documents.

That's all very fine and good, but is there anything which even
remotely implies that Beethoven approved of and/or liked to have his
pianos tuned in Kirnberger? The above quote demonstrates nothing other
than Beethoven wanted to present his student with a good selection of
current literature on composition. It says nothing about whether or
not B subscribed to any of the ideas contained therein.
>
>
> Paul: Kinrnberger's KRS is a big work covering lots of stuff. I
don't doubt
> Beethoven could have used some aspects of it.
>
> Johnny: You are somewhat unfair. You lambaste the fellow, and
then begin
> hedging.

I don't see what is unfair about drawing a distinction between
different aspects of Kirnberger's work. I don't take an "all or
nothing" approach to anybody's work. I have always held that
Kirnberger was an amateur in the field of temperament ONLY. His level
of sophistication/expertise in the aspects of aesthetics and
composition I leave to other musicologists who are better versed in
the historical conditions in these fields. I agree completely with
Sorge and Marpurg that anyone who needed a monochord in the 1770's to
set ET was seriously lacking in tuning/tempering skills, and by no
means could be considered to be any sort of expert on the topic.
Kirnberger's assertion that the ear can only "know" pure intervals is
absolutely true, but any accomplished tuner knows various tricks to
accurately define tempered intervals. In this respect, he was simply
incompetent.

The only thing "unfair" about this is you trying to create a point of
contention where there is none.

> You really just don't know.
> Kirnberger III is only a bow towards just,and this serves
Beethoven's piano
> music well.

> As an example, listen to the Moonlight Sonata in the radio archive
> of the AFMM web site _www.afmm.org_ (http://www.afmm.org)
> played by Joshua Pierce. You have to be tone deaf not to listen to
that
> performance and then to say the tuning is "amateurish and of little
or no
> practical value."

I did listen to it, and it sounds about the same as any other
circulating temperament performance, of which I have tuned for quite a
few. So what?

I'm not talking about KIII, which represents Kirnberger backing down
on his own principles after Marpurg, Sorge, and others heaped scorn
upon his two 12 comma fifths. I think we must treat any application of
KIII in regards to late 18th/early 19th c music with great suspicion,
since we have absolutely no idea if anybody knew about it expect
Forkel until its publication in the 1880's. And as for KII, I can
assure I am not tone deaf, I just don't like the sound of that many
Pythagorean thirds, nor can I believe that anybody excepted so many of
them in this time period precisely because of the number of sources
complaining about the ugly sound of ET thirds. But you are free to
like what you will, and believe what you will, of course, since beauty
is in the ear of the beholder. You can rest assured I won't accuse you
of being "tone deaf". . . historically suspect, perhaps, but tone
deaf?! No way. . .

>
> Paul: Woodwinds were built at many different pitches at many different
> times. They sound different, I hesitate to say "better".
>
> Johnny: That's because you are not a woodwind player. Even a half
step
> flatter, and I'm sure Mr. Haynes would agree, makes for a warmer
sound in the
> instrument. In this regard, Frederick the Great had been pushing
for the
> lowest pitches so he would sound best. Warmer is better.
>
> Paul: Depends on
> what you are after; a mellow, warm sound, or a bright piquant sound.
> It just so happens that bassons and clarinets were largely of French
> origins, and French pitch was low
>
> Johnny: Actually, there is a different between the German "fagot"
and the
> French "basson" in that they are completely different instruments
to play with
> the German instrument played dark due to a heavy spine running down
the
> reed, and the French emphasizing the higher register with a reedier
or "buzzier"
> sound (an little if any spine).

There is nothing German about the word "Fagot" nor French about the
word "Bassoon" in regards to the early forms of the bassoon. Both
words are of French origin, and Mersenne spoke of a breakdown dulcian
as being "fagoted". The first use of the term "fagot" to name an
instrument directly is in Italy, the Choristfagot. In the 18th
century, bassoon was also used in German-speaking lands. The sort of
nationalistic differences in terminology, sound, and reed making of
which you speak are aspects of periods beyond that which we are
discussing, and certainly have no relevance whatsoever to the origins
of the instrument.

> I'm sure the oboes that were longer sounded
> better than otherwise and were associated with lower kammer pitch.

Ah, well the first and last part of your statement go automatically
together, unless we include French opera pitch in the gamma of
possibilities. As to the middle part, "better" is subjective, but if
we accept your defention of warmer=better, than French Opera pitch
wins over kammer pitch.

>
> > Secondly, organ parts are cheaper when the pipes are smaller.
>
> This idea was floating around the Netherlands 15 years ago or so, but
> I don't know anybody who believes it.
>
> Johnny: The people who wrote about it, like Werckmeister, believed it.

I don't remember reading a passage in Werckmeister saying such, but
then I haven't read all of Werckmeister. Could you give us the quote,
please?

> Johnny: Are you saying that the wooden cornetti couldn't have
their pitched
> changed more easily than organs? Are you saying they were only
able to be
> constructed smaller rather than larger?

Yep, that seems to be the historical evidence. Trumpets and Cornetti
in Cornetton, or Trompetton, which is why they are so called (didn't
you ever wonder about that?). Don't ask me why, that's just the way it
was. And yes, the organs were raised in pitch to match them. Don't ask
me why, that's just the historical fact. Go read Haynes/Story of A.

> This seem really odd.

Man, there are so many odd things about the history of instrument
making it'll make your head spin if you ever start to get into it.

In general, your understanding of the origins and development of wind
instruments seems to be flawed, perhaps based on incomplete
information or a series of assumptions which have little basis in
historical fact. For instance, in your first post in this exchange you
stated that:

"Firstly, woods sound better when they can be elongated a bit (note
the early music fixation on 415 hertz). This is why English Horn and
bass clarinet developed."

Wrong on all counts.

Count 1. The adoption of 415 as the modern "Baroque" pitch comes from
a mid-20th c misunderstanding of the complex nature of pitch levels in
the 16th, 17th, and 18th centuries. Many people believed that pitch
rose, being lowest in old times and highest at the end of the 19th
century before musicians put there foot down and demanded a fixed
standard at 440 - or so the story goes. So 415 was seen as being "the"
old pitch, before it rose a lot. Nothing could be further from the
truth, of course, and now we know that 415 was but one of 4 common and
5 less common general pitch levels, from 390 to 490. it just so
happens that since the modern early music movement comes largely from
Dutch and German speaking lands, they gravitated towards the normal
Cammerton pitch, because that is what the artifacts of their musical
tradition indicated was the case. Nowadays, some people argue that we
really ought to be performing at 390, 440 and 465 as well, and
occasionally it is done, though the economics of needing to buy
instruments pitched at these levels is far too often prohibitive.

Count 2. The modern English Horn is nothing other than the direct
descendent of the tenor oboe in F, which was just another member of
the oboe consort or oboe band. These consorts of interments were made
at whatever pitch level was locally required. A warmer sound due to a
longer length has absolutely nothing to do with either the appearance
or the development of the instrument.

Count 3. Ditto the bass clarinet. The earliest examples, from the end
of the 18th century, were probably intended to replace the bassoon in
military marching bands, and in fact they resemble the bassoon in
structure. Again, they simply performed a musical function in
producing the bass voice for that particular family; warmth of sound
had nothing to do with the introduction of a longer version of the
instrument.
>
> Johnny: Um, don't you realize that this is true _only_ in meantone?
Man,
> there is another part to music making that you seem to be missing.
Slide
> manuals were made by Silbermann for use with his meantone
keyboards. In
> contradistinction, irregular tunings (where each key is distinctive
sounding) can not
> make transpositions without changing character, even sentiment.
That's
> another valuable Kirnberger lesson; modulation outside of a close
key is harmful
> to the meaning of the music. It would be much like Tom has warned,
changing
> the meaning of the music by changing one meantone variant for another.

I'm not talking about a seamless transposition in which the music
sounds exactly the same except for pitch level. I'm not stupid, and I
realize just as well as anyone that distribution is different for
every key in any circulating temperament. I just don't think it was
all that important to them. The did what they had to with the tools
the had at hand, and if on this Sunday it required the organist to
transpose, than so be it. I seriously doubt that precise key quality
was of any importance to any composer. Naturally they were aware of
the traditional location of harsh sounding keys, and they could be
avoided or used for effect. But the precise quality of any particular
chord was something they simply could not know with a any general
consistency, only with regards to a particular instrument they knew or
tuned themselves. To suggest that Bach or anyone else would write any
particular piece with an absolutely specific chord quality for each
and every triad in mind is ludicrous. I know a number of people on
this list are of the opposite opinion, and it provides a wonderful
outlet for endless intellectual speculation, and endless debate, as
each person constructs ever more convincing arguments to support their
particular vision of "Bach's temperament" and compiles endless lists
of particular moments in particular piece that "prove" that this
temperament was used by Bach. That's all very fine and good, though I
don't think the surviving comments support such hair splitting, and I
think it is more a disease of our modern times than any historically
accurate attitude towards music. I doubt anybody in the 18th century,
upon sitting down to play through the WTC, would have thought, "Ah,
but first I must tune Bach's temperament, otherwise the music will not
sound as it should." Any circulating temp will more or less do.

>
> Paul: Or works like the Krebs Fantasy for organ and oboe, with
> the manuscript organ part written in f minor (!) and the oboe in g
> minor (could be a case of subsemitones).
>
> Johnny: Um, with all respect, you are missing out on a different but
> concurrent musical aesthetic. It's like the lute, or the bagpipes,
something
> outside the conservative status quo.

How so? The lute was central to "status quo" music making, a common
member of the continuo ensemble. To say nothing of the various "status
quo" composers for the lute, like Weiss. Bagpipes, yes, definitely a
folk instrument, often straight JI, considering they played in a very
limited number of keys - like the hurdy-gurdy.

> The Krebs situation was likely in
> Werckmeister III.

How do you know? Have you got some evidence for the tuning of Kreb's
organ when he wrote this piece?

> If so, then the oboe would be a whole step flat to the organ.

Temperament choice has absolutely nothing to do with respective pitch
levels. The organ was at Cornetton and the oboe in Cammerton,
obviously. The oboe simply WAS a whole step flat relative to the
organ; "if so" has no meaning here, as the case would be the same no
matter what temperament the organ was in.

> For
> them to match, but only as close as possible to non-church pitch
interactions,
> the oboe reads g to the organist playing in f.

Duh! That's the way Krebs wrote it.

> The oboist still has to
> match the actual pitch of g minor no matter what notation is on the
paper.

Wrong. The oboist, reading/fingering/playing g, or b-flat, or d simply
has to match whatever sounds come out of the organ when the organist
plays a f, a-flat, or c. And so on. No big mystery, hardly worthy the
effort of explaining, even if you explain it correctly.
>
> written
> f
> 0 90 198 294 390 498 594 702 792 894 996 1092
>
> sounding
> g
> 0 96 192 300 396 504 594 696 798 894 1002 1092
>
> Accordingly, the oboist has a large difference in fifth. Other
intervals
> are six cents apart.

What are you on about, here? If the organ is tuned in WIII at
Cornetton, the "written" values you give are also the "sounding"
values when the organist plays in f-minor. There is no discrepancy.

What values the oboist can produce when he plays in g minor is
anybody's guess, though normally g minor is a "good" minor key on the
oboe, having proper high flats and low sharps, and relatively pure
fifths in the basic triads. But as you yourself stated, the idea of
fixing hard and fast numbers to wind intonation is highly suspect. I
don't see the point of this exercise.

> The composer Krebs must have wanted the key of g minor,
> transposing the oboe as best as possible to allow them to play
together.

Neither part transposes. They both play as written. What key he
"wanted" is not really a valid question, unless the background
assumption is regular meantone with 12 notes to the octave, which
seems unlikely, in which case the difference between g minor and f
minor would be extreme. Sure, if it's a piece about wailing and
gnashing of teeth, that's another story, but this fantasy doesn't seem
to have that sort of feel to it. I've played it in f-minor in
meantone, it just sounds like shit.

There's no great mystery here, the only question is what kind of
temperament the organ had. There are only two realistic possibilities.
First, some sort of circulating temperament which lessened the
meantone sourness of f minor. In this case, the oboist just matched
the less than ideal intonation as much as possible, or more likely,
deviated from it as much as tolerable towards flexible JI, as Rameau
said good singers did, and as Ibo's latest research suggests was
common practice for solo melody instruments with flexible intonation.
The main problem with this hypothesis is that Ibo's work strongly
implies there is little support for the application of circulating
temperaments to organs of this period, including your assumed WIII. In
any case, IF the organ WAS tuned in some circulating temperament, then
what key Kreb's "wanted" was whatever sort of mishmash comes out of
combining that particular fixed f-minor with whatever the oboe was
capable of getting away with in his somewhat flexible g minor. About
all we can say is that it is obviously he was letting the oboe play in
one of the easiest and best-intoned minor keys on the instrument.

The other possibility is that the organ was tuned in regular meantone
and had subsemitones ("split sharps"), as many did, although by Kreb's
time they were becoming less common. If the instrument had separate
keys for a real Ab and Db, there would have been no problem, as there
would have been no conflict between oboe and organ.

Either solution is musically acceptable, considering the aesthetics of
the time, and if the organ part was played by a small positive, the
circulating temperament solution is perfectly believable. Brad's
temperament, for example, or Werckmeister's continuo temperament, when
pushed to the limits of circulating (1/7 comma tempering, for
example), both make f-minor tolerable if not ideal. I've tried it both
ways, it sounds fine. But then, I'm tone deaf, so what do I know?
Registration can also help soften sour intonation, as a stopped flute
hides a multitude of sins compared to an open diapason, though asking
it to make 1/4 meantone f-minor sound like anything other than sucking
on lemons is pushing it.

> Mattheson wrote about how funny this process normally sounds. But
it didn't stop
> Bach's St. Matthew's Passion, perhaps Bach's greatest chromatic
achievement,
> still unheard outside of ET. If it was truly in ET, it would never
have
> needed to have retained both church and chamber pitch.

In the manuscript parts, I guess you mean. On the contrary, once
again, the choice of temperament has nothing to do with relative
scoring of the parts. If your winds are at Cammerton and the organ in
Cornetton, they simply are written a step apart, ET or otherwise. No
big mystery, no great hidden significance. Over and over and over
again in the historical literature we read that the pitch level of the
winds is the arbiter of everything. This is because they were not yet
fully chromatic instruments, and playing in certain keys was not an
option. Keyboards, on the other hand, can be tuned and/or constructed
in such a manner that far more keys are possible, so it is the
keyboard player who must transpose, either at sight, or written out
(like Krebs), or with the assistance of mechanical devices.

I just had an experience last spring when tuning for the Nederlands
Bachvereeniging. They did a program of all Buxthude in a number of
churches around the country. The central piece was Schlagt, Künstler,
die Pauken (BuxWV 122), in which they did the large choruses with the
large organ. The organs were tuned at either 440 or 465, and a variety
of temperaments, though none was in meantone. The orchestra tuned 415,
so the organist had to transpose by either a half or a whole step. My
job was to tune the harpsi and the traveling positive organ in a
temperament that satisfied the wind players intonation and was not too
far from the various organ temperaments. After much discussion and
deliberation between the organist, the director, and the winds, we
decided on a Neidhardt 1724 Kleine/1732 Grosse at 415. It was a
wonderful lesson in what sort of problems they were dealing with, and
it was also a great lesson about how much actual intonation
disagreement you can get away with in that sort of acoustic. Anybody
who thinks that it is important to define things down to 4 or 5 cents
up or down is simply not being realistic.
>
> Have you read Haynes/Story of A?

> Johnny: I know of Haynes's writing, but not Story of A. I tried to
contact
> him in Canada, but I must have had an old address.

I am not surprised to hear that you haven't read this, because so many
of your statements seem to contradict the historical record. I suggest
you get it as soon as possible. You can buy the book on Amazon, or
directly from Scarecrow press. You don't have to order it from Bruce
directly. I would also suggest you read Rita Steblin's excellent book
on the history of key affects, it might cause you to reconsider many
of your assertions and attitudes.

Happy Trails!

🔗Afmmjr@aol.com

Hi Paul,
It was previosly not possible to divine clearly that the “only” thing you
didn’t like about Kirnberger was his second temperament. I am relieved. J

Paul: I agree completely with
Sorge and Marpurg that anyone who needed a monochord in the 1770's to
set ET was seriously lacking in tuning/tempering skills, and by no
means could be considered to be any sort of expert on the topic.
Kirnberger's assertion that the ear can only "know" pure intervals is
absolutely true, but any accomplished tuner knows various tricks to
accurately define tempered intervals. In this respect, he was simply
incompetent.
Johnny: um, you may have to eat your words, Paul. While Kirnberger did say
what he said, he actually did not have any difficultly tuning ET without a
monochord. As reported by Claude Palisca in his translation, and published by
Kirnberger in his Vermischte Musicalien (1769):
“Kirnberger had discovered a tempered fifth whose ratio (10935:16384) is
almost exactly 1/12 of a comma smaller than the pure fifth. Kirnbeger arrived
at this ratio in the process of developing his own temperament. The method by
which he arrived at this figure was to take seven pure fifths in succession
plus one pure major third. This gives the tempered fourth 8192:10935, which
is larger than the pure fourth by 1/12 of a comma; its inversion is the
tempered fifth 10935:16384. According to Kirnbeger, he showed this discovery to
the mathematician Leonahard Euler in 1766 a, and remarked to him that this
tempered fifth could be used to determine all ratios of equal temperament.
This idea was subsequently developed by Johann Heinrich Lambert and published in
the Memoires de l’academie royale des sciences et belle letters (Berlin,
1774), pp. 64ff.
“Lambert’s method of determining the fifth 10935:16384, which indeed does
bear a remarkable resemblance to Kinrberger’s was reported by Marpurg in his
Versuch… (He [Marpurg] later translated Lambert’s essay into German and …
Marpurg praised Lambert’s method and pointed out that its deviations from exact
equal temperament never exceed .00001. Another advantage of this method is
that it does not require a monochord. Marpurg, who supported equal
temperament and was violently opposed to Kirnberger’s views on temperament as well as
on harmony, was only too eager to point out that this feature of Lambert’s
method invalidated one of Kirnberger’s objections to equal temperament (p. 20).”

Paul: The only thing "unfair" about this is you trying to create a point of
contention where there is none.

Johnny: Crocodile tears! You call the man incompetent and mischaracterize
his ears and intellect.
Kirnberger’s expose on tuning ET by more secretive methods is now out before
your eyes.
….

Paul: I'm not talking about KIII, which represents Kirnberger backing down
on his own principles after Marpurg, Sorge, and others heaped scorn
upon his two 12 comma fifths.
Johnny: F#!@ scorn. I scorn Marpurg. Vicentino was scorned. Timotheus was
scorned. Harry Partch was corned. Who cares if an arse like Marpurg scorns
Kirnberger? It was all about agenda. Maybe still now?
Paul: I just don't like the sound of that many
Pythagorean thirds, nor can I believe that anybody excepted so many of
them in this time period precisely because of the number of sources
complaining about the ugly sound of ET thirds.
Johnny: Yes, it is clear you have stated a bias against the ditone. There
were Pythagorean ditones still being sung as hymns in churches, and ditones
existed in Werckmeister III, and there is a falling ditone when the NYC subway
doors close. Rasch hates them, too. Truth is, if you perform Kirnberger’s
music, as we have, you won’t find a ditone being heard in that awful way you
object to. The composer Kirnberger favors the intervals that are just or
nearby.
Paul: But you are free to
like what you will, and believe what you will, of course, since beauty
is in the ear of the beholder. You can rest assured I won't accuse you
of being "tone deaf". . . historically suspect, perhaps, but tone
deaf?! No way. . .

Johnny: Hah, Hah, what did you hear…?
Paul: There is nothing German about the word "Fagot" nor French about the
word "Bassoon" in regards to the early forms of the bassoon. Both
words are of French origin, and Mersenne spoke of a breakdown dulcian
as being "fagoted". The first use of the term "fagot" to name an
instrument directly is in Italy, the Choristfagot. In the 18th
century, bassoon was also used in German-speaking lands. The sort of
nationalistic differences in terminology, sound, and reed making of
which you speak are aspects of periods beyond that which we are
discussing, and certainly have no relevance whatsoever to the origins
of the instrument.

Johnny: I’m not sure when the two instrument diverged, mechanically, when
keys were included. One thing is for sure, no one playing a German (Heckel)
bassoon can play a French (Buffet) bassoon, or vice versa.
More importantly, the fagot being French will likely surprise a lot of
Germans.

Paul: I don't remember reading a passage in Werckmeister saying such, but
then I haven't read all of Werckmeister. Could you give us the quote,
please?

Johnny: I’ll try to find it for you. I should be able to come across it
again by the end of the week.

> Johnny: Are you saying that the wooden cornetti > This seem really odd.

Paul: In general, your understanding of the origins and development of wind
instruments seems to be flawed, perhaps based on incomplete
information or a series of assumptions which have little basis in
historical fact. For instance, in your first post in this exchange you
stated that:

Johnny: You might have tried, “’may’ be flawed” and stop assuming you are
all that. I am trying to weigh every new thing I learn from you, and
elsewhere. Pomposity does not earn you any credits.

"Firstly, woods sound better when they can be elongated a bit (note
the early music fixation on 415 hertz). This is why English Horn and
bass clarinet developed."

Wrong on all counts.

Johnny: Count 1. What crap! Yeah, it’s all a big understanding. So let’s
change back, or do they like segregation in a modern mimicry of the old
church (your conettone) and chamber pitch. Um, could it be the instruments even
a half step lower enrich the warmth of the instruments as they play in
resonant venues? Makes sense to me. However 415 started, it is maintained by
what I have just alluded to.

Johnny: You got me there. I got carried away. Almost wanted to say
contrabassoon. But there is something to warmer climates allowing for the lower
pitched instruments. It may have come from Hanes, somewhere.

Paul: Count 3. Ditto the bass clarinet. The earliest examples, from the end
of the 18th century, were probably intended to replace the bassoon in
military marching bands, and in fact they resemble the bassoon in
structure. Again, they simply performed a musical function in
producing the bass voice for that particular family; warmth of sound
had nothing to do with the introduction of a longer version of the
instrument.
Johnny: Well, I can’t resist; do you know the Quintfagot? I was hunting
one down in Europe to play on, but after listening to every tenoroon I could
find, I passed on it. they sound to small they have no tone, they went to far
in the smaller side of things (maybe great for youngsters with small hands
and fingers). The reason bassoons are not in marching bands is because if
someone ever stopped suddenly in front of a bassoonist, his reed would slash
through his or her throat. That is the reason the bassoon is outlawed in
marching bands in certain states.
Surprised you didn’t bring up the Sarusophone family, or even the
Ophecleides. Were they replacements for bassoons, too?
>
….

Paul: I'm not talking about a seamless transposition in which the music
sounds exactly the same except for pitch level. I'm not stupid, and I
realize just as well as anyone that distribution is different for
every key in any circulating temperament. I just don't think it was
all that important to them. The did what they had to with the tools
the had at hand, and if on this Sunday it required the organist to
transpose, than so be it. I seriously doubt that precise key quality
was of any importance to any composer.
Johnny: Disagree.
Paul: To suggest that Bach or anyone else would write any
particular piece with an absolutely specific chord quality for each
and every triad in mind is ludicrous.
Johnny: As an extreme, yes. But to choose D minor as a starting point for
the Chromatic Fantasy and Fugue because it is the most just key in the
circular well temperament, as it would be in Werckmeister III, is certainly to be
aware of key character.
Paul: ….more a disease of our modern times than any historically
accurate attitude towards music. I doubt anybody in the 18th century,
upon sitting down to play through the WTC, would have thought, "Ah,
but first I must tune Bach's temperament, otherwise the music will not
sound as it should." Any circulating temp will more or less do.

Johnny: The Well-tempered Clavier is as you describe. More to teach the
fingers the lay of the land than to supplement the ear with a deeper dimension
through distinction. But a piece the St. Matthew’s Passion might make much
more of a difference, regardless of the titles.
As for your transposition idea, the Mattheson quote I mentioned earlier has
been located, about how he never liked the “abominations” caused by force
fitting a chamber pitched instrument with a church pitched instrument, always a
risk when transposing in a well temperament.
But consider the abomination when instruments tuned in Cammerton such as
horns, bassoons, etc. are sometimes expected to play with others that are in
Chorton, so that either one group or the other must be transposed; is that sound
any different than if the composer or Cantor wished to portray the quarrel
of the hounds over the fallen body of Jesabel? (Haynes, dissertation, p. 257).

By this quote, it seems Mr. Haynes certainly see the other side.
…..

Paul: How so? The lute was central to "status quo" music making, a common
member of the continuo ensemble. To say nothing of the various "status
quo" composers for the lute, like Weiss. Bagpipes, yes, definitely a
folk instrument, often straight JI, considering they played in a very
limited number of keys - like the hurdy-gurdy.

Johnny: The lute is an oddball which is why Dowland could create a tuning
just for it. The lute is used when in a good lute key, not just any key. Aren’
t certain keys better for lute as they are for guitar? As for bagpipes,
check out Podnos’s book on bagpipe intonation for a huge variety of
microtonally different bagpipe tunings found throughout Europe. Maybe I should have
added alphorn, but it is all too often played today in equal temperament.

> The Krebs situation was likely in
> Werckmeister III.

Paul: How do you know? Have you got some evidence for the tuning of Kreb's
organ when he wrote this piece?

Johnny: Right now, I don’t know. I don’t have the music in my hands and
am functionally blind. Besides, your using the term cornettone when I had
been using Chor or church pitch blurs a bit. My assumption of Krebs having
Werckmeister III simply stems from WIII being the first published circular
tuning in Bach’s neck of the woods, and Krebs is a name associated with Bach. Too
general for me to get really analytical, I’m sure you’ll agree. I’d want
to see which notations he uses, check out the town and organ he wrote for,
etc.
As for Ibo, we do not agree about a number of things, and they have been
discussed on the List in years past. It makes sense that you would seem
vehement about some of the same things. One difference is that because there are
no extant records that detail meantone to well temperament, you fellows deduce
it didn’t happen, like there was an express from meantone to ET. I don’t
want to put words in your mouth, but that’s the sense I get.

We disagree as to whether d minor is any different in sound and sentiment
from e minor. I think it does matter and you do not. And I think it mattered
to Bach. As you were so generous to allow me to hear whatever I want to
hear, I graciously allow you to believe whatever you want to believe. Only, I
do not believe the story as it has been handed down. To that end, I am re
ordering the categories into meantone (regular) and irregular. To this end, a
modified meantone is better stated as an irregular.
We don’t agree that there is such a thing as a Thuringian aesthetic toward
unequalness, variegation, and simple irregularity. It is a bit more in the
ethnomusicological realm. It will be my task to explain that some wanted the
regularity of meantone, like they like they equalness of Vicenzo Gallilei’s
archlute. There were democrats and republicans, or another version thereof.
Each thinks they are the whole enchilada.

Paul: I would also suggest you read Rita Steblin's excellent book
on the history of key affects, it might cause you to reconsider many
of your assertions and attitudes. Happy Trails! P
Johnny: Thank you. I am happy to have my own copy of Ms. Steblin’s book
which I read very carefully. I have quoted from it on occasion in my new book
(my second). It is worth noting that, like Barbour, Rita Steblin admits she
had not heard any of the tunings that her book seeks to describe. Still, it
is a valuable addition to scholarship. Happy tuning, Johnny

************************************** See what's new at http://www.aol.com

🔗Paul Poletti <paul@polettipiano.com>

--- In tuning@yahoogroups.com, Afmmjr@... wrote:
>
>
> Hi Paul,
> It was previosly not possible to divine clearly that the âonlyâ
thing you
> didnât like about Kirnberger was his second temperament. I am
relieved.

I consider his only valid temperament for 18th century consideration
to be that which he himself called "How I tune", which we now call
KII. KIII did not become public until the 1880's. Not only that, but
KIII represents him grudgingly backing down on his principles (read
the letter to Forkel). Sort of undermines his whole shtick...

> Paul: I agree completely with
> Sorge and Marpurg that anyone who needed a monochord in the 1770's to
> set ET was seriously lacking in tuning/tempering skills, and by no
> means could be considered to be any sort of expert on the topic.
> Kirnberger's assertion that the ear can only "know" pure intervals is
> absolutely true, but any accomplished tuner knows various tricks to
> accurately define tempered intervals. In this respect, he was simply
> incompetent.
> Johnny: um, you may have to eat your words, Paul.

What a pity! Wouldn't sit well after the delicious roast lamb I've
just polished off for lunch.

> While Kirnberger did say
> what he said,

I'm glad to hear that! For a moment, I thought the poor guy was
possesed by evil spirits forcing him to speak in tongues against his
weill! But now that we have established that Kirnberger said what he
said, we are still left with the issue of whether or not he actually
said what *I* attributed to him, to wit: ET can't be tuned without a
monochord.

> he actually did not have any difficultly tuning ET without a
> monochord. As reported by Claude Palisca in his translation, and
published by
> Kirnberger in his Vermischte Musicalien (1769):
> âKirnberger had discovered a tempered fifth whose ratio
(10935:16384) is
> almost exactly 1/12 of a comma smaller than the pure fifth.
Kirnbeger arrived
> at this ratio in the process of developing his own temperament. The
method by
> which he arrived at this figure was to take seven pure fifths in
succession
> plus one pure major third. This gives the tempered fourth
8192:10935, which
> is larger than the pure fourth by 1/12 of a comma; its inversion is
the
> tempered fifth 10935:16384. According to Kirnbeger, he showed this
discovery to
> the mathematician Leonahard Euler in 1766 a, and remarked to him
that this
> tempered fifth could be used to determine all ratios of equal
temperament.
> This idea was subsequently developed by Johann Heinrich Lambert and
published in
> the Memoires de lâacademie royale des sciences et belle letters
(Berlin,
> 1774), pp. 64ff.

Wow!!! So Kirnberger, all by his little old self, "discovers" the
schisma in 1766, and then further "discovers" that it is equal to 1/12
of a P comma! Oooo! Cutting edge stuff indeed! What did he do for an
encore, "discover" the wheel? I mean, do you really think this
represented a "discovery" in 1766? If anything, it just proves to me
yet again how incompetent K was in this matter.

> Another advantage of this method is
> that it does not require a monochord.

Ahhhhh, how does that follow. Yes, you can set this one fifth without
a monchord, but how does one then proceed to use it to finish the
other 11 fifths? I suppose that having set up the initial fifth you
could try to go all the way back around the circle doing the old 2:3
beat rate shuffle, but the possiblity for cumulative error doing a 2:3
comparison for a chain of 11 fifths is pretty extreme.

But of course, the main objection is this proves nothing, unless there
is missing text you neglected to quote. Did Kirnberger ever SAY he
uses this method to set ET on an instrument entirely by ear? The quote
you give only says he could use this ratio to determine the ratios of
all the other fifths. Now you know as well as I that knowing ratios is
pointless when you sit down with tuning hammer in hand. The only time
they come in handy is, ah, gee, well, when you are marking out a
monochord. That's precisely when K "could use this fifth to determine
the RATIOS of all the other fifths." So we are back to square one, K
using a monochord to tune ET. Your proof that it was otherwise proves
nothing. I'm heading for dessert of chocolate fondeau.

>
> Paul: The only thing "unfair" about this is you trying to create a
point of
> contention where there is none.
>
> Johnny: Crocodile tears! You call the man incompetent and
mischaracterize
> his ears and intellect.

Uh, there you go again, making a mountain out of molehill. If I had
some wackadoodle theory about brain surgery, some brain surgeon could
well call me incompetent, which I am as a brain surgeon, and I
wouldn't see it as an attack on my intellect. Only on my knowledge of
brain surgery.

> Kirnbergerâs expose on tuning ET by more secretive methods is now
out before
> your eyes.

Get off! I've known this trick for at least 20 years or so. Always
thought it was common knowledge: the Schisma = 1/12 comma. As old as
the hills! "Secret methods" indeed!

> Paul: I just don't like the sound of that many
> Pythagorean thirds, nor can I believe that anybody excepted so many of
> them in this time period precisely because of the number of sources
> complaining about the ugly sound of ET thirds.
> Johnny: Yes, it is clear you have stated a bias against the
ditone. There
> were Pythagorean ditones still being sung as hymns in churches, and
ditones
> existed in Werckmeister III, and there is a falling ditone when the
NYC subway
> doors close.

Ah yes, but the relevant question is, what sonic signal did the Vienna
subway use in Beethoven's time? Answer me that, if you will.

;-)

> Rasch hates them, too. Truth is, if you perform Kirnbergerâs
> music, as we have, you wonât find a ditone being heard in that
awful way you
> object to. The composer Kirnberger favors the intervals that are
just or
> nearby.

Ah, excuse me, but I thought we were talking about that OTHER composer
guy, what was his name again?

Beethoven, that's it!

BTW, now that we are back on topic, what ever happened to you
providing some evidence that Beethoven expressed a preference for K's
temperament? Let alone the question of WHICH K temperament that might
have been...

> One thing is for sure, no one playing a German (Heckel)
> bassoon can play a French (Buffet) bassoon, or vice versa.

Irrelevant to the discussion.

> More importantly, the fagot being French will likely surprise a lot of
> Germans.

Ach, die Deutscher! Also irrelevant to the discussion. Lots of things
would surprise modern Germans, such as the fact that "schweben" didn't
mean "beating" for Neidhardt to name but one. So what? Are modern
Germans the repository of all wisdom and knowledge?

>
> Count 1. The adoption of 415 as the modern "Baroque" pitch comes from
> a mid-20th c misunderstanding of the complex nature of pitch levels in
> the 16th, 17th, and 18th centuries.
>
> Johnny: Count 1. What crap! Yeah, itâs all a big understanding.
So letâs
> change back, or do they like segregation in a modern mimicry of the
old
> church (your conettone) and chamber pitch.
> Um, could it be the instruments even
> a half step lower enrich the warmth of the instruments as they play in
> resonant venues? Makes sense to me. However 415 started, it is
maintained by
> what I have just alluded to.

I quote from Bruce Haynes introduction to A History of A, page xxi:

"Thus history itself thorws down an uncomfortable challenge to the
historically oriented performance movement. If we are interested in
original sonorities, if we want our instruments to act and feel as
they did for the composers who concieved their parts, it seems we have
no choice but to renounce the luxory of a single hard-earned pitch
standard. As in so many other issues of historical performance, when
the outer layer of this onion is peeled away, it becomes evident that
there is more beyond A-415, waiting to be revealed."

> But there is something to warmer climates allowing for the lower
> pitched instruments. It may have come from Hanes, somewhere.

I doubt it, as Bruce certainly knows that Italy was by and large
pitched high, circa 465, and here in Spain even higher yet, 490 or so
in many cases.

> As for your transposition idea,

It's not MY idea, it's just what the old authors complain about and
talk about endlessly, and what drove a lot of the circulating
temperament development at the time.

> the Mattheson quote I mentioned earlier has
> been located, about how he never liked the âabominationsâ
caused by force
> fitting a chamber pitched instrument with a church pitched
instrument, always a
> risk when transposing in a well temperament.
> But consider the abomination when instruments tuned in Cammerton
such as
> horns, bassoons, etc. are sometimes expected to play with others
that are in
> Chorton, so that either one group or the other must be transposed;
is that sound
> any different than if the composer or Cantor wished to portray the
quarrel
> of the hounds over the fallen body of Jesabel? (Haynes,
dissertation, p. 257).

Yep, it was a big problem. Doesn't mean they didn't do it. Quite the
opposite. There's quite a few more quotes in Bruce's book making this
undeniable.

>
> By this quote, it seems Mr. Haynes certainly see the other side.

What other side? Nobody denies it was a BIG pain in the butt. Doesn't
mean they didn't have to deal with it.
>
> > The Krebs situation was likely in
> > Werckmeister III.
>
> Paul: How do you know? Have you got some evidence for the tuning
of Kreb's
> organ when he wrote this piece?
>
> Johnny: Right now, I donât know. I donât have the music in my
hands and
> am functionally blind. Besides, your using the term cornettone
when I had
> been using Chor or church pitch blurs a bit.

There is no such thing as "Church pitch". Churches used different
pitches at different times. Chorton changed from 415-sih to 465-ish,
so that term means nothing definite either. Only Cornetton/Trompetton
stayed fixed at 465-ish.

My assumption of Krebs having
> Werckmeister III simply stems from WIII being the first published
circular
> tuning in Bachâs neck of the woods, and Krebs is a name associated
with Bach.

My first computer was an Atari 800, with 2 kb of memory. I now have a
MacBook, limping along on a mere 1 Gb. Krebs wasn't even born when
Neidhardt published his first work, he was 11 when N published his
second, and 19 when N published his 1732 treatise. Why would yo assume
that Krebs would only use something so out of date as Werckmeister III?

Ciao,

🔗Afmmjr@aol.com

Well, this is the wrap up!
Paul: I consider his only valid temperament for 18th century consideration
to be that which he himself called "How I tune", which we now call
KII. KIII did not become public until the 1880's. Not only that, but
KIII represents him grudgingly backing down on his principles (read
the letter to Forkel). Sort of undermines his whole shtick...

Johnny: Kirnberger had a very interesting coterie of friends. It seems
reasonable to assume that once Forkel knew, others would know. It was likely
passed through the Mendellsohn family through Felix’s mother and aunt. Surely
Beethoven could have said something disparaging about Kirnberger while
feverishly copying K’s writings in his own hand to look more impressive to his
student(s). What you consider is limited to a single publication, and that is not
the full picture.

> While Kirnberger did say
> what he said,

Paul: I'm glad to hear that! For a moment, I thought the poor guy was
possesed by evil spirits forcing him to speak in tongues against his
weill! But now that we have established that Kirnberger said what he
said, we are still left with the issue of whether or not he actually
said what *I* attributed to him, to wit: ET can't be tuned without a
monochord.

Johnny: Who do you know in music in so mono-dimensionable? Incidentally, he
has his name attached to an ET paper…but Marpurg attributes it to Moses
Mendelssohn. And many of Kirnberger’s contributions to various encyclopedias
were signed by Sulzer, and others. In other words, get off your high horse and
deal with KIII. Rather than a retreat from KII, it is also a retro return to
WIII. Checking the numbers, they are not that different, but almost 100
years apart…with JS Bach in between.

Paul: Wow!!! So Kirnberger, all by his little old self, "discovers" the
schisma in 1766, and then further "discovers" that it is equal to 1/12
of a P comma! Oooo! Cutting edge stuff indeed! What did he do for an
encore, "discover" the wheel? I mean, do you really think this
represented a "discovery" in 1766? If anything, it just proves to me
yet again how incompetent K was in this matter.

Johnny: You sound like the people who condemn Al Gore, over the top. You
are way too critical. Really, I don’t even think tuning is a category that
deserves appendages like inferior and superior, let alone incompetent. Lou
Harrison has proved that Kirnberger II is not incompetent. If you want to
suggest that K stole his ideas, please, proceed. Otherwise, you are blowing hot
air. There is too much great music accomplished in this tuning. Whether K,
and W, etc. lived in a vacuum in which no one paid attention, or they
influenced everyone in their time in one direction or another, it is more likely it
was something in between. Ibo studying every extant contract does not inform
on the majority of contracts that are not extant and tunings where no
contract was every drawn up.

> Another advantage of this method is
> that it does not require a monochord.

Paul: Ahhhhh, how does that follow. Yes, you can set this one fifth without
a monchord, but how does one then proceed to use it to finish the
other 11 fifths? I suppose that having set up the initial fifth you
could try to go all the way back around the circle doing the old 2:3
beat rate shuffle, but the possiblity for cumulative error doing a 2:3
comparison for a chain of 11 fifths is pretty extreme.

Johnny: Me, with my inferior ear, has often tuned keyboards “melodically.”
Read an interesting discussion by Jorgensen (Tuning) and a Handel tuning
that was performed by ear. (I am not trying to bring the authenticity of this
Handelian well temperament into the discussion…really.) People will tune
differently.

Paul: But of course, the main objection is this proves nothing, unless there
is missing text you neglected to quote. Did Kirnberger ever SAY he
uses this method to set ET on an instrument entirely by ear?
Johnny: Kirnberger did not tune ET because he didn’t like it. Do you need
me to post his vehement statements against ET? Hence, the contrary
statements. He is discouraging others from tuning in ET by saying a monochord is
required (while all the time passing on the trivia that it is indeed possible to a
foreign, only for it to get back to Marpurg under another’s name). Get it
now?
Paul: Uh, there you go again, making a mountain out of molehill. If I had
some wackadoodle theory about brain surgery, some brain surgeon could
well call me incompetent, which I am as a brain surgeon, and I
wouldn't see it as an attack on my intellect. Only on my knowledge of
brain surgery.
Johnny: Guess this is you being clear that my ideas are wackadoodle. That’s
fine. We’ve covered enough after this exchange. I need to finish editing.
After the ideas are fully fleshed out, if not flushed out, criticism will
based on complete explanations, and not e-mail postings which are always found
wanting (as you have regularly demonstrated).

Paul: BTW, now that we are back on topic, what ever happened to you
providing some evidence that Beethoven expressed a preference for K's
temperament? Let alone the question of WHICH K temperament that might
have been...

Johnny: It is you that has divorced K’s tuning from his book. K’s tuning
is throughout the beginning of the book and throughout. Who are you to cut it
out from Beethoven’s interest? Regardless, the connection has not been made
explicitly. Only, the music works better in K tuning than it does in ET
when the piano is involved. Unless you know of something better, I continue to
recommend a K tuning. KII worked beautifully for the Trio of piano, clarinet,
and cello. We have a nice recording. Basically, it had to be something….so
why not K for B?

Paul: Ach, die Deutscher! Also irrelevant to the discussion. Lots of things
would surprise modern Germans, such as the fact that "schweben" didn't
mean "beating" for Neidhardt to name but one. So what? Are modern
Germans the repository of all wisdom and knowledge?

Johnny: no comment.

Paul: I quote from Bruce Haynes introduction to A History of A, page xxi:

Johnny: Thank you for the quote. I agree.
> By this quote, it seems Mr. Haynes certainly see the other side.

Paul: There is no such thing as "Church pitch". Churches used different
pitches at different times. Chorton changed from 415-sih to 465-ish,
so that term means nothing definite either. Only Cornetton/Trompetton
stayed fixed at 465-ish.

Johnny: Wow. You are the first person I know of that has refuted the use
of the term chor (church) pitch for organ tuning a whole step higher than that
of a kammer (chamber) pitch. I’m going to have redo the book to use Paul
Poletti terms.

My assumption of Krebs having
> Werckmeister III simply stems from WIII being the first published
circular
> tuning in Bachâ€™s neck of the woods, and Krebs is a name associated
with Bach.

Paul: My first computer was an Atari 800, with 2 kb of memory. I now have a
MacBook, limping along on a mere 1 Gb. Krebs wasn't even born when
Neidhardt published his first work, he was 11 when N published his
second, and 19 when N published his 1732 treatise. Why would yo assume
that Krebs would only use something so out of date as Werckmeister III?

Johnny: Ah, we end with the Krebs cycle. ;)
The Bach circle was not on board with what N was selling, at least in 1706
Jena. I’m not sure which Krebs you mean (did you ever give a first name?).
There are several Krebs’s. As Leopold Mozart mentions Werckmeister’s tuning
in his violin book, it was certainly still around, and I maintain, used past
the Baroque period. While you doubt at every turn, it is you who may have
actually been sold a bill of goods.
By 1681 W’s tuning (WIII) was a new line in the sand. All 24 keys could be
used. The last 3 chapters of Musical Temperament (1691) tell of the hell he
received from a critic of his, whose identity is anonymous. It was about a
war between Werckmeister’s III and his IV (Trost irregular temperament).
Fanciful ideas of alternatives to this fact of W III’s singularity leave no
known alternatives. Neidhardt is too late for the formative JS Bach, and
the mature JS Bach. Research, visiting the different cities, studying
everything pertinent, while experimenting publicly in NYC through concerts, will
yield a product. You may want to rip into it with the relish you reserve for
your roast beef sandwiches. I hope that other readers will be more open
minded.
Yours, Johnny Reinhard

************************************** See what's new at http://www.aol.com

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

🔗Paul Poletti <paul@polettipiano.com>

--- In tuning@yahoogroups.com, Afmmjr@... wrote:
>
> Paul: I consider his only valid temperament for 18th century
consideration
> to be that which he himself called "How I tune", which we now call
> KII. KIII did not become public until the 1880's. Not only that, but
> KIII represents him grudgingly backing down on his principles (read
> the letter to Forkel). Sort of undermines his whole shtick...
>
> Johnny: Kirnberger had a very interesting coterie of friends. It
seems
> reasonable to assume that once Forkel knew, others would know.

I agree completely that it is not outside the realm of possibility,
though it is very strange that with all the press Kirnberger got,
there is no mention in later texts of the "new and improved" version
of the controversial Kirnberger tuning, which resolves the problem of
the two ugly fifths in important natural keys. Not one single mention
to my knowledge.

>Really, I donât even think tuning is a category that
> deserves appendages like inferior and superior, let alone
incompetent. Lou
> Harrison has proved that Kirnberger II is not incompetent.

You mix apples and oranges, Johnny. I think KII is an incompetent
solution to the musical and practical challenges of the last quarter
of the 18th century, for the music of Mozart, Beethoven, Haydn, CPE
Bach, etc. Lou Harrison may well like the structure of KII, but once
again, the likes/dislikes of a 20th c composer and/or musician have
nothing to do with the 18th c. It's likes/dislikes of 18th century
composers/musicians I'm interested in.

> If you want to
> suggest that K stole his ideas, please, proceed. Otherwise, you
are blowing hot
> air.

I don't know of any direct quote mentioning this factoid (schisma=1/12
P comma), but both Neidhardt and Werckmeister had calculated the
values for ET, though W failed to provide them, blaming an
uncooperative engraver who balked at the precision required for ET,
which is believable when you consider than Neidhardt chose NOT to
offer a diagram but rather values on a scale of 1000 to 2000 and to
the second decimal place no less. I would be extremely surprised if
both men did not know what the schisma was and how big it was. In
their presentation of their temperaments, both did conflate the two
different commas in their numerical analysis of the tempering of
thirds, simply adding up the total of P comma tempering over four
adjacent fifths and pronouncing a cumulative tempering of one complete
P comma as resulting in a "pure" third. The minute aberration created
by so doing certainly has no relevance in a realistic musical setting,
and a major third over-tempered by 1/12 P comma sounds every bit the
pure third. I suspect they this did this merely to keep from getting
bogged down in the schisma and confusing the reader. Ultimately one is
only interested in relative third quality, not absolute, and since
none of the temperaments they presented contained any chain of 4
adjacent fifths with a total tempering of 12/12ths, playing fast and
loose with the commas makes no real difference in that respect. I know
that in my acoustics classes, I have the kids do it right, summing up
the total tempering over four fifths and then subtracting it from 11
instead of 12 to get the actual tempering. But it always confuses
them, I have to explain it two or three times, and I know some of them
never really get it - exactly the sort of confusion W & N were wishing
to avoid, I would guess.

All that aside, both W & N were pretty sophisticated mathematically,
both worked out their temperaments in decimal values for string
lengths rather than just doing fractions, and Neidhardt even
demonstrated the minute difference between a geometrical (log) and
arithmetical (linear) division of the comma. I find it hard to believe
that this level of sophistication would allow someone to be unaware
that there is a difference between correcting for a pure third and
correcting to close the circle, and that doing the former leaves over
the last bit of doing the latter equally. But since this factoid it is
of little or no practical value, there was no need to mention it. No
practical value in tuning because simply establishing one correct
fifth when there are still 11 more to do in exactly the same fashion
is of little assistance. As far as calculating string lengths goes, it
is much better to simply work it out, proceeding by successive
division of the comma into halves, the halves into quarters, and
finally the quarters into thirds, using the standard ancient geometric
calculation techniques to derive the geometric means, or simply taking
the arithmetic mean, at every step.

The critical question in the whole case is, how did Kirnberger know
that the number he came up with was indeed exceedingly close to a
1/12th P comma fifth? Did he work it out mathematically, or did he
just do the division of his ratio derived from 3:2 to the 7th power
times 5:4 plus the octave corrections, and then go look up that value
in Neidhardt's table of ET values from the 1724 publication?

>
> Johnny: Me, with my inferior ear, has often tuned keyboards
âmelodically.â

I'll believe that when I hear it. If you can accurately tune an entire
temperament MELODICALLY, than you are an absolute marvel, a dramatic
exception to the rest of humanity, and should get yourself directly to
the nearest acoustician to have your super-human capability documented
before age sets in and your ability begins to erode, because in order
to set tempeaments with any reasonable degree of accuracy, you are
consistently functioning well below what is considered to be the JND.
And standard JND test are of course just that, the smallest difference
between two adjacent tones. I don't know if there are any JND test
involving larger intervals, but I'd bet the value gets worse.

BTW, after tuning these keyboards melodically, did you verify the
tuning in any reliable way? Comparison to a set of electronically
generated tones? Check it with a strobe? Beat rates? Or did you just
assume that because it felt right, it was right?

> Read an interesting discussion by Jorgensen (Tuning) and a Handel
tuning
> that was performed by ear.

I've read Jorgenson, and I don't believe his melodic tuning hypothesis
has any substance whatsoever. Neither does anybody who is steeped in
the original literature, which talks about the beating an interval
makes when you are tuning it when it is tempered and how it gets
faster or slower depending upon the distance from purity of the interval.

I know that Jorgenson is very popular in the states, mostly among the
modern piano tech crowd, and much less so among Early Music types. The
general consensus here in Europe is that his work is seriously flawed
due to a number of misinterpretations and failure to examine many
important sources. This is not only my opinion, but that of the
majority of temperament scholars enthusiasts (including Mark Lindley,
who has said as much on the harpsichord list), who DO have a wider
knowledge of the sources in a wide variety of languages.

> People will tune
> differently.

And with different degrees of accuracy.

>
> Paul: Did Kirnberger ever SAY he
> uses this method to set ET on an instrument entirely by ear?
> Johnny: Kirnberger did not tune ET because he didnât like it.

Ah, so now you are saying he DIDN'T tune ET. Again, try to stay on
point, Johnny. We were discussing whether or not the man was CAPABLE,
not what he liked to do. I don't like ET either, but I can tune it,
and without a monchord, using a variety of historical and
self-invented means, none of which require any knowledge and/or
technology foreign to the 18th century.

> Do you need
> me to post his vehement statements against ET?

Nope, I'm well aware of his and others.

> Paul: Uh, there you go again, making a mountain out of molehill. If
I had
> some wackadoodle theory about brain surgery, some brain surgeon could
> well call me incompetent, which I am as a brain surgeon, and I
> wouldn't see it as an attack on my intellect. Only on my knowledge of
> brain surgery.
> Johnny: Guess this is you being clear that my ideas are wackadoodle.

Uh, why do you think this was directed at you when we were talking
about Kirnberger? It's Kirnberger who had wacadoodle ideas about how
to make a temperament which was well-suited for the music of the late
18th century. I will grant that one possibility is that Kirnberger did
NOT intend his temperament to have wide application to the music being
written by others at the time, that he only wanted to invent a new
complete integrated system of temperament AND composition, and that
only HIS music works well in his temperament. That is quite credible.
If so, then it argues against applying it generally to other composers
music, unless they are post-K and their music can be demonstrated to
exhibit whatever principles of composition K was expounding in his thesis.

>
> Paul: There is no such thing as "Church pitch". Churches used
different
> pitches at different times.
>
> Johnny: Wow. You are the first person I know of that has refuted
the use
> of the term chor (church) pitch for organ tuning a whole step
higher than that
> of a kammer (chamber) pitch. Iâm going to have redo the book to
use Paul
> Poletti terms.

These are not "Poletti" terms, it is just correct translation of the
German words. Chor="choir", not "church". As far as I know, no
original source uses any term which can be translated as "church
pitch". Grant, Choir Pitch, which was sometimes higher and sometimes
lower by a whole tone than Chamber Pitch, is usually that pitch which
was used in the church, but most people I know who choose to translate
it all (Haynes chooses not to) do so correctly. I have seen
exceptions, but I see no need for *intentionally* confusing the issue.
Look at all the grief that the mistranslation of "schweben" has
caused, creating an entire fantasy world of "historical" equal-beating
(literally) temperaments. It's this misinterpretation which underlies
Jorgenson's most ridiculous claim, that 1/4 S comma meantone was
impossible before modern times because everybody thought that equal
beating meant equally-tempered, and so they tempered all their fifths
to beat the same.

> Iâm not sure which Krebs you mean (did you ever give a first
name?).
> There are several Krebsâs.

Johann Ludwig Krebs, 1713-1780

> As Leopold Mozart mentions Werckmeisterâs tuning
> in his violin book, it was certainly still around, and I maintain,
used past
> the Baroque period.

Leopold does not mention Werckmeister's "tuning" (by this I assume you
mean his temperament, the one we call WIII). He only refers to
Werckmeister in a list of masters who have written extensively on the
topic of temperament which the reader should consult to gain further
understanding:

"Now that the divided keys on the organ are abolished, if one tuned
everything in pure fifths, then with the progression of remaining
notes, there would arise an intolerable dissonance. They must
therefore be tempered. That is, one has to take something from the one
consonance and add something to the other. They must be so distributed
and the notes so balanced with each other that they are all tolerable
to the ear. And this is called Tempered Tuning. It would be too wide a
subject to cite here all the mathematical researches of many learned
men. Read Sauver, Bümler, Henfling, Werckmeister, and Neidhardt."
(footnote 2 to I-III-25, trans. Knocker)

Need I point out the last fellow on the list? But in any event, this
in no way proves that WIII was in use, neither when Leopold was
writing or ever.

> While you doubt at every turn, it is you who may have
> actually been sold a bill of goods.

I'm trying not to suffer that fate, which is exactly why I'm examining
your statements with a fine-toothed comb.

;-)

> By 1681 Wâs tuning (WIII) was a new line in the sand. All 24
keys could be
> used. The last 3 chapters of Musical Temperament (1691) tell of
the hell he
> received from a critic of his, whose identity is anonymous. It was
about a
> war between Werckmeisterâs III and his IV (Trost irregular
temperament).
> Fanciful ideas of alternatives to this fact of W IIIâs
singularity leave no
> known alternatives.

Ah, beg pardon, but Werckmeister V is quite similar to WIII, except
the good thirds are not as good. But you can play in all 24 keys no prob.

> Neidhardt is too late for the formative JS Bach, and
> the mature JS Bach.

...and the mature JS Bach what? Certainly one cannot argue that
Neidhardt's publications are "too late" chronologically for the mature
JS Bach, so there must some thought that completes the sentence in a
hypothetically credible though possibly debatable manner. Suggestions:
The mature JS Bah was a conservative fuddy-duddy so set in his ways
that he never grew beyond what he had learned in his youth. Or: The
mature JS Bach was a reclusive who never read any contemporary
literature on music or organ construction. Or: The mature JS Bach,
living behind the Iron Curtain as he did in East Germany, could not
get ahold of anything published in the West.

> Research, visiting the different cities, studying
> everything pertinent, while experimenting publicly in NYC through
concerts, will
> yield a product.

Undoubtedly, but what sort of product?

> You may want to rip into it with the relish you reserve for
> your roast beef sandwiches.

Roast lamb. Don't care for roast beef, and it wasn't a sandwich, but
rather a full hot meal with potatos and gravy.

> I hope that other readers will be more open
> minded.

I will be plenty open minded, I just hope your final arguments are
better thought out and better documented than what you have so far
presented here. Believe me, if you can make a convincing argument for
JSB liking/using WIII, I'll be the first to recommend your work as
offering a credible resolution to one of the most intriguing questions
regarding historical temperament practice. Ditto Beethoven and
Kirnberger. So far, however, it has been little more than wishful
thinking, vague insinuations, questionable conclusions, and subjective
judgments by you and other moderns.

In any event, do let me know when your book is available.

Ciao,

🔗Afmmjr@aol.com

Paul: I agree completely that it is not outside the realm of possibility,
though it is very strange that with all the press Kirnberger got,
there is no mention in later texts of the "new and improved" version
of the controversial Kirnberger tuning, which resolves the problem of
the two ugly fifths in important natural keys. Not one single mention
to my knowledge.

Johnny: Kirnberger died in 1683. People were not so generous to advertise
a dead man’s tuning innovation. Forkel likely spread the word to the extended
Bach circle.

Johnny: In the sense that Kirnberger insisted as early as KI to have at
least one just 386 cent major third in the key of C, and throughout two later
innovations, he was innovative. Notice, I trace the derivation of KIII from
KII, in which case KII is less incompetent and more pioneering. On this list we
support pioneers.
Paul: I don't know of any direct quote mentioning this factoid (schisma=1/12
P comma), but both Neidhardt and Werckmeister had calculated the
values for ET, though W failed to provide them, blaming an
uncooperative engraver who balked at the precision required for ET,
which is believable when you consider than Neidhardt chose NOT to
offer a diagram but rather values on a scale of 1000 to 2000 and to
the second decimal place no less.
Johnny: Kirnberger has nothing to prove regarding his intelligence, math or
otherwise. He is not a tuner in the sense you wished him to be. I don’t
know who he might represent to you, but his numbers do add up. The apochryphal
story reported by Mark Lindley, if true, is sarcastic at best. Reading
Werckmeister is an intensely personal experience. He’s outright paranoid at
times. Werckmeister is always looking over his soldier for the next lobbed
criticism. Throughout W’s entire life he preferred that his diatonic major thirds
were more just than others. The very tuning of WIII was his solution to
present day tuning. W. definitely wanted to influence everyone to 24 major and
minor keys, unequal. Any fair reading of W’s work indicates that WIII is
the basis of Musicalische Temperatur. WIV is the old Trost tuning with its 2
dogs. WV is more ET than WIII, but not, but with a 690 and a 708 fifths. I
think the variegation was important to W as it was to those that favored WIV
(Trost, Thayssner). There isn’t even a half page about WV in Musicalische
Temperatur. And VI, I haven’t ever heard of a piece of music in WVI.
Paul: The critical question in the whole case is, how did Kirnberger know
that the number he came up with was indeed exceedingly close to a
1/12th P comma fifth? Did he work it out mathematically, or did he
just do the division of his ratio derived from 3:2 to the 7th power
times 5:4 plus the octave corrections, and then go look up that value
in Neidhardt's table of ET values from the 1724 publication?

Johnny: See, that’s not critical to me. I couldn’t care less. However, I
see you want to know if Kirnberger was a thief, partly due to his mental
retardation. ;)

>
> Johnny: Me, with my inferior ear, has often tuned keyboards
â€œmelodically.â€

Paul: I'll believe that when I hear it. If you can accurately tune an entire
temperament MELODICALLY, than you are an absolute marvel, blah, blah, blah,

Johnny: My life has been an open book. You pretty much come off like I
should submit to experimentation in a lab, maybe with needles and electric
shocks! Please, no rendition. Instead, please attempt to recognize I am coming
from a different direction than you are. It may well be our hemispheres are
reversed, for example. Also, do you sing? Play a flexible pitched instrument
for early music performance? For now we should at least take each other as
it is said, without sarcasm if at possible (and less it is really obvious).
> Read an interesting discussion by Jorgensen (Tuning) and a Handel
tuning
> that was performed by ear.

Paul: I've read Jorgenson, and I don't believe his melodic tuning hypothesis
has any substance whatsoever. Neither does anybody who is steeped in
the original literature, which talks about the beating an interval
makes when you are tuning it when it is tempered and how it gets
faster or slower depending upon the distance from purity of the interval.

Johnny: Call me a liar. I don’t tune by counting beats, and it is the
basis of the AFMM.
Paul: I know that Jorgenson is very popular in the states, mostly among the
modern piano tech crowd, and much less so among Early Music types. The
general consensus here in Europe is that his work is seriously flawed
due to a number of misinterpretations and failure to examine many
important sources. This is not only my opinion, but that of the
majority of temperament scholars enthusiasts (including Mark Lindley,
who has said as much on the harpsichord list), who DO have a wider
knowledge of the sources in a wide variety of languages.
Johnny: I agree about Jorgenson, about several different matters. But I am
talking to you now, not Jorgenson. It is Mark Lindley who gave me my
translation of W’s Musical Temperament (Hehr). I helped him with his PhD at Columbia
University by informing the professors on the committee as to his
importance, often translating his ideas for them. I am most supportive of him and his
eminent career. But he is not the final answer to any question that I am
raising.

> Johnny: Kirnberger did not tune ET because he didnâ€™t like it.

Ah, so now you are saying he DIDN'T tune ET. Again, try to stay on
point, Johnny. We were discussing whether or not the man was CAPABLE,
not what he liked to do.
Johnny: Yes, his calculations and commentary demonstrate he was capable of
tuning ET. No, didn’t do it often because he didn’t like it. But what does
this matter? Stay on point? Why does Kirnberger have to prove to you,
hundreds of years later, that he had mojo to a keyboard builder and tuner?
History owes him so very much. And you started off by slandering him, now just
trying to land as gently as possible. Paul, you have no business being part of
the destruction of a personage such as Kirnberger. Go after someone meaner,
like Rameau (the skinflint).
Paul: I don't like ET either, but I can tune it,
and without a monchord, using a variety of historical and
self-invented means, none of which require any knowledge and/or
technology foreign to the 18th century.

Johnny: I shall accept you at your word. …

Paul: Uh, why do you think this was directed at you when we were talking
about Kirnberger? It's Kirnberger who had wacadoodle ideas about how
to make a temperament which was well-suited for the music of the late
18th century. I will grant that one possibility is that Kirnberger did
NOT intend his temperament to have wide application to the music being
written by others at the time, that he only wanted to invent a new
complete integrated system of temperament AND composition, and that
only HIS music works well in his temperament. That is quite credible.
If so, then it argues against applying it generally to other composers
music, unless they are post-K and their music can be demonstrated to
exhibit whatever principles of composition K was expounding in his thesis.

Johnny: Yes, it is quite credible, and about KII. KIII has wider
applications because it was around long enough for other people to write in it. D’oh,
when KII was published there were no other composers composing in it. So,
only K composed in KII, unless someone was influenced by the book. KIII,
another story.

Paul: These are not "Poletti" terms, it is just correct translation of the
German words. Chor="choir", not "church". As far as I know, no
original source uses any term which can be translated as "church
pitch". Grant, Choir Pitch, which was sometimes higher and sometimes
lower by a whole tone than Chamber Pitch, is usually that pitch which
was used in the church, but most people I know who choose to translate
it all (Haynes chooses not to) do so correctly. I have seen
exceptions, but I see no need for *intentionally* confusing the issue.
Johnny: I am not inclined to confuse issues. I have seen cornettone to
mean a minor third above chamber pitch, with Chor pitch in between. Yes,
confusing. I’d like to use a proper English translation for every German word.
Maybe Church organ pitch?

Paul: Leopold does not mention Werckmeister's "tuning" (by this I assume you
mean his temperament, the one we call WIII). He only refers to
Werckmeister in a list of masters who have written extensively on the
topic of temperament which the reader should consult to gain further
understanding:

Need I point out the last fellow on the list? But in any event, this
in no way proves that WIII was in use, neither when Leopold was
writing or ever.

Paul: Perspective. Leopold did not write in alphabetical order.
Werckmeister’s book on tuning is really about WIII. There would be no point to even
mention Werckmeister by name if he didn’t have a circular well temperament
ideal for Mozart’s music.
According to Michiyo Uchida, compiled in Notes by Edward Seckerson (Hi-Fi
News and Record
Review, august 1984). Mozart specialist Uchida wrote:
“Actually, the tuning we have used is slightly different to the usual. We
experimented with meantone, Werckmeister, and equal temperament, and I chose
Werckmeister for Mozart. (I’ve even played a Brahms concerto with it and
none of the orchestral players noticed!) It is quite playable in most keys,
actually, and when it is distorted, as it must be, it is in the keys that you
need that distortion. It was rather interesting with the B-minor Adagio. The
ending of this is a magical change to B-major, and to have it as different,
as remote as possible from the minor is difficult to manage unless you are in
absolute top form. It’s particularly difficult in the recording studio at
the end of a tiring day. With this tuning it is just that tiny bit easier to
achieve the effect you want.”

> While you doubt at every turn, it is you who may have
> actually been sold a bill of goods.

I'm trying not to suffer that fate, which is exactly why I'm examining
your statements with a fine-toothed comb.

;-)

Johnny: I think it is fine time to reevaluate what has been said. That is
what I have planned. As long as the facts are correct, I should be entitled
to my interpretation.

Paul: Ah, beg pardon, but Werckmeister V is quite similar to WIII, except
the good thirds are not as good. But you can play in all 24 keys no prob.

Johnny: As my book is called Bach’s Tuning, I have trouble assuming WV
because of 2 dogs of 690 and 708 cents. There is also the matter of variegation;
except for the dogs, WIII keeps hold of the perceivable distinctiveness of key
character better. I believe there is evidence that W preferred the variety
of keys. One of W’s sons continued to perform in WIV, an irregular tuning
that was not a well temperament.
> Neidhardt is too late for the formative JS Bach, and
> the mature JS Bach.

Paul:...and the mature JS Bach what? Certainly one cannot argue that
Neidhardt's publications are "too late" chronologically for the mature
JS Bach, so there must some thought that completes the sentence in a
hypothetically credible though possibly debatable manner. Suggestions:
The mature JS Bah was a conservative fuddy-duddy so set in his ways
that he never grew beyond what he had learned in his youth. Or: The
mature JS Bach was a reclusive who never read any contemporary
literature on music or organ construction. Or: The mature JS Bach,
living behind the Iron Curtain as he did in East Germany, could not
get ahold of anything published in the West.

Johnny: I believe JS was imprinted with WIII by 1703 when he adjudicated
and won the Arnstadt position. I think the way cousin Walther treats
Werckmeister in his Lexicon trumps his treatment of Neidhardt, who is included. While
I have no problems whatsoever with Mr. Neidhardt, I point out that he was
not a Thuringian which in relative in that all those in favor of non-ET were
indeed Thuringians, and I would include the Bachs, who were more of a guild, if
not a phenomenon, than just a family of composers.

> I hope that other readers will be more open
> minded.

Paul: I will be plenty open minded, I just hope your final arguments are
better thought out and better documented than what you have so far
presented here….
Johnny: I think I have piqued your interest. The rest is to follow.
There is more, Horatio, than you are yet aware (to paraphrase). Sometimes a
tempered interval is meant as a tempered interval, even an exact one. Ignore
this at your peril. Choos

************************************** See what's new at http://www.aol.com

Re: Re Meantone, was Metastable intervals, re Paul

🔗Afmmjr@aol.com

🔗Paul Poletti <paul@polettipiano.com>

🔗Afmmjr@aol.com

🔗Paul Poletti <paul@polettipiano.com>

🔗Afmmjr@aol.com

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

🔗Paul Poletti <paul@polettipiano.com>

🔗Afmmjr@aol.com

🔗Brad Lehman <bpl@umich.edu>

🔗Afmmjr@aol.com