Yahoo Tuning Groups Ultimate Backup mills-tuning-list Tetrachords

Eratosthenes: The simplest explanation of E's enharmonic and chromatic
tunings is that he took an open string of 120 parts and subtracted
Aristoxenos's parts in order. For the enharmonic, this procedure results
in 120 117 114 90 and 114/90 is 19/15. In the case of the chromatic,
120 114 108 90, and 108/90 is 6/5. His diatonic is the old Sumero-
Babylonian "Pythagorean" tuning.

I think E chose 120 for the open string because astronomers were
used to using base-60 notation (actually a form of decimally-coded
sexagesimal). Ptolemy's string lengths are all in base-60 as it would
have been the most familiar notation for expressing fractions,
which are cumbersome in the Greek and Roman numeral systems.

As for computing Aristo's parts as 30th roots of 4/3, I agree that
it would be difficult to extract the 5th root of 4/3 and I think
you may have the solution to why no known Greek theorist recorded
A's genera in ET. Still, use might have been made of the Mesolabium,
an instrument invented by Eratosthenes himself (Heath) (or Archimedes,
elsewhere) for just this type of problem. After the 5th roots were
approximated, it would have been relatively easy to compute the square
and cube roots of the segments.

Barbera also thinks that Aristoxenos meant 4/3 for the tetrachord
span and listed the roots for Aristo's main genera. However, if one
simply tries to approximate the divisions, it is possible to do so
with just square roots. The enharmonic may be described as two
successive intervals of the square root of 256/243 followed by a ditone
of 81/64 and the Intense chromatic as two square roots of 9/8 and 32/27.

One further thought, I am inclined to agree with you about Ptolemy
having observed 1:2 divisions in practice. His enharmonic still looks
artificial to me, mostly because the enharmonic was supposed to be
extinct in practice by his time. I suspect it was supplied by analogy
with the Soft Chromatic, which greatly resembles Archytas's chromatic,
and the Soft Diatonic and Intense Chromatic, both of which may be
descended from Aristoxenos's Soft Diatonic (100 150 250 cents) versus
22/21 x 12/11 x 7/6 (81 151 267) and 21/20 x 10/9 x 8/7 (85 182 231).
(I believe Winnington-Ingram had similar thoughts, but I don't have
his paper on Aristoxenos and musical intervals at hand). In 500 years
musical taste could have changed sufficiently that the 1:2 pyknon was
to be prefered in the non-diatonic genera (the stereotype soft diatonic
is just on the border between Chromatic and diatonic and sounds chromatic
to me most of the time).

Polychronios: Thanks for the source of the 68-tet division formerly
used in the Greek Orthodox liturgical music. I hadn't heard of
Archbishop Chrysanthos before.

--John

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Ed Foote (A440A@aol.com) wrote:

> Interesting viewpoint concerning the evolution of ET, however,
> after reading the McGeary article, I remain unconvinced of your
> main points and dates. This follows from several reasonings.

I should probably mention that I have no vested interest in promoting
well-tempered systems over ET for music of this period (nor vice
versa), so I'll accept whatever arguments seem most convincing for the
dating of ET's emergence as a universally employed keyboard
temperament. (Since I haven't carried out any independent research in
this area, I'd prefer if someone else took over the discussion.)

> I would agree that the Kirnberger II was not so popular. It would
> seem that Kirnberger's temperament was touted mainly by his
> personal supporters, and their agenda is suspect, (it was not that
> smooth of a temperament). To some extent, I believe McGeary
> has drawn his sights on a "straw man".

Kirnberger II is cruder (in construction and aesthetic effect) than
many other well-tempered schemes, so I would certainly agree that the
surprising degree of support it seemed to have gained in contemporary
treatises is suspect. Your comment about McGeary attacking a straw man
isn't quite fair though: McGeary's article is based on a survey of
contemporary keyboard treatises and related literature; most of these
writers endorse ET as a keyboard standard, but a substantial minority
mention Kirberger II. McGeary thus set himself the task of showing how
this was not such strong evidence in favour of K II's practical
popularity as it might seem -- indeed, far from it. If the treatises
McGeary examined had also included a wealth of material on other
well-tempered systems, then his focussing on K II would have been an
attack on a straw man; but on other well-tempered systems there is a
general silence among McGeary's sources, so I can't see that your
charge stands. You could of course argue that McGeary's sources might
have represented an ET vanguard, and that ordinary musicians (i.e. of
the kind which doesn't compile treatises) remained content with the
resources of their preferred well-tempered systems. This might well be
so, but the trouble is precisely that no written evidence was left by
such musicians, so McGeary has all the advantage here. Or do you have
any hard, historical evidence to counter his claims?

> >From Jorgenson's research, I offer the following quotes from
> Johann Joseph Loehr's book,> Uber die Scheibler'sche Erfindung
> uberhaupt und dessen Pianoforte-und Orgel-Stimmung insbesondere<
>
> "There never was a man capable of tuning by a ear a pianoforte or
> an organ so as not to leave some inequality of temperament, and
> there never will be" and " {Equal Temperament} hitherto has not
> been possible....Before Mr. Scheibler's invention no
> such means existed by which even a tolerable
> equality of temperament could be obtained" [1836]

There is a logical problem here, which would seem to undermine its
usefulness for your argument: namely, we can always choose a margin
of error small enough to wrong-foot the best tuner, and if we set the
margin at 0, then of course no-one can tune flawless ET. But then
no-one can tune flawlessly according to any given well-tempered
system, or any given meantone system, or any tuning system whatsoever.
The argument above would thus appear to be quite empty. What you would
have to find is not evidence to show that ET was not achieved to
perfection, but rather that musicians/tuners were not _aiming_ for the
equality of all keys, but continued, rather, to favour some over
others. In other words, you will need to find contemporary evidence
not for inequality of results, but inequality of _intention_.

> Again, according to Jorgenson, in 1850, ( I think), A.J. Hipkins
> stated that the best tuners at the Broadwood factory "didn't tune l
> anything like equal temperament"

Same again (and I'll assume you aren't using documentation of English
practices to make any point concerning German practices at this time
or earlier). Was it simply that the Broadwood tuners were aiming at,
but not achieving equality, or were they deliberately retaining a
well-tempered scheme (or schemes)? Random inequalities don't count;
the required inequalities would tend towards 1/4-comma meantone at the
"white-keys" side of the "circle of fifths", and towards Pythagorean
at the other end (you would agree with this, I trust as a fair summary
of the general tendencies of well-tempered systems?).

> > ... in Germany during the second half of the 18th is because
> >piano tuning had largely become a task of professionals by the time
> >of theFrench/English changes, whereas many or most players were
> >capable of tuning their own pianos at the time the changes occured
> >in Germany. Am I right?
>
> Yes, but that doesn't mean that they were using ET, which is
> about the most difficult temperament to tune. There were much easier
> ways to tune a non-restrictive temperament.

That wasn't my point. I was saying that I thought this was why we
should have evidence from professionals in the piano industry
concerning the changeover to ET in England and France, whereas our
evidence concerning Germany/Austria is less direct, and gives us only
a vague idea of dates. This _wasn't_ being offered as an argument to
establish the case that the Germans adopted ET earlier -- it was only
an auxiliary point.

> There is no evidence that there was knowledge of test intervals
> in the 1700's. Without the normal tests for interval width, I
> would not be able to tune an acceptable ET today, and I have been
> tuning ET for the last 20 years at recording and broadcast n
> standards.

Doesn't the very name *gleichschwebende* suggest how they approached
the practicalities of tuning ET? I doubt, of course that they would
have achieved anything approaching the "recording and broadcast
standards" of the past twenty years, but as I have said, the random
inequalities resulting from tuning methods designed to achieve
something like ET will not serve your argument; only the intended
inequalities of well-temperament will do this.

> I don't think ET was possible, given the state of
> science in the 1700's. ( Mersenne ratios didn't give much
> information that was of any use to piano tuning, did he?)

You could make much the same argument for meantone temperaments being
impossible in the 15th century, but we have corroborating evidence to
show that they did indeed employ such tunings then; the fact that no
adequate mathematical description of even 1/4-comma meantone appeared
until long after is irrelevant. Likewise, a knowledge of how to
extract the twelfth root of 2 (and the persistence to make the
burdensome calculations) didn't bother lute and viol players during
the following century: they were quite happy to use rational
approximations such as 18/17 for ET semitones, and relied upon natural
margins of error, and a little fudging to make (18/17)^12 arrive at
2/1. Something like ET was thus being practiced centuries earlier than
the time we are talking about (late 18th/early 19th centuries); the
reason why it wasn't adopted for keyboards at this stage was that
unlike lutes and viols, keyboard instruments were capable of tunings
which better suited musical needs: first the meantone systems, and
then the well-tempered. Even the fretted clavichord offered better
tuning possibilities than lutes or viols.

> > Beethoven's well-known pronouncements on key
> > characteristics should not be taken to imply that he preferred
> >some variety of well-temperament, because he is also on record as
> >claiming that he could distinguish between C# major and Db major
> >(!), which, I need hardly say, is not a distinction that can be
> >made in any well-temperament.
>
> Hmmmm. Is this a distinction that can be made in ET? I am not sure
> by what mechanism this last statement is supposed to support either
> direction.

I'm afraid you've missed my point entirely. I was saying that the
pronouncements Beethoven made, well into his career, upon key
characteristics, might seem to indicate his continuing preference for
and use of well-temperament (for as long as he could hear any
difference). My point was that he also claimed to be able to
distinguish between Db and C# (and other such "enharmonic
equivalents") and this renders his other statements on key
characteristics useless as a supposed Beethovenian endorsement of
well-temperament. Of course the distinction can't be made in ET -- it
can't be made on any 12-notes per octave keyboard; this has nothing to
do with my point here. Whatever Beethoven's grounds for discerning key
characteristics might have been, they were more nebulous than the
intervallic patterns produced by well-temperaments (though as I said,
I don't doubt that well-temperament would have played a part in
suggesting such distinctions, alongside such matters as which keys
were most congenial for the various orchestral instruments, etc.).

> >I would accept that Beethoven's notions concerning
> >key characteristics originated in part from the well-temperament
> >that he must have been familiar with in his youth, but if the
> >connection between the two had been of great importance to him, we
> >should expect him to have made statements on the lamentable erosion
> >of key characteristics through the adoption of equal temperament
>
> This seems to beg the question. If, in fact, there was no ET
> actually being tuned, he would have made no statements about the
> erosion.

I'm not begging any questions because, as I thought I had made clear,
I was ASSUMING the consensus position, and contenting myself with
making various auxiliary points; I left it to McGeary to argue for the
main issue of when ET emerged in Germany/Austria. My discussion of
Beethoven was intended to make the point that _whatever was happening
tuning-wise in Germany/Austria_, these statements on key
characteristics will not serve to establish that Beethoven demanded
well-temperament for his pianos. This is a purely negative point: it
says nothing either way about which tuning he actually used.

> These are all academic points, There is very little hope of
> arriving at a definitive answer, However, There is another
> compelling reason to doubt that Mozart and Beethoven composed their
> keyboard work on equally tempered pianos.
> To investigate their music, one should play it in ET and well
> temperament, side by side, and compare. I am presently working
> with several artists that are intimately familiar with this music.
> When I introduced them to Well temperament, the one common response
> I got from them is that the music of the classical Germans makes a
> lot more sense when played in well temperament.

Look, lest we start arguing at cross-purposes, I think I should point
out that there are, as far as I'm concerned, two separate issues here.

1. The first is the factual matter of what temperaments were actually
used at different times and places; insofar as we can arrive at any
conclusions on this, it will be on the basis of documentary sources.
On this issue, I am at present prepared to accept the view that ET had
become entrenched as a keyboard tuning in Germany/Austria by about
1800. If I ever encounter more powerful arguments to the contrary --
again based on documentary evidence -- I shall change my views; but to
date I have not seen any such arguments. This has nothing to do with
any blind pro-ET bigotry; for instance, I do not doubt that, for
example, the "48" were intended for well-temperament and not for equal
temperament; because the evidence (including the title WTK!) clinches
the issue, I can afford to dismiss anyone who thinks otherwise as
simply ignorant of the relevant facts.

2. The second issue is that of our aesthetic preferences _today_ for
one tuning system over another for a given repertory. I am very glad
to hear of your work concerning the promotion of well-tempered systems
among professional pianists, and I wish you every success. However,
the fact that someone today, like yourself, is prepared to state a
preference for well-tempered Mozart or Beethoven neither establishes
any historical case, nor does such a preference require any historical
evidence to justify it. There are all sorts of ways we can devise to
refresh our perceptions of familiar repertoire -- and a change of
tuning is as good as any. Many of the performance practices labelled
as authentic have no historical basis; but this doesn't mean we should
abandon them (they may have succeeded in reawakening us to
over-familiar pieces) -- only that we should abandon the spurious
historical arguments we used to justify these performance practices.

Say we were to discover (not that I expect it) irrefutable evidence
that Beethoven (for as long as his hearing lasted) never encountered a
single well-tempered piano after 1800. What then? Should this stop us
from experimenting with well-temperaments in performances of his
post-1800 sonatas? Not in the least! If pianists need their confidence
bolstered by foundationless arguments, concerning the tunings
Beethoven would have heard, before they are prepared to try out a
well-tempered tuning, then I regret this. The only argument that
should be needed is that Beethoven would certainly have played on
well-tempered instruments at least in his youth, and that they are
certainly not blatantly inappropriate to his sonatas of the 1800s in
the way that, say, pelog and slendro tunings would. The performance of
the sonatas today in well-temperaments ought not to be made
conditional upon any historical proof that they actually _were_ ever
performed in well-temperaments;

In any case, Beethoven would have been used to hearing a variety of
tunings: well-temperament and ET on pianos, also probably 1/6-comma
meantone on organs; various efforts on strings running from
quasi-Pythagorean to 1/6-comma meantone, depending on the preferences
and proficiency of the players ... by the time we come to aging
cimbalom players in taverns, we should be able to accept that
Beethoven could not have fastened his designs upon any single tuning.
What of arrangements, such as that of the op.14/1 piano sonata for
string quartet? Would anyone suggest that Beethoven must have required
the quartet players to switch to well-temperament or ET? In any case,
orchestras are a graveyard for fine tuning sensibilities, and all the
more so in Beethoven's time. The musician in Beethoven's time could be
expected to display a tolerance for a variety of tuning systems,
reproduced with varying degrees of accuracy. Play Beethoven in a
well-temperament if you wish, but more importantly, play him well.

> I prefer the Young or Prinz temperaments for the modern pianos, as
> the greatly increased overtones renders many of the other well temps
> to be harsh when more than three accidentals are involved.

Barbour says that Young's account of his No.1 includes a
miscalculation of Eb, which should form a true 4/3 with G#; do you
incorporate this correction?

> ... presently in the works is
> a CD of historical tunings on the modern concert grand. The record
> company has requested that I not divulge exactly what, yet, but this
> list will be the first place I will notify when given the go-ahead.

Ah, now this is interesting -- historical tunings divorced from
historical instruments. A welcome development, as far as I'm
concerned. I'm looking forward to seeing further details. Good luck
with the project.

--
Jonathan Walker
Queen's University Belfast
mailto:kollos@cavehill.dnet.co.uk
http://www.music.qub.ac.uk/~walker/

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Tetrachords

🔗John Chalmers <non12@...>

🔗Allen <STRANGE@...>

🔗kollos@cavehill.dnet.co.uk (Jonathan Walker)