Re: [tuning] tuning paradigm shift and _sturm und drang_ (was:: orchestral tu...

Monz, could you clarify the evidence that Mozart's extended meantone was
sixth comma and not quarter comma? Thanks,

Johnny

--- In tuning@yahoogroups.com, Afmmjr@a... wrote:
> Monz, could you clarify the evidence that Mozart's extended
meantone was
> sixth comma and not quarter comma? Thanks,
>
> Johnny

johnny, the word "comma" was used by mozart's father and his
contemporaries to mean three different quantities: the pythagorean
comma (23.5 cents), the syntonic comma (21.5 cents), and the
meantone "enharmonic" or diminished second. in 1/4-comma meantone,
the latter is twice the size of the other two, while in 55-equal or
thereabouts, it's right in-between the other two, and all three are
almost identical. so the use of the single "comma" term for all three
suggests something close to 55-equal, such as 1/6-comma (under either
of the first two definitions of comma) meantone. also, mozart's
probable keyboard well-temperament began with six fifths tuned 1/6-
pythagorean comma flat (and closed with six pure fifths); thus the
white notes would have been identical between the keyboard and the
strings had 1/6-pythagorean-comma meantone been used on the latter.

hi Johnny and paul,

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, June 13, 2003 12:38 PM
> Subject: [tuning] Re: tuning paradigm shift and _sturm und drang_ (was::
orchestral tu...
>
>
> --- In tuning@yahoogroups.com, Afmmjr@a... wrote:
> > Monz, could you clarify the evidence that Mozart's extended
> > meantone was sixth comma and not quarter comma? Thanks,
> >
> > Johnny
>
>
> johnny, the word "comma" was used by mozart's father and his
> contemporaries to mean three different quantities: the pythagorean
> comma (23.5 cents), the syntonic comma (21.5 cents), and the
> meantone "enharmonic" or diminished second. in 1/4-comma meantone,
> the latter is twice the size of the other two, while in 55-equal or
> thereabouts, it's right in-between the other two, and all three are
> almost identical. so the use of the single "comma" term for all three
> suggests something close to 55-equal, such as 1/6-comma (under either
> of the first two definitions of comma) meantone.

that's a great explanation, paul. but there's also
an additional one: the business about a whole-tone
equaling 9 commas.

in 55edo, a whole-tone is 9 degrees of 55edo.

31edo, which is the smallest-cardinality EDO which can
represent 1/4-comma meantone well, has a whole-tone
which equals 5 degrees of 31edo.

so 55edo, 1/6-comma meantone, and other meantones
which have a "5th" size of around 698 cents, fit the
descriptions given by Leopold Mozart (W.A.'s father)
and Thomas Atwood (W.A.'s student), of "9 commas per
whole-tone".

> also, mozart's
> probable keyboard well-temperament began with six fifths tuned 1/6-
> pythagorean comma flat (and closed with six pure fifths); thus the
> white notes would have been identical between the keyboard and the
> strings had 1/6-pythagorean-comma meantone been used on the latter.

now *that's* interesting!

what tuning is this? one of Niedhart's, perhaps?
what exactly was the placement of the tempered and
untempered 5ths among the letter-names?

i've figured out the following based on your
description; the chain F-C-G-D-A-E-B has the
1/6-Pythagorean comma tempering and the two
chains B-F#-C#-G# and F-Bb-Eb-Ab are tuned to
3/2 ratios; G# and Ab are thus exactly the same.
1/6-syntonic-comma meantone is given for comparison:

------------ cents ----------------
note gen. keyboard tuning 1/6-c MT difference

G# 8 796.0899983 786.9649541 +9.125044142
C# 7 94.1349974 88.59433486 +5.540662542
F# 6 592.1799965 590.2237156 +1.956280943
B 5 1090.224996 1091.853096 -1.628100657
E 4 392.1799965 393.4824771 -1.302480525
A 3 894.1349974 895.1118578 -0.976860394
D 2 196.0899983 196.7412385 -0.651240263
G 1 698.0449991 698.3706193 -0.325620131
C 0 0 0 0
F -1 501.9550009 501.6293807 0.325620131
Bb -2 1000 1003.258761 -3.258761468
Eb -3 298.0449991 304.8881422 -6.843143068
Ab -4 796.0899983 806.5175229 -10.42752467

only the black keys Ab/G#, Eb, Bb, and C# show an
appreciable difference from the meantone.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> that's a great explanation, paul. but there's also
> an additional one: the business about a whole-tone
> equaling 9 commas.

If c is the comma, and if the mean tone is nine commas, then the
corresponding meantone fifth will be sqrt(2) c^(9/2). If we put the
Pythagorean comma into this formula we get a sharp fifth, so clearly
that isn't the comma in question. If we put 81/80 for c in the above
expression, we get a fifth of 696.778 cents, which is close to 1/4
comma meantone. Are we really sure Mozart was a 1/6-comma, not a
1/4-comma supporter?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> If c is the comma, and if the mean tone is nine commas, then the
> corresponding meantone fifth will be sqrt(2) c^(9/2). If we put the
> Pythagorean comma into this formula we get a sharp fifth, so clearly
> that isn't the comma in question. If we put 81/80 for c in the above
> expression, we get a fifth of 696.778 cents, which is close to 1/4
> comma meantone. Are we really sure Mozart was a 1/6-comma, not a
> 1/4-comma supporter?

It's even closer to 31-equal meantone; in fact, if we look at the
convergents of the continued fraction, we get

1, 1/2, 3/5, 4/7, 7/12, 18/31, 5461/9405, ...

so "nine commas" is a very precise description of 31-equal, which has a
fifth of 696.774, only .004 cents flatter than the 9-comma fifth.
While it's hardly likely Mozart pere or fils would have made such a
computation, it was easily within the scope of the highly sophisicated
mathematics of the time, and someone might have done so.

----- Original Message -----
From: "Gene Ward Smith" <gwsmith@svpal.org>
To: <tuning@yahoogroups.com>
Sent: Friday, June 13, 2003 11:25 PM
Subject: [tuning] Mozart 31-equal??

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> > If c is the comma, and if the mean tone is nine commas, then the
> > corresponding meantone fifth will be sqrt(2) c^(9/2). If we put the
> > Pythagorean comma into this formula we get a sharp fifth, so clearly
> > that isn't the comma in question. If we put 81/80 for c in the above
> > expression, we get a fifth of 696.778 cents, which is close to 1/4
> > comma meantone. Are we really sure Mozart was a 1/6-comma, not a
> > 1/4-comma supporter?
>
> It's even closer to 31-equal meantone; in fact, if we look at the
> convergents of the continued fraction, we get
>
> 1, 1/2, 3/5, 4/7, 7/12, 18/31, 5461/9405, ...
>
> so "nine commas" is a very precise description of 31-equal, which has a
> fifth of 696.774, only .004 cents flatter than the 9-comma fifth.
> While it's hardly likely Mozart pere or fils would have made such a
> computation, it was easily within the scope of the highly sophisicated
> mathematics of the time, and someone might have done so.

hmmm ... this is curious and intriguing, because it's
patently obvious when you look at the whole scale of
31edo that there are 5 equal divisions to each whole-tone.

so if someone was willing to split each of those,
thereby resulting in 62edo, there would be 10 per whole-tone.
but not 9.

in their description of "9 commas per whole-tone",
i don't think either of the Mozarts were interested
_per se_ in any particular comma. they just knew
that the logarithmic relationship between "chromatic
semitone" and "diatonic semitone" was about 4:5,
and that together those 9 parts made a whole-tone.

55edo, 1/6-comma meantone, and other tunings with
a "5th" size in that neighborhood, are the ones
that fit the bill.

Gene (or whomever), i'd appreciate it if what you're
saying could be explained in more detail so that i
can understand it. algebra is nice and elegant,
but i need concrete examples.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> hmmm ... this is curious and intriguing, because it's
> patently obvious when you look at the whole scale of
> 31edo that there are 5 equal divisions to each whole-tone.

Right. So the whole tone is 5/31, and we have

(5/31)/log2(81/80) = 8.99961782

which is remarkable for a mere coincidence, if it is one.

> in their description of "9 commas per whole-tone",
> i don't think either of the Mozarts were interested
> _per se_ in any particular comma.

(1) Do we know that?

(2) Could they have gotten this from someone else, who *was*
interested in a particular comma?

(3) How many particular commas could they possibly have been
interested in? We eliminate the Pythagorean comma, and what's left?

they just knew
> that the logarithmic relationship between "chromatic
> semitone" and "diatonic semitone" was about 4:5,
> and that together those 9 parts made a whole-tone.

Again, we know this how? What do we know about what they knew, and
what someone who might have influenced them knew?

> 55edo, 1/6-comma meantone, and other tunings with
> a "5th" size in that neighborhood, are the ones
> that fit the bill.

> Gene (or whomever), i'd appreciate it if what you're
> saying could be explained in more detail so that i
> can understand it. algebra is nice and elegant,
> but i need concrete examples.

It simply boils down to this--the 31-equal whole tone is
8.9996 Didymus commas, and that's very damned close to 9 Didymus commas.

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> Gene (or whomever), i'd appreciate it if what you're
> saying could be explained in more detail so that i
> can understand it. algebra is nice and elegant,
> but i need concrete examples.

Here's a short table giving some meantone equal temperaments, and how
big the tone is measured in terms of Didymus and Pythagorean commas:

12 9.299605 8.525145
19 8.810152 8.076453
31 8.999618 8.250141
43 9.083335 8.326886
50 8.927621 8.184139
55 9.130521 8.370143

hi Gene,

replies here to both of your posts.

> From: "Gene Ward Smith" <gwsmith@svpal.org>
> To: <tuning@yahoogroups.com>
> Sent: Friday, June 13, 2003 11:58 PM
> Subject: [tuning] Re: Mozart 31-equal??
>
>
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
>
> > hmmm ... this is curious and intriguing, because it's
> > patently obvious when you look at the whole scale of
> > 31edo that there are 5 equal divisions to each whole-tone.
>
> Right. So the whole tone is 5/31, and we have
>
> (5/31)/log2(81/80) = 8.99961782
>
> which is remarkable for a mere coincidence, if it is one.
>
> > in their description of "9 commas per whole-tone",
> > i don't think either of the Mozarts were interested
> > _per se_ in any particular comma.
>
> (1) Do we know that?
>
> (2) Could they have gotten this from someone else, who *was*
> interested in a particular comma?
>
> (3) How many particular commas could they possibly have been
> interested in? We eliminate the Pythagorean comma, and what's left?
>
> they just knew
> > that the logarithmic relationship between "chromatic
> > semitone" and "diatonic semitone" was about 4:5,
> > and that together those 9 parts made a whole-tone.
>
> Again, we know this how? What do we know about what they knew, and
> what someone who might have influenced them knew?

Leopold Mozart wrote violin exercises which distinguish
between "enharmonically-equivalent" sharps and flats,
and he taught the 4+5=9 division.

Chesnut suggests that Mozart Sr. got the idea from Tosi,
because he wrote to Mozart Jr. praising Tosi's division
of the whole-tone into 9 parts.

there was discussion of this a while back too, and
i recall Quantz was also an advocate of this tuning.
there's quite a bit about it in the list archives.

i tried to emphasize that it was my *opinion* that they
weren't really considering particular commas. i can't
back that up. i just have the sense that they worked
within a meantone framework, and knew the 9-part division,
which actually goes way, way back to Pythagorean days,
in which context 53edo would be the appropriate EDO
which gives 9 commas to the whole-tone.

53edo gives the Pythagorean version (in which the
diatonic semitone is 4 parts and the chromatic is 5),
and 55edo gives the meantone version. those are the
only two EDOs which have a step-size which measures
intervals this way.

53edo is actually a curious case, because the chromatic
and diatonic semitones can be other sizes too, depending
on whether 53edo is emulating Pythagorean or 5-limit JI.

for example, the Pythagorean chromatic semitone 2187:2048
(i.e., 3^7, the apotome) is 5 steps of 53edo, and the
Pythagorean diatonic semitone 256:243 (3^-5, the limma)
is 4 steps of 53edo.

but, the 5-limit chromatic semitone 135:128 is 4 steps,
and the 5-limit chromatic semitone 25:24 is 3 steps. of 53edo.
interesting...

> From: "Gene Ward Smith" <gwsmith@svpal.org>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, June 14, 2003 12:08 AM
> Subject: [tuning] Re: Mozart 31-equal??
>
>
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > Gene (or whomever), i'd appreciate it if what you're
> > saying could be explained in more detail so that i
> > can understand it. algebra is nice and elegant,
> > but i need concrete examples.
>
> Here's a short table giving some meantone equal
> temperaments, and how big the tone is measured in
> terms of Didymus and Pythagorean commas:
>
> 12 9.299605 8.525145
> 19 8.810152 8.076453
> 31 8.999618 8.250141
> 43 9.083335 8.326886
> 50 8.927621 8.184139
> 55 9.130521 8.370143

so we see that for basically *all* the meantones,
the whole-tone is approximately 9 Didymus (syntonic)
commas.

but the point is: of the meantones with cardinalities
under 100, only 55edo actually *has* intervals that are
approximately the size of a comma. so that's the only
one where they could actually have a sense of measuring
the diatonic semitone as 4 steps, the chromatic semitone
as 5 steps, and the whole-tone as 9 steps.

-monz

Hi,

I joined this list looking for tips as to how to get people who sing either a cappella or with a few non-tempered instruments to sing more in tune. I have recently read Gerals Eskelin's _Lies My Music Teacher Told Me..._ and have ordered others of his products. But I was hoping here to learn more about what others think, with regards to the interest that prompted me to join the list.

Whereas such comments as

>It's even closer to 31-equal meantone; in fact, if we look at the
>convergents of the continued fraction, we get
>
>1, 1/2, 3/5, 4/7, 7/12, 18/31, 5461/9405, ...
>
>so "nine commas" is a very precise description of 31-equal, which has a
>fifth of 696.774, only .004 cents flatter than the 9-comma fifth.
>While it's hardly likely Mozart pere or fils would have made such a
>computation, it was easily within the scope of the highly sophisicated
>mathematics of the time, and someone might have done so.

(chosen at random from a plethora of other such esoteric mathematically oriented remarks) are of obviously keen interest to those of you who are interested in fine-tuning, say, a keyboard instrument or a digital-music program; they do not speak at all to matters of practical concern to me, such as: What exercises might best be used by small groups of people who seek to lock in harmonics of their music?

I hesitated to "chime in" like that, as I don't want to give the impression that I am trying to suppress any of the dialog that is already flying by so charmingly! PLEASE keep all that going, okay?

But leads to Web pages that address the concerns I have raised here would be greatly appreciated, too.

In passing, I wonder how many choral groups really work effectively with tunings of various kinds. Do any of them change tuning modes from piece to piece? Does anybody physically measure how well they succeed in singing the various tuning targets?

Thanks!

Steve Langford

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

/tuning/topicId_44573.html#44605

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
>
> > hmmm ... this is curious and intriguing, because it's
> > patently obvious when you look at the whole scale of
> > 31edo that there are 5 equal divisions to each whole-tone.
>
> Right. So the whole tone is 5/31, and we have
>
> (5/31)/log2(81/80) = 8.99961782
>
> which is remarkable for a mere coincidence, if it is one.
>
> > in their description of "9 commas per whole-tone",
> > i don't think either of the Mozarts were interested
> > _per se_ in any particular comma.
>
> (1) Do we know that?
>
> (2) Could they have gotten this from someone else, who *was*
> interested in a particular comma?
>
> (3) How many particular commas could they possibly have been
> interested in? We eliminate the Pythagorean comma, and what's left?
>
> they just knew
> > that the logarithmic relationship between "chromatic
> > semitone" and "diatonic semitone" was about 4:5,
> > and that together those 9 parts made a whole-tone.
>
> Again, we know this how? What do we know about what they knew, and
> what someone who might have influenced them knew?
>
> > 55edo, 1/6-comma meantone, and other tunings with
> > a "5th" size in that neighborhood, are the ones
> > that fit the bill.
>
> > Gene (or whomever), i'd appreciate it if what you're
> > saying could be explained in more detail so that i
> > can understand it. algebra is nice and elegant,
> > but i need concrete examples.
>
> It simply boils down to this--the 31-equal whole tone is
> 8.9996 Didymus commas, and that's very damned close to 9 Didymus
commas.

***This is *very* interesting, and it seems that Gene has *deflated*
the whole idea that Mozart was in 1/6th comma meantone or 55-EDO,
yes??

J. Pehrson

--- In tuning@yahoogroups.com, Steve Langford <s@T...> wrote:

/tuning/topicId_44573.html#44611

>
> I hesitated to "chime in" like that, as I don't want to
give the
> impression that I am trying to suppress any of the dialog that is
already
> flying by so charmingly! PLEASE keep all that going, okay?
>
> But leads to Web pages that address the concerns I have
raised
> here would be greatly appreciated, too.
>
> In passing, I wonder how many choral groups really work
> effectively with tunings of various kinds. Do any of them change
tuning
> modes from piece to piece? Does anybody physically measure how
well they
> succeed in singing the various tuning targets?
>

***Hi Steve,

This list waxes and wanes; sometimes it's on one kind of topic,
sometimes on another. Sometimes the posts are esoteric; sometimes
they are very practical. It covers about anything regarding tuning.

I would search for the name "Bob Wendell" among some of the previous
posts, as well as Gerald Eskelin. They both talked a *lot* about
your interest. I would say it would be anywhere from a 1 to 3 years
ago on this list, and all of the posts, to my knowledge, are still
there.

best,

Joe Pehrson

>***Hi Steve,
>
>This list waxes and wanes; sometimes it's on one kind of topic,
>sometimes on another. Sometimes the posts are esoteric; sometimes
>they are very practical. It covers about anything regarding tuning.
>
>I would search for the name "Bob Wendell" among some of the previous
>posts, as well as Gerald Eskelin. They both talked a *lot* about
>your interest. I would say it would be anywhere from a 1 to 3 years
>ago on this list, and all of the posts, to my knowledge, are still
>there.
>
>best,
>
>Joe Pehrson
--
Thanks, Joe! I'll get cookin' at lookin'.

Steve

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

> > It simply boils down to this--the 31-equal whole tone is
> > 8.9996 Didymus commas, and that's very damned close to 9 Didymus
> commas.
>
> ***This is *very* interesting, and it seems that Gene has *deflated*
> the whole idea that Mozart was in 1/6th comma meantone or 55-EDO,
> yes??

That's one for the experts in music history, of which I am not one;
but I think the point at least needs to be considered. One thing I can
say with some assurance is that Mozart sounds *fine* in 1/4 comma
meantone; the temperment suits him.
.
.
.
.
.

/tuning/files/rainbow/k4071.mid

It has been great to hear Gene and Joe Monzo go back and forth on this, which
is what I had hoped. What I got out of it is:

Why would Leopold accept enharmonic identities (even mentioning Werckmeister
in a list of writers for further tuning info in his book on violin playing),
but his son is in extended meantone?

Why would a composer using extended meantone not want to use pure thirds
since there are extensions available for any harmonic need?

Why would Mozart, under the influence of the Johann Christian Bach (the
youngest son of JS Bach) use sixth comma meantone when the England was so notorious
for its use of quarter comma meantone (and JS was so negative about
Silbermann's reputed use of sixth comma)?

best, Johnny Reinhard

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> > also, mozart's
> > probable keyboard well-temperament began with six fifths tuned
1/6-
> > pythagorean comma flat (and closed with six pure fifths); thus the
> > white notes would have been identical between the keyboard and the
> > strings had 1/6-pythagorean-comma meantone been used on the
latter.
>
>
> now *that's* interesting!

that's simply what i remember from the chesnut article. is it correct?

> what tuning is this?

valotti or young, i believe . . .

> what exactly was the placement of the tempered and
> untempered 5ths among the letter-names?
>
>
> i've figured out the following based on your
> description; the chain F-C-G-D-A-E-B has the
> 1/6-Pythagorean comma tempering and the two
> chains B-F#-C#-G# and F-Bb-Eb-Ab are tuned to
> 3/2 ratios; G# and Ab are thus exactly the same.
> 1/6-syntonic-comma meantone is given for comparison:
>
>
> ------------ cents ----------------
> note gen. keyboard tuning 1/6-c MT difference
>
> G# 8 796.0899983 786.9649541 +9.125044142
> C# 7 94.1349974 88.59433486 +5.540662542
> F# 6 592.1799965 590.2237156 +1.956280943
> B 5 1090.224996 1091.853096 -1.628100657
> E 4 392.1799965 393.4824771 -1.302480525
> A 3 894.1349974 895.1118578 -0.976860394
> D 2 196.0899983 196.7412385 -0.651240263
> G 1 698.0449991 698.3706193 -0.325620131
> C 0 0 0 0
> F -1 501.9550009 501.6293807 0.325620131
> Bb -2 1000 1003.258761 -3.258761468
> Eb -3 298.0449991 304.8881422 -6.843143068
> Ab -4 796.0899983 806.5175229 -10.42752467
>
>
> only the black keys Ab/G#, Eb, Bb, and C# show an
> appreciable difference from the meantone.

why would you compare with 1/6-*syntonic* comma meantone?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > that's a great explanation, paul. but there's also
> > an additional one: the business about a whole-tone
> > equaling 9 commas.
>
> If c is the comma, and if the mean tone is nine commas, then the
> corresponding meantone fifth will be sqrt(2) c^(9/2). If we put the
> Pythagorean comma into this formula we get a sharp fifth, so clearly
> that isn't the comma in question.

this already shows that your equation is irrelevant, because the
pythagorean comma and the syntonic comma, being so close in size,
were both simply called "the comma", and there was a third quantity
called this same "comma" as well, as i just explained.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> (3) How many particular commas could they possibly have been
> interested in? We eliminate the Pythagorean comma, and what's left?

you eliminated it on false premises. consider that instead of the
equation you wrote, monz could have said that the diatonic semitone
was considered 5 commas or that the chromatic semitone was considered
4 semitones and you would have come up with a different equation, and
different solutions. clearly you're treating as *sharp* certain
quantities that were only *fuzzy* in the practical-minded but
scientifically primitive 18th century.

> It simply boils down to this--the 31-equal whole tone is
> 8.9996 Didymus commas, and that's very damned close to 9 Didymus
>commas.

the "9 commas per whole tone" idea is far, far older than meantone,
and goes back to pythagorean lore. it resurfaced in late meantone
days, but the expression of the two types of semitones in terms of 4
and 5 commas was, in the 18th century, always reversed relative to
the pythagorean spec.

--- In tuning@yahoogroups.com, Afmmjr@a... wrote:
> It has been great to hear Gene and Joe Monzo go back and forth on
this, which
> is what I had hoped. What I got out of it is:
>
> Why would Leopold accept enharmonic identities (even mentioning
Werckmeister
> in a list of writers for further tuning info in his book on violin
playing),
> but his son is in extended meantone?

where does leopold's teaching or writing show more acceptance of
enharmonic identities than his son? we certainly haven't seen *that*
side of leopold in the documents that have been quoted on this list.

> Why would a composer using extended meantone not want to use pure
thirds
> since there are extensions available for any harmonic need?

pure which thirds? major? minor? how about the purity of the fifths?
the incisiveness of the leading tones? the ability to use a mildly
wolfy fifth in passing?

> Why would Mozart, under the influence of the Johann Christian Bach
(the
> youngest son of JS Bach) use sixth comma meantone when the England
was so notorious
> for its use of quarter comma meantone (and JS was so negative about
> Silbermann's reputed use of sixth comma)?
>
> best, Johnny Reinhard

not sure what "the england" has to do with mozart, and js bach would
have been even more negative about a quarter-comma meantone
instrument had it been tested in the same way.

hi paul and Gene,

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, June 15, 2003 1:26 AM
> Subject: [tuning] Re: Mozart 31-equal??

i was writing a detailed post on this last night,
but decided to hunt down the Chesnut article an
read it again before proceeding. stay tuned,
i'll be posting it soon.

-monz

hi paul,

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, June 15, 2003 12:57 AM
> Subject: [tuning] Re: tuning paradigm shift and
> _sturm und drang_ (was:: orchestral tu...
>
>
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > > [paul erlich:]
> > > also, mozart's probable keyboard well-temperament
> > > began with six fifths tuned 1/6-pythagorean comma
> > > flat (and closed with six pure fifths); thus the
> > > white notes would have been identical between the
> > > keyboard and the strings had 1/6-pythagorean-comma
> > > meantone been used on the latter.
> >
> >
> > now *that's* interesting!
>
> that's simply what i remember from the chesnut article.
> is it correct?

i've just re-read the Chesnut article. he based his
study on two primary sources: Leopold Mozart's violin
method (written the year his son was born), and
the manuscript of Thomas Attwood's study-notes from
his lessons with W. A. Mozart, with some information
in Mozart's own handwriting.

it is very clear from Leopold Mozart's violin
method that keyboards had a different tuning
from other instruments.

it is also clear from both Mozart's and Attwood's
notations in the study-notes exactly which notes
were the approximately the same for all instruments
and which ones non-keyboard instruments could play
but keyboards could not.

Chesnut remarks that there were many circulating
(i.e., "well") temperaments current in Mozart's
time, and that there is no way to know, from the
materials Chesnut had available, exactly which tuning
would have been preferred by Mozart for keyboards.

Chesnut's solution, for purposes of comparing
keyboard tuning with the extended meantone obviously
meant for all other instruments, was to average
the cents-values for a selection of well-temperaments
tabulated in Barbour's book, to represent a Mozartian
keyboard tuning.

the meantone chain extends 21 "5ths" from Ebb to A#,
but as Chesnut remarks, Mozart left Cb out of all
of his tables of pitches.

Chesnut plots the cents values of this "composite"
well-temperament on a graph, but otherwise
doesn't specify those values more accurately.

anyway, it's clear from his graph that C, G, D,
A, and E are the same in both keyboard and
non-keyboard tunings.

B, F#, and C# in the keyboard tuning are closer
to the sharp notes of the extended meantone.

the keyboard G#/Ab is essentially right between
the meantone version of both notes.

the keyboard Eb, Bb, and F are closer to the flats
of the extended meantone.

> > what tuning is this?
>
> valotti or young, i believe . . .

i'd really have to dig back into Barbour's book to
research this, and i don't have time now.

> > what exactly was the placement of the tempered and
> > untempered 5ths among the letter-names?
> >
> >
> > i've figured out the following based on your
> > description; the chain F-C-G-D-A-E-B has the
> > 1/6-Pythagorean comma tempering and the two
> > chains B-F#-C#-G# and F-Bb-Eb-Ab are tuned to
> > 3/2 ratios; G# and Ab are thus exactly the same.
> > 1/6-syntonic-comma meantone is given for comparison:
> >
> >
> > ------------ cents ----------------
> > note gen. keyboard tuning 1/6-c MT difference
> >
> > G# 8 796.0899983 786.9649541 +9.125044142
> > C# 7 94.1349974 88.59433486 +5.540662542
> > F# 6 592.1799965 590.2237156 +1.956280943
> > B 5 1090.224996 1091.853096 -1.628100657
> > E 4 392.1799965 393.4824771 -1.302480525
> > A 3 894.1349974 895.1118578 -0.976860394
> > D 2 196.0899983 196.7412385 -0.651240263
> > G 1 698.0449991 698.3706193 -0.325620131
> > C 0 0 0 0
> > F -1 501.9550009 501.6293807 0.325620131
> > Bb -2 1000 1003.258761 -3.258761468
> > Eb -3 298.0449991 304.8881422 -6.843143068
> > Ab -4 796.0899983 806.5175229 -10.42752467
> >
> >
> > only the black keys Ab/G#, Eb, Bb, and C# show an
> > appreciable difference from the meantone.
>
> why would you compare with 1/6-*syntonic* comma meantone?

since the Pythagorean and syntonic commas are so similar
in size, i knew there wouldn't be much difference between
the 1/6-syntonic-comma meantone and this well-temperament
for the "white key" notes. taking human tuning error
into consideration (generally considered to be +/- ~5 cents),
the only notes which wouldn't be a decent match are the
two sharps and two flats at either end, which are only
3 different notes (Eb, C#, Ab/G#).

i'm not aware of any "meantone" in which the Pythagorean
comma is divided, rather than the syntonic comma. but
anyway, my point was to show that if 1/6-syntonic-comma
meantone was used on non-keyboard instruments, and
keyboards were tuned as i calculated above, both types
of instruments could play together with the "white-key"
notes and the listener wouldn't hear anything too offensive.

-monz

hi Johnny and Gene,

> From: <Afmmjr@aol.com>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, June 14, 2003 11:05 AM
> Subject: Re: [tuning] Re: Mozart 31-equal??
>
>
> It has been great to hear Gene and Joe Monzo
> go back and forth on this, which is what I had
> hoped. What I got out of it is:
>
> Why would Leopold accept enharmonic identities
> (even mentioning Werckmeister in a list of writers
> for further tuning info in his book on violin playing),
> but his son is in extended meantone?

i haven't read Leopold Mozart's book, but from your
description and from the info given in the Chesnut
article, it sounds like Leopold may have mentioned
Werckmeister's book merely for the reader with a
further interest in tuning for keyboards. Leopold
gives a list of several well-known writers on
well-temperament, including Sauveur and Neidhardt
as well as Werckmeister.

i *do* know that Leopold's book contains exercises
in which the violinist is specifically instructed
to play the sharps lower than the flats, and the
exercises are specifically to develop facility in
playing all scales correctly in a 19-note extended
meantone, which is what he recommended for the violin.
the exercises quoted in the Chesnut article use a
meantone chain of 5ths from Gb to B#.

in W.A. Mozart's handwritten notes preserved by his
student Thomas Attwood, Mozart gives a table of
chromatic pitches, always extending to at least
17 different notes but sometimes as many as 21,
and he labels the diatonic and chromatic semitones
accordingly, always indicating that the flats should
be higher than their "enharmonically-equivalent" sharps.

so most likely both Leopold and Wolfgang would have
recommended a well-temperament such as Werckmeister III
for keyboards. but it is clear, from descriptions in
both Leopold's violin method and in the Attwood notes,
that for non-keyboard instruments a 17 to 21-note
chain of extended meantone was assumed as "correct
intonation", or, as Leopold described it, "right ratio
tuning".

> Why would a composer using extended meantone not want
> to use pure thirds since there are extensions available
> for any harmonic need?
>
> Why would Mozart, under the influence of Johann
> Christian Bach (the youngest son of JS Bach) use
> sixth comma meantone when the England was so notorious
> for its use of quarter comma meantone (and JS was so
> negative about Silbermann's reputed use of sixth comma)?

well, Gene has been making the argument for 31edo
or 1/4-comma meantone as a good Mozart tuning. there
is no basis for asserting that 55edo or 1/6-comma was
"the" Mozart tuning, and in fact any variety of meantone
would work.

the crucial point is that Leopold describes flats as
being "a comma higher" than their enharmonically-equivalent
sharps.

(see L. Mozart 1951: ch. 1, sec. 3, par. 15;
ch. 1, sec. 3, par. 25 and its footnote; and
ch. 3, par. 6 footnote).

for purposes of comparision, let's examine three very
common and popular meantones: 2/7-, 1/4-, and 1/6-comma.

we see that the difference between, say, G# and Ab,
is about 50.27583966, 41.05885841, and 19.55256881 cents,
respectively, for these three meantones.

the last one is the only one that looks like a "comma"
to me.

the difference between G# and Ab in 1/4-comma meantone
is in fact nearly double the size of any of the typical
commas, and is actually a form of diesis.

as i've said before, only 55edo, 1/6-comma meantone,
or any of the other meantones with a "5th"-size of
approximately 698 cents, fits Leopold Mozart's description.

in answer to Johnny's question about "pure thirds":

he is of course referring to the fact 1/4-comma meantone
is the only member of the meantone family which gives
absolutely "pure" 5/4 ratios (~386.3137139 cents)
for the "major-3rds".

however, the "major-3rds" of all meantones are fairly
close to this value, that being the whole point of
using meantone. the "major-3rd" of 2/7-comma meantone
is a bit narrower, ~383.2413868 cents, while that
of 1/6-comma meantone is a bit wider, ~393.4824771 cents.
the differences are -3.072327085 and +7.168763199 cents,
respectively.

those differences are nowhere near as divergent as
those i cited above for the enharmonic "comma".

and even if W. A. Mozart lived in London and was
influenced by J. C. Bach, i find it likely that he'd
give much more weight to the opinions of his own
father than to those of the younger Bach's father.

so now, can we finally agree that 55edo and its
relatives were the tunings intended by Mozart for
his orchestral and chamber works?

REFERENCES
----------

Mozart, Leopold.
_A Treatise on the Fundamental Principles of Violin Playing_
translated by Editha Knocker, 2nd edition, London, 1951.

Heartz, Daniel; Mann, Alfred,; Oldman, Cecil B.; Hertzmann, Erich. 1965.
_Thomas Attwoods Theorie- und Kompositionsstudien bei Mozart_
Wolfgang Amadeus Mozart: Neue Ausgabe s�mtlicher Werke,
Ser. X, Werkgruppe 30, Bd. 1. Kassel.

PS to Manuel: neither of these books are listed in
the "Tuning and Temperament Bibliography".

-monz

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> > (3) How many particular commas could they possibly have been
> > interested in? We eliminate the Pythagorean comma, and what's left?
>
> you eliminated it on false premises. consider that instead of the
> equation you wrote, monz could have said that the diatonic semitone
> was considered 5 commas or that the chromatic semitone was considered
> 4 semitones and you would have come up with a different equation, and
> different solutions.

(1) Did someone specifically say that a tone was exactly nine commas?

(2) Did they also specifically say the diatonic semitone was five commas?

(3) Do we know, if they said this, where they took the idea from?

clearly you're treating as *sharp* certain
> quantities that were only *fuzzy* in the practical-minded but
> scientifically primitive 18th century.

Eh? You must be talking about a different 18th century. It was not
fuzzy-minded, and it was far from primitive. As I remarked, the
computations involved would have been trivial for a mathematician of
the time.

> > It simply boils down to this--the 31-equal whole tone is
> > 8.9996 Didymus commas, and that's very damned close to 9 Didymus
> >commas.
>
> the "9 commas per whole tone" idea is far, far older than meantone,
> and goes back to pythagorean lore.

In that case, which whole tone are we talking about? This doesn't make
much sense.

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> well, Gene has been making the argument for 31edo
> or 1/4-comma meantone as a good Mozart tuning.

The argument for that can be made by listening; I've been pointing out
that if you give a very specific meaning to "nine commas", you end up
with 31-et.

> the crucial point is that Leopold describes flats as
> being "a comma higher" than their enharmonically-equivalent
> sharps.

If we take a Didymus comma, that means a fifth f satisfying

128/f^12 = 81/80,

which fives an f of 698.208 cents, or 4/23 comma meantone. If it is a
comma of Pythagoras instead, we get 698.045 cents, or 2/11 comma
meantone almost precisely.

It's interesting to see how meantones can be classified by what they
do with the Pythagorean comma, by the way:

7: -1
12: 0
19: -1
26: -2
31: -1
33: -3
43: -1
50: -2
55: -1
74: -2
81: -3

> so now, can we finally agree that 55edo and its
> relatives were the tunings intended by Mozart for
> his orchestral and chamber works?

Still speculative. It would help to find some passages where a 698
cent fifth works better than one a cent flatter.

In a message dated 6/15/03 3:26:57 PM Eastern Daylight Time,
monz@attglobal.net writes:

> in answer to Johnny's question about "pure thirds":
>
> he is of course referring to the fact 1/4-comma meantone
> is the only member of the meantone family which gives
> absolutely "pure" 5/4 ratios (~386.3137139 cents)
> for the "major-3rds".
>
> however, the "major-3rds" of all meantones are fairly
> close to this value, that being the whole point of
> using meantone. the "major-3rd" of 2/7-comma meantone
> is a bit narrower, ~383.2413868 cents, while that
> of 1/6-comma meantone is a bit wider, ~393.4824771 cents.
> the differences are -3.072327085 and +7.168763199 cents,
> respectively.
>

As 2/7ths commas is pure theory and not applicable to the discussion, lets
move on to +7 cents of 1/6-comma. I had mentioned Werckmeister being mentioned
by Leopold Mozart was significant, not for his tuning systems, but because he
is the beginning of enhramonic identities. It seems surprising that someone
encouraging a distinction between diatonic and chromatic semitones would
mention the originator of enharmonic identities.

> those differences are nowhere near as divergent as
> those i cited above for the enharmonic "comma".
>
> and even if W. A. Mozart lived in London and was
> influenced by J. C. Bach, i find it likely that he'd
> give much more weight to the opinions of his own
> father than to those of the younger Bach's father.
>
>

Monz, check out Heinz Gartner's book "John Christian Bach: Mozart's Friend
and Mentor" translated by Reinhard G. Pauly, Amadeus Press, Portland, Oregon.
It tells you more than that Mozart lived in London.

Also, J. Christian Bach was the most famous musician of his time, AND
considered his father the "Old Wig," and that he recieved no direct music lessons
from his father as he was too young when JS died in 1750. It was CPE Bach that
was the big influence.

>
> so now, can we finally agree that 55edo and its
> relatives were the tunings intended by Mozart for
> his orchestral and chamber works?
>
>

Why rush too judgment? There are still many things to work out first.

>
>
>

best, Johnny Reinhard

hi Gene and paul,

> From: "Gene Ward Smith" <gwsmith@svpal.org>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, June 15, 2003 3:36 PM
> Subject: [tuning] Re: Mozart 31-equal??
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" wrote
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" wrote:
> >
> > > (3) How many particular commas could they possibly have been
> > > interested in? We eliminate the Pythagorean comma, and what's left?
> >
> > you eliminated it on false premises. consider that instead of the
> > equation you wrote, monz could have said that the diatonic semitone
> > was considered 5 commas or that the chromatic semitone was considered
> > 4 semitones and you would have come up with a different equation, and
> > different solutions.
>
> (1) Did someone specifically say that a tone was exactly nine commas?
>
> (2) Did they also specifically say the diatonic semitone was five commas?
>
> (3) Do we know, if they said this, where they took the idea from?

yes, lots of people. we've discussed this point
much in the past, and it's something in which i took
a particular interest. Boethius, in his famous treatise
written c. 505 AD, shows a mathematical proof that the
whole-tone is larger than 8 commas but smaller than 9,
with the purpose of deflating the statement that
"whole-tone = 9 commas". so that idea must have
already been current in his day or earlier.

(unfortunately, Boethius's method contains the
mathematical error of multiplying a linear difference,
rather than multiplying the ratio by itself over
and over again. today, we do it by either taking
the log and multiplying *that*, or by using
prime-factor notation and adding/subtracting the
exponents.

Boethius was a very erudite scholar, and probably was
aware of the error in his method and used it anyway,
because he's dealing with the Pythagorean comma here,
and multiplying the ratios would have resulted in
monstrous numbers of up to 52 digits -- i know,
because i did the calculations myself -- and don't
forget, he was using Roman numerals! but the
discrepancy between his erroneous method and the
correct one is so small that it hardly matters, and
he probably knew that too.)

in this particular Mozart case, Tosi described
tone = 9 commas = diatonic + chromatic semitone,
diatonic semitone = 5 commas, and
chromatic semitone = 4 commas; and Tosi was cited
by Leopold Mozart, in a letter to Wolfgang, as an
"authoritative source" on the matter of intonation.

as i said, the crucial point in determining Mozart's
(senior and junior) intonation is Leopold's statement
that "the flats are a comma higher than the sharps".

i just did some calculations, and i see that the
range of meantones which give an "enharmonic" difference
about the size of any of the usual commas, are those
which flatten each 5th by between about 0.165 and .0183
of a syntonic comma. the enharmonic difference between
flats and sharps for these two examples is ~19.12244302
and ~23.76780157 cents, respectively.

1/6-comma is 0.166666...-comma, and 55edo is
~0.175445544-comma.

now, if the Mozarts were fond of tossing about the
term "comma" in a general sense to mean any number
of small intervals, as some people are here on this
list, then perhaps a good argument could be made for
other varieties of meantone.

but my guess is that the well-read Mozarts knew enough
about intonation to know the difference between a
comma and a diesis, which would limit their preferred
non-keyboard tunings to something in the range i describe
above.

(but then again, those people on this list who use
"comma" in a general sense also know quite a bit about
intonation. oh well ...)

anyway, if they had in mind the general conception
conforming to Tosi's description, then 55edo is the winner.

-monz

hi Gene,

> From: "Gene Ward Smith" <gwsmith@svpal.org>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, June 15, 2003 4:06 PM
> Subject: [tuning] Re: Mozart 31-equal??
>
>
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > the crucial point is that Leopold describes flats as
> > being "a comma higher" than their enharmonically-equivalent
> > sharps.
>
> If we take a Didymus comma, that means a fifth f satisfying
>
> 128/f^12 = 81/80,
>
> which gives an f of 698.208 cents, or 4/23 comma meantone.
> If it is a comma of Pythagoras instead, we get 698.045 cents,
> or 2/11 comma meantone almost precisely.

thanks! that exactly what i was hoping to get from you.
(in lieu of trying to figure out how to do the algebra myself)

but i'm convinced that the Mozarts were using the term
"comma" with at least a *little* specificity, to mean
something around 21 to 24 cents, and not anything much
bigger or smaller than that range, but at the same time,
they were probably *not* being more specific than that.

i.e., their use of the word "comma" could have referred
to either the syntonic (~21.5 cents), the Pythagorean
(~23.5 cents), or -- and i think most likely -- one degree
of 55edo (2^(1/55) = ~21.8 cents).

-monz

hi Gene,

> From: "monz" <monz@attglobal.net>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, June 15, 2003 4:30 PM
> Subject: Re: [tuning] Re: Mozart 31-equal??
>

>
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > > the crucial point is that Leopold describes flats as
> > > being "a comma higher" than their enharmonically-equivalent
> > > sharps.
> >
> > If we take a Didymus comma, that means a fifth f satisfying
> >
> > 128/f^12 = 81/80,
> >
> > which gives an f of 698.208 cents, or 4/23 comma meantone.
> > If it is a comma of Pythagoras instead, we get 698.045 cents,
> > or 2/11 comma meantone almost precisely.
>
>
>
> thanks! that exactly what i was hoping to get from you.
> (in lieu of trying to figure out how to do the algebra myself)

can you (or anyone else who knows) please explain this is
more detail?

i understand that the formula you give above
involves the equation of the tempered Pythagorean
comma with the syntonic comma, correct?

but then how do you manipulate the terms to isolate f?

-monz

hmmm ...

> From: "monz" <monz@attglobal.net>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, June 15, 2003 4:56 PM
> Subject: Re: [tuning] Re: Mozart 31-equal??
>

>
> > > > the crucial point is that Leopold describes flats as
> > > > being "a comma higher" than their enharmonically-equivalent
> > > > sharps.
> > >
> > > If we take a Didymus comma, that means a fifth f satisfying
> > >
> > > 128/f^12 = 81/80,
> > >
> > > which gives an f of 698.208 cents, or 4/23 comma meantone.
> > > If it is a comma of Pythagoras instead, we get 698.045 cents,
> > > or 2/11 comma meantone almost precisely.

> can you (or anyone else who knows) please explain this is
> more detail?
>
> i understand that the formula you give above
> involves the equation of the tempered Pythagorean
> comma with the syntonic comma, correct?
>
> but then how do you manipulate the terms to isolate f?

never mind. i've become such an expert at using
prime-factor notation that after i made the comment
about the Pythagorean comma, i tried prime-factoring
the whole equation, and i see how it works now.

more accurate cents values for the tempered meantone
5ths for the syntonic and Pythagorean commas are
~698.2078092 and ~698.0449991, respectively.

-monz

for those keeping track ...

the prime-factor notation for the size of
the "5th" in the meantone which gives an exact
syntonic comma as the enharmonic difference
between a sharp and its associated (higher) flat:

2^(11/12) * 3^(-1/3) * 5^(1/12)

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> in this particular Mozart case, Tosi described
> tone = 9 commas = diatonic + chromatic semitone,
> diatonic semitone = 5 commas, and
> chromatic semitone = 4 commas; and Tosi was cited
> by Leopold Mozart, in a letter to Wolfgang, as an
> "authoritative source" on the matter of intonation.

This does give three equations in three unknowns leading to
55-equal; did Tosi know this was a consequence?

meantone equation: -4a + 4b - c = 0
diatonic semitone equation: 4a - b - c = 5
chromatic semitone equation: -3a - b + 2c = 4

Solving this for a, b, and c leads to
a = 55, b = 87, c = 128, which is 55-equal.

Gene Ward Smith,

I think you are making something simple into something complicated.

In meantone and meantone-like tunings, the sum of five tones and two
semitones is one octave. The relevant calculation here is five whole
tones at nine "commas" each plus two diatonic semitones with
five "commas" each, or (5x9)+(2*5) = 55.

Your comparison of tunings in terms of the rational commas (syntonon,
pythagorean) is interesting, but besides the point. In a scale like
55 or 53, the smallest unit is the "comma" however close it may be to
one or the other rational commas.

Gabor

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > in this particular Mozart case, Tosi described
> > tone = 9 commas = diatonic + chromatic semitone,
> > diatonic semitone = 5 commas, and
> > chromatic semitone = 4 commas; and Tosi was cited
> > by Leopold Mozart, in a letter to Wolfgang, as an
> > "authoritative source" on the matter of intonation.
>
> This does give three equations in three unknowns leading to
> 55-equal; did Tosi know this was a consequence?
>
> meantone equation: -4a + 4b - c = 0
> diatonic semitone equation: 4a - b - c = 5
> chromatic semitone equation: -3a - b + 2c = 4
>
> Solving this for a, b, and c leads to
> a = 55, b = 87, c = 128, which is 55-equal.

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> i'm not aware of any "meantone" in which the Pythagorean
> comma is divided, rather than the syntonic comma.

check out jorgenson's book.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"

> (1) Did someone specifically say that a tone was exactly nine
commas?

*exactly*? this was meant as a practical instruction for
instrumentalists.

> (2) Did they also specifically say the diatonic semitone was five
commas?

yes, that went with the above.

> (3) Do we know, if they said this, where they took the idea from?

you mean the original source of the idea? like i said, it was a
modification of earlier pythagorean doctrine, modified so that it fit
the then-current meantone system.

>> clearly you're treating as *sharp* certain
> > quantities that were only *fuzzy* in the practical-minded but
> > scientifically primitive 18th century.
>
> Eh? You must be talking about a different 18th century. It was not
> fuzzy-minded, and it was far from primitive. As I remarked, the
> computations involved would have been trivial for a mathematician of
> the time.

computations? we're talking about adjustments that a violinist must
implement in real time! there were no oscilloscopes, etc., anyway --
the instrumentalists had to trust their ears and that of their
directors/teachers, not sit down and do computations!

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> > so now, can we finally agree that 55edo and its
> > relatives were the tunings intended by Mozart for
> > his orchestral and chamber works?
>
> Still speculative. It would help to find some passages where a 698
> cent fifth works better than one a cent flatter.

though i agree that the evidence favors monz's viewpoint, i don't
believe you'll find any smoking guns in the music. mozart sounds fine
in *any* meantone tuning! also in adaptive variants of meantone.

--- In tuning@yahoogroups.com, Afmmjr@a... wrote:
> In a message dated 6/15/03 3:26:57 PM Eastern Daylight Time,
> monz@a... writes:
>
>
> > in answer to Johnny's question about "pure thirds":
> >
> > he is of course referring to the fact 1/4-comma meantone
> > is the only member of the meantone family which gives
> > absolutely "pure" 5/4 ratios (~386.3137139 cents)
> > for the "major-3rds".
> >
> > however, the "major-3rds" of all meantones are fairly
> > close to this value, that being the whole point of
> > using meantone. the "major-3rd" of 2/7-comma meantone
> > is a bit narrower, ~383.2413868 cents, while that
> > of 1/6-comma meantone is a bit wider, ~393.4824771 cents.
> > the differences are -3.072327085 and +7.168763199 cents,
> > respectively.
> >
>
> As 2/7ths commas is pure theory and not applicable to the
discussion, lets
> move on to +7 cents of 1/6-comma.

how is that less "pure theory"?

> I had mentioned Werckmeister being mentioned
> by Leopold Mozart was significant, not for his tuning systems, but
because he
> is the beginning of enhramonic identities. It seems surprising
that someone
> encouraging a distinction between diatonic and chromatic semitones
would
> mention the originator of enharmonic identities.

not surprising at all, as there was a specific purpose intended for
both systems -- please re-read monz's post.

In a message dated 6/16/03 4:18:48 PM Eastern Daylight Time,
wallyesterpaulrus@yahoo.com writes:

> As 2/7ths commas is pure theory and not applicable to the
> discussion, lets
> > move on to +7 cents of 1/6-comma.
>
> how is that less "pure theory"?

Do you know of any example of use of 2/7ths comma in the Baroque or after?

>
> > I had mentioned Werckmeister being mentioned
> > by Leopold Mozart was significant, not for his tuning systems, but
> because he
> > is the beginning of enhramonic identities. It seems surprising
> that someone
> > encouraging a distinction between diatonic and chromatic semitones
> would
> > mention the originator of enharmonic identities.
>
> not surprising at all, as there was a specific purpose intended for
> both systems -- please re-read monz's post.
>

Please aid me here if you can. Johnny

--- In tuning@yahoogroups.com, Afmmjr@a... wrote:
> In a message dated 6/16/03 4:18:48 PM Eastern Daylight Time,
> wallyesterpaulrus@y... writes:
>
>
> > As 2/7ths commas is pure theory and not applicable to the
> > discussion, lets
> > > move on to +7 cents of 1/6-comma.
> >
> > how is that less "pure theory"?
>
>
> Do you know of any example of use of 2/7ths comma in the Baroque or
>after?

no, if that's what "pure theory" means, certainly you're correct. but
monz was simply making an illustration, and it's always easier to see
a linear trend if you extend the line further in one or both
directions.

if you want to be precise about what was actually used in the baroque
period, there was really no way to tune a chain of logarithmically
identical fifths back then anyway. at best, equally-beating fifths
could have been used, but these produce logarithmic intervals that
are far from equal. regular meantone tunings were an ideal, and a
very valuable one at that.

> > > I had mentioned Werckmeister being mentioned
> > > by Leopold Mozart was significant, not for his tuning systems,
but
> > because he
> > > is the beginning of enhramonic identities. It seems surprising
> > that someone
> > > encouraging a distinction between diatonic and chromatic
semitones
> > would
> > > mention the originator of enharmonic identities.
> >
> > not surprising at all, as there was a specific purpose intended
for
> > both systems -- please re-read monz's post.
> >
>
> Please aid me here if you can. Johnny

the use of split keys had pretty much disappeared, so well-
temperament was a necessity for the keyboards. not so for the
strings, which were to be played in "true intonation" (what today is
known as "meantone" in the general sense). leopold was quite clear
about this.

In a message dated 6/16/03 5:18:15 PM Eastern Daylight Time,
wallyesterpaulrus@yahoo.com writes:

> if you want to be precise about what was actually used in the baroque
> period, there was really no way to tune a chain of logarithmically
> identical fifths back then anyway. at best, equally-beating fifths
> could have been used, but these produce logarithmic intervals that
> are far from equal. regular meantone tunings were an ideal, and a
> very valuable one at that.
>

Since you mention the Baroque in the above, and not the Classical which is
where Mozart in concerned, I have a question. How does the above apply to the
tuning of Werckmeister III?

> > > > I had mentioned Werckmeister being mentioned
> > > > by Leopold Mozart was significant, not for his tuning systems,
> but
> > > because he
> > > > is the beginning of enhramonic identities. It seems surprising
> > > that someone
> > > > encouraging a distinction between diatonic and chromatic
> semitones
> > > would
> > > > mention the originator of enharmonic identities.
> > >
> > > not surprising at all, as there was a specific purpose intended
> for
> > > both systems -- please re-read monz's post.
> > >
> >
> > Please aid me here if you can. Johnny
>
> the use of split keys had pretty much disappeared, so well-
> temperament was a necessity for the keyboards. not so for the
> strings, which were to be played in "true intonation" (what today is
> known as "meantone" in the general sense). leopold was quite clear
> about this.

Has Leopold been quoted to the list on this? Maybe I missed it.

Johnny

--- In tuning@yahoogroups.com, "wallyesterpaulrus"

/tuning/topicId_44573.html#44717
>
> computations? we're talking about adjustments that a violinist must
> implement in real time! there were no oscilloscopes, etc., anyway --

> the instrumentalists had to trust their ears and that of their
> directors/teachers, not sit down and do computations!

***This is funny. Well, maybe Gene means they had a
little "theoretical bird" at their ear, giving them ideas, just as
some of our tuning theorists here have given some composers and
performers ideas... :)

J. Pehrson

--- In tuning@yahoogroups.com, Afmmjr@a... wrote:
> In a message dated 6/16/03 5:18:15 PM Eastern Daylight Time,
> wallyesterpaulrus@y... writes:
>
>
> > if you want to be precise about what was actually used in the
baroque
> > period, there was really no way to tune a chain of
logarithmically
> > identical fifths back then anyway. at best, equally-beating
fifths
> > could have been used, but these produce logarithmic intervals
that
> > are far from equal. regular meantone tunings were an ideal, and a
> > very valuable one at that.
> >
>
> Since you mention the Baroque in the above, and not the Classical
which is
> where Mozart in concerned, I have a question. How does the above
apply to the
> tuning of Werckmeister III?

the tempered fifths resulting from an intention of tuning werckmeister
iii could have been rather different from one another in result.

> > > > both systems -- please re-read monz's post.
> > > >
> > >
> > > Please aid me here if you can. Johnny
> >
> > the use of split keys had pretty much disappeared, so well-
> > temperament was a necessity for the keyboards. not so for the
> > strings, which were to be played in "true intonation" (what today
is
> > known as "meantone" in the general sense). leopold was quite clear
> > about this.
>
> Has Leopold been quoted to the list on this? Maybe I missed it.

again, monz's post.

In a message dated 6/17/03 1:19:00 AM Eastern Daylight Time,
wallyesterpaulrus@yahoo.com writes:

> > Since you mention the Baroque in the above, and not the Classical
> which is
> > where Mozart in concerned, I have a question. How does the above
> apply to the
> > tuning of Werckmeister III?
>
> the tempered fifths resulting from an intention of tuning werckmeister
> iii could have been rather different from one another in result.
>

Paul, I do not understand you at all. Maybe it can help if you reread
yourself. Capital letters could help as well, especially in long paragraphs as you
sometimes do.

> > > > > both systems -- please re-read monz's post.
> > > > >
> > > >
> > > > Please aid me here if you can. Johnny
> > >
> > > the use of split keys had pretty much disappeared, so well-
> > > temperament was a necessity for the keyboards. not so for the
> > > strings, which were to be played in "true intonation" (what today
> is
> > > known as "meantone" in the general sense). leopold was quite clear
> > > about this.
> >
> > Has Leopold been quoted to the list on this? Maybe I missed it.
>
> again, monz's post.
>

I'll try to see if I can find it. Too bad you are not as clear as you say
"leopold" was.

Johnny

--- In tuning@yahoogroups.com, Afmmjr@a... wrote:
> In a message dated 6/17/03 1:19:00 AM Eastern Daylight Time,
> wallyesterpaulrus@y... writes:
>
>
> > > Since you mention the Baroque in the above, and not the
Classical
> > which is
> > > where Mozart in concerned, I have a question. How does the
above
> > apply to the
> > > tuning of Werckmeister III?
> >
> > the tempered fifths resulting from an intention of tuning
werckmeister
> > iii could have been rather different from one another in result.
> >
>
> Paul, I do not understand you at all. Maybe it can help if you
reread
> yourself. Capital letters could help as well, especially in long
paragraphs as you
> sometimes do.

The tempered fifths resulting from an intention of tuning
Werckmeister III could have been rather different from one another in
result.

on 6/16/03 2:15 PM, wallyesterpaulrus <wallyesterpaulrus@yahoo.com> wrote:

> if you want to be precise about what was actually used in the baroque
> period, there was really no way to tune a chain of logarithmically
> identical fifths back then anyway. at best, equally-beating fifths
> could have been used, but these produce logarithmic intervals that
> are far from equal. regular meantone tunings were an ideal, and a
> very valuable one at that.

It seems to me something better than equally-beating fifths would have been
possible. If you tune a fifth in one octave to a certain number of beats,
and then tune the octaves above both of those notes to zero beat, it will be
obvious that the corresponding fifth tuned a zero-beat octave higher will
beat twice as fast as the fifth an octave below. A reasonably intelligent
self-tuning musician would recognize this fact and make some kind of
adjustment in how a series of fifths are tuned, no? It seems to me trying
to tune equally-beating fifths would lead to contradictions. Trial and
error resolution of those contradictions would lead (according to available
time and patience) arbitrarily close to a logarithmically ideal tuning,
unless inharmonicity is getting in the way - and maybe that's what you
meant, in which case ... _never_mind_!

-Kurt Bigler

on 6/15/03 10:53 AM, monz <monz@attglobal.net> wrote:

> i'm not aware of any "meantone" in which the Pythagorean
> comma is divided, rather than the syntonic comma.

FWIW, Padgham (which is admittely a cursory overview of tunings, and only
12-tone tunings at that) has three mean-tone tunings in his The
Well-Tempered Organ book, and they are as follows:

"Quarter Comma Mean Tone" based on 1/4 Syntonic Comma
"Fifth Comma Mean Tone" based on 1/5 Syntonic Comma
"Silberman (Sixth Comma Mean Tone)" based on 1/6 Pythagorean Comma

but he notes that the 1/6-P (3.91 cents) is pretty close to 1/5-S (4.30
cents).

-Kurt

hi paul,

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, June 16, 2003 1:17 PM
> Subject: [tuning] Re: Mozart 31-equal??
>
>
> > [me, monz, asking about Gene's algebra:]
> >
> > but then how do you manipulate the terms to isolate f?
>
> ultra-straightforward algebra.
>
> 128/f^12 = 81/80
>
> f^12/128 = 80/81

aha! the rest of it i knew, but this i didn't.

see how sometimes math that's so simple trips
me up, simply because i don't know some real
basic stuff.

anyway, as i stated in another post, i already
figured out how Gene's formula works, by using
prime-factor notation instead of regular integer
terms.

seems kind of weird ... huge gaps in my knowledge
of *basic* algebra, but i'm comfortable enough with
vector-addition that i can use *it* to solve problems
that otherwise stump me.

see? ... i knew there was some value in my
prime-factorization theories! ;-)

-monz

hi Johnny,

> From: <Afmmjr@aol.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, June 16, 2003 11:20 PM
> Subject: Re: [tuning] Re: Mozart 31-equal??
>
>
> <snip>
>
> > > > [paul]
> > > > the use of split keys had pretty much disappeared,
> > > > so well-temperament was a necessity for the keyboards.
> > > > not so for the strings, which were to be played in
> > > > "true intonation" (what today is known as "meantone"
> > > > in the general sense). leopold was quite clear
> > > > about this.
> > >
> > > [johnny]
> > > Has Leopold been quoted to the list on this?
> > > Maybe I missed it.
> >
> > [paul]
> > again, monz's post.
> >
>
> [johnny]
> I'll try to see if I can find it. Too bad you are
> not as clear as you say "leopold" was.

i quote (from the post in which i originally
answered your question about Leopold advocating
both well-temperament and meantone):

/tuning/topicId_44573.html#44662

>> From: "monz" <monz@attglobal.net>
>> To: <tuning@yahoogroups.com>
>> Sent: Sunday, June 15, 2003 12:24 PM
>> Subject: Re: [tuning] Re: Mozart 31-equal??
>>
>> <snip>
>>
>> i *do* know that Leopold's book contains exercises
>> in which the violinist is specifically instructed
>> to play the sharps lower than the flats, and the
>> exercises are specifically to develop facility in
>> playing all scales correctly in a 19-note extended
>> meantone, which is what he recommended for the violin.
>> the exercises quoted in the Chesnut article use a
>> meantone chain of 5ths from Gb to B#.

Leopold gave Werckmeister and Neidhardt as references
for well-temperament, but also included excercises
in a 19-tone meantone chain for the violin. he was
clearly advocating well-temperaments for keyboards
and extended 55edo-style meantone for all other
instruments ... or at least for the string family,
anyway.

-monz

> From: "monz" <monz@attglobal.net>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, June 17, 2003 1:09 AM
> Subject: Re: [tuning] Re: Mozart 31-equal??
>
>
> hi paul,
>
>
> > From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> > To: <tuning@yahoogroups.com>
> > Sent: Monday, June 16, 2003 1:17 PM
> > Subject: [tuning] Re: Mozart 31-equal??
> >
> >
> > > [me, monz, asking about Gene's algebra:]
> > >
> > > but then how do you manipulate the terms to isolate f?
> >
> > ultra-straightforward algebra.
> >
> > 128/f^12 = 81/80
> >
> > f^12/128 = 80/81
>
>
> aha! the rest of it i knew, but this i didn't.
>
> see how sometimes math that's so simple trips
> me up, simply because i don't know some real
> basic stuff.
>
> anyway, as i stated in another post, i already
> figured out how Gene's formula works, by using
> prime-factor notation instead of regular integer
> terms.
>
> seems kind of weird ... huge gaps in my knowledge
> of *basic* algebra, but i'm comfortable enough with
> vector-addition that i can use *it* to solve problems
> that otherwise stump me.
>
> see? ... i knew there was some value in my
> prime-factorization theories! ;-)

i thought i'd share my method for anyone
who's interested.

Gene made it easy for me to check that i
was doing things correctly, since he gave
a cents-value for "f". the tiny error in
my final result in the check, was the result
of Gene rounding his cents-value to 3
decimal places.

(on the Yahoo web interface, use "Expand Messages"
mode to view with correct formatting)

using vector-addition:

f^ 2^ 3^ 5^ f^ 2^ 3^ 5^

[-12 7 0 0] = 128/f^12 = [ 0 -4 4 -1 ] = 81/80
- [ 0 7 0 0] / 128 - [ 0 7 0 0 ] / 128
------------------- ---------- ------------------------ ---------
---
[-12 0 0 0] 1/f^12 = [ 0 -11 4 -1 ]
81/10240
/ [-12 -12 -12 -12] ^ -(1/12) / [-12 -12 -12 -12 ]
^ -(1/12)
------------------- ---------- ------------------------ ---------
---
[ 1 0 0 0] f = [ 0 11/12 -1/3 1/12]
~1.496756817

here's an illustration of the steps paul used:

> 128/f^12 = 81/80
>
> f^12/128 = 80/81
>
> f^12 = 80/81*128 = 10240/81
>
> f = (10240/81)^(1/12) = 1.49675681696853 = 698.207809200274 cents.

f^ 2^ 3^ 5^ f^ 2^ 3^ 5^

[-12 7 0 0] = 128/f^12 = [ 0 -4 4 -1 ] =
81/80
* [ -1 -1 -1 -1] [ -1 -1 -1 -1 ]
------------------- ---------- ------------------------ ---------
---
[ 12 -7 0 0] f^12/128 [ 0 4 -4 1 ]
80/81
+ [ 0 7 0 0] * 128 - [ 0 7 0 0 ] / 128
------------------- ---------- ------------------------ ---------
---
[ 12 0 0 0] f^12 = [ 0 11 -4 1 ]
10240/81
/ [ 12 12 12 12] ^ (1/12) / [ 12 12 12 12 ] ^
(1/12)
------------------- ---------- ------------------------ ---------
---
[ 1 0 0 0] f = [ 0 11/12 -1/3 1/12]
~1.496756817

i know it doesn't mean much to you, paul, but
i get great satisifaction out of knowing that
i can precisely plot the answer to Gene's
equation on a 5-limit lattice. to me, it's
interesting to look at a meantone lattice where
the -12th generator is exactly a syntonic comma!

:)

-monz

-monz

In a message dated 6/17/03 2:43:59 AM Eastern Daylight Time,
wallyesterpaulrus@yahoo.com writes:

> The tempered fifths resulting from an intention of tuning
> Werckmeister III could have been rather different from one another in
> result.
>

JR:
> Ah, I understand now what you are saying. But I do disagree. The quarter
> comma flattened fifth was in the ear of the musician by the time of
> Werckmeister. It's a very clear interval to hear. If a musician couldn't hear this
> interval by Werckmeister's time, they couldn't hear any interval, which may be
> your point. I do not believe this to be true.

Thanks, Monz, for readdressing the Leopold issues. However, I had asked
about a "quote" from Leopold (in translation would be fine). I appreciate your
summarizing Leopold's intentions and such, but I would love to see specific,
determinant quotes.

Thanks, Johnny

----- Original Message -----
From: <Afmmjr@aol.com>
To: <tuning@yahoogroups.com>
Sent: Tuesday, June 17, 2003 9:39 AM
Subject: Re: [tuning] Re: Mozart 31-equal??

> Thanks, Monz, for readdressing the Leopold issues.
> However, I had asked about a "quote" from Leopold
> (in translation would be fine). I appreciate your
> summarizing Leopold's intentions and such, but I
> would love to see specific, determinant quotes.

sorry, Johnny ... i wrote in that post that i don't
have a copy of Leopold's book, but that one of his
violin exercises *is* quoted in the Chesnut article,
and it's plain to see that it contains all 19 pitches
which are notated as a chain of 5ths from Gb to B#.

so yes, i have specific evidence that Leopold Mozart
advocated the use of a 19-note scale. no, the score
of the exercise does not indicate the tuning -- only
the notation shows 19 different pitches.

but of course, the whole point of Chesnut's article
is Leopold's statement that "the flats are a comma
higher than the sharps", which indicates meantone.

so that quoted exercise is direct evidence that he
advocated a 19-tone meantone tuning for the string family.

now, as for L. Mozart's advocacy of well-temperament
on keyboards, i only have what Chesnut says.
... but *you're* the one defending Werckmeister! :)

my whole emphasis has been on the *particular variety*
of meantone that the Mozarts had in mind for
non-keyboard instruments, and i'm convinced that
it was 55edo.

i'd like to have a look at Leopold's book myself,
but right now i'm busy tracking down a copy of the
Attwood studies to see what W.A. Mozart himself
wrote about intonation.

-monz

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 6/16/03 2:15 PM, wallyesterpaulrus <wallyesterpaulrus@y...>
wrote:
>
> > if you want to be precise about what was actually used in the
baroque
> > period, there was really no way to tune a chain of logarithmically
> > identical fifths back then anyway. at best, equally-beating fifths
> > could have been used, but these produce logarithmic intervals that
> > are far from equal. regular meantone tunings were an ideal, and a
> > very valuable one at that.
>
> It seems to me something better than equally-beating fifths would
have been
> possible. If you tune a fifth in one octave to a certain number of
beats,
> and then tune the octaves above both of those notes to zero beat,
it will be
> obvious that the corresponding fifth tuned a zero-beat octave
higher will
> beat twice as fast as the fifth an octave below.

yup!

> A reasonably intelligent
> self-tuning musician would recognize this fact and make some kind of
> adjustment in how a series of fifths are tuned, no?

yes, but what kind of adjustment? by ear?

> It seems to me trying
> to tune equally-beating fifths would lead to contradictions.

and yet this was a common misunderstanding in tuning instructions of
the day. see jorgenson's tome for more info.

> Trial and
> error resolution of those contradictions would lead (according to
available
> time and patience) arbitrarily close to a logarithmically ideal
tuning,

arbitrarily close? i don't think so. all you can do is make sure that
the several fifths that are supposed to be tempered by the same
amount beat faster and faster as you go up in register. that won't
get you arbitrarily close, but it'll get you pretty close. this is
known as "progressive beating" or something. though it seems obvious,
it was actually unknown to the tuning community in those days, and
equal-beating tended to be used instead until the "obvious" law of
progressive beating was published. see jorgenson for details.

--- In tuning@yahoogroups.com, Afmmjr@a... wrote:
> In a message dated 6/17/03 2:43:59 AM Eastern Daylight Time,
> wallyesterpaulrus@y... writes:
>
>
> > The tempered fifths resulting from an intention of tuning
> > Werckmeister III could have been rather different from one
another in
> > result.
> >
>
> JR:
> > Ah, I understand now what you are saying. But I do disagree.
The quarter
> > comma flattened fifth was in the ear of the musician by the time
of
> > Werckmeister.

how did it get there? it was only an ideal, and only achieved in an
*average* sense by tuners aiming for 1/4-comma meantone and tuning
all the major thirds just.

> > It's a very clear interval to hear. If a musician couldn't hear
this
> > interval by Werckmeister's time, they couldn't hear any interval,
which may be
> > your point. I do not believe this to be true.

they could hear just intervals. tempered intervals, if truly
identical in logarithmic measurement, would beat faster in higher
registers, and hence sound different. thus, for four fifths that
added up to a just major third, logarithmically identical measurement
could only have been achieved by accident. this point is made over
and over on page after page in jorgensen, and i firmly believe it to
be true having toiled over tuning my piano in 1/4-comma meantone in
the past. you will of course say that one can hear these fifths
*melodically* and make them melodically identical, to within 1 cent.
i will of course retort that the just noticeable difference for
melodic intervals is over 5 cents. you claim you can do better. let
me make a CD of various tempered fifths in various registers and see
if you can identify them to the cent. maybe you've trained yourself
to do so (i'll believe it when i see it), but you've had access to
precise tuning devices that were not available in werckmeister's
time . . . let's put it this way. it would take a rare ear, perhaps a
reinhard but certainly not a jorgensen, to be able to reliably
tune "identical" (to within a cent, say) tempered fifths without the
aid of techniques and technologies introduced after werckmeister's
time.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

though it seems obvious,
> it was actually unknown to the tuning community in those days, and
> equal-beating tended to be used instead until the "obvious" law of
> progressive beating was published. see jorgenson for details.

It seems to me one obvious starting point is no beating of major
thirds; if you fill in the four fifths using equal beating, your
error wouldn't be so bad.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> though it seems obvious,
> > it was actually unknown to the tuning community in those days,
and
> > equal-beating tended to be used instead until the "obvious" law
of
> > progressive beating was published. see jorgenson for details.
>
> It seems to me one obvious starting point is no beating of major
> thirds; if you fill in the four fifths using equal beating, your
> error wouldn't be so bad.

right, this is what i was talking about to begin with. but the fifths
would end up being tempered by quite different amounts -- for
example, for two fifths a perfect fifth apart which beat at the same
rate, the lower one is tempered about 50% more than the higher one,
right? jorgensen's book contains exact figures for what such a
strategy would have produced using the typical bearing plans (which
notes are tuned first, before proceeding up and down by octaves) of
the time.

In a message dated 6/17/03 3:16:46 PM Eastern Daylight Time,
wallyesterpaulrus@yahoo.com writes:

> how did it get there? it was only an ideal, and only achieved in an
> *average* sense by tuners aiming for 1/4-comma meantone and tuning
> all the major thirds just.
>

Over a hundred years of practice is one way. The art of organ building took
a long time to get there as well, but it did. The monochordi certainly aided
it getting there. The habit of a regular tuner would seem to cement it
getting there. The few individuals with genetic perfect pitch could certainly get
there. It was the ONLY fifth other than pure that was recognized, a Praetorian
fifth you might say. I'm sorry if this does not explain enough why my gut
tells me that it is a recognizable interval. It is for me from having played
Jon Catler's "Hey Sailor" so many times with Brad Catler outlining the 696 cents
fifths in the opening so many times. Why couldn't someone else not learn to
hear it from rote? Jorgensen is speaking from a different life experience,
that of counting beats and deducing past practice. My take is different. I
await your visit. Until then, let's give it a rest, this poor dead horse.

Johnny

--- In tuning@yahoogroups.com, Afmmjr@a... wrote:
> In a message dated 6/17/03 3:16:46 PM Eastern Daylight Time,
> wallyesterpaulrus@y... writes:
>
>
> > how did it get there? it was only an ideal, and only achieved in
an
> > *average* sense by tuners aiming for 1/4-comma meantone and
tuning
> > all the major thirds just.
> >
>
> Over a hundred years of practice is one way. The art of organ
building took
> a long time to get there as well, but it did. The monochordi
certainly aided
> it getting there. The habit of a regular tuner would seem to
cement it
> getting there. The few individuals with genetic perfect pitch
could certainly get
> there. It was the ONLY fifth other than pure that was recognized,
a Praetorian
> fifth you might say. I'm sorry if this does not explain enough why
my gut
> tells me that it is a recognizable interval. It is for me from
having played
> Jon Catler's "Hey Sailor" so many times with Brad Catler outlining
the 696 cents
> fifths in the opening so many times. Why couldn't someone else not
learn to
> hear it from rote? Jorgensen is speaking from a different life
experience,
> that of counting beats and deducing past practice. My take is
different. I
> await your visit. Until then, let's give it a rest, this poor dead
horse.
>
> Johnny

i don't recall ever discussing this before . . .

Le 17/06/03 4:16,J --- in the digest 2579 --- message 7

>
>
>
>
> Afmmjr@aol.com writes :
>> > As 2/7ths commas is pure theory and not applicable to the
>> > discussion, lets
>>> > > move on to +7 cents of 1/6-comma.
>
> then wallyesterpaulrus@yahoo.com writes:
>> > how is that less "pure theory"?
>
> then Afmmjr@aol.com writes :
> Do you know of any example of use of 2/7ths comma in the Baroque or after?
>
>
> As meantone with 1/4 C synt. give a pure major third, 1/3 C synt. gave pure
> minor third. A meantone with 2/7 C synt. have been mentioned (or proposed) by
> Zarlino. In between, we could call him "supermeantone", with this averaged
> third. I don't know if it had been usedŠ
> but I don't trust than an absence of trace mean an absence of use.
> For example, I have an ancestor who played flute, about 20 000 years ago, I am
> certain about that, but there is few chance to find his flute to prove itŠ
> Don¹t tell me that my demonstration is not scientificŠ I know it.
>
> Gilles Patrat

--- In tuning@yahoogroups.com, Afmmjr@a... wrote:
> It was the ONLY fifth other than pure that was recognized,
a Praetorian
> fifth you might say.

it better not have been recognized *too* accurately, since the
werckmeister tempered fifth is 1/2 cent different, which means a 2
cent error would have landed on one of the "just" fifths had the
preatorian fifth been used instead.

In a message dated 6/19/03 12:45:22 AM Eastern Daylight Time,
wallyesterpaulrus@yahoo.com writes:

> --- In tuning@yahoogroups.com, Afmmjr@a... wrote:
> > It was the ONLY fifth other than pure that was recognized,
> a Praetorian
> > fifth you might say.
>
> it better not have been recognized *too* accurately, since the
> werckmeister tempered fifth is 1/2 cent different, which means a 2
> cent error would have landed on one of the "just" fifths had the
> preatorian fifth been used instead.

While the mathematics of meantone indicate half-cent alterations to just
thirds, musicians would go for the pure thirds and let the tempered fifths fall
where they may in meantone tuning (or as it was called, Praetorian tuning).
Specifically, Werckmeister felt that a tuner could keep the tempered C-G, G-D,
and D-A fifths and retune the others for Werckmeister III tuning, or as he put
it,
"our tuning."

best, Johnny

--- In tuning@yahoogroups.com, Afmmjr@a... wrote:
> In a message dated 6/19/03 12:45:22 AM Eastern Daylight Time,
> wallyesterpaulrus@y... writes:
>
>
> > --- In tuning@yahoogroups.com, Afmmjr@a... wrote:
> > > It was the ONLY fifth other than pure that was recognized,
> > a Praetorian
> > > fifth you might say.
> >
> > it better not have been recognized *too* accurately, since the
> > werckmeister tempered fifth is 1/2 cent different, which means a
2
> > cent error would have landed on one of the "just" fifths had the
> > preatorian fifth been used instead.
>
> While the mathematics of meantone indicate half-cent alterations to
just
> thirds,

no they don't -- 1/4-comma meantone has absolutely pure major thirds.

> musicians would go for the pure thirds and let the tempered fifths
fall
> where they may in meantone tuning (or as it was called, Praetorian
tuning).

this was my whole original point, johnny! hence these tempered fifths
cannot be presumed to have come out exactly equal to one another!