Yahoo Tuning Groups Ultimate Backup tuning Re: [tuning] Re: Werckmeister etc.

In a message dated 2/21/02 10:38:02 PM Eastern Standard Time,
paul@stretch-music.com writes:

> i'll buy this . . . but i'm still wondering (in case you missed this
> question today), what were the status of werckmeister iv, v, and vi
> with respect to bach?
>
>
>

Sorry, I wrote a message on this and it did not get through. Werckmeister
IV, V, and VI are diatonic variants, and have no perceivable connection with
Bach. I believe Bach used a single tuning his entire adult life, and it was
chromatic (as was his uncles).

The bulk of material of Werckmeister's book "Musicalische Temperatur"
(published in Quedlinburg in the Harz) is on the circular chromatic which
would be III. Werckmeister I is JI and W. II is quarter-comma meantone.

So now it appears to me, since Werckmeister chromatic was used substantially
in Central Germany during the Baroque, that the increased dissonance of
sixth-comma meantone (from earlier quarter comma) may be in reaction to the
success of well-temperament, best exemplified by Werckmeister III tuning.
This is because sixth comma better maps the diatonic keys of Werckmeister III
and allow for more performances between systems.

Best, Johnny Reinhard

🔗monz <joemonz@yahoo.com>

> From: <Afmmjr@aol.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, February 22, 2002 5:51 AM
> Subject: Re: [tuning] Re: Werckmeister etc.
>
>
> So now it appears to me, since Werckmeister chromatic was used
substantially
> in Central Germany during the Baroque, that the increased dissonance of
> sixth-comma meantone (from earlier quarter comma) may be in reaction to
the
> success of well-temperament, best exemplified by Werckmeister III tuning.
> This is because sixth comma better maps the diatonic keys of Werckmeister
III
> and allow for more performances between systems.

wow, Johnny, this is fascinating! and i can buy it.

as Barbour would have it, well-temperament and 1/6-comma meantone
were both stages along the way in the evolution from the hegemony
of 1/4-comma meantone to that of 12edo ... and i'm willing to
grant that there's at least a little truth to his scenario.

at any rate, 1/6-comma meantone and the well-temperaments both
resemble 12edo much more than does 1/4-comma meantone.

the important difference between well-temperaments and meantones
is that meantones depend on the disappearance of the syntonic
comma 81:80 = [2 3 5]**[-4 4 -1] = ~22 cents to turn the flat JI
lattice into a closed cylinder, whereas well-temperaments depend
on the ~2-cents fudging allowed by the skhisma = 32805:32768 =
[2 3 5]**[-15 8 1], to do the same thing.

thus, for example (i'm using Kirnberger III here instead of
Werckmeister IV), if a well-temperament is tuned as a chain
of "5ths" (some Pythagorean, some tempered) from Db to F#,
the C# which would be a 3:2 above F# is only a skhisma away from
the Db which started the chain. this ~2-cent difference is
outside the tuning resolution of any human tuner of Bach's time,
so that in effect the open well-tempered chain becomes a
closed well-tempered cycle, and F# = ~Gb, Db = ~C#, Ab = ~G#,
Eb = ~D#, Bb = ~A#, etc.

this audibly-identical equivalence of the chromatic notes in
a well-temperament is quite different from the implied ratios
of the tempered notes (which in Kirnberger III are all "white
key" notes), which employ tempering by a fraction of a comma
(either Pythagorean or syntonic, depending on the WT),
which is a much larger interval than a skhisma, to use as
a unison-vector.

-monz

_________________________________________________________
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🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:

> the important difference between well-temperaments and meantones
> is that meantones depend on the disappearance of the syntonic
> comma 81:80 = [2 3 5]**[-4 4 -1] = ~22 cents to turn the flat JI
> lattice into a closed cylinder, whereas well-temperaments depend
> on the ~2-cents fudging allowed by the skhisma = 32805:32768 =
> [2 3 5]**[-15 8 1], to do the same thing.

if you'll permit me to correct you, monz, the latter is true of
schismic temperaments (such as van zwolle and 17-tone medieval arabic
and 24-tone helmoltzian and 53-tone gabat-saribaldi), not well-
temperaments. well-temperaments turn the flat lattice into a torus,
and the simplest pair of commas for this purpose are the syntonic
comma and the diesis. though you could just as well use the syntonic
comma and the schisma if you wish -- it's equivalent.

> thus, for example (i'm using Kirnberger III here instead of
> Werckmeister IV), if a well-temperament is tuned as a chain
> of "5ths" (some Pythagorean, some tempered) from Db to F#,
> the C# which would be a 3:2 above F# is only a skhisma away from
> the Db which started the chain.

this is _only_ true for kirnberger iii. you're making implicit use of
the certain intervals in kirnberger iii, which need not hold in any
well-temperament. in terms of fifths, well-temperaments eliminate the
pythagorean comma, of the 'semicolon' variety: 531441;524288.

the analogy you tried to make in the first paragraph is also flawed
because while meantone does its cylinder thing through *eliminating*
81;80
(semicolon used because the actual value of the ratio 81:80 is
irrelevant -- it's just the result of stacking the constituent
consonant intervals that is relevant), the kirnberger iii schisma
you're pointing to is a *literal* ratio 32805:32768.

> this audibly-identical equivalence of the chromatic notes in
> a well-temperament is quite different from the implied ratios
> of the tempered notes (which in Kirnberger III are all "white
> key" notes),

where are you getting that from?

> which employ tempering by a fraction of a comma
> (either Pythagorean or syntonic, depending on the WT),

why does it depend on the well-temperament?

> which is a much larger interval than a skhisma, to use as
> a unison-vector.

any 12-tone well-temperament can be understood as having the unison
vectors {syntonic comma, diesis} or {syntonic comma, schisma} or
{syntonic comma, diaschisma} or {syntonic comma, pythagorean comma}.
all these ways are equivalent. what differs is how the temperament
errors are distributed among the various intervals, not which commas
are tempered out.

🔗monz <joemonz@yahoo.com>

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, February 22, 2002 11:47 AM
> Subject: [tuning] Re: Werckmeister etc.
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > the important difference between well-temperaments and meantones
> > is that meantones depend on the disappearance of the syntonic
> > comma 81:80 = [2 3 5]**[-4 4 -1] = ~22 cents to turn the flat JI
> > lattice into a closed cylinder, whereas well-temperaments depend
> > on the ~2-cents fudging allowed by the skhisma = 32805:32768 =
> > [2 3 5]**[-15 8 1], to do the same thing.
>
> if you'll permit me to correct you, monz, the latter is true of
> schismic temperaments (such as van zwolle and 17-tone medieval arabic
> and 24-tone helmoltzian and 53-tone gabat-saribaldi), not well-
> temperaments. well-temperaments turn the flat lattice into a torus,
> and the simplest pair of commas for this purpose are the syntonic
> comma and the diesis. though you could just as well use the syntonic
> comma and the schisma if you wish -- it's equivalent.

ok, apparently much of what i tried to generalize about well-temperaments
only holds for Kirnberger III and not the others. in Kirnberger
it's definitely the syntonic comma and the schisma that will be
the most noticeable to disappear, it seems to me. more below...

> > thus, for example (i'm using Kirnberger III here instead of
> > Werckmeister IV), if a well-temperament is tuned as a chain
> > of "5ths" (some Pythagorean, some tempered) from Db to F#,
> > the C# which would be a 3:2 above F# is only a skhisma away from
> > the Db which started the chain.
>
> this is _only_ true for kirnberger iii. you're making implicit use of
> the certain intervals in kirnberger iii, which need not hold in any
> well-temperament.

precisely why i specified that i was using Kirnberger III as my
example and not the Werckmeister III that was already under discussion.

> in terms of fifths, well-temperaments eliminate the
> pythagorean comma, of the 'semicolon' variety: 531441;524288.

ah ... not all of them. Kirnberger eliminates the syntonic comma.

> the analogy you tried to make in the first paragraph is also flawed
> because while meantone does its cylinder thing through *eliminating*
> 81;80
> (semicolon used because the actual value of the ratio 81:80 is
> irrelevant -- it's just the result of stacking the constituent
> consonant intervals that is relevant), the kirnberger iii schisma
> you're pointing to is a *literal* ratio 32805:32768.

that's exactly right ... and it's because the schisma is s o
small that it disappears on its own, without any help from
the tuner. as i pointed out, the <2-cent size of the schisma
lies outside the resolution of a human piano tuner.

paul, this has something to do with the confusion over
exactly what i mean by a xenharmonic bridge. in the case
of Kirnberger III as i'm describing here, the schisma is
d e f i n i t e l y a xenaharmonic bridge. it forms
a bridge between different parts of the lattice without
any artificial human construction; the latter is the case
with meantones and other temperaments where a perceptible
interval as large as a comma or diesis is tempered out.

> > this audibly-identical equivalence of the chromatic notes in
> > a well-temperament is quite different from the implied ratios
> > of the tempered notes (which in Kirnberger III are all "white
> > key" notes),
>
> where are you getting that from?

that's how Kirnberger III works: in "8ve"-equivalent terms,

if C = n^0,

- the 4 "5ths" between C and E are tuned in 1/4-comma meantone:
G = 5^(1/4), D = 5^(1/2), A = 5^(3/4), E = 5^1;

- all the rest of the "5ths" are tuned Pythagorean down from
C and up from E: Db...C = 3^(-5...0) and E...F# = 3^(0...2) * 5 .

So the next "5th" above F# would be C# 33 * 5 = ~92.17871646 cents,
which is only a skhisma (~2 cents) higher than Db 3-5 = ~90.22499567 cents.

> > which employ tempering by a fraction of a comma
> > (either Pythagorean or syntonic, depending on the WT),
>
> why does it depend on the well-temperament?

as i explained here, Kirnberger III tempers out the syntonic
comma, while most other well-temperaments temper out the
Pythagorean comma.

> > which is a much larger interval than a skhisma, to use as
> > a unison-vector.
>
> any 12-tone well-temperament can be understood as having the unison
> vectors {syntonic comma, diesis} or {syntonic comma, schisma} or
> {syntonic comma, diaschisma} or {syntonic comma, pythagorean comma}.
> all these ways are equivalent. what differs is how the temperament
> errors are distributed among the various intervals, not which commas
> are tempered out.

right. somewhere in my webpages i really need to start emphasizing
that in the case of temperaments, there are many equivalent sets
of unison-vectors like this.

-monz

_________________________________________________________
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Get your free @yahoo.com address at http://mail.yahoo.com

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:

> ok, apparently much of what i tried to generalize about well-
temperaments
> only holds for Kirnberger III and not the others. in Kirnberger
> it's definitely the syntonic comma and the schisma that will be
> the most noticeable to disappear, it seems to me.

that's completely dependent on the music, not the tuning.

> > in terms of fifths, well-temperaments eliminate the
> > pythagorean comma, of the 'semicolon' variety: 531441;524288.
>
>
> ah ... not all of them. Kirnberger eliminates the syntonic comma.

monz, *all* of them eliminate the syntonic comma, and *all* of them
eliminate the pythagorean comma.
>
> paul, this has something to do with the confusion over
> exactly what i mean by a xenharmonic bridge. in the case
> of Kirnberger III as i'm describing here, the schisma is
> d e f i n i t e l y a xenaharmonic bridge.

well, it's a unison vector in *any* well temperament.

> it forms
> a bridge between different parts of the lattice without
> any artificial human construction; the latter is the case
> with meantones and other temperaments where a perceptible
> interval as large as a comma or diesis is tempered out.

ok, so it's a special instance of a unison vector, where its
constituent intervals are all just (yet another putative definition
of xenharmonic bridge). but this only happens in one place in
kirnberger iii. functionally, there are lots of other schismas
vanishing, in this and all other well temperaments.

> > > this audibly-identical equivalence of the chromatic notes in
> > > a well-temperament is quite different from the implied ratios
> > > of the tempered notes (which in Kirnberger III are all "white
> > > key" notes),
> >
> > where are you getting that from?
>
>
> that's how Kirnberger III works: in "8ve"-equivalent terms,
>
> if C = n^0,

aha! *that's* the key assumption you have to make. but isn't it just
as valid to use G, or any other pitch as the 'center'?

> > > which employ tempering by a fraction of a comma
> > > (either Pythagorean or syntonic, depending on the WT),
> >
> > why does it depend on the well-temperament?
>
>
> as i explained here, Kirnberger III tempers out the syntonic
> comma, while most other well-temperaments temper out the
> Pythagorean comma.

again, *all* 12-tone well-temperaments temper out *both* the syntonic
comma and the pythagorean comma. gene, what do you think?

> > > which is a much larger interval than a skhisma, to use as
> > > a unison-vector.
> >
> > any 12-tone well-temperament can be understood as having the
unison
> > vectors {syntonic comma, diesis} or {syntonic comma, schisma} or
> > {syntonic comma, diaschisma} or {syntonic comma, pythagorean
comma}.
> > all these ways are equivalent. what differs is how the temperament
> > errors are distributed among the various intervals, not which
commas
> > are tempered out.
>
>
> right.

whew!

🔗genewardsmith <genewardsmith@juno.com>

🔗paulerlich <paul@stretch-music.com>

🔗Afmmjr@aol.com

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., Afmmjr@a... wrote:
> In a message dated 2/22/02 4:35:50 PM Eastern Standard Time,
> paul@s... writes:
>
>
> > >
> > > that's how Kirnberger III works: in "8ve"-equivalent terms,
> > >
> > > if C = n^0,
> >
> > aha! *that's* the key assumption you have to make. but isn't it
just
> > as valid to use G, or any other pitch as the 'center'?
> >
> >
>
> Actually, it not musically valid to shift the starting pitch for a
well
> tempered tuning. Changing the starting note for a tuning
completely
> disregards the composer's choice of intervals (as each key is
different).

johnny, you're completely misinterpreting me. i'm *not* talking about
transposing werckmeister up a fifth, or anything like that. with what
i'm suggesting, the intervals between any pair of pitches would
remain the same. all the pitches remain the same. the structure of
the tuning, and the sound of the music, remains unchanged. it's just
that monz is pinning the 1/1 ratio on c, and drawing far reaching
conclusions from that, which i'm arguing don't have much meaning.

> Re Werckmeister IV, V, & VI, perhaps Paul, you can do some focus
on these 3
> orphan scales. To my knowledge, they are not meant to be a full
circle. Do
> you disagree?

from the data i presented, they sure seem to be full circles,
*literally*. they do have a few very slightly *sharpened* fifths but
this was common for example in french temperament ordinaire. please
present alternative data if you have any. perhaps werckmeister said
he intended them only for use in certain keys? please fill us in.

> Remember, Werckmeister invented the circle in music regarding
> 12 major and minor keys. He doesn't focus on these 3 scales much
at all. Of
> course, anything you find out about them will be of the utmost
interest.

i'd be happy to produce interval matrices for these or anything else
you'd like to see. manuel, can you confirm the numbers for these
tunings? it seems the page i presented did not bother to distinguish
between the size of the just syntonic comma and the size of the just
pythagorean comma, but they are slightly different.

Re: [tuning] Re: Werckmeister etc.

🔗Afmmjr@aol.com

🔗monz <joemonz@yahoo.com>

🔗paulerlich <paul@stretch-music.com>

🔗paulerlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

🔗paulerlich <paul@stretch-music.com>

🔗genewardsmith <genewardsmith@juno.com>

🔗paulerlich <paul@stretch-music.com>

🔗Afmmjr@aol.com

🔗paulerlich <paul@stretch-music.com>

🔗manuel.op.de.coul@eon-benelux.com