Yahoo Tuning Groups Ultimate Backup tuning Kirnberger III (was: Werckmeister etc.)

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, February 22, 2002 1:33 PM
> Subject: [tuning] Re: Werckmeister etc.
>
>
> --- 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.

i'd have to disagree: i'd say that it's l a r g e l y
dependent on the music, but that the tuning also plays
a significant role. the disappearance of the skhisma
and the syntonic comma are built right into Kirnberger III,
so there's no way that could be missed.

> > > 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.

somehow i just knew that you were going to take that the wrong
way, and i almost reworded it and am now sorry i didn't.

what i meant is that the t u n i n g procedure of Kirnberger III
flattens a few of the 5ths by a fraction of the s y n t o n i c
comma, whereas the vast majority of other well-temperaments
flattens the 5ths by a fraction of 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.

right, i understand that. that's part of my own explanation
of Kirnberger III. see below.

> > > > 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'?

ok, paul, i see now that we weren't on the same wavelength here.

with this business about diatonic/white-key and chromatic/black-key,
i'm not talking about C as a key center, i'm simply assuming the
traditional method of tuning a piano in Kirnberger III.

The tuner starts by tuning C, then the four tempered 5ths/4ths
which result in the major 3rd C:E = 5:4, then a series of 4:3s
and 3:2s from C to Db, then finally from E up a 3:2 to B then
down a 4:3 to F#.

the Db is only a schisma wider than a 3:2 C# above F#, and
assuming a tuning resolution by the human tuner no greater
than ~2 cents, the Db and C# a r e the same, and the cycle
is closed at 12 tones.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, February 22, 2002 3:04 PM
> Subject: [tuning] Re: Werckmeister etc.
>
>
> > > > [me, monz]
> > > > that's how Kirnberger III works: in "8ve"-equivalent terms,
> > > >
> > > > if C = n^0,
> > >
> > > [paul]
> > > 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'?
> > >
> > >
> >
> > [Johnny Reinhard]
> > 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).
>
> [paul]
> 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.

but as johnny says, paul, my approach here is the valid one
for well-temperament. i'm not sure what you're suggesting,
because composers who used well-temperaments expected each key
to sound a certain way, and that only works if the tuning has
a well-defined a b s o l u t e structure.

it's not the same as with meantone, where the generator is
always the same and a linear chain results. well-temperaments
are irregular, and in Kirnberger III the keys only have the
effect that composers expected them to have if you tune it
the way i described, with specific intervals associated with
each letter-name and/or letter-name plus # or b.

well-temperaments have a rich variety of specific sizes
for generic interval-classes. For example, Kirnberger III has :

Minor 3rds
----------

~cents [3 5] vector instances

310 [ 0 -3/4] E:G, A:C
305 [-1 -1/2] D:F, B:D
300 [-2 -1/4] F#:A, G:Bb
296 [ 5 1 ] Db:E, Eb:F#, Ab:B
294 [-3 0 ] C:Eb, F:Ab, Bb:Db

because of the irrelevance of the skhisma, the 296-cent interval,
[5 1], which is the JI ratio 1215:1024, can be interpreted as
several different instances of the Pythagorean minor 3rd 32:27
= [-3 0] = ~294 cents, as follows:

Db:E = ~C#:E,, Db:~Fb
Eb:F# = ~D#:F#, Eb:~Gb
Ab:B = ~G#:B, Ab:~Cb

note that the intervals F#:A and G:Bb are basically identical
to the 12edo minor 3rds.

we can see that there's an even richer variety of major 3rds:

Major 3rds
----------

~cents [3 5] vector instances

408 [ 4 0 ] Db:F, Ab:C
406 [-4 -1 ] E:Ab, F#:Bb, B:Eb
402 [ 3 1/4] Eb:G
400 [-5 -3/4] A:Db
397 [ 2 1/2] D:F#, Bb:D
392 [ 1 3/4] F:A, G:B
386 [ 0 1 ] C:E

because of the irrelevance of the skhisma, the 406-cent interval,
[-4 -1], which is the JI ratio 512:405, can be interpreted as
several different instances of the Pythagorean major 3rd 81:64
= [4 0] = ~408 cents, as follows:

E:Ab = E:~G#, ~Fb:Ab
F#:Bb = F#:~A#, ~Gb:Bb
B:Eb = B:~D#, ~Cb:Eb

note that the interval A:Db is basically identical to the
12edo major 3rd. also note that in the case of the major 3rd,
there are also intervals which are only +2 and -3 cents different
from this one: Eb:G and (D:F#, Bb:D) respectively. i would
say that all of this would automatically be equivalent as well,
because they are also near the limit of tuning resolution.

so, equivalences come into play depending on which note
you pick as a key center, but the template for tuning
Kirnberger III doesn't change, so the true interval
relationships b e t w e e n n a m e d n o t e s
will always be as i describe here (plus or minus a few
cents due to limits of human tuning resolution).

the fact that the size of the skhisma, as well as the
other very small differences which arise in connection with
the tempered major 3rds, are all already below the possible
tuning resolution, is why i can confidently invoke the
equivalences in the tables above.

-monz

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Kirnberger III (was: Werckmeister etc.)

🔗monz <joemonz@yahoo.com>

🔗paulerlich <paul@stretch-music.com>