Werkmeister #3

[Mike Keener]

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Hello group.  This is my first entry to pianotech.  According to G.C. =
Klop, in his Harpsichord Tuning, page 18, in Werckmeister III, also =
known as the Correct Temperament No. 1, the four tempered fifths, c-g, =
g-d, d-a, b-f#, are each 1/4 comma narrow (Klop calls these "meantone =
fifths").  The remaining fifths are just.
=20
This temperament also appears in Jorgensen's The Equal-Beating =
Temperaments on page25.  Being an aural tuner, and not having the tables =
in front of me, I can't quote the cents deviation from ET.  I hope this =
helps.
=20
Mike
    -----Original Message-----
    From: Richard Moody <remoody@easnet.net>
    To: pianotech@ptg.org <pianotech@ptg.org>
    Date: Thursday, September 02, 1999 1:07 PM
    Subject: Re: Werkmeister #3
   =20
   =20
    Curious as to his source.  Does he  mean Kirnberger III? That is =
mentioned in New
    Groves.
   =20
    For Werckmeister they give "For the first four 5ths (C--G, G--D, =
D--A, B--F#) he
    used a tuning slightly tempered by a third of a comma, while the =
eight other 5ths
    remained pure....the 3rds varied from those of a quarter of a comma =
to high to
    others equalling Pythagorean tuning."   {New Groves}=20
   =20
    ** Note "the first four 5ths" are not the first four fifths of the =
cycle.  A--E, and
    E--B are passed by so they must be  of the "eight others"  thus  =
pure.  But the=20
    order of the eight others is not mentioned.  At any rate, 1/3 =
syntonic comma is
    is a Fifth 7.17 cents narrow from pure. or 5.2 cents from ET. (695)  =
That is not=20
    exactly "slightly tempered".   The narrowest third -would- seem to =
be wide by a=20
    quarter comma  as tempering three 5ths by 1/3 comma gives a full =
comma (3/3) at A,=20
    the third 5th.  Going a pure fifth to E  to make the C--E third,  =
would seem to to
    take it back to 3/4.  or 1/4 comma wider than pure.(full comma =3D =
4/4 =3D pure Third
    when=20
    Fifths are tempered 1/4 comma)
   =20
    HOWEVER when the cents are added up, The C--E comes out at 386--387 =
which=20
    is a pure Third.  Also the A makes a pure Third with F within a =
cent. If they are
    suppposed to be 1/4 comma "high" that would make them 391.
     =20
    On the aural side, I am wondering if they thought three 1/3 comma =
Fifths gives a=20
    pure Sixth? 5/3.  Tuning a 5/3 sixth from C, then tuning a pure 5th =
up to E, then an
    octave down should give a 5/4 Third (pure)(this would be C--E).  A =
pure Fifth from C
    also works out to a pure Third at A.  The only  tempering in this =
scheme is two
    notes G and D to "fit" into the 5/3 C--A . The other Fifths with one =
execption are
    pure.  THAT doesn't seem like too bad a tuning
    task. =20
    So adding up the cents,  the 1/3 comma fifth is aprox 695 cents.  =20
    Three of these minus 1200 gives 885. A 5/3 Sixth is 884.3587.  Thats =
"99 1/2 pure"=20
   =20
    From this A a pure Fifth (702!!)  (E) gets 885 + 702 - 1200 =3D 387 =
cents. (C--E)=20
    again close enough to call it pure. C--F is supposed to be pure, =
that would give
    498.   885 - 498 =3D 387.=20
   =20
    Perhaps I did some  math wrong, or the assumption is wrong that a =
pure fifth from a
    full comma gives a quarter comma off. I am hoping Dr. Coleman will =
have an Rx for my
    dilemma.---ric
   =20
    ps  A comma (syntonic) is how much wider from pure a Third is after =
a series of four
    pure (Pythagorean) Fifths.  But all tuners know that..... ; )=20
    ----------
    > From: dale  r fox <foxpiano@juno.com>
    > To: pianotech@ptg.org
    > Subject: Werkmeister #3
    > Date: Wednesday, September 01, 1999 8:47 PM
    >=20
    > List,  especially those HT specialists.  I have a customer who =
does some
    > concert work here in town who would like something similar to a
    > Werkmeister#3.  Is there a generic equivalent included on RCT's
    > historical temperments?
    >=20
    > Thanks in advance.
    >=20
    > Dale Fox
    >=20
    > =
___________________________________________________________________
    > Get the Internet just the way you want it.
    > Free software, free e-mail, and free Internet access for a month!
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    >=20


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