Werkmeister #3

[Richard Moody]

Curious as to his source.  Does he  mean Kirnberger III? That is mentioned in New
Groves.
 
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} 

** 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 
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 
exactly "slightly tempered".   The narrowest third -would- seem to be wide by a 
quarter comma  as tempering three 5ths by 1/3 comma gives a full comma (3/3) at A, 
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 = 4/4 = pure Third
when 
Fifths are tempered 1/4 comma)

HOWEVER when the cents are added up, The C--E comes out at 386--387 which 
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.
  
On the aural side, I am wondering if they thought three 1/3 comma Fifths gives a 
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.  
 So adding up the cents,  the 1/3 comma fifth is aprox 695 cents.   
 Three of these minus 1200 gives 885. A 5/3 Sixth is 884.3587.  Thats "99 1/2 pure" 

	From this A a pure Fifth (702!!)  (E) gets 885 + 702 - 1200 = 387 cents. (C--E) 
again close enough to call it pure. C--F is supposed to be pure, that would give
498.   885 - 498 = 387. 

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

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..... ; ) 
----------
> From: dale  r fox <foxpiano@juno.com>
> To: pianotech@ptg.org
> Subject: Werkmeister #3
> Date: Wednesday, September 01, 1999 8:47 PM
> 
> 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?
> 
> Thanks in advance.
> 
> Dale Fox
> 
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