historical tuning 1805

[Richard Moody]

I think the "old style tuning" he is describing is what today is called Quarter
Comma Meantone. When the term "quarter comma" is used, usually it can be attributed
to the practice of tuning four fifths flat 1/4 comma to produce a pure third. Four
fifths tuned pure exceed a pure third (+ 2 octaves) by the ratio 81/80.  This ratio
is known as the comma of Didymus. (also called the syntonic comma).   The
other comma is the Comma of Pythagorus which is the amount 12  5ths exceed 7
octaves.
 Now if the fifth is "weakened" by one twelth of a comma, we assume it is the
Pythagorean comma. This is 23.46 cents...divided by 12 = 1.955 cents, or the "two
cents" flat fifths of ET. 
	The two tunings are not on a par because of the the wolf Fifth, pure thirds with
their accomanying ultra wide thirds or augmented fourths in Meantone; compared to
the  less flattened fifths giving wider thirds in ET. A quarter of the syntonic
comma (21.5/4 cents) would make the fifths 5+  cents flat more than twice as flat as
the ET fifth.  
	When he refers to a "whole tone" it means the just intonation whole tone of 
which there are two, (9/8 and 10/9)  which might be why he mentions two divisions,  
1/32 and 1/40.   However the quarter comma  also produces a ratio that is equal to 
 the mean of the whole tones, (hence Meantone) or the square root of 9/8 times 10/9.
  (1.25^1/2) = 1.118, * so I am not sure why
he is concerned with 1/32 or 1/40. It might be though the comparison of the
reduction of the fifth in ET to the meantone fifth, but I don't know how the figures
are gotten.    ric.

* meantone fifth = 1.49535 .  Two of these fifths produce a Whole Tone + an octave.
This divided by two = 1.118.   (9/8=1.25.........10/9=1.1111111)


----------
> From: Stephen Birkett <birketts@wright.aps.uoguelph.ca>
> To: pianotech@ptg.org
> Subject: Re: historical tuning 1805
> Date: Wednesday, August 18, 1999 2:25 PM
> 
> Here's an interesting snatch. I'm working on the old-style German in the 
> rest.
> 
> Gall, Clavierstimmbuch, 1805, P.62
> 
> [...general discussion about the need for tempering...then goes on]
> "There are varying systems of temperament, particularly two: an old and a 
> new one. The old system tempers the fifths in an irregular (unequal) way: 
> in order to make eight large thirds exact, the old system weakens eleven 
> fifths by a quarter comma...and carries everything which, because of the 
> weakened fifths would still be lacking for the correctness of the octave 
> onto a single fifth ("Wolfsquinte"). This exaggerated fifth which one 
> surrenders lies on a not very common (frequently used) tone. It is 
> maintained that the fifths can be weakened by a quarter comma (1/32 or 
> 1/40 tone) and even more without forfeiting their harmony and it is 
> claimed that the old system is on a par with the new. The new temperament 
> system weakens all the fifths equally and, to be exact, weakens each by 
> only one twelfth of a comma (the 120th part of a whole tone). However all 
> large thirds are, of course, exaggerated and inevitably offend the ear. 
> For this reason, many do not like the new system - they find it too harsh 
> and less harmonious than the old one."
> 
> Food for thought here. N.B. "new", "old" and "present" are with respect to 
> 1805. 
> 
> Stephen
> 
> Stephen Birkett Fortepianos
> Authentic Reproductions of 18th and 19th Century Pianos
> 464 Winchester Drive
> Waterloo, Ontario
> Canada N2T 1K5
> tel: 519-885-2228
> email: birketts@wright.aps.uoguelph.ca
>