Re: Werckmeister and maximal proximity of intervals Digest Number 1915

Johnny, Manuel, Marc, Everyone,

Is it not the case that the largest major third in Werkmeister III
is a Pythagorean major third of 81 : 64 or about 408 cents
(e.g. Db to F) and the smallest is a bit more than a just major
third of 5 : 4 or about 386 cents (e.g. 390c F to A) ? The
difference between these is the syntonic comma, about 22 cents.
The four different major thirds, 390c, 396c, 402c, and 408c adopt
approximate quarter comma positions between these two extremes.
Similarly minor thirds adopt approximate quarter comma positions
of 294c, 300c, 306c and 312c between a Pythagorean 32 : 27 (294c)
and a just 6 : 5 (316c). I have always assumed that Werkmeister
planned it that way, seeking to improve on quarter comma mean tone
temperament.
(Perhaps that's what you just said, Manuel.)

Margo Schulter discusses this in more detail at
http://www.medieval.org/emfaq/harmony/pyth5.html#5

Peter Frazer,
www.midicode.com

--- In tuning@y..., Peter Frazer <paf@e...> wrote:

> Johnny, Manuel, Marc, Everyone,
>
> Is it not the case that the largest major third in Werkmeister III
> is a Pythagorean major third of 81 : 64 or about 408 cents
> (e.g. Db to F) and the smallest is a bit more than a just major
> third of 5 : 4 or about 386 cents (e.g. 390c F to A) ? The
> difference between these is the syntonic comma, about 22 cents.
> The four different major thirds, 390c, 396c, 402c, and 408c adopt
> approximate quarter comma positions between these two extremes.

this is exactly what everyone's been saying. so yes, it's the case.

> I have always assumed that Werkmeister
> planned it that way,

i'm sure he was aware of all these characteristics, yes.