source file: mills2.txt
Date: Tue, 14 Nov 1995 08:54:57 -0800

Subject: How common was the quasi-pythagorean `intonation'

From: Lars Frandsen <lafrance@musik.auc.dk>

The quasi-pythagorean `intonation' (note the quote-marks) is not
really an intonation, but more the formation of certain chords, scales
etc. within the usual pythagorean intonation.

Tuning twelve pure fifths from Gb to H one has acces to four
close-to-pure major thirds:

     Gb   Db   Ab   Eb   Bb    F    C    G    D    A    E   B
    (F#) (C#) (G#) (D#)                       |    |    |   |
     |_____|____|____|________________________|    |    |   |
           |____|____|_____________________________|    |   |
                |____|__________________________________|   |
                     |______________________________________|

These thirds (actualy diminished fourths) differ from the 5/4-thirds
by a skisma, that is the ratio pyth. vs. synth. comma, which is a very
small interval, less than 2 ct.

So when the need/heed for consonant triads evolved, and the pure pythagorean 
intonation still governed, one may expect to find certain scales and
chords more `in use' than others.

Now, according to Mark Lindley, there is `coroboration musical
evidence, mostly from the first half of the [15.] century', that
`exploit the especially resonant triads of the F#><B-disposition'
(shown above).

This quote is from The New Grove Dict. of Music and Musicians, in his
article on the pythagorean intonation. Lindley also mentions this
`intonation' in the book `Mathematical models of musical scales'
(Verlag f\"ur systematische Musikwissenschaft GmbH., Bonn 1993), but
none of these resources seems to cover, how common this choice of
tonality (which the evidence must show) was.

Lindley diplays some examples, but it is not clear, to what extend
there is statistical inference on the matter. So I would like some
feedback on:

 -   Was it a `well-known phenomenon' by then, and
 -   just how common was this practice among composers and musicians?

Sincerely,

Lars Frandsen.
lafrance@musik.auc.dk

Institut for Musik og Musikterapi
Kroghstraede 6
DK-9220 Aalborg Ost
Tlf. +45 98 15 42 11 - lokal 3111


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