Re: Digest Number 303

On 4 Sep 1999 tuning@onelist.com wrote:

> Message: 6
> Date: Fri, 3 Sep 1999 20:59:56 +0000
> From: kopiez@hmt-hannover.de
> Subject: ratios of intervals in different tunings
>

Hello, there, and here's a revision of your table showing the Pythagorean
intervals in relation to 12-tone equal temperament (12-tet):

> Interval JI5 pyth JI7
> --------------------------------------------------------------
maj3 5/4 81/64 5/4
vs. 12-tet -13.7c 7.8c

min3 6/5 32/27 6/5
15.6c -5.9c

min7 9/5 16/9 7/4
17.6c 4.1c -31.2c

Please note that these are regular intervals in these tunings -- in
Pythagorean or 3-limit JI, for example, an augmented sixth is very close
to the 9/5 minor seventh of 5-limit JI, while the augmented second and
diminished fourth are likewise close to the 5-limit minor and major
thirds. In a 5-limit tuning, either 9/5 or 16/9 (the latter shared with
Pythagorean) may be used for the minor seventh, but I'd say that 9/5 (a
3/2 fifth plus a 6/5 minor third) is more typical, and also somewhat more
tense.

(Interestingly, an augmented sixth in 5-limit is very close to 7/4.)

Thus the more active thirds of 3-limit (81:64, 32:27) fit the unstable
role of these intervals in Gothic European music (up to around 1450),
while the rather tense minor seventh of 5-limit (9:5) serves nicely as a
dissonance in Renaissance music.

Most respectfully,

Margo Schulter
mschulter@value.net