source file: m1437.txt
Date: Thu, 4 Jun 1998 10:45:07 -0500

Subject: Re: magic chord

From: "Benjamin Tubb" <brtubb@cybertron.com>

On Wed, 3 Jun 1998 14:37:35 -0400, Paul H. Erlich wrote:

>I was playing around with a harmonica sound on my Ensoniq VFX-SD tuned
as close as possible to JI (the scale being the JI version of the
pentachordal decatonic scale of my paper, hence 12 notes per octave with
two 50:49 "commas"). Playing the chord 1/1 5/4 3/2 7/4, and dropping the
3/2 down to 7/5, I didn't hear much of an increase in dissonance. After
some confusion, I realized that the only potentially dissonant interval
in this chord is the 28:25 (196 cents) between 5/4 and 7/5.

The difference between 5/4 and 7/5 is 23:20 which is 241.961 cents.

>Now the
Ensoniq's tuning tables allow cents values for each note, and I had
approximated the other intervals in such a way that the 28:25 was
nominally represented by 199 cents.

28:25 coverts to 196.198 cents.

>Since the true tuning resolution of
the Ensoniq VFX-SD is 512 notes per octave, this interval was probably
represented by 85/512 octave, or 199.2 cents.

85[+512] : 512 converts to 265.905 cents.

>This is only 4.7 cents off
a just 9/8, which explains the relative consonance of the chord.

9:8 converts to 203.91 cents which is a difference from 265.903 of 61.993
cents.

>It is a
near miss to a saturated 9-limit chord. A true 28/25 would be 7.7 cents
off a just 9/8, which would be quite a bit more dissonant.

28:25 converts to 196.98 cents which is a difference from 203.91 (for 9:8) of
6.93 cents.

>The difference between 28/25 and 9/8 is 225/224, a fundamental
comma-like interval in Fokker's writings.

The difference between 28:25 and 9:8 is 201:200.

I use the following Mathematica formulas and its Rationalize[] function for all
conversions.

cent2rat[cents_]:=10.^((Log[10,2]/1200)  cents)
rat2cent[ratio_]:=1200/Log[10,2] Log[10,ratio]

-------------
Benjamin Tubb
brtubb@cybertron.com
http://home.cybertron.com/~brtubb