Path: utzoo!utgpu!jarvis.csri.toronto.edu!rutgers!cs.utexas.edu!sun-barr!decwrl!megatest!djones
From: djones@megatest.UUCP (Dave Jones)
Newsgroups: comp.music,ba.music
Subject: Re: Tuning (e.g. pianos)
Message-ID: <11435@goofy.megatest.UUCP>
Date: 21 Dec 89 03:03:59 GMT
References: <15132@well.UUCP>
Organization: Megatest Corporation, San Jose, Ca
Lines: 26

From article <15132@well.UUCP>, by rbp@well.UUCP (Bob Pasker):

> 
> The dissonance in these distant keys is introduced because of the
> requirement in western tuning that the dominant (or the "fifth") be 2/3rds
> the frequency of the tonic (i.e. A must be 2/3rds the frequency of C) and
> the octave above must be 1/2 the frequency (C' must be 1/2 the frequency of
> C).

Bob, Bob, Bob.

In Western, Eastern, and every other kind of music, the dominant is 3/2
the frequency of the tonic, not 2/3. "A" is not the dominant of "C", nor is
"C" the dominant of "A".  And C' is twice the frequency of of C, not half.

[Taking a deep breath...] The slight dissonance of well-tempered
chords is not due to the approximation of the fifth, which is virtually
perfect: 1.4983 as opposed to 3/2. It is mostly due to the bad approximation
of the the major third, 1.2599 -- it should be 5/4 -- and of
the minor third, 1.1892 -- it should be either 7/6 or 6/5, depending
on its function. Other intervals are also off somewhat, but those are
the ones that matter. In a major-7 chord, for example, you mostly hear the
seven in relation to the three and the five, not to the root. The seven,
having a ratio of 15/8 of the root, combines with the root to produce only
"buzz" and subsonic undertones. It combines with the third and the fifth
to produce resonance.