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From: rbp@well.UUCP (Bob Pasker)
Newsgroups: comp.music,ba.music
Subject: Re: Tuning (e.g. pianos)
Message-ID: <15132@well.UUCP>
Date: 20 Dec 89 21:40:58 GMT
References: <129476@sun.Eng.Sun.COM>
Reply-To: rbp@well.UUCP (Bob Pasker)
Organization: Bob Pasker, Software & Communications Consultant
Lines: 77

There's a problem with talking about C vs. C# because C# is "naturally" a
minor key, i.e. C#-minor is the relative minor of the key A-major and
A-major is a minor sixth above the C-major.  This means that they are
relatively distant from each other.  We'll see what that means in a minute. 

The method of well-tempered tuning gives keys which are "distant" to the
tuned-key (usually C) "better" sound. Originally, keyboards (harpsichords,
fortepianos and pianos) that were not well-tempered only sounded "good" in a
few keys, those "close" to the key which they were _usually_ tuned for:
C-major. 

In order of distance to the key of C-major we have:

	  / sharps: C, G, D, A, E, B, F#/Gb   \
closest -<				       > - furthest
	  \ flats:  F, Bb, Eb, Ab, Db, Gb, Cb /

So, on non-well-tempered keyboards, you could probably play in the keys of:
C, G, F, Bb and sometimes D and Eb without too much dissonance. 

The minor keys also sound fine if their relative major key was close to the
tuning: 

	Major	Relative Minor
	C	A-minor
	G	E-minor
	D	B-minor
	F	D-minor
	Bb	G-minor
	Eb	C-minor

With well-tempered tuning, a harpsichordist coud play in distant keys, like
F#, and Cb without "too much" dissonance. 

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).  It is easy to see how the effects of the mathematics of this are
multiplied (pun intended!) when the tuning gets further away from the key
which the piano is tuned for, since each successive key is a fifth away from
the previous one.

For example:

(note: the "prime" symbol (') indicates the number of octaves above middle
C. D' is the D note in the first octave above the octive containing middle
C.) 

    Suppose C     is of frequency    x
    then    G     is of frequency   2x/3    since it's a fifth above C.
    and     D'    is of frequency   4x/9    similarly
	    A'    is of frequency   8x/27   similarly
	    E''   is of frequency  16x/81   similarly
	    B''   is of frequency  32x/243  similarly
	    F'''  is of frequency  64x/729  similarly
	    C'''' is of frequency 128x/2187 similarly

    therefore
	    C''' must be of frequency  256x/2187, twice C'''' (128x/2187)
    and     C''  must be of frequency  512x/2187
    and     C'   must be of frequency 1024x/2187
    and     C    must be of frequency 2048x/2187

    but yet we know that C is x, not 2048x/2187 (.93644262x)

So, if C is x, C'''' should be 4x on an instrument tuned in the key of C not
some approximation. By adjusting the frequency of the notes so they are not
exactly what the harmonic progression wants, but close enough, you get a
keyboard that is out of tune in all keys but sounds "good" in all of them.

Hope this clears things up.
-- 
- bob
;-----------------------------------------------------------------
; Bob Pasker                            | rbp@well.sf.ca.us
; San Francisco, CA			| +1 415-695-8741