Newsgroups: rec.music.classical
Date: Wed, 27 Jan 1993 17:47:45 -0500 
From: "Ned S. VanderVen" <nsv+@andrew.cmu.edu>
Subject: Re: Define "Well Tempered"

Excerpts from netnews.rec.music.classical: 23-Jan-93 Re: Define "Well
Tempered" by William Alves@calvin.usc
 
>Inharmonicity is the tendency for the harmonics of stretched strings to
>get gradually more sharp due to the stiffness of the material. 

True enough, but it might be better to call them "higher frequency
partials" and say that they become gradually sharper than the harmonics
of the fundamental, reserving the term "harmonic" for integer multiples
of the fundamental. 

>Because our ear determines pitch both through the pitch of the
fundamental >and the rest of the partials, inharmonicity can make a
difference to the
>perceived pitch, depending on the relative loudness of the partials.

Also correct, and the perceived pitch will be slightly sharp compared
with a pure tone of the same fundamental frequency.
 
>If the fundamental is relatively weak and higher partials are relatively
>strong, the string will have to be tuned slightly sharp of the fundamental
>pitch in order to sound in tune with other strings.

Intervals, particularly octaves, are indeed tuned sharp, but *not*
because of the perceived pitch. If it were really a question of the
perceived pitch of a cluster of increasingly sharper partials, one would
have to conclude that octaves should be compressed rather than
stretched. Suppose that, because of inharmonicity, a string tuned to a
fundamental of A4 = 440 Hz has a perceived pitch that is matched to a
pure tone of 441 Hz. Because inharmonicity gradually increases for the
higher notes, an A5 tuned to a fundamental of 880 Hz will have a
perceived pitch somewhat higher than twice the frequency of the
perceived pitch of the 440 Hz A4, i.e, slightly greater than 882 Hz. The
perceived pitch explanation suggests that the A5 fundamental should be
tuned a little flatter than 880 Hz in order to bring the perceived pitch
down to 882 Hz. 

Octaves are stretched because the primary tuning cue comes from the
beating of the second partial of the lower note with the fundamental of
the higher note. Because the second partial of a 440 Hz A4 is slightly
greater than 880 Hz, the fundamental of A5 must be tuned slightly sharp
to decrease the beat between these two components when the notes are
sounded together. Of course there will be beats between other components
as well, so the net effect of the tuner's effort, which is to achieve
the best acoustical result, may not tune the fundamental of A5 exactly
to the (sharp) second partial of A4. Nevertheless, the fundamental
frequency of A5 will definitely be greater than twice the fundamental
frequency of A4.    


>On the piano this means that the higher pitches are tuned increasingly
sharp >of true equal temperament ... as one moves up the keyboard.

Depends on where you start moving "up the keyboard". Relative to the
notes in the middle of the keyboard, successively higher notes are tuned
increasingly sharp. If one moves up from the low end of the keyboard,
the stretching of the octaves decreases so that, although the higher
pitches are tuned sharp, the sharpness decreases as one approaches the
middle of the keyboard. Once the gamut is set in the middle of the
keyboard the tuner works outward in both directions. It may be better to
say that the stretching is least in the middle of the keyboard where the
inharmonicity is least, and that the stretching increases as you move
away from the middle in either direction. Moving upward from the middle
the treble is tuned increasingly sharp; moving downward from the middle
the bass is tuned increasingly flat.  

>The effect exists but is far less noticeable on the harpsichord and other
>string instruments.

Smaller diameter strings reduce the inharmonicity, lower tension
increases it, but the reduced diameter dominates for harpsichords and
other 
instruments.
  
Ned VanderVen
Physics
Carnegie Mellon U.