From: alves@calvin.usc.edu (William Alves)
Newsgroups: rec.music.classical
Subject: Re: Define "Well Tempered"
Date: 23 Jan 1993 18:13:09 -0800
Organization: University of Southern California, Los Angeles, CA
NNTP-Posting-Host: calvin.usc.edu

In article <20k3!jq@rpi.edu> mccomt@aix.rpi.edu (Todd Michel McComb) writes:

>Indeed.  Many well-tempered usages in modern early music recordings have
>concentrated on versions in which thirds are given preference; as this
>relates to specific intervals in a large group of keys, would you say
>that this fits into your definition of well temperament or perhaps as a
>variation on your definition of circle temperament?

Most composers in the baroque seem to have preferred very consonant thirds,
especially major thirds. (Note that the major and minor thirds in equal
temperament are pretty far away from just thirds, while fourths and fifths
are pretty close). This is why probably the most popular tuning system of
the time was 1/4-comma meantone.

Meantone is a type of temperament in which the tuner progresses by fifths
or fourths, but "tempers" the fifth by a certain amount. The 1/4-comma
refers to the amount of temperament - 1/4 of the syntonic comma or the
difference between 4 consecutive 3/2's (fifths) and a 5/4 (major third).
If you start at C and add just fifths until you get to E, your E will be
about 21.5 cents sharp of a just major third C-E. If instead you subtract
about 5 cents every time you tune a fifth, you will end up with slightly
flat fifths, but a near-perfect third. Thus in 1/4-comma meantone, the
thirds of the most commonly used keys are really sweet. However, by the
time you get around the circle of fifths, you have to pay the piper. You
will have a severely out of tune fifth - the so-called "wolf" - usually
put between C# and Gb. Keys which use that fifth are thus pretty unusable.

Well, circular, and meantone temperaments as classes don't say anything
about the preference for certain intervals. Instead, well simply means
all keys playable, but some the more common more playable. Circular means
all keys playable, but the differences are spread throughout the circle
of fifths.

>Perhaps you can go on to explain enharmonicty as it relates to practical
>tuning?  I am still not entirely clear on this.

Inharmonicity is the tendency for the harmonics of stretched strings to
get gradually more sharp due to the stiffness of the material. 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.

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. On the piano this
means that the higher pitches are tuned increasingly sharp of true equal
temperament relative to the fundamental as one moves up the keyboard. The
effect exists but is far less noticeable on the harpsichord and other
string instruments.

Because tuning systems are normally defined in the way that pitches relate
to one another within the octave, the piano is spoken of as having equal
temperament. Indeed, to our ears it does. However, if one defines equal
temperament in a much narrower, more acoustical way, then the piano is
not precisely equally tempered. That doesn't mean that it has any other
kind of named temperament, because they, too, are defined within a single
octave. Nor does it mean that there is more than one kind of equal tempera-
ment. It simply means that the piano has equal temperament with some
adjustments across the range of the keyboard.

Bill Alves
Newsgroups: rec.music.classical