new FAQ item

Fresh out of the flames on MMM comes this new FAQ entry
by yours truly. Comments welcome.

-----------------------------------------------------------------

Q: What is "just intonation"?

A: Generally, it means tuning instruments so that pitches are
related to one another by simple ratios. For example, A = 440 Hz
and E = 660 Hz are related by the ratio 3/2. In equal
temperament, this A-E "perfect fifth" is very close to 3/2, but
differs by a precise amount so that the circle of fifths returns
us to the pitch we started on.

There are three sources of disagreement over the precise meaning
of the term, however. First, What qualifies as a simple ratio?
Is 19/16 simple? Historically, only ratios that can be factored
with primes no greater than 5 were considered simple.
Contemporary authors tend to be more inclusive, but the phrase
"extended just intonation" is sometimes used to indicate that
primes greater than 5 are being allowed.

Second, What tuning accuracy is required? If we have 440 Hz and
660.2 Hz, does it count as 3/2? One answer is that if the tuning
error is being applied intentionally (in a systematic way), then
it should not be considered just intonation. If instead the
error is a result of the limited accuracy of an instrument (and
is therefore somewhat random over the range of the instrument),
we can still consider the result to be just intonation.

Finally, Which intervals should be considered when more than two
pitches are involved? The classical "just intonation" scale has
the pitches: 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1. It contains
chords like 4/3 5/3 2/1, which simplify to 4:5:6 and which are
certainly just intonation chords. But it also contains the chord
9/8 4/3 5/3, which simplifies to 27:32:40 and sounds quite harsh
if a consonant minor triad is expected. Nevertheless, it is
often implied that whatever combination of pitches we pluck from
a scale like the one above, we are still in just intonation.

An alternative is to consider just intonation to be a property of
chords (simultaneous pitches) only. Scales can then be discussed
in terms of how many justly intoned chords they contain, what
portion of their triads are justly intoned, etc. Except for the
harmonic series itself (and only the first few octaves thereof),
all rational scales with more than a handful of pitches/octave
contain chords that fail classical tests of just intonation, such
as being easy to tune by ear.

-Carl