Subject: RE: Unequal temperament
From:
Peter K (Fionn)
Date: 15 Dec 04 - 11:09 AM
s&r's selective history implies that this is a relatively recent issue, but in fact the problem was well understood by Pythagoras. It comes down to fairly basic maths (OK math) which I will try briefly to explain, since I don't think other posts have really dealt with it.
In western music the basic building block is the semitone. A semitone is one twelfth of an octave. Thus, according to the currently prevailing standard for concert pitch, the A above "middle" C vibrates at 440Hz; and the A one octave higher than that vibrates at 880; the gap between these two As is filled with 11 other notes, the intervals between which are known as semitones.
A string which sounds A (440) when open will sound an octave higher - A'- if stopped halfway along its length. (Just as a 4ft organ pipe will be pitched one octave higher than an 8ft pipe, other things being equal.) Stop the string at the 2/3 point and the note produced will be E.
The interval from A to E is known as a fifth. After the octave itself, the fifth is the second strongest interval in the harmonic sequence - the pattern of overtones heard when any note is sounded, or the notes that can be derived from a fixed-length tube. The fifth has a "pure" quality about it that sounds almost hollow. So much so that classical and romantic composers for orchestra, organ, piano etc, usually sought to avoid successive fifths in harmony (except for special effect, as in Vaughan- Williams' orchestration of Greensleeves for instance). This interval is instantly recognisable to the tuned ear, and to many untrained ears too. Indeed violinists - once they have tuned the A string to an external source (tuning fork, piano, oboe or whatever) - use the unique sound of the fifth to tune the three other strings, G, D and E. (A feat they usually accomplish within a second or two for each note.)
But from A, move to its fifth, then the fifth above that and so on, until you come to the next A. The note you arrive at will have a frequency that is not divisible by the 440 you started from, because it is an unequal number of octaves removed from the original A. You have arrived at a note that would sound noticeably out of tune.
For an instrument confined to a range of an octave or so, this problem is not likely to be a factor, in the same way as it would suffice to divide a year into 365 days if there was only one year. But as the range extends, the problem becomes increasingly evident and is particular evident on piano and organ, with their ranges of several octaves.
If a sequence of notes is tuned in conformance with harmonic intervals, there will come a point where a significant correction is needed - an interval so unnatural to the ear as to be unusable. Such an interval is known as a "wolf tone". But if the mathematical discrepency is shared equally across all intervals, harmonies will be compromised.
Tempering, be it well tempering, meantime tempering or whatever, are all attempts to square the circle. In effect they are musical equivalents of the leap year. Some systems give priority to the most crucial intervals - fifths and major thirds. Some favour particular keys at the expense of others. Hence a tendency for some instruments to sound better in one key than another, and in the classical era for subjective values such as "warm," "brilliant," "plangent" etc to be attached to specific keys.
Instruments of infinite pitch - the strings and trombones in an orchestra, and of course the human voice - can arrive at compromises almost on a note-by-note basis, but other instruments are contrained to their pre-determined semitones, and it is between these, as someone observed, that discrepencies will sometimes be noticed (but not by me, because my ear is not good enough).
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sj
