Technical Library

TEMPERAMENTS XXIX: Equal Temperament

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Equal Temperament ©2004 CBH 4K gif
I guess we’ve come so far, we ought not be so dismissive of Equal Temperament.

I used to write that the concept of Equal Temperament is so old that the Chinese were talking about it in 2000BC: That may or may not be true. Jean Joseph Marie Amiot (1718–1793) was a Jesuit priest who spent the final four decades of his life in Beijing. His fifteen-volume Mémoires concernant l’histoire, les sciences et les arts des Chinois was progressively published in Paris. [Alexander Ellis, the verbose English translator of Hermann Helmholz’s On the Sensations of Tone noted “Amiot reports equal temperament from China long previously even to Pythagoras.”] It is now generally safer to ascribe the first true mathematical basis of Equal Temperament to Zhu Zaiyu (朱載堉, 1536–1611) in 1584—which is early enough!

The various early meantones limit modulation, making most of the remote keys with more than say three sharps or flats sound quite harsh or even unusable. The well temperaments will let you play in any key, with excellent intonation in frequently-used keys, and some differences of flavor from key to key as you move about.

Equal Temperament is regarded as the ultimate temperament by some because of the complete freedom to play in any key with impunity. Old as the concept may be, Owen Jorgensen believed that pianos have only been able to be accurately tuned in true Equal Temperament since the beginning of the twentieth-century, when Jerry Cree Fischer added minor thirds as test intervals. It’s really the mellow timbre of the modern piano which has allowed musicians to tolerate Equal Temperament’s rapidly beating Major thirds. By contrast, Equal temperament always sounds a bit rough on the harpsichord, and in my opinion is best avoided.

In theory, if we squeeze every fifth in our circle equally by a twelfth of a comma, every interval in the octave would be out of tune, the fifths all slightly narrow, the Major thirds all quite wide. Every chord would have exactly the same color, every key would sound like every other, and you could modulate without restriction from key to key. Mathematically very neat, but aurally dull. In Europe, our “modern” equal temperament was supposedly advocated early in the Renaissance, especially for fretting lutes and guitars.

To tune equal temperament, simply tune twelve fifths forward on the sharp side from your tuning fork a', making each fifth very slightly narrow of perfect (not so narrow that you can hear too much of a beat, but not exactly perfect either). When you come out at your final fifth, D–A, the A should be the same as the a' you started with. Piano tuners know that if they’re tuning at A440, the fifth d'a' beats at exactly 0.9 times per second.

It can take some practice to hear a perfect fifth, because there is a bit of room on each side of perfect where there are no perceptible beats—or the beats are so slow that the notes die out before the beat can be heard. Split the interval across the octave, helping make the slow but perceptible wave or beat of the equal-tempered fifth audible.

Equal temperament is not easy to tune with anything approaching absolute mathematical accuracy. To help, you might like to split the circle in three parts, establishing reference points at f and c' from the a an octave below the A440 (or A415) you set with your tuning fork. When you consider the help of the triangle, the process—not unlike what I’ve asked you to do in the various previous temperaments, dividing a third into its four contained fifths—becomes a possibility. Following the rule that beat speed increases as pitch raises, if you do this right, your fa will beat quite rapidly, ac' some more, and c'f' more so again. These reference points are represented by the dotted lines in the diagram: Your ff' must obviously be a beatless octave. Now it becomes a case of just dividing up each of the three segments of the circle in turn, not moving your reference points unless you have to.

Piano tuners make a great to-do about counting the beats, and have an elaborate system of various checks and adjustments as they proceed around the circle. They’ve spent a good deal of time memorizing the quality of each interval—sometimes so much so, that it is impossible for them to “turn off” Equal Temperament and approach the earlier tuning systems with open ears and mind.

The simplest test to verify your Equal Temperament is the playing of chains of chromatic ascending intervals: No matter what interval you choose, the beats should get uniformly faster as you come up the scale. Try it in your tenor octave with fifths, then thirds. But as long as none of your fifths are wide instead of narrow, and as long as you come out even at the end of your circle, your Equal Temperament will probably be equal enough. If your fifths are not tempered with perfect mathematical consistency, some of your chords will be more in tune than others, which—depending on the music played—some people would find all to the good on the harpsichord. To a sensitive ear, Equal Temperament is colorless and bland, and not much fun to listen to.


Further discussion
Anonymous [Kayano, Moxzan] Dodecagon — Chi-s akt temo Tokyo 2012, p108
Asselin, Pierre-Yves Musique et Tempérament Éditions Costallat, Paris 1985, p123
Barbour, J Murray Tuning and Temperament Michigan State College Press, East Lansing 1951, p45
Helmholz, Hermann On the Sensations of Tone [English translation by Alexander Ellis] Dover, New York 1954, p548
Jorgensen, Owen H Tuning — Containing The Perfection of Eighteenth-Century Temperament, The Lost Art of Nineteenth-Century Temperament and The Science of Equal Temperament Michigan State University Press, East Lansing 1991, p645
Klop, G C Harpsichord Tuning Werkplaats voor Clavecimbelbouw, Garderen 1974, p28
Kottick, Edward L Harpsichord Owner’s Guide University of North Carolina Press, Chapel Hill & London 1987, p154
Padgham, Charles The Well-Tempered Organ Positive Press, Oxford 1986, p48
Veroli, Claudio di Unequal Temperaments Artes Graficas Farro, Buenos Aires 1978, p112
Zuckermann, Wolfgang Joachim The Modern Harpsichord, New York 1969, p245


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