Technical Library TEMPERAMENTS XX: Equal Temperament Entire Contents Copyright ©2006 CBH

I guess we’ve come so far, we ought not be so dismissive of Equal Temperament.

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. It allows complete freedom to play in every key, and pianos have been accurately tuned this way since the beginning of the twentieth-century. Equal temperament always sounds a bit rough on the harpsichord. In contrast, it’s really the mellow timbre of the modern piano which has allowed musicians to tolerate Equal Temperament’s rapidly beating Major thirds.

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. This idea is so old that the Chinese were talking about it 2000BC, and in Europe it was supposedly advocated early in the Renaissance, especially for fretting lutes and guitars. This is our “modern” equal temperament.

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 f–a will beat quite rapidly, a–c^#' some more, and c^#'–f' more so again. These reference points are represented by the dotted lines in the diagram: Your f–f' 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.

	Tuning Bibliography
	Technical Library overview
	Harpsichords Australia Home Page