Technical Library TEMPERAMENTS XXXII: Equal Temperament Entire Contents Copyright © 2020 CBH

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!

Equal Temperament did rear its ugly head from time to time. In his Sectio Canonis Harmonici zur völligen Richtigkeit der Generum Modulandi of 1724, Johann Georg Neidhardt (1680–1739) went so far as to recommend which of his temperaments might be suitable for a village, a small city, a large city, or for making music in the Court itself—the latter in fact our familiar Equal Temperament.

And in the first chapter of his Principes de clavecin published in Berlin in 1756, Friedrich Wilhelm Marpourg (1718–1795) gave directions for tuning l‘égalité du tempérament (Equal Temperament), dividing the octave into four equally-sounding Major thirds C–E, E–G♯, & A♭–C and proceeding from there. [We will use this idea below.]

I will say again, that despite what you might have been taught, Bach neither invented nor used Equal Temperament. He probably used many different tuning systems—and if he had one particular one in mind for any of his works, he never chose to write clear directions for setting it. Note that his great opus is called the Well-tempered Clavier in English, not the “Equal Tempered Clavichord”, as it has too often been mistranslated. You will find several possible Bach temperaments discussed in earlier pages in this series.

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 it allows 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 on the circle from your tuning fork a'. To ensure you are indeed tuning the fifths sightly narrow and not wide, tune each fifth pure first, then squeeze it by making it very slightly narrow of perfect. When you come out at your final fifth, D–A, the A should obviously match 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.

Equal temperament is not easy to tune with anything approaching absolute mathematical accuracy.

To help, you might like to split the circle of fifths into three parts similar to what Marpourg proposed in 1756. Seeing as you are probably tuning from the A440 (or A415) of your pitch source, establish reference points at f and c♯' from the a an octave below that. When you consider the help of the triangle, the process—not unlike what I’ve often asked you to do in various previous temperaments, dividing a third into its four contained fifths—becomes fathomable. Your wide f–a third will beat a quite rapid seven beats a second at A440. Following the rule that beat speed of same theoreticallly-sized instervals increases as pitch rises, if you do this right, your a–c♯' will beat about nine times a second, and c♯'–f' eleven—if you can count that fast. These reference points are represented by the dotted lines in the diagram: Your f–f' must obviously be a beatless octave. Then it becomes a case of just dividing up each of the three segments of the circle in turn, without moving those fixed reference points.

Piano tuners love counting beats, and have an elaborate system of various checks and balances 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 to play chains of chromatic ascending intervals: No matter what interval you choose, the beats should get uniformly faster as you ascend. Try it in your tenor octave with fifths, then Major thirds. As my teacher D. Jacques Way used to say, “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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