Summary of Equal-Temperament, Meantone, and Well-Temperament Systems
If you tune twelve perfect C → G → D → A → E → B → F# → C# → G# → D# → A# → E# → B# The problem is that the B# thus obtained is not a C seven octaves above the root C but 23.46 cents An additional problem is that when the E thus tuned is dropped two octaves, the major third, C—E is wide from perfect by 21.5 cents (syntonic comma). Similar statements hold for G—B, D—F#, etc. Note that tuning up a fifth is equivalent to tuning down a fourth and then up an octave. So, in actual practice, the above tuning sequence would utilize both fourths and fifths in order to group the tuned notes within one octave called the Whenever an interval varies from perfect, beats are heard. When beats are more than a few per second, they become quite unpleasant to hear. Next, tuning twelve perfect fifths downward, starting on C produces the following notes: etc. ← G-flat ← D-flat ← A-flat ← E-flat ← B-flat ← F ← C The problem now is that the G-flat thus obtained is 23.46 cents The following systems deal with these problems in different ways and to varying degrees. Note that each method of tuning given here is for purposes of clarity, and tuning methods may vary in actual practice.
The goal is to close the Pythagorean comma by narrowing each of the twelve fifths by 1/12 of a Pythagorean comma, or 23.46 ÷ 12 = 1.96 cents. Since this narrowing is equally distributed in the four fifths required to tune E, the interval C—E is now wide by 21.5 – 4/12 of 23.46 = 13.68 cents. Because all fifths are tempered by the same amount (such a temperament is termed a
The goal is to have perfect major thirds. As mentioned above, tuning four perfect fifths upward from C produces a major third C—E that is wide by 21.5 cents, so in order to produce a perfect major third C—E, these four fifths are each narrowed by one fourth of 21.5 cents = 5.375 cents: C → G → D → A → E Then B is tuned up a perfect major third from G, and F is tuned down a perfect major third from A. That takes care of the naturals. Next, sharps are tuned upward a perfect third, and flats are tuned downward a perfect third. For example, G# is tuned upward a perfect major third from E, and A-flat is tuned downward a perfect third from C. It can be shown that tuning a G# up am perfect third is equivalent to continuing to tune up from E by using fifths each narrowed by 5.375 cents. E → B → F# → C# → G# Similarly, tuning an A-flat down a perfect third is equivalent to continuing to tune down from C by using fifths all narrowed by 5.375 cents: A-flat ← E-flat ← B-flat ← F ← C The discrepancy between an associated sharp and flat then is 21.5(3) – 23.46 = 41 cents. Because this temperament is regular, all associated sharps and flats differ by 41 cents (sharps are 41 cents flatter than associated flats). Meantone is, therefore, not a closed temperament, and only certain keys are usable. The notorious meantone wolf occurs in fifths such as C#—G#, where the upper accidental is tuned to A-flat instead of a G#. The resulting 41 – 5 = 36-cent discrepancy has been likened to the out-of-tune howling of wolves. Download a simple scheme for tuning in Meantone Temperament
The goal is to have one perfect major third and to be able to play in all keys. In this particular version of well temperament, the third C—E will be perfect. We start out as in meantone, tuning up four fifths each narrowed by one fourth of 21.5 cents = 5.375 cents to produce a perfect major third C—E. That uses up 21.5 cents of a required amount of tempering of 23.46 cents. From there on, perfect fifths are tuned downward from C: C-flat ← G-flat ← D-flat ← A-flat ← E-flat ← B-flat ← F ← C Since there is only 23.46 – 21.5 = 1.96 cents left of tempering, C-flat will differ from B by 1.96 cents, and the interval E—C-flat will be a “fifth,” narrowed by 1.96 cents. This narrowing, called a Well temperament is closed but irregular. So you can play in all keys, and each key sounds different. As you move away from the key of C (G, D, A, E, etc.), the beat rates of the major thirds increase. Close to the key of C, the beat rates are slow, but far from the key of C, the beat rates are much greater than in equal temperament. No intervals other than octaves, six fifths, and one major third are perfect. Download a simple scheme for tuning in Well temperament
©2011 by Robert Chuckrow |