Thomas Donahue is the author of The Modern Classical Organ and the editor of the forthcoming Gerhard Brunzema: His Work and His Influence.
For modern harpsichords and continuo organs built with transposing keyboards, a problem arises with respect to temperament. Many unequal temperaments are designed to favor the keys of F, C, and G at the expense of C#, F#, and B. In transposing down a semitone, one is actually traveling almost halfway around the circle of 5ths; as a result, many temperaments do not sound acceptable when transposed. This requires that equal temperament be used. This article will present a transposable temperament that is an alternative to equal temperament, that has the characteristics of an unequal temperament, is easily set by ear, and does not clash with the tuning of string instruments.
Description
The transposable temperament consists of four 5ths each tempered by one quarter of the Pythagorean comma and eight 5ths tuned pure. Its main feature is its symmetry: the tempered 5ths are C, G, F#, and C#, so that the tuning of the 5ths C, G, D, A, E, and B is mirrored by the tuning of the 5ths F#, C#, G#, Eb , Bb , and F. There are three different sizes of major 3rds and two different sizes of 5ths, resulting in five different triad combinations. The compromise with this temperament is the occurrence of two Pythagorean major 3rds. Two versions will be presented, which vary according to the location of the Pythagorean 3rds.
Setting the Temperament
In the following instructions, italics indicate the note that is tuned.
Version I: Pythagorean major 3rds on D and G#
tune a' to 440 Hertz
tune pure: a-a' d-a e-a e-b f#-b
tune c-e wide, 4 beats per second
tune pure: c-c' f-c' f-bb eb-bb eb-ab
tune c#-f# wide, 2 beats per second
tune c-g narrow to beat as c#-g#
test: d-g beats as c#-f#, 2 beats per second
Version II: Pythagorean major 3rds on A and Eb
tune a' to 440 Hertz
tune pure: a-a' e- a e-b f#-b c#-f#
tune c-e wide, 4 beats per second
tune pure: c-c' c-g f-c' f-bb eb-bb
tune d#-g# wide, 2 beats per second
tune d-a narrow to beat as c#-g#
test: d-g beats as d#-g#, 2 beats per second
Major 3rds
Table 1 (above) lists the size of the major 3rds in cents. For reference, a pure major 3rd is 386 cents, an equal temperament major 3rd is 400 cents, and a Pythagorean major 3rd is 408 cents.
Beat Frequencies with String Instruments
Table 3 (above) lists the number of beats per second that would occur between the transposable temperament and the open strings of the violin, viola, and cello. Generally speaking, beat rates less than 4 beats per second are not objectionable, and beat rates less than 2 beats per second are either unnoticeable or produce a pleasant "céleste" effect. This table assumes that the reference pitch is a' = 440 Hertz.
