Temperaments & Tuning, Part IV
(continued from Part III)
As explained earlier, the ear cannot tolerate the entire Syntonic comma of 22 cents, and it has to be divided up somehow. In standard quarter-comma meantone the Syntonic comma is distributed equally among the 4 major 3rds built upon Db, F#, Ab, and B at the bottom of the Circle. The 4 "wolf" intervals thus created eventually became so annoying that, by the late 17th century, meantone tuning was being modified substantially in practice. Simply put, the 8 purest major 3rds were tuned very slightly away from pure in order to try to lessen the extreme nastiness of these 4 wolf intervals. In fifth-comma meantone each of the 4 major 3rds built upon Db, F#, Ab, and B were adjusted to absorb only 1/5 of the comma each, with the remaining 8 purest major 3rds adjusted accordingly to absorb the remaining 1/5. In sixth-comma meantone (so-called "Silbermann temperament") each of the 4 major 3rds built upon Db, F#, Ab, and B were adjusted to absorb only 1/6 of the comma each, again, with the remaining 8 purest major 3rds adjusted to absorb the remaining 2/6. This allowed the composer a little more freedom still, to modulate a bit more boldly and frequently, perhaps permiting a brief excursion into the keys with as many as 5 sharps or flats. The wolves were still there however and just as unusuable, just not quite as nasty as before.
In 16th century German organs tuned in quarter comma meantone, where half of the triads of modern music could not be used, organ builders developed the short octave, which was a special means of extending the lower compass of the keyboards by eliminating those keys for notes which were seldom, if ever, used in meantone tuning. The bass octave was built to end on the low E key, and to this key were assigned the pipes which sounded low C. The low F# and low G# keys in the bass octave were built to sound low D and low E, respectively. This meant that the pipes which were supposed to sound the low C#, D#, F#, and G# notes in the bass octave were omitted entirely from the organ.
Those builders who tuned their instruments in meantone forms sometimes provided another arrangement of the bass octave known as a broken octave (photo). These manuals were built and worked exactly like a short octave keyboard but were further provided with split sharps, or subsemitones, for the low F# and low G# keys. These 2 black keys were built with a short portion placed on top of, and to the rear of, the long portion. The short portions, when played, sounded F# and G#, while the long portions continued to sound the low D and low E of the short octave. Using a short or broken octave manual, intervals in the bass as wide as a 10th could be easily reached. Above this bass octave, sometimes the builder would provide split sharps for the Ab and Eb keys in every octave of the manual's range. The long portions of these keys would sound Ab and Eb, with the short portions sounding G# and D#, respectively.
In an attempt to better hide the wolves a modified temperament based upon standard (quarter-comma) meantone developed in France during the late 17th century. This was called Chaumont temperament and was based upon 6 pure major 3rds (D-F#, E-G#, F-A, G-B, A-C#, C-E, the last one being slightly diminished). This kept most of the old meantone sound while achieving expanded, but not unlimited, modulation scope and enharmonic capability.
Around the same time in Germany organ builders were experimenting and using various irregular or circular temperaments, also known as well temperaments, with the object of hiding the wolves completely, thereby making all keys usuable. This could have be done by distributing the Pythagorean comma equally across the entire scale, but this was NOT the path the Germans took, except as an academic exercise. The reason was, the circulating (no-wolf) temperaments are those which allow the widest exploration of key color, and so, the Pythagorean comma was distributed around the entire circle in an irregular manner. Well temperaments in this category included various tunings. Werckmeister temperament, invented by organist and music theorist Andreas Werckmeister, is recommended for performing German organ repertoire of the late 1600's. Kirnberger temperament, developed by Johann Philipp Kirnberger, pupil of J.S. Bach, is suitable for playing the German Baroque composers and the works of Bach. Kellner temperament, is named after Herbert Anton Kellner, whose research enabled him to identify the unequal temperament of the same name used by Bach for his Well-Tempered Klavier, Books I and II, and is therefore suitable for 18th century German organ music and the music of Bach in particular. Valloti temperament is an Italian temperament first invented by Francescantonio Valloti and later taken up in England by Thomas Young, which can be effectively used for Italian and English repertoire of the 18th century.
In all these circular systems it is possible to play in any key, though the more remote keys may sound a bit unpleasant, and enharmonic modulation is not always happy. Werckmeister, for example, is notable for its purity in the best keys and its suitability for organs with large quint mixtures (many of the perfect 4ths and 5ths are in tune), but it's irregular and bumpy in the way it deals with modulation and key color. Most keys however are better than with equal temperament, with only the most remote keys at the bottom of the Circle being slightly worse. The actual pitches are very close to equal temperament. There are NO wolf intervals, and the unevenness of the chromatic scale is not apparent. Except for the omission of the low C# key, which was almost universal in those days, the bass octaves of early organs tuned in a well temperament were usually made fully chromatic and not the short octave found in so many organs of preceding generations (See blog, Bach d minor, Part IV).
(continued in Part V)