Meantone temperament
Meantone temperaments are musical temperaments;[1] that is, a variety of tuning systems constructed, similarly to Pythagorean tuning, as a sequence of equal fifths, both rising and descending, scaled to remain within the same octave.
, in order to bring the major or minor thirds closer to the just intonation ratio of
Twelve-tone equal temperament is almost exactly the same as 1/ 11 syntonic comma meantone tuning (1.955 cents vs. 1.95512).
Four ascending fifths (as C G D A E) tempered by 1 / 4 comma (and lowered by two octaves) produce a just major third (C E) (with ratio 5 : 4), which is one syntonic comma (or about 22 cents) narrower than the Pythagorean third that would result from four perfect fifths.
For church organs and some other keyboard purposes, it continued to be used well into the 19th century, and is sometimes revived in early music performances today.
It proceeds in the same way as Pythagorean tuning; i.e., it takes the fundamental (say, C) and goes up by six successive fifths (always adjusting by dividing by powers of 2 to remain within the octave above the fundamental), and similarly down, by six successive fifths (adjusting back to the octave by multiplying by powers of 2 ).
This results in the interval C E being a just major third 5 / 4 , and the intermediate seconds (C D, D E) dividing C E uniformly, so D C and E D are equal ratios, whose square is 5 / 4 .
This is in the sense opposite to the Pythagorean comma (i.e. the upper end is flatter than the lower one) and nearly twice as large.
It follows that in 1 / 4 comma meantine the whole tone is exactly half of the just major third (in cents) or, equivalently, the square root of the frequency ratio of 5 / 4 .
[1] Mention of tuning systems that could possibly refer to meantone were published as early as 1496 (Gaffurius).
The first mathematically precise meantone tuning descriptions are to be found in late 16th century treatises by Zarlino[5] and de Salinas.
Marin Mersenne described various tuning systems in his seminal work on music theory, Harmonie universelle,[7] including the 31 tone equitempered one, but rejected it on practical grounds.
For example, in 1691 Huygens[8] advocated the use of the 31 tone equitempered system (31 TET) as an excellent approximation for the 1 / 4 comma meantone system, mentioning prior writings of Zarlino and Salinas, and dissenting from the negative opinion of Mersenne (1639).
How tuners could identify a "quarter comma" reliably by ear is a bit more subtle.
Since this amounts to about 0.3% of the frequency which, near middle C (~264 Hz), is about one hertz, they could do it by using perfect fifths as a reference and adjusting the tempered note to produce beats at this rate.
Although meantone is best known as a tuning system associated with earlier music of the Renaissance and Baroque, there is evidence of its continuous usage as a keyboard temperament well into the 19th century.
Historically, commonly used meantone temperaments, discussed below, occupy a narrow portion of this tuning continuum, with fifths ranging from approximately 695 to 699 cents.
Meantone temperaments can be specified in various ways: By what fraction of a syntonic comma the fifth is being flattened (as above), the width of the tempered perfect fifth in cents, or the ratio of the whole tone (in cents) to the diatonic semitone.
This last ratio was termed "R" by American composer, pianist and theoretician Easley Blackwood.
is the closest approximation to the corresponding meantone tempered fifth within the equitempered division of the octave into
The first column gives the fraction of the systonic comma by which the perfect fifths are tempered in the meantone system.
The third gives the fraction of an octave, within the corresponding equitempered microinterval system, that best approximates the meantone fifth.
(+1.16371×10−4) (tritones) 16/ 15 and 15/ 8 (diatonic semitone and major seventh) 5 / 4 and 8 / 5 (just major third and minor sixth) 25 / 24 and 48 / 25 (chromatic semitone and major seventh ) 6/ 5 and 5/ 3 (just minor third and major sixth) (large limma) (just minor tone and diminished seventh) In neither the twelve tone equitemperament nor the quarter-comma meantone is the fifth a rational fraction of the octave, but several tunings exist which approximate the fifth by such an interval; these are a subset of the equal temperaments ( "N TET" ), in which the octave is divided into some number (N) of equally wide intervals.
This can be overcome by tempering the partials to match the tuning, which is possible, however, only on electronic synthesizers.
Even edge conditions produce wolf intervals only if the isomorphic keyboard has fewer buttons per octave than the tuning has enharmonically-distinct notes (Milne 2007 harvnb error: no target: CITEREFMilne2007 (help)).
Many musical instruments are capable of very fine distinctions of pitch, such as the human voice, the trombone, unfretted strings such as the violin, and lutes with tied frets.
On the other hand, the piano keyboard has only twelve physical note-controlling devices per octave, making it poorly suited to any tunings other than 12 ET.
When choosing which notes to map to the piano's black keys, it is convenient to choose those notes that are common to a small number of closely related keys, but this will only work up to the edge of the octave; when wrapping around to the next octave, one must use a "wolf fifth" that is not as wide as the others, as discussed above.
Throughout the Renaissance and Enlightenment, theorists as varied as Nicola Vicentino, Francisco de Salinas, Fabio Colonna, Marin Mersenne, Christiaan Huygens, and Isaac Newton advocated the use of meantone tunings that were extended beyond the keyboard's twelve notes,[1][13][14] and hence have come to be called "extended" meantone tunings.
All of these alternative instruments were "complicated" and "cumbersome" (Isacoff 2009), due to which can significantly reduce the number of note-controlling buttons needed on an isomorphic keyboard (Plamondon 2009 harvnb error: no target: CITEREFPlamondon2009 (help)).
