Meantone temperament
Meantone temperaments are musical temperaments; 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. But rather than using perfect fifths, consisting of frequency ratios of value, these are tempered by a suitable factor that narrows them to ratios that are slightly less than, in order to bring the major or minor thirds closer to the just intonation ratio of or, respectively. Among temperaments constructed as a sequence of fifths, a regular temperament is one in which all the fifths are chosen to be of the same size.
Twelve-tone equal temperament is obtained by making all semitones the same size, with each equal to one-twelfth of an octave; i.e. with ratios. Relative to Pythagorean tuning, it narrows the perfect fifths by about 2 cents or of a Pythagorean comma to give a frequency ratio of. This produces major thirds that are wide by about 13 cents, or of a semitone. Twelve-tone equal temperament is almost exactly the same as syntonic comma meantone tuning.
Notable meantone temperaments
, which tempers each of the twelve perfect fifths by of a syntonic comma, is the best known type of meantone temperament, and the term meantone temperament is often used to refer to it specifically. Four ascending fifths tempered by produce a just major third , which is one syntonic comma narrower than the Pythagorean third that would result from four perfect fifths.It was commonly used from the early 16th century till the early 18th, after which twelve-tone equal temperament eventually came into general use. 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. Quarter-comma meantone can be well approximated by a division of the octave into 31 equal steps.
It proceeds in the same way as Pythagorean tuning; i.e., it takes the fundamental and goes up by six successive fifths, and similarly down, by six successive fifths dividing uniformly, so and are equal ratios, whose square is. The same is true of the major second sequences and.
However, there is a residual gap in quarter-comma meantone tuning between the last of the upper sequence of six fifths and the last of the lower sequence; e.g. between and if the starting point is chosen as, which, adjusted for the octave, are in the ratio of or This is in the sense opposite to the Pythagorean comma and nearly twice as large.
In third-comma meantone, the fifths are tempered by of a syntonic comma. It follows that three descending fifths produce a just minor third of ratio, which is one syntonic comma wider than the minor third resulting from Pythagorean tuning of three perfect fifths. Third-comma meantone can be very well approximated by a division of the octave into 19 equal steps.
The tone as a mean
The name "meantone temperament" derives from the fact that in all such temperaments the size of the whole tone, within the diatonic scale, is somewhere between the major and minor tones of just intonation, which differ from each other by a syntonic comma. In any regular system the whole tone is reached after two fifths , while the major third is reached after four fifths . It follows that in comma meantone the whole tone is exactly half of the just major third or, equivalently, the square root of the frequency ratio of.Thus, one sense in which the tone is a mean is that, as a frequency ratio, it is the geometric mean of the major tone and the minor tone:
equivalent to 193.157 cents: the quarter-comma whole-tone size. However, any intermediate tone qualifies as a "mean" in the sense of being intermediate, and hence as a valid choice for some meantone system.
In the case of quarter-comma meantone, where the major third is made narrower by a syntonic comma, the whole tone is made half a comma narrower than the major tone of just intonation, or half a comma wider than the minor tone. This is the sense in which quarter-tone temperament is often considered "the" exemplary meantone temperament since, in it, the whole tone lies midway between its possible extremes.
History of meantone temperament and its practical implementation
Mention of tuning systems that could possibly refer to meantone were published as early as 1496. Pietro Aron was unmistakably discussing quarter-comma meantone. Lodovico Fogliani mentioned the quarter-comma system, but offered no discussion of it. The first mathematically precise meantone tuning descriptions are to be found in late 16th century treatises by Zarlino and de Salinas. Both these authors described the and meantone systems.Marin Mersenne described various tuning systems in his seminal work on music theory, Harmonie universelle, including the 31 tone equitempered one, but rejected it on practical grounds.
Meantone temperaments were sometimes referred to under other names or descriptions. For example, in 1691 Huygens advocated the use of the 31 tone equitempered system as an excellent approximation for the meantone system, mentioning prior writings of Zarlino and Salinas, and dissenting from the negative opinion of. He made a detailed comparison of the frequency ratios in the system and the quarter-comma meantone temperament, which he referred to variously as temperament ordinaire, or "the one that everyone uses".
Of course, the quarter-comma meantone system could not have been implemented with high accuracy until much later, since devices that could accurately measure all pitch frequencies didn't exist until the mid-19th century. But tuners could apply the same methods that "by ear" tuners have always used: Go up by fifths, and down by octaves, or down by fifths, and up by octaves, tempering the fifths so they are slightly smaller than the just ratio. 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 , 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. However, the frequency of the beats would have to be slightly adjusted, proportionately to the frequency of the note. Alternatively the diatonic scale major thirds can be adjusted to just major thirds, of ratio, by eliminating the beats.
For 12 tone equally-tempered tuning, the fifths have to be tempered by considerably less than a , since they must form a perfect cycle, with no gap at the end. For meantone tuning, if one artificially stops after filling the octave with only 12 pitches, one has a residual gap between sharps and their enharmonic flats that is slightly smaller than the Pythagorean one, in the opposite direction. Both quarter-comma meantone and the Pythagorean system do not have a circle but rather a spiral of fifths, which continues indefinitely. Slightly tempered versions of the two systems that do close into a much larger circle of fifths are 31 equal temperament| for meantone, and 53 equal temperament| for Pythagorean.
Although meantone is best known as a tuning system associated with earlier music of the Renaissance and Baroque, there is evidence of its continuous use as a keyboard temperament well into the 19th century.
It has had a considerable revival for early music performance in the late 20th century and in newly composed works specifically demanding meantone by some composers, such as Adams, Ligeti, and Leedy.
Meantone temperaments
A meantone temperament is a regular temperament, distinguished by the fact that the correction factor to the Pythagorean perfect fifths, given usually as a specific fraction of the syntonic comma, is chosen to make the whole tone intervals equal, as closely as possible, to the geometric mean of the major tone and the minor tone. 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, the width of the tempered perfect fifth in cents, or the ratio of the whole tone to the diatonic semitone. This last ratio was termed by American composer, pianist and theoretician Easley Blackwood. If happens to be a rational number then is the closest approximation to the corresponding meantone tempered fifth within the equitempered division of the octave into equal parts. Such divisions of the octave into a number of small parts greater than 12 are sometimes refererred to as microtonality, and the smallest intervals called microtones.
In these terms, some historically notable meantone tunings are listed below, and compared with the closest equitempered microtonal tuning. The first column gives the fraction of the syntonic comma by which the perfect fifths are tempered in the meantone system. The second lists 5 limit rational intervals that occur within this tuning. The third gives the fraction of an octave, within the corresponding equitempered microinterval system, that best approximates the meantone fifth. The fourth gives the difference between the two, in cents. The fifth is the corresponding value of the fraction and the fifth is the number of equitempered microtones in an octave.
| Meantone fraction of comma | 5-limit rational intervals | Size of fifths as fractions of an octave | Error between meantone fifths and fifths | Blackwood’s ratio = | Number of equal temperament| microtones |
| For all practical purposes, the fifth is a "perfect" | +0.000066 | 2.25 | |||
= | +0.000116 | 2.00 | |||
| −0.188801 | 1.80 | ||||
| +0.0206757 | 1.75 | ||||
| +0.195765 | 1.66 | ||||
| +0.189653 | 1.60 | ||||
| −0.0493956 | 1.50 | ||||
| +0.0958 | 1.40 | ||||
| | −0.292765 | 1.25 |