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 (~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. 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 (composer)|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

Equal temperaments

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 in which the octave is divided into some number of equally wide intervals.
Equal temperaments that are useful as approximations to meantone tunings include [12 equal temperament|] and . almost perfectly fits both Pythagorean tuning and 5 limit just intonation, with a few 7 limit and 11 limit intervals. The farther the tuning gets away from quarter-comma meantone, however, the less related the tuning is to harmonic ratios. This can be overcome by tempering the partials to match the tuning, which is possible, however, only on electronic synthesizers. The following table gives various meantone temperaments

Wolf intervals

A whole number of just perfect fifths will never add up to a whole number of octaves, because is an irrational number. If a whole number of perfect fifths is stacked-up, then in order to close that stack to fit an octave, at least one of the intervals that is enharmonically equivalent to a fifth must have a different width than all the other fifths. For example, to make a 12 note chromatic scale in Pythagorean tuning close at the octave, one of the fifth intervals must be lowered by the Pythagorean comma; this altered fifth is called a "wolf fifth" because it sounds similar to a fifth in its interval size and seems like an out-of-tune fifth, but is actually a diminished sixth. Likewise, 11 of the 12 perfect fourths are also in tune, but the remaining fourth is actually an augmented third.
Wolf intervals are not inherent to a complete tuning system, rather they are an artifact of inadequate keyboards that do not have enough keys for all of the in-tune notes used in any given piece. Keyboard players then create a "wolf" by substituting a key that is actually in-tune with a different pitch, nearby the actual notated pitch, but not quite near enough to pass.
Image:Isomorphic Note Layout.jpg|thumb|bottom|550px|Figure 2: Kaspar Wicki's isomorphic keyboard, invented in 1896.
The issue can be most easily shown by using an isomorphic keyboard, with many more than just 12 keys per octave, such as that shown in Figure 2. Here's an example: On the keyboard shown in Figure 2, from any given note, the note that's a perfect fifth higher is always upward-and-rightward adjacent to the given note. There are no wolf intervals within the note-span of this keyboard. The problem is at the edge, on the note. The note that's a perfect fifth higher than is, which is not included on the keyboard shown. Because there is no button, when playing an power chord, one must choose some other note, such as, to play instead of the missing.
Even edge conditions produce wolf intervals only if the isomorphic keyboard has fewer buttons per octave than the tuning has enharmonically-distinct notes. For example, the isomorphic keyboard in Figure 2 has 19 buttons per octave, so the above-cited edge-condition, from to, is not a wolf interval in 12 tone equal temperament, 17 , or 19 ; however, it is a wolf interval in 26 , 31 , and 50 ET. In these latter tunings, using electronic transposition could keep the current key's notes on the isomorphic keyboard's white buttons, such that these wolf intervals would very rarely be encountered in tonal music, despite modulation to exotic keys.
Isomorphic keyboards expose the invariant properties of the meantone tunings of the syntonic temperament isomorphically because both the isomorphic keyboard and temperament are two-dimensional entities. One-dimensional keyboards can expose accurately the invariant properties of only a single one-dimensional tuning in hence, the one-dimensional piano-style keyboard, with 12 keys per octave, can expose the invariant properties of only one tuning: 12 .
When the perfect fifth is exactly 700 cents wide then the tuning is identical to the familiar 12 tone equal temperament. This appears in the table above when
Because of the compromises forced on meantone tunings by the limitation of having only 12 key per octave on a conventional piano-style keyboard, well temperaments and eventually equal temperament became more popular.
Using standard interval names, twelve fifths equal six octaves plus one augmented seventh; seven octaves are equal to eleven fifths plus one diminished sixth. Given this, three "minor thirds" are actually augmented seconds, and four "major thirds" are actually diminished fourths. Several triads contain both these intervals and have normal fifths.

Extended meantones

All meantone tunings fall into the valid tuning range of the syntonic temperament, so all meantone tunings are syntonic tunings. All syntonic tunings, including the meantones and the various just intonations, conceivably have an infinite number of notes in each octave, that is, seven natural notes, seven sharp notes, seven flat notes ; then double sharp notes, double flat notes, triple sharps and flats, and so on. In fact, double sharps and flats are uncommon, but still needed, but triple sharps and flats are almost never seen, so might be skipped or compromised. In any syntonic tuning that happens to divide the octave into a small number of equally wide smallest intervals.
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, and hence these are now called "extended" meantone tunings.
Such efforts required a corresponding extension of keyboard instruments to provide means of producing more than 12 notes per octave; examples include Vincento's archicembalo, Mersenne's harpsichord, Colonna's sambuca rota, and Huygens's harpsichord.
Other instruments extended the keyboard by only a few notes. Some period harpsichords and organs have split / keys, such that both E major / C minor and E major / C minor can be played with no wolf fifths. Many of those instruments also have split / keys, and a few have all the five accidental keys split.
All of these alternative instruments were "complicated" and "cumbersome", due to
which can significantly reduce the number of note-controlling buttons needed on an isomorphic keyboard. Both of these criticisms could be addressed by electronic isomorphic keyboard instruments, which could be simpler, less cumbersome, and more expressive than existing keyboard instruments.