Meantone temperament: Difference between revisions

Latest revision as of 19:50, 18 October 2025

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. But rather than using perfect fifths, consisting of frequency ratios of value $3 : 2$ , these are tempered by a suitable factor that narrows them to ratios that are slightly less than $3 : 2$ , in order to bring the major or minor thirds closer to the just intonation ratio of $5 : 4$ or $6 : 5$ , 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 (Template:Nobr) is obtained by making all semitones the same size, with each equal to one-twelfth of an octave; i.e. with ratios Template:Nobr. Relative to Pythagorean tuning, it narrows the perfect fifths by about 2 cents or Template:Nobr of a Pythagorean comma to give a frequency ratio of $2^{7 / 12} : 1$ . This produces major thirds that are wide by about 13 cents, or Template:Nobr of a semitone. Twelve-tone equal temperament is almost exactly the same as Template:Nobr syntonic comma meantone tuning (1.955 cents vs. 1.95512).

Notable meantone temperaments

File:Meantone.jpg

Figure 1. Comparison between Pythagorean tuning (blue), equal-tempered (black), quarter-comma meantone (red) and third-comma meantone (green). For each, the common origin is arbitrarily chosen as C. The values indicated by the scale at the left are deviations in cents with respect to equal temperament.

Quarter-comma meantone, which tempers each of the twelve perfect fifths by Template:Sfrac 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 (as Template:Sc) tempered by Template:Nobr (and then lowered by two octaves) produce a just major third (Template:Sc) (with ratio Template:Nobr), which is one syntonic comma (or about 22 cents) 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 equitemperament 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 (say, Template:Sc) and goes up by six successive fifths (always adjusting by dividing by powers of Template:Math to remain within the octave above the fundamental), and similarly down, by six successive fifths (adjusting back to the octave by multiplying by Template:Nobr However, instead of using the Template:Sfrac ratio, which gives perfect fifths, this must be divided by the fourth root of Template:Sfrac, which is the syntonic comma: The ratio of the Pythagorean third Template:Sfrac to the just major third Template:Sfrac. Equivalently, one can use Template:Math instead of Template:Sfrac, which produces the same slightly reduced fifths. This results in the interval Template:Sc being a just major third Template:Sfrac, and the intermediate seconds (Template:Sc, Template:Sc) dividing Template:Sc uniformly, so Template:Sc and Template:Sc are equal ratios, whose square is Template:Sfrac. The same is true of the major second sequences Template:Sc and Template:Sc.

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 Template:Sc^{Template:Music} and Template:Sc Template:Music if the starting point is chosen as Template:Sc, which, adjusted for the octave, are in the ratio of Template:Sfrac or Template:Nobr 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.

In third-comma meantone, the fifths are tempered by Template:Nobr of a syntonic comma. It follows that three descending fifths (such as Template:Sc) produce a just minor third (Template:Sc) of ratio Template:Sfrac, 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 (9:8 and 10:9 respectively) of just intonation, which differ from each other by a syntonic comma. In any regular system ^[1] the whole tone (as Template:Sc) is reached after two fifths (as Template:Sc) (lowered by an octave), while the major third is reached after four fifths Template:Nobr (lowered by two octaves). It follows that in Template:Sfrac comma meantone the whole tone is exactly half of the just major third (in cents) or, equivalently, the square root of the frequency ratio of Template:Sfrac.

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: $\sqrt{\frac{10}{9} \cdot \frac{9}{8}} = \sqrt{\frac{5}{4}} = 1.1180340,$ 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 (9:8), or half a comma wider than the minor tone (10:9). This is the sense in which quarter-tone temperament is often considered "the" exemplary meantone temperament since, in it, the whole tone lies midway (in cents) between its possible extremes.^[1]

History of meantone temperament and its practical implementation

Mention of tuning systems that could possibly refer to meantone were published as early as 1496 (Gaffurius).^[2] Pietro Aron^[3] (Venice, 1523) was unmistakably discussing quarter-comma meantone. Lodovico Fogliani^[4] 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^[5] and de Salinas.^[6] Both these authors described the Template:Nobr Template:Nobr and Template:Nobr meantone systems. 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.

Meantone temperaments were sometimes referred to under other names or descriptions. For example, in 1691 Huygens^[8] advocated the use of the 31 tone equitempered system Template:Nobr as an excellent approximation for the Template:Nobr meantone system, mentioning prior writings of Zarlino and Salinas, and dissenting from the negative opinion of Template:Harvp. He made a detailed comparison of the frequency ratios in the Template:Nobr system and the quarter-comma meantone temperament, which he referred to variously as temperament ordinaire, or "the one that everyone uses". (See references cited in the article Temperament Ordinaire.)

Of course, the quarter-comma meantone system (or any other 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 Template:Sfrac 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 C|middle Template:Sc (~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 Template:Sfrac, by eliminating the beats.

For 12 tone equally-tempered tuning, the fifths have to be tempered by considerably less than a Template:Nobr (very close to a Template:Nobr, or a Template:Nobr), since they must form a perfect cycle, with no gap at the end ("circle of fifths"). For Template:Nobr 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|Template:Nobr]] for meantone, and [[53 equal temperament|Template:Nobr]] 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.

"The mode of tuning which prevailed before the introduction of equal temperament, is called the Meantone System. It has hardly yet died out in England, for it may still be heard on a few organs in country churches. According to Don B. Yñiguez, organist of Seville Cathedral, the meantone system is generally maintained on Spanish organs, even at the present day." — Template:Harvp^[9]

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

File:Meantone spectrum.svg

For a tuning to be meantone, its fifth must be between Template:Sfrac and 700 ¢ in size. Note that Template:Nobr is on the flatmost extreme, Template:Nobr is on the sharpmost extreme, and Template:Nobr forms the midpoint of the spectrum.

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 (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 Template:Nobr by American composer, pianist and theoretician Easley Blackwood. If Template:Mvar happens to be a rational number $R = \frac{N}{D},$ then $2^{\frac{3 R + 1}{5 R + 2}}$ is the closest approximation to the corresponding meantone tempered fifth within the equitempered division of the octave into $5 N + 2 D$ 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 $R = \frac{N}{D},$ and the fifth is the number $5 N + 2 D$ of equitempered (Template:Sc ) microtones in an octave.

Meantone vs. Equitempered tunings
Meantone fraction of (syntonic) comma	5-limit rational intervals	Size of Template:Sc fifths as fractions of an octave	Error between meantone fifths and Template:Sc fifths (in cents)	Blackwood’s ratio Template:Mvar = $\frac{N}{D}$	Number of [[equal temperament\|Template:Sc]] microtones $5 N + 2 D$
REDIRECT Template:Large Template:Redirect category shell (very nearly Pythagorean tuningScript error: No such module "Check for unknown parameters".)	For all practical purposes,Script error: No such module "Check for unknown parameters". the fifth is a "perfect" Template:Nobr Script error: No such module "Check for unknown parameters".	Template:Sfrac	+0.000066 (+6.55227×Template:10^Script error: No such module "Check for unknown parameters".)	Template:Sfrac = 2.25	REDIRECT Template:Large Template:Redirect category shell
REDIRECT Template:Large Template:Redirect category shell ( or Template:Sfrac Pythagorean commaScript error: No such module "Check for unknown parameters".)	Template:Sfrac =Template:Sfrac ( Kirnberger fifth: a just fifth flattened by a schisma. Equals meantone to 6 significant figures.)Script error: No such module "Check for unknown parameters".	Template:Sfrac	+0.000116 (+1.16371×Template:10^Script error: No such module "Check for unknown parameters".)	Template:Sfrac = 2.00	REDIRECT Template:Large Template:Redirect category shell
REDIRECT Template:Large Template:Redirect category shell	Template:Sfrac and Template:Sfrac Script error: No such module "Check for unknown parameters". (tritonesScript error: No such module "Check for unknown parameters".)	Template:Sfrac	−0.188801	Template:Sfrac = 1.80	REDIRECT Template:Large Template:Redirect category shell
REDIRECT Template:Large Template:Redirect category shell	Template:Sfrac and Template:SfracScript error: No such module "Check for unknown parameters". (diatonic semitone and major seventhScript error: No such module "Check for unknown parameters".)	Template:Sfrac	+0.0206757	Template:Sfrac = 1.75	REDIRECT Template:Large Template:Redirect category shell
REDIRECT Template:Large Template:Redirect category shell	Template:Sfrac and Template:SfracScript error: No such module "Check for unknown parameters". (just major third and minor sixthScript error: No such module "Check for unknown parameters".)	Template:Sfrac	+0.195765	Template:Sfrac = 1.66	REDIRECT Template:Large Template:Redirect category shell
REDIRECT Template:Large Template:Redirect category shell	Template:Sfrac and Template:SfracScript error: No such module "Check for unknown parameters". (chromatic semitone and major seventh Script error: No such module "Check for unknown parameters".)	Template:Sfrac	+0.189653	Template:Sfrac = 1.60	REDIRECT Template:Large Template:Redirect category shell
REDIRECT Template:Large Template:Redirect category shell	Template:Sfrac and Template:SfracScript error: No such module "Check for unknown parameters". (just minor third and major sixthScript error: No such module "Check for unknown parameters".)	Template:Sfrac	−0.0493956	Template:Sfrac = 1.50	REDIRECT Template:Large Template:Redirect category shell
REDIRECT Template:Large Template:Redirect category shell	Template:Sfrac (large limmaScript error: No such module "Check for unknown parameters".)	Template:Sfrac	+0.0958	Template:Sfrac = 1.40	REDIRECT Template:Large Template:Redirect category shell
REDIRECT Template:Large Template:Redirect category shell	Template:Sfrac and Template:SfracScript error: No such module "Check for unknown parameters". (just minor tone and diminished seventhScript error: No such module "Check for unknown parameters".)	Template:Sfrac	−0.292765	Template:Sfrac = 1.25	REDIRECT Template:Large Template:Redirect category shell

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 Template:Nobr in which the octave is divided into some number (Template:Mvar) of equally wide intervals.

Equal temperaments that are useful as approximations to meantone tunings include (in order of increasing generator width) Template:Nobr Template:Nobr Template:Nobr Template:Nobr Template:Nobr Template:Nobr Template:Nobr Template:Nobr Template:Nobr Template:Nobr [[12 equal temperament|Template:Nobr]] Template:Nobr and Template:Nobr Template:Nobr. Template:Nobr 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.^[10] The following table gives various meantone temperaments

File:Meantone fifths vs thirds comparison.svg

Comparison of perfect fifths, major thirds, and minor thirds in various meantone tunings with just intonation

**Approximation of just intervals in equal temperaments**
*in order of increasing number of scale tones*
[[12 equal temperament\|Template:Nobr]]	[[19 equal temperament\|Template:Nobr]]	[[31 equal temperament\|Template:Nobr]]	[[43 equal temperament\|Template:Nobr]]	[[50 equal temperament\|Template:Nobr]]	[[53 equal temperament\|Template:Nobr]]	[[55 equal temperament\|Template:Nobr]]
Template:Sfrac comma	Template:Sfrac comma	Template:Sfrac comma	Template:Sfrac comma	Template:Sfrac comma	Template:Sfrac comma	Template:Sfrac comma
File:12ed2-5Limit.svg	File:19ed2.svg	File:31ed2.svg	File:43ed2.svg	File:50ed2.svg	File:53ed2.svg	File:55ed2.svg

Wolf intervals

A whole number of just perfect fifths will never add up to a whole number of octaves, because Template:Nobr 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 ("out-of-tune") 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 (e.g. between Template:Sc^{Template:Music} and Template:Sc^{Template:Music}). Likewise, 11 of the 12 perfect fourths are also in tune, but the remaining fourth is actually an augmented third (rather than a true fourth).

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.^[11]

File:Isomorphic Note Layout.jpg

Script error: No such module "anchor".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 (on an isomorphic keyboard, any given musical interval has the same shape wherever it appears, except at the edges). 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 Template:Sc^{Template:Music}. The note that's a perfect fifth higher than Template:Sc^{Template:Music} is Template:Sc^{Template:Music}, which is not included on the keyboard shown (although it could be included in a larger keyboard, placed just to the right of Template:Sc^{Template:Music}, hence maintaining the keyboard's consistent note-pattern). Because there is no Template:Sc^{Template:Music} button, when playing an Template:Sc^{Template:Music} power chord (open fifth chord), one must choose some other note, such as Template:Sc, to play instead of the missing Template:Sc^{Template:Music}.

Even edge conditions produce wolf intervals only if the isomorphic keyboard has fewer buttons per octave than the tuning has enharmonically-distinct notes.^[12] For example, the isomorphic keyboard in Figure 2 has 19 buttons per octave, so the above-cited edge-condition, from Template:Sc^{Template:Music} to Template:Sc, is not a wolf interval in 12 tone equal temperament (Template:Sc), 17 Template:Sc, or 19 Template:Sc; however, it is a wolf interval in 26 Template:Sc, 31 Template:Sc, 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.^[13]

Isomorphic keyboards expose the invariant properties of the meantone tunings of the syntonic temperament isomorphically (that is, for example, by exposing a given interval with a single consistent inter-button shape in every octave, key, and tuning) because both the isomorphic keyboard and temperament are two-dimensional (i.e., rank 2) entities.Template:Sfn One-dimensional Template:Nobr keyboards (where Template:Mvar is some number) can expose accurately the invariant properties of only a single one-dimensional tuning in Template:Nobr hence, the one-dimensional piano-style keyboard, with 12 keys per octave, can expose the invariant properties of only one tuning: 12 Template:Sc.

When the perfect fifth is exactly 700 cents wide (that is, tempered by almost exactly Template:Sfrac of a syntonic comma, or exactly Template:Sfrac of a Pythagorean comma) then the tuning is identical to the familiar 12 tone equal temperament. This appears in the table above when Template:Nobr

Because of the compromises (and wolf intervals) 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 (for example, Template:Sc^{Template:Music} to Template:Sc^{Template:Music}), and four "major thirds" are actually diminished fourths (for example, Template:Sc to Template:Sc^{Template:Music}). Several triads (like Template:Sc Template:Sc^{Template:Music} Template:Sc^{Template:Music} and Template:Sc^{Template:Music} Template:Sc^{Template:Music} Template:Sc) 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 (Template:Sc^{Template:Music} to Template:Sc^{Template:Music}), seven flat notes (Template:Sc^{Template:Music} to Template:Sc^{Template:Music}) (which is the limit of the orchestral harp, which allows 21 distinct pitches per octave); then double sharp notes (Template:Sc^{Template:Music} to Template:Sc^{Template:Music}), double flat notes (Template:Sc^{Template:Music} to Template:Sc^{Template:Music}), 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 (such as 12, 19, or Template:Nobr this extended set of notes still exists, but is not infinite, since some notes will be equivalent. For example, in Template:Nobr Template:Sc^{Template:Music} and Template:Sc^{Template:Music} are the same pitch; in Template:Nobr Template:Sc^{Template:MusicTemplate:Music} and Template:Sc^{Template:Music} are identical, as are Template:Sc^{Template:Music} and Template:Sc^{Template:Music}; and in just intonation for C major, Template:Sc^{Template:Music} Template:Sc^{Template:Music} are within 8.1 ¢, and so can be tempered to be identical, with the compromise note being only a tolerable 4 ¢ off for each.

Many musical instruments are capable of very fine distinctions of pitch, such as the human voice, the trombone, unfretted strings such as the violin family and fretless guitars, and lutes with movable frets. These instruments are well-suited to the use of meantone tunings.

On the other hand, the conventional piano keyboard only has twelve note-producing keys per octave, making it poorly suited to any tunings other than Template:Nobr or well temperaments. Almost all of the historic problems with the meantone temperament are caused by the failure to map meantone's infinite number of notes per octave to a finite number of piano keys. This is, for example, the source of the "wolf fifth" discussed above. 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 crossing up or down to an adjacent octave, for some of the intervals must be a "wolf fifth" – that is, slightly flatter than the others, as described above.

The existence of the "wolf fifth" is one of the reasons why, before the introduction of well temperament, instrumental music generally stayed in a number of "safe" tonalities that did not involve the "wolf fifth" (which was generally put between Template:Sc^{Template:Music} and Template:Sc^{Template:Music}).

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]^[14]^[15] 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 Template:Nobr harpsichord, Colonna's Template:Nobr sambuca rota, and Huygens's Template:Nobr harpsichord.^[16]

Other instruments extended the keyboard by only a few notes. Some period harpsichords and organs have split Template:Sc^{Template:Music} / Template:Sc^{Template:Music} keys, such that both E major / [[C-sharp minor|C^{Template:Music} minor]] (4 sharps) and [[E-flat major|E^{Template:Music} major]] / C minor (3 flats) can be played with no wolf fifths. Many of those instruments also have split Template:Sc^{Template:Music} / Template:Sc^{Template:Music} keys, and a few have all the five accidental keys split.

All of these alternative instruments were "complicated" and "cumbersome" (Script error: No such module "Footnotes".), due to

(a) not being isomorphic, and

(b) not having a transposing mechanism,

which can significantly reduce the number of note-controlling buttons needed on an isomorphic keyboard (Script error: No such module "Footnotes".). Both of these criticisms could be addressed by electronic isomorphic keyboard instruments (such as the open-source hardware jammer keyboard), which could be simpler, less cumbersome, and more expressive than existing keyboard instruments.^[17]

References

Template:Reflist

External links

An explanation of constructing Quarter Comma Meantone Tuning
How to tune quarter-comma meantone
Template:Webarchive Music fragments played in different temperaments - mp3s not archived
Kyle Gann's Introduction to Historical Tunings has an explanation of how the meantone temperament works.
Willem Kroesbergen, Andrew cruickshank: Meantone, unequal and equal temperament during J.S. Bach's life https://www.academia.edu/9189419/Blankenburg_Equal_or_unequal_temperament_during_J.S._Bach_s_life
Meantone Temperament | Definition & Meaning | M5 Music - M5 Music

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@@ Line 1: / Line 1: @@
 {{Short description|Musical tuning system}}
-'''Meantone temperaments''' are [[musical temperaments]];<ref name = "Barbour">{{cite book |last=Barbour |first=James Murray |year=1951 |title=Tuning and Temperament: A historical survey, Chapters III, IV and VII |publisher=Dover Publications (2013)| series =Dover Books On Music: History |isbn=9780486434063 }}</ref> that is, a variety of [[Musical tuning#Tuning systems|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 fifth]]s, consisting of frequency ratios of value <math>3:2</math>, these are ''tempered'' by a suitable factor that narrows them to ratios that are slightly less than <math>3:2</math>, in order to bring the major or minor thirds closer to [[Just intonation|the just intonation ratio]] of <math> 5:4 </math> or <math> 6:5 </math>, 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.
+'''Meantone temperaments''' are [[musical temperaments]];<ref name=Barbour>{{cite book |last=Barbour |first=James Murray |year=1951 |title=Tuning and Temperament: A historical survey |at=Chapters&nbsp;III, IV and VII |publisher=Dover Publications (2013) |series=Dover Books On Music: History |isbn=978-0-4864-3406-3 }}</ref> that is, a variety of [[Musical tuning#Tuning systems|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 fifth]]s, consisting of frequency ratios of value <math>3:2</math>, these are ''tempered'' by a suitable factor that narrows them to ratios that are slightly less than <math>3:2</math>, in order to bring the major or minor thirds closer to [[Just intonation|the just intonation ratio]] of <math> 5:4 </math> or <math> 6:5 </math>, 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]]'''  ({{nobr|12 {{sc|TET}}}}) is obtained by making all semitones the same size, with each equal to one-twelfth of an octave; i.e. with ratios {{nobr|{{math|{{radic|2 |12}} : 1}}}}. Relative to  [[Pythagorean tuning]], it narrows the [[perfect fifth]]s by about 2&nbsp;[[cents (music)|cents]] or {{nobr|{{sfrac| 1 | 12 }}}} of a [[Pythagorean comma]] to give a frequency ratio of <math> 2^{7/12}:1</math>. This produces major thirds that are wide by about 13&nbsp;[[cents (music)|cents]], or {{nobr|{{sfrac| 1 | 8 }}}} of a semitone. Twelve-tone equal temperament is almost exactly the same as {{nobr|{{sfrac| 1 | 11 }}}} [[syntonic comma]] meantone tuning (1.955 [[cents (music)|cents]] vs. 1.95512).
+'''Twelve-tone [[equal temperament]]''' ({{nobr|12 {{sc|TET}}}}) is obtained by making all semitones the same size, with each equal to one-twelfth of an octave; i.e. with ratios {{nobr|{{math|{{radic|2 |12}} : 1}}}}. Relative to  [[Pythagorean tuning]], it narrows the [[perfect fifth]]s by about 2&nbsp;[[cents (music)|cents]] or {{nobr|{{sfrac| 1 | 12 }}}} of a [[Pythagorean comma]] to give a frequency ratio of <math> 2^{7/12}:1</math>. This produces major thirds that are wide by about 13&nbsp;[[cents (music)|cents]], or {{nobr|{{sfrac| 1 | 8 }}}} of a semitone. Twelve-tone equal temperament is almost exactly the same as {{nobr|{{sfrac| 1 | 11 }}}} [[syntonic comma]] meantone tuning (1.955 [[cents (music)|cents]] vs. 1.95512).
 ==Notable meantone temperaments==
-[[Image:Meantone.jpg|500px|thumb|right|Figure 1. Comparison between [[Pythagorean tuning]] (blue), [[equal-tempered]] (black), [[quarter-comma meantone]] (red) and third-comma meantone (green). For each, the common origin is arbitrarily chosen as C. The values indicated by the scale at the left are deviations in cents with respect to equal temperament.]]
+[[Image:Meantone.jpg|500px|thumb|right|Figure&nbsp;1. Comparison between [[Pythagorean tuning]] (blue), [[equal-tempered]] (black), [[quarter-comma meantone]] (red) and third-comma meantone (green). For each, the common origin is arbitrarily chosen as C. The values indicated by the scale at the left are deviations in cents with respect to equal temperament.]]
-[[Quarter-comma meantone]], which tempers each of the twelve [[perfect fifths]] by {{sfrac| 1 | 4 }} 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 (as {{sc|'''C G D A E'''}}) tempered by {{nobr|{{sfrac| 1 | 4 }} comma}} (and lowered by two octaves) produce a just [[major third]] ({{sc|'''C E'''}}) (with ratio {{nobr|{{math|5 : 4}}}}), which is one syntonic comma (or about 22 [[Cent (music)|cents]]) narrower than the Pythagorean third that would result from four [[perfect fifth]]s.
+[[Quarter-comma meantone]], which tempers each of the twelve [[perfect fifths]] by {{sfrac| 1 | 4 }} 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 (as {{sc|'''C G D A E'''}}) tempered by {{nobr|{{sfrac| 1 | 4 }} comma}} (and then lowered by two octaves) produce a just [[major third]] ({{sc|'''C E'''}}) (with ratio {{nobr|{{math|5 : 4}}}}), which is one syntonic comma (or about 22&nbsp;[[cent (music)|cents]]) narrower than the Pythagorean third that would result from four [[perfect fifth]]s.
-It was commonly used from the early 16th century till the early 18th, after which twelve-tone equitemperament 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 temperament|31&nbsp;equal steps]].
+It was commonly used from the early 16th&nbsp;century till the early&nbsp;18th, after which twelve-tone equitemperament eventually came into general use. For church organs and some other keyboard purposes, it continued to be used well into the 19th&nbsp;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 temperament|31&nbsp;equal steps]].
-It proceeds in the same way as [[Pythagorean tuning]]; i.e., it takes the fundamental (say, {{sc|'''C'''}}) and goes up by six successive fifths (always adjusting by dividing by powers of {{math| 2 }} to remain within the octave above the fundamental), and similarly down, by six successive fifths (adjusting back to the octave by multiplying by {{nobr|powers of {{math|2}} ).}} However, instead of using the {{sfrac| 3 | 2 }} ratio, which gives [[perfect fifths]], this must be divided by the fourth root of {{sfrac| 81 | 80 }}, which is the [[syntonic comma]]: the ratio of the Pythagorean third {{sfrac| 81 | 64 }} to the just major third
+It proceeds in the same way as [[Pythagorean tuning]]; i.e., it takes the fundamental (say, {{sc|'''C'''}}) and goes up by six successive fifths (always adjusting by dividing by powers of {{math| 2 }} to remain within the octave above the fundamental), and similarly down, by six successive fifths (adjusting back to the octave by multiplying by {{nobr|powers of {{math|2}} ).}} However, instead of using the {{sfrac| 3 | 2 }} ratio, which gives [[perfect fifths]], this must be divided by the fourth root of {{sfrac| 81 | 80 }}, which is the [[syntonic comma]]: The ratio of the Pythagorean third {{sfrac| 81 | 64 }} to the just major third
 {{sfrac| 5 | 4 }}. Equivalently, one can use {{math|{{radic|5 |4}}}} instead of {{sfrac| 3 | 2 }}, which produces the same slightly reduced fifths. This results in the interval {{sc|'''C E'''}} being a [[Just intonation|just major third]] {{sfrac| 5 | 4 }}, and the intermediate seconds ({{sc|'''C D'''}}, {{sc|'''D E'''}}) dividing {{sc|'''C E'''}} uniformly, so {{sc|'''D C'''}} and {{sc|'''E D'''}} are equal ratios, whose square is {{sfrac| 5 | 4 }}. The same is true of the major second sequences {{sc|'''F G A'''}} and {{sc|'''G A B'''}}.
-However, there is a residual gap in [[Quarter-comma meantone | quarter-comma meantone tuning]] between the last of the upper sequence of six fifths and the last of the lower sequence; e.g. between {{sc|'''F'''}}{{music|#}} and {{sc|'''G'''}}{{music|b}} if the starting point is chosen as  {{sc|'''C'''}}, which, adjusted for the octave, are in the ratio of {{sfrac| 125 | 128 }} or {{math | -41.06}} cents. 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.
+However, there is a residual gap in [[Quarter-comma meantone | quarter-comma meantone tuning]] between the last of the upper sequence of six fifths and the last of the lower sequence; e.g. between {{sc|'''F'''}}{{sup|{{music|#}}}} and {{sc|'''G'''}}{{music|b}} if the starting point is chosen as {{sc|'''C'''}}, which, adjusted for the octave, are in the ratio of {{sfrac| 125 | 128 }} or {{nobr|{{math| −41.06}} cents.}} 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.
-In [[third-comma meantone]], the fifths are tempered by {{nobr|{{sfrac| 1 | 3 }}}} of a syntonic comma. It follows that three descending fifths (such as {{sc|'''A D G C'''}}) produce a [[Just intonation|just minor third]] ({{sc|'''A C'''}}) of ratio {{sfrac| 6 | 5 }}, which is nearly one syntonic comma wider than the minor third resulting from Pythagorean tuning of three [[perfect fifth]]s. Third-comma meantone can be very well approximated by a division of the octave into [[19 equal temperament|19&nbsp;equal steps]].
+In [[third-comma meantone]], the fifths are tempered by {{nobr|{{sfrac| 1 | 3 }}}} of a syntonic comma. It follows that three descending fifths (such as {{sc|'''A D G C'''}}) produce a [[Just intonation|just minor third]] ({{sc|'''A C'''}}) of ratio {{sfrac| 6 | 5 }}, which is one syntonic comma wider than the minor third resulting from Pythagorean tuning of three [[perfect fifth]]s. Third-comma meantone can be very well approximated by a division of the octave into [[19 equal temperament|19&nbsp;equal steps]].
 ==The tone as a mean==
@@ Line 23: / Line 23: @@
 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:
-<math>\ \sqrt{ \tfrac{ 10 }{\ 9\ } \cdot \tfrac{\ 9\ }{ 8 }\ } = \sqrt{ \tfrac{\ 5\ }{ 4 }\ } = 1.1180340 </math>,
+<math>\ \sqrt{ \tfrac{ 10 }{\ 9\ } \cdot \tfrac{\ 9\ }{ 8 }\ } = \sqrt{ \tfrac{\ 5\ }{ 4 }\ } = 1.1180340\ ,</math>
 equivalent to 193.157&nbsp;[[musical cent|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 (9:8), or half a comma wider than the minor tone (10:9). This is the sense in which quarter-tone temperament is often considered "the" exemplary meantone temperament since, in it, the whole tone lies midway (in [[musical cents|cents]]) between its possible extremes.<ref name ="Barbour"/>
+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 (9:8), or half a comma wider than the minor tone (10:9). This is the sense in which quarter-tone temperament is often considered "the" exemplary meantone temperament since, in it, the whole tone lies midway (in [[musical cents|cents]]) between its possible extremes.<ref name =Barbour/>
 ==History of meantone temperament and its practical implementation==
-Mention of tuning systems that could possibly refer to meantone were published as early as 1496 ([[Franchinus Gaffurius|Gaffurius]]).<ref>{{cite book |first=Franchinus |last=Gaffurius |author-link=Franchinus Gaffurius |year=1496 |title=Practicae musica |publisher = Gulielmum signer Rothomagensem|location = Milan |language=it|url=https://imslp.org/wiki/Practica_musicae_(Gaffurius,_Franchinus)}}</ref> [[Pietro Aron]]<ref>{{cite book |first=Pietro |last=Aron |author-link=Pietro Aron |year=1523 |location =Venice |title=Thoscanello de la musica |publisher= Marchio Sessa|language=it|url=https://imslp.org/wiki/Thoscanello_de_la_musica_(Aron%2C_Pietro)}}</ref>  (Venice, 1523) was unmistakably discussing quarter-comma meantone.  Lodovico Fogliani <ref name="Fogliani">{{cite book |
+Mention of tuning systems that could possibly refer to meantone were published as early as 1496 ([[Franchinus Gaffurius|Gaffurius]]).<ref>{{cite book |first=Franchinus |last=Gaffurius |author-link=Franchinus Gaffurius |year=1496 |title=Practicae musica |publisher = Gulielmum signer Rothomagensem|location = Milan |language=it|url=https://imslp.org/wiki/Practica_musicae_(Gaffurius,_Franchinus)}}</ref> [[Pietro Aron]]<ref>{{cite book |first=Pietro |last=Aron |author-link=Pietro Aron |year=1523 |location =Venice |title=Thoscanello de la musica |publisher= Marchio Sessa|language=it|url=https://imslp.org/wiki/Thoscanello_de_la_musica_(Aron%2C_Pietro)}}</ref> (Venice, 1523) was unmistakably discussing quarter-comma meantone. Lodovico Fogliani<ref name=Fogliani>{{cite book |first= Lodovico |last = Fogliani |title= Musica theorica |publisher = Civico Museo Bibliografico Musicale |series = Bibliotheca Musica Bononiensis |volume = II/13, 88 pp. |location = Bologna, IT | edition = Line-cut of the Venice, 1529 }}</ref> 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&nbsp;century treatises by [[Gioseffo Zarlino|Zarlino]]<ref>{{cite book |first=Gioseffo |last=Zarlino |author-link=Gioseffo Zarlino |year=1558 |title=Le istitutioni harmoniche |lang=it | location = Venice | url=https://imslp.org/wiki/Le_Istitutioni_Harmoniche_(Zarlino%2C_Gioseffo) }}</ref> and [[Francisco de Salinas|de&nbsp;Salinas]].<ref>{{cite book |first=Francisco |last=de&nbsp;Salinas |author-link=Francisco de Salinas |year=1577 |title=De musica libri septem |location = Salamanca | publisher = Mathias Gastius | url= https://imslp.org/wiki/De_musica_libri_septem_(Salinas%2C_Francisco_de)}}</ref> Both these authors described the {{nobr|{{sfrac| 1 | 4 }} comma,}} {{nobr|{{sfrac| 1 | 3 }} comma,}} and {{nobr|{{sfrac| 2 | 7 }} comma}} meantone systems.
-first= Lodovico |last = Fogliani |title= Musica theorica |publisher = Civico Museo Bibliografico Musicale | series= Bibliotheca Musica Bononiensis|volume= II/13, 88 pp. |location =Bologna | edition = Line-cut of the Venice, 1529}}</ref> 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 [[Gioseffo Zarlino|Zarlino]]<ref>{{cite book |first=Gioseffo |last=Zarlino |author-link=Gioseffo Zarlino |year=1558 |title=Le istitutioni harmoniche |language=it | location = Venice | url=https://imslp.org/wiki/Le_Istitutioni_Harmoniche_(Zarlino%2C_Gioseffo)}}</ref> and [[Francisco de Salinas|de&nbsp;Salinas]].<ref>{{cite book |first=Francisco |last=de&nbsp;Salinas |author-link=Francisco de Salinas |year=1577 |title=De musica libri septem |location = Salamanca | publisher = Mathias Gastius | url= https://imslp.org/wiki/De_musica_libri_septem_(Salinas%2C_Francisco_de)}}</ref> Both these authors described the {{nobr|{{sfrac| 1 | 4 }} comma,}} {{nobr|{{sfrac| 1 | 3 }} comma,}} and {{nobr|{{sfrac| 2 | 7 }} comma}} meantone systems.
+[[Marin Mersenne]] described various tuning systems in his seminal work on [[music theory]], ''[[Harmonie universelle]]'',<ref>{{cite book |first = M. |last = Mersenne |author-link=Marin Mersenne |year = 1639 |title = Harmonie universelle |edition = 1st online |place = Paris, FR |publisher = [[Bibliothèque nationale de France#Gallica|Gallica]] |url = http://gallica.bnf.fr/ark:/12148/bpt6k5471093v |quote = Translation to English by Roger E. Chapman (The&nbsp;Hague, 1957) }}</ref> including the 31&nbsp;tone equitempered one, but rejected it on practical grounds.
-[[Marin Mersenne]] described various tuning systems in his seminal work on [[music theory]], ''[[Harmonie universelle]]'',<ref>{{cite book | first = Marin | last = Mersenne| title= Harmonie universelle | author-link=Marin Mersenne|date = 1639 | location = Paris |url= http://gallica.bnf.fr/ark:/12148/bpt6k5471093v |publisher = First edition online from [[Bibliothèque nationale de France#Gallica|Gallica]] | trans-title = Translation to English by Roger E. Chapman (The Hague, 1957)}}</ref> 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 [[Christiaan Huygens|Huygens]]<ref name=Huygens-1691>{{cite book |first=Christiaan |last=Huygens |author-link=Christiaan Huygens |year=1691 |title=Lettre à Henri Basnage de Beauval touchant le cycle harmonique, citée dans: "Histoire des Ouvrages des Sçavans" |language=fr |trans-title=Letter concerning the harmonic cycle|location = Rotterdam |url=https://www.huygens-fokker.org/docs/lettre.html}}</ref> advocated the use of the 31 tone equitempered system {{nobr|([[31 equal temperament|31 {{sc|TET}}]])}} as an excellent  approximation for the {{sfrac| 1 | 4 }} comma meantone system, mentioning prior writings of [[Gioseffo Zarlino|Zarlino]] and [[Francisco de Salinas|Salinas]], and dissenting from the negative opinion of [[Mersenne]] (1639). He made a detailed comparison of the frequency ratios in the {{nobr|31 {{sc|TET}}}} system and the [[quarter-comma meantone]] temperament, which he referred to variously as "temperament ordinaire", or "the one that everyone uses". (See references cited in the article [[Temperament Ordinaire]].)
+Meantone temperaments were sometimes referred to under other names or descriptions. For example, in 1691 [[Christiaan Huygens|Huygens]]<ref name=Huygens-1691>{{cite book |first=Christiaan |last=Huygens |author-link=Christiaan Huygens |year=1691 |title=Lettre à Henri Basnage de Beauval touchant le cycle harmonique, citée dans: "Histoire des Ouvrages des Sçavans" |lang=fr |trans-title=Letter concerning the harmonic cycle |location = Rotterdam, NL |url=https://www.huygens-fokker.org/docs/lettre.html |via=huygens-fokker.org }}</ref> advocated the use of the 31&nbsp;tone equitempered system {{nobr|([[31 equal temperament|31 {{sc|TET}}]])}} as an excellent  approximation for the {{nobr|{{sfrac| 1 | 4 }} comma}} meantone system, mentioning prior writings of [[Gioseffo Zarlino|Zarlino]] and [[Francisco de Salinas|Salinas]], and dissenting from the negative opinion of {{harvp|Mersenne|1639}}. He made a detailed comparison of the frequency ratios in the {{nobr|31 {{sc|TET}}}} system and the [[quarter-comma meantone]] temperament, which he referred to variously as ''temperament ordinaire'', or "the one that everyone uses". (See references cited in the article [[Temperament Ordinaire]].)
-Of course, the quarter-comma meantone system (or any other meantone system) could not have been implemented with complete 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 tuning|just]] {{sfrac| 3 | 2 }} 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 C|middle {{sc|'''C'''}} (~264&nbsp;Hz)]], is about one [[Hertz (unit)|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 tuning|just]] major thirds, of ratio {{sfrac| 5 | 4 }}, by eliminating the beats.
+Of course, the quarter-comma meantone system (or any other 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&nbsp;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 tuning|just]] {{sfrac| 3 | 2 }} 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 C|middle {{sc|'''C'''}} (~264&nbsp;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 tuning|just]] major thirds, of ratio {{sfrac| 5 | 4 }}, by eliminating the beats.
-For 12&nbsp;tone [[equal temperament|equally-tempered tuning]], the fifths have to be tempered by considerably less than a {{nobr|{{sfrac|1|4}} comma}} (very close to a {{nobr|{{sfrac|1|11}} [[syntonic comma]]}}, or a {{nobr|{{sfrac|1|12}} [[Pythagorean comma]]}}), since they must form a perfect cycle, with no gap at the end, whereas {{nobr|{{sfrac|1|4}} comma}} meantone tuning, as mentioned above, has a residual gap that is twice as large as the Pythagorean one, in the opposite direction.
+For 12&nbsp;tone [[equal temperament|equally-tempered tuning]], the fifths have to be tempered by considerably less than a {{nobr|{{sfrac|1|4}} comma}} (very close to a {{nobr|{{sfrac|1|11}} [[syntonic comma]]}}, or a {{nobr|{{sfrac|1|12}} [[Pythagorean comma]]}}), since they must form a perfect cycle, with no gap at the end (''"[[circle of fifths]]"''). For {{nobr|{{sfrac|1|4}} comma}} meantone tuning, if one artificially stops after filling the octave with only 12&nbsp;pitches, one has a [[syntonic comma|residual gap]] between sharps and their enharmonic flats that is slightly smaller than the [[Pythagorean comma|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|{{nobr|31 {{sc|tet}}}}]] for meantone, and [[53 equal temperament|{{nobr|53 {{sc|tet}}}}]] for [[Pythagorean tuning|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 usage as a keyboard temperament well into the 19th&nbsp;century.
+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&nbsp;century.
-: "The mode of tuning which prevailed before the introduction of equal temperament, is called the Meantone System. It has hardly yet died out in England, for it may still be heard on a few organs in country churches. According to Don B. Yñiguez, organist of Seville Cathedral, the meantone system is generally maintained on Spanish organs, even at the present day." — [[George Grove|G. Grove]] (1890)<ref>{{cite encyclopedia |first=G. |last=Grove |author-link=George Grove |year=1890 |title={{grey|[no title cited]}} |encyclopedia=[[A Dictionary of Music and Musicians]] |publisher=Macmillan |place=London, UK |volume=IV |edition=1st |page=72}}</ref>
+: "The mode of tuning which prevailed before the introduction of equal temperament, is called the ''Meantone System''. It has hardly yet died out in England, for it may still be heard on a few organs in country churches. According to Don B. Yñiguez, organist of [[Seville Cathedral]], the meantone system is generally maintained on Spanish organs, even at the present day." — {{harvp|Grove|1890}}<ref>{{cite encyclopedia |editor-first=G. |editor-last=Grove |editor-link=George Grove |year=1890 |title={{grey|[no title cited]}} |encyclopedia=[[A Dictionary of Music and Musicians]] |publisher=Macmillan |place=London, UK |volume=IV |edition=1st |page=72}}</ref>
 It has had a considerable revival for early music performance in the late 20th&nbsp;century and in newly composed works specifically demanding meantone by some composers, such as [[John Adams (composer)|Adams]], [[György Ligeti|Ligeti]], and [[Douglas Leedy|Leedy]].
@@ Line 49: / Line 48: @@
 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&nbsp;to 699&nbsp;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 {{nobr|"{{mvar|R}}"}} by American composer, pianist and theoretician [[Easley Blackwood Jr.|Easley Blackwood]]. If {{mvar|R}} happens to be a rational number {{mvar|R}} = <math>{\scriptstyle\frac{N}{D}}</math>, then  <math>2^{\frac{3R+1}{5R+2}}</math> is the  closest approximation to the corresponding meantone tempered fifth within the equitempered division of the octave into <math>5N +2D</math> 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]].
+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 {{nobr|"{{mvar|R}}"}} by American composer, pianist and theoretician [[Easley Blackwood Jr.|Easley Blackwood]]. If {{mvar|R}} happens to be a rational number <math>\ R = \scriptstyle{ \frac{\ N\ }{ D } }\ ,</math> then <math>\ 2^{\frac{\ 3 R + 1\ }{ 5 R + 2} }\ </math> is the  closest approximation to the corresponding meantone tempered fifth within the equitempered division of the octave into <math>\ 5\ N + 2\ D\ </math> 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 [[Five-limit tuning|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 [[Cent (music)|cents]]. The fifth is the corresponding value of the fraction {{mvar|R}} = <math>\scriptstyle{\frac{N}{D}}</math>, and the fifth is the number <math>5N+2D</math> of equitempered ({{sc|ET}} ) microtones in an octave.
+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 [[Five-limit tuning|5&nbsp;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 [[Cent (music)|cents]]. The fifth is the corresponding value of the fraction <math>\ R = \scriptstyle{ \frac{\ N\ }{ D } }\ ,</math> and the fifth is the number <math>\ 5\ N + 2\ D\ </math> of equitempered ({{sc|ET}} ) microtones in an octave.
 {| class="wikitable" style="text-align: center;"
@@ Line 162: / Line 161: @@
 ===Equal temperaments===
 In neither the twelve tone equitemperament nor the quarter-comma meantone is the fifth a [[rational number|rational]] fraction of the octave, but several tunings exist which approximate the fifth by such an interval; these are a subset of the [[equal temperament]]s {{nobr|( "{{mvar|N}} {{sc|TET}}" ),}} in which the octave is divided into some number ({{mvar|N}}) of equally wide intervals.
-Equal temperaments that are useful as approximations to meantone tunings include (in order of increasing [[Generated collection|generator]] width) {{nobr|[[19 equal temperament|19 {{sc|TET}}]]}} {{nobr|({{sfrac| ~ | 1 | 3 }} comma),}} {{nobr|50 {{sc|TET}}}} {{nobr|({{sfrac| ~ | 2 | 7 }} comma),}} {{nobr|[[31 equal temperament|31 {{sc|TET}}]]}} {{nobr|({{sfrac| ~ | 1 | 4 }} comma),}} {{nobr|43 {{sc|TET}}}} {{nobr|({{sfrac| ~ | 1 | 5 }} comma),}} and {{nobr|55 {{sc|TET}}}} {{nobr|({{sfrac| ~ | 1 | 6 }} comma).}} 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 [[dynamic tonality|tempering the partials]] to match the tuning, which is possible, however, only on electronic synthesizers.<ref>
+Equal temperaments that are useful as approximations to meantone tunings include (in order of increasing [[Generated collection|generator]] width) {{nobr|[[19 equal temperament|19 {{sc|TET}}]]}}  {{nobr|({{sfrac| ~ | 1 | 3 }} comma),}} {{nobr|50 {{sc|TET}}}} {{nobr|({{sfrac| ~ | 2 | 7 }} comma),}} {{nobr|[[31 equal temperament|31 {{sc|TET}}]]}} {{nobr|({{sfrac| ~ | 1 | 4 }} comma),}} {{nobr|43 {{sc|TET}}}} {{nobr|({{sfrac| ~ | 1 | 5 }} comma),}} {{nobr|55 {{sc|TET}}}} {{nobr|({{sfrac| ~ | 1 | 6 }} comma),}} [[12 equal temperament|{{nobr|12 {{sc|TET}}}}]] {{nobr|({{sfrac| {{=}} | 1 | 11 }} comma),}} and {{nobr|[[53 equal temperament|53 {{sc|TET}}]]}} {{nobr|({{sfrac| ~ | 1 | 315 }} comma)}}. {{nobr|53 {{sc|TET}}}} almost perfectly fits both [[Pythagorean tuning]] and 5&nbsp;limit [[just intonation]], with a few 7&nbsp;limit and 11&nbsp;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 [[dynamic tonality|tempering the partials]] to match the tuning, which is possible, however, only on electronic synthesizers.<ref>
 {{cite journal
-  | last1 = Sethares | first1 = W.A. | author1-link = William Sethares
+  | last1 = Sethares | first1 = W.A.  | author1-link = William Sethares
   | last2=Milne      | first2 = A.       | last3=Tiedje    | first3 = S.
   | last4=Prechtl    | first4 = A.       | last5=Plamondon | first5 = J.
@@ Line 178: / Line 176: @@
   | id = {{Project MUSE|266411}}
 }}
-</ref>
+</ref> The following table gives various meantone temperaments
 [[File:Meantone_fifths_vs_thirds_comparison.svg|800px|thumb|center|Comparison of perfect fifths, major thirds, and minor thirds in various meantone tunings with just intonation]]
 <br />
-{{Center | '''Approximation of just intervals in equal temperaments'''<br />{{Gallery items | itemalign="center" | captionalign="center"
+{| style="text-align:center;vertical-align:top;"
-| [[File:12ed2-5Limit.svg|150px|thumb|center]] | 12-ET
+|+ '''Approximation of just intervals in equal temperaments'''<br/>''in order of increasing number of scale tones''
-| [[File:19ed2.svg|150px|thumb|center]] | 19-ET
+|- style="vertical-align:center;"
-| [[File:31ed2.svg|150px|thumb|center]] | 31-ET
+! [[12 equal temperament|{{nobr|12 {{sc|ET}}}}]]
-| [[File:43ed2.svg|150px|thumb|center]] | 43-ET
+! [[19 equal temperament|{{nobr|19 {{sc|ET}}}}]] !! [[31 equal temperament|{{nobr|31 {{sc|ET}}}}]]
-| [[File:50ed2.svg|150px|thumb|center]] | 50-ET
+! [[43 equal temperament|{{nobr|43 {{sc|ET}}}}]] !! [[50 equal temperament|{{nobr|50 {{sc|ET}}}}]]
-| [[File:55ed2.svg|150px|thumb|center]] | 55-ET
+! [[53 equal temperament|{{nobr|53 {{sc|ET}}}}]] !! [[55 equal temperament|{{nobr|55 {{sc|ET}}}}]]
-}}}}
+|- style="vertical-align:center;"
+| {{sfrac| {{=}} | 1 | 11 }} comma
+| {{sfrac| ~ | 1 | 3 }} comma    || {{sfrac| ~ | 1 | 4 }} comma
+| {{sfrac| ~ | 1 | 5 }} comma    || {{sfrac| ~ | 2 | 7 }} comma
+| {{sfrac| ~ | 1 | 315 }} comma  || {{sfrac| ~ | 1 | 6 }} comma
+|- style="vertical-align:top;"
+| [[File:12ed2-5Limit.svg|125px|thumb|center]]
+| [[File:19ed2.svg|125px|thumb|center]] || [[File:31ed2.svg|125px|thumb|center]]
+| [[File:43ed2.svg|125px|thumb|center]] || [[File:50ed2.svg|125px|thumb|center]]
+| [[File:53ed2.svg|125px|thumb|center]] || [[File:55ed2.svg|125px|thumb|center]]
+|}
 ==Wolf intervals==
-A whole number of just perfect fifths will never add up to a whole number of octaves, because {{nobr| {{math| log{{sub|2}} 3}} }} is an irrational number. If a stacked-up whole number of perfect fifths is too close to the octave, then one of the intervals that is enharmonically equivalent to a fifth must have a different width than the other fifths. For example, to make a 12&nbsp;note chromatic scale in [[Pythagorean tuning]] close at the octave, one of the fifth intervals must be lowered ("out-of-tune") 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 (e.g. between {{sc|'''G'''}}{{music|#}} and {{sc|'''E'''}}{{music|b}}). Likewise, 11 of the 12&nbsp;perfect fourths are also in tune, but the remaining fourth is an augmented third (rather than a true fourth).
+A whole number of just perfect fifths will never add up to a whole number of octaves, because {{nobr| {{math| log{{sub|2}} 3}} }} 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&nbsp;note chromatic scale in [[Pythagorean tuning]] close at the octave, one of the fifth intervals must be lowered ("out-of-tune") 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 (e.g. between {{sc|'''G'''}}{{sup|{{music|#}}}} and {{sc|'''E'''}}{{sup|{{music|b}}}}). Likewise, 11 of the 12&nbsp;perfect fourths are also in tune, but the remaining fourth is ''actually'' an augmented third (rather than a true fourth).
-Wolf intervals are an artifact of keyboard design, and keyboard players using a key that is actually in-tune with a different pitch than intended.<ref>
+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.<ref>
 {{cite journal
   | last1 = Milne     | first1 = Andrew
@@ Line 208: / Line 216: @@
   }}
 </ref>
-This can be shown most easily using an [[isomorphic keyboard]], such as that shown in Figure&nbsp;2.
-[[Image:Isomorphic Note Layout.jpg|thumb|bottom|550px|Figure&nbsp;2: [[Kaspar Wicki]]'s [[isomorphic keyboard]], invented in 1896.]]
-On an [[isomorphic keyboard]], any given musical interval has the same shape wherever it appears, except at the edges. Here's an example. On the keyboard shown in figure&nbsp;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 {{sc|'''E'''}}{{music|#}}. The note that's a perfect fifth higher than {{sc|'''E'''}}{{music|#}} is {{sc|'''B'''}}{{music|#}}, which is not included on the keyboard shown (although it could be included in a larger keyboard, placed just to the right of {{sc|'''A'''}}{{music|#}}, hence maintaining the keyboard's consistent note-pattern). Because there is no {{sc|'''B'''}}{{music|#}} button, when playing an {{sc|'''E'''}}{{music|#}} [[power chord]] (open fifth chord), one must choose some other note, such as {{sc|'''C'''}}, to play instead of the missing {{sc|'''B'''}}{{music|#}}.
-Even edge conditions produce wolf intervals only if the isomorphic keyboard has fewer buttons per octave than the tuning has [[enharmonic]]ally-distinct notes ({{harvnb|Milne|2007}}). For example, the isomorphic keyboard in figure&nbsp;2 has 19&nbsp;buttons per octave, so the above-cited edge-condition, from {{sc|'''E'''}}{{music|#}} to {{sc|'''C'''}}, is ''not'' a wolf interval in [[12 equal temperament|12&nbsp;tone equal temperament]] ({{sc|TET}}), 17&nbsp;{{sc|TET}}, or 19&nbsp;{{sc|TET}}; however, it ''is'' a wolf interval in 26&nbsp;{{sc|TET}}, 31&nbsp;{{sc|TET}}, and 50&nbsp;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.<ref>
+[[Image:Isomorphic Note Layout.jpg|thumb|bottom|550px|{{anchor|fig2_anchor}}Figure&nbsp;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&nbsp;keys per octave, such as that shown in [[#fig2_anchor|Figure&nbsp;2]] (on an ''[[isomorphic keyboard]]'', any given musical interval has the same shape wherever it appears, except at the edges). Here's an example: On the keyboard shown in [[#fig2_anchor|Figure&nbsp;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 {{sc|'''E'''}}{{sup|{{music|#}}}}. The note that's a perfect fifth higher than {{sc|'''E'''}}{{sup|{{music|#}}}} is {{sc|'''B'''}}{{sup|{{music|#}}}}, which is not included on the keyboard shown (although it could be included in a larger keyboard, placed just to the right of {{sc|'''A'''}}{{sup|{{music|#}}}}, hence maintaining the keyboard's consistent note-pattern). Because there is no {{sc|'''B'''}}{{sup|{{music|#}}}} button, when playing an {{sc|'''E'''}}{{sup|{{music|#}}}} [[power chord]] (open fifth chord), one must choose some other note, such as {{sc|'''C'''}}, to play instead of the missing {{sc|'''B'''}}{{sup|{{music|#}}}}.
+Even edge conditions produce wolf intervals only if the isomorphic keyboard has fewer buttons per octave than the tuning has [[enharmonic]]ally-distinct notes.<ref>
+{{cite journal
+ | last1 = Milne     | first1 = Andrew
+ | last2 = Sethares  | first2 = W.A. |author2-link=William Sethares
+ | last3 = Plamondon | first3 = J.
+ | date  = Winter 2007
+ | title = Isomorphic controllers and dynamic tuning: Invariant fingering over a tuning continuum
+ | journal = Computer Music Journal
+ | volume = 31  | issue = 4  | pages = 15–32
+ | doi = 10.1162/comj.2007.31.4.15
+}}
+</ref> For example, the isomorphic keyboard in Figure&nbsp;2 has 19&nbsp;buttons per octave, so the above-cited edge-condition, from {{sc|'''E'''}}{{sup|{{music|#}}}} to {{sc|'''C'''}}, is ''not'' a wolf interval in [[12 equal temperament|12&nbsp;tone equal temperament]] ({{sc|TET}}), 17&nbsp;{{sc|TET}}, or 19&nbsp;{{sc|TET}}; however, it ''is'' a wolf interval in 26&nbsp;{{sc|TET}}, 31&nbsp;{{sc|TET}}, and 50&nbsp;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.<ref>
 {{cite conference
   | last1 = Milne     | first1 = Andrew
@@ Line 225: / Line 244: @@
 </ref>
-Isomorphic keyboards expose the invariant properties of the meantone tunings of the [[syntonic temperament]] isomorphically (that is, for example, by exposing a given interval with a single consistent inter-button shape in every octave, key, and tuning) because both the isomorphic keyboard and temperament are two-dimensional (''i.e.'', [[Rank of an abelian group|rank&nbsp;2]]) entities ({{harvnb|Milne|2007}}). One-dimensional {{nobr|{{mvar|N}} key}} keyboards (where {{mvar|N}} is some number) can expose accurately the invariant properties of only a single one-dimensional tuning in {{nobr|{{mvar|N}} {{sc|TET}};}} hence, the one-dimensional piano-style keyboard, with 12&nbsp;keys per octave, can expose the invariant properties of only one tuning: 12&nbsp;{{sc|TET}}.
+Isomorphic keyboards expose the invariant properties of the meantone tunings of the [[syntonic temperament]] isomorphically (that is, for example, by exposing a given interval with a single consistent inter-button shape in every octave, key, and tuning) because both the isomorphic keyboard and temperament are two-dimensional (''i.e.'', [[Rank of an abelian group|rank&nbsp;2]]) entities.{{sfn|Milne|Sethares|Plamondon|2007}} One-dimensional {{nobr|{{mvar|N}} key}} keyboards (where {{mvar|N}} is some number) can expose accurately the invariant properties of only a single one-dimensional tuning in {{nobr|{{mvar|N}} {{sc|TET}};}} hence, the one-dimensional piano-style keyboard, with 12&nbsp;keys per octave, can expose the invariant properties of only one tuning: 12&nbsp;{{sc|TET}}.
-When the perfect fifth is exactly 700&nbsp;[[cent (music)|cents]] wide (that is, tempered by approximately {{sfrac|11}} of a syntonic comma, or exactly {{sfrac|12}} of a Pythagorean comma) then the tuning is identical to the familiar 12&nbsp;tone [[equal temperament]]. This appears in the table above when {{nobr|{{math| ''R'' {{=}} 2:1}} .}}
+When the perfect fifth is exactly 700&nbsp;[[cent (music)|cents]] wide (that is, tempered by almost exactly {{sfrac|11}} of a [[syntonic comma]], or exactly {{sfrac|12}} of a [[Pythagorean comma]]) then the tuning is identical to the familiar 12&nbsp;tone [[equal temperament]]. This appears in the table above when {{nobr|{{math| ''R'' {{=}} 2:1}} .}}
-Because of the compromises (and wolf intervals) forced on meantone tunings by the one-dimensional piano-style keyboard, [[well temperament]]s and eventually equal temperament became more popular.
+Because of the compromises (and wolf intervals) forced on meantone tunings by the limitation of having only 12&nbsp;key per octave on a conventional piano-style keyboard, [[well temperament]]s 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 second]]s (for example, {{sc|'''B'''}}{{music|b}} to {{sc|'''C'''}}{{music|#}}), and four "major thirds" are actually [[diminished fourth]]s (for example, {{sc|'''B'''}} to {{sc|'''E'''}}{{music|b}}). Several triads (like {{sc|'''B'''}} {{sc|'''E'''}}{{music|b}} {{sc|'''F'''}}{{music|#}} and {{sc|'''B'''}}{{music|b}} {{sc|'''C'''}}{{music|#}} {{sc|'''F'''}}) contain both these intervals and have normal fifths.
+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 second]]s (for example, {{sc|'''B'''}}{{sup|{{music|b}}}} to {{sc|'''C'''}}{{sup|{{music|#}}}}), and four "major thirds" are actually [[diminished fourth]]s (for example, {{sc|'''B'''}} to {{sc|'''E'''}}{{sup|{{music|b}}}}). Several triads (like {{sc|'''B'''}} {{sc|'''E'''}}{{sup|{{music|b}}}} {{sc|'''F'''}}{{sup|{{music|#}}}} and {{sc|'''B'''}}{{sup|{{music|b}}}} {{sc|'''C'''}}{{sup|{{music|#}}}} {{sc|'''F'''}}) contain both these intervals and have normal fifths.
 ==Extended meantones==
 <!-- [[WP:NFCC]] violation: [[Image:Tonraum30.11-14 1.jpg|thumb|left|Figure&nbsp;3: The archicembalo, a harpsichord-like instrument built to use an extended quarter-comma meantone tuning]] -->
-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 intonation]]s, conceivably have an infinite number of notes in each octave, that is, seven natural notes, seven sharp notes ({{sc|'''F'''}}{{music|#}} to {{sc|'''B'''}}{{music|#}}), seven flat notes ({{sc|'''B'''}}{{music|b}} to {{sc|'''F'''}}{{music|b}}) (which is the limit of the [[orchestral harp]], which allows 21&nbsp;pitches in an octave); 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; triple sharps and flats are almost never seen. In any syntonic tuning that happens to divide the octave into a small number of equally wide smallest intervals (such as [[12 equal temperament#Twelve-tone equal temperament|12]], [[19 equal temperament|19]], or [[31 equal temperament|31]]), this infinity of notes still exists, although some notes will be equivalent. For example, in 19&nbsp;ET, {{sc|'''E'''}}{{music|#}} and {{sc|'''F'''}}{{music|b}} are the same pitch; and in [[just intonation]] for [[C major]], {{sc|'''C'''}}{{music|#}} {{sc|'''D'''}}{{music|bb}} are within 8.1&nbsp;[[Music|¢]], and so can be tempered to be identical.
+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 intonation]]s, conceivably have an infinite number of notes in each octave, that is, seven natural notes, seven sharp notes ({{sc|'''F'''}}{{sup|{{music|#}}}} to {{sc|'''B'''}}{{sup|{{music|#}}}}), seven flat notes ({{sc|'''B'''}}{{sup|{{music|b}}}} to {{sc|'''F'''}}{{sup|{{music|b}}}}) (which is the limit of the [[orchestral harp]], which allows 21&nbsp;distinct pitches per octave); then double sharp notes ({{sc|'''F'''}}{{sup|{{music|x}}}} to {{sc|'''B'''}}{{sup|{{music|x}}}}), double flat notes ({{sc|'''F'''}}{{sup|{{music|bb}}}} to {{sc|'''B'''}}{{sup|{{music|bb}}}}), 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 (such as [[12 equal temperament#Twelve-tone equal temperament|12]], [[19 equal temperament|19]], or {{nobr|[[31 equal temperament|31]] {{sc|ET}}),}} this extended set of notes still exists, but is ''not'' infinite, since some notes will be equivalent. For example, in {{nobr|19 {{sc|ET}},}} {{sc|'''E'''}}{{sup|{{music|#}}}} and {{sc|'''F'''}}{{sup|{{music|b}}}} are the same pitch; in {{nobr|31 {{sc|ET}},}} {{sc|'''C'''}}{{sup|{{music|x}}{{music|#}}}} and {{sc|'''E'''}}{{sup|{{music|bb}}}} are identical, as are {{sc|'''E'''}}{{sup|{{music|x}}}} and {{sc|'''G'''}}{{sup|{{music|bb}}}}; and in [[just intonation]] for [[C major]], {{sc|'''C'''}}{{sup|{{music|#}}}} {{sc|'''D'''}}{{sup|{{music|bb}}}} are within 8.1&nbsp;[[cents (music)|¢]], and so can be tempered to be identical, with the compromise note being only a tolerable 4&nbsp;¢ off for each.
-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. These instruments are well-suited to the use of meantone tunings.
+Many musical instruments are capable of very fine distinctions of pitch, such as the [[singing|human voice]], the [[trombone]], unfretted strings such as the [[violin family]] and [[fretless guitar]]s, and [[oud|lutes with movable frets]]. These instruments are well-suited to the use of meantone tunings.
-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&nbsp;ET. Almost all of the historic problems with the meantone temperament are caused by the attempt to map meantone's infinite number of notes per octave to a finite number of piano keys. This is, for example, the source of the "wolf fifth" discussed above. 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.
+On the other hand, the conventional piano keyboard only has twelve note-producing keys per octave, making it poorly suited to any tunings other than {{nobr|12 {{sc|ET}}}} or [[well temperament]]s. Almost all of the historic problems with the meantone temperament are caused by the failure to map meantone's infinite number of notes per octave to a finite number of piano keys. This is, for example, the source of the "wolf fifth" discussed above. 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 crossing up or down to an adjacent octave, for some of the intervals must be a "wolf fifth" – that is, slightly flatter than the others, as described above.
-The existence of the "wolf fifth" is one of the reasons why, before the introduction of [[well temperament]], instrumental music generally stayed in a number of "safe" tonalities that did not involve the "wolf fifth" (which was generally put between {{sc|'''G'''}}{{music|#}} and {{sc|'''E'''}}{{music|b}}).
+The existence of the "wolf fifth" is one of the reasons why, before the introduction of [[well temperament]], instrumental music generally stayed in a number of "safe" tonalities that did not involve the "wolf fifth" (which was generally put between {{sc|'''G'''}}{{sup|{{music|#}}}} and {{sc|'''E'''}}{{sup|{{music|b}}}}).
-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,<ref name = "Barbour"/><ref name = "Duffin">{{cite book |last=Duffin |first=Ross W. |year=2007 |title=How Equal Temperament Ruined Harmony (and why you should care) |publisher=W. W. Norton & Company |isbn=978-0-393-06227-4 }}{{page needed|date=January 2024}}</ref><ref>{{cite book |last1=Isacoff |first1=Stuart |title=Temperament: How music became a battleground for the great minds of western civilization |year=2009 |publisher=Knopf Doubleday Publishing Group |isbn=978-0-307-56051-3 }}{{page needed|date=January 2024}}</ref> and hence have come to be called "extended" meantone tunings. These efforts required a corresponding extension of keyboard instruments to offer means of controlling more than 12&nbsp;notes per octave, including Vincento's [[Archicembalo]],<!-- (shown in Figure 3) --> Mersenne's 19&nbsp;ET harpsichord, Colonna's 31&nbsp;ET sambuca, and Huygens's 31&nbsp;ET harpsichord.<ref>
+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,<ref name=Barbour/><ref name=Duffin>
+{{cite book
+ |last=Duffin |first=Ross W.
+ |year=2007
+ |title=How Equal Temperament Ruined Harmony (and why you should care)
+ |publisher=W.W. Norton & Company
+ |isbn=978-0-393-06227-4
+}}{{page needed|date=January 2024}}
+</ref><ref>
+{{cite book
+ |last=Isacoff |first=Stuart
+ |year=2009
+ |title=Temperament: How music became a battleground for the great minds of western civilization
+ |publisher=Knopf Doubleday Publishing Group
+ |isbn=978-0-307-56051-3
+}}{{page needed|date=January 2024}}
+</ref> 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&nbsp;notes per octave; examples include [[Nicola Vicentino|Vincento's]] [[archicembalo]],<!-- (shown in Figure&nbsp;3) --> [[Marin Mersenne|Mersenne's]] {{nobr|19 {{sc|ET}}}} [[harpsichord]], [[Fabio Colonna|Colonna]]'s {{nobr|31 {{sc|ET}}}} [[hurdy-gurdy|sambuca rota]], and [[Christiaan Huygens|Huygens's]] {{nobr|31 {{sc|ET}}}} harpsichord.<ref>
 {{cite journal
   | first = Christopher | last = Stembridge
@@ Line 253: / Line 289: @@
 }}
 </ref>
-Other instruments extended the keyboard by only a few notes. Some period harpsichords and organs have split {{sc|'''D'''}}{{music|#}} / {{sc|'''E'''}}{{music|b}} keys, such that both [[E major]] / [[C-sharp minor|C{{music|#}} minor]] (4&nbsp;sharps) and [[E-flat major|E{{music|b}} major]] / [[C minor]] (3&nbsp;flats) can be played with no wolf fifths. Many of those instruments also have split {{sc|'''G'''}}{{music|#}} / {{sc|'''A'''}}{{music|b}} keys, and a few have all the five accidental keys split.
+Other instruments extended the keyboard by only a few notes. Some period [[harpsichord]]s and [[organ (music)|organs]] have split {{sc|'''D'''}}{{sup|{{music|#}}}} / {{sc|'''E'''}}{{sup|{{music|b}}}} keys, such that both [[E major]] / [[C-sharp minor|C{{sup|{{music|#}}}} minor]] (4&nbsp;sharps) and [[E-flat major|E{{sup|{{music|b}}}} major]] / [[C minor]] (3&nbsp;flats) can be played with no wolf fifths. Many of those instruments also have split {{sc|'''G'''}}{{sup|{{music|#}}}} / {{sc|'''A'''}}{{sup|{{music|b}}}} keys, and a few have all the five accidental keys split.
 All of these alternative instruments were "complicated" and "cumbersome" ({{harvnb|Isacoff|2009}}), due to
@@ Line 265: / Line 302: @@
   | year = 2007
   | title = The Thummer mapping project (ThuMP)
-  | conference = 7th International Conference on New Interfaces for Musical Expression (NIME 07)
+  | conference = 7th&nbsp;International Conference on New Interfaces for Musical Expression (NIME 07)
   | pages = 70–77
   | url = http://www.activatedspace.com/Papers/PaineThuMP.pdf

Meantone temperament: Difference between revisions

Latest revision as of 19:50, 18 October 2025

Contents

Notable meantone temperaments

The tone as a mean

History of meantone temperament and its practical implementation

Meantone temperaments

Equal temperaments

Wolf intervals

Extended meantones

See also

References

External links

Navigation menu

Meantone temperament: Difference between revisions

Latest revision as of 19:50, 18 October 2025

Notable meantone temperaments

The tone as a mean

History of meantone temperament and its practical implementation

Meantone temperaments

Equal temperaments

Wolf intervals

Extended meantones

See also

References

External links

Navigation menu

Search