# 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 $3:2$, 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. A regular temperament is one in which all the fifths are chosen to be of the same size .
Twelve-tone equal temperament (12 TET) is obtained by making all semitones the same size, with each equal to one-twelfth of an octave; i.e. with ratios $^{12}$√2  : 1. Relative to Pythagorean tuning, it narrows the perfect fifths by about 2 cents or ⁠1/ 12 ⁠th 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 ⁠1/ 8 ⁠th of a semitone. Twelve-tone equal temperament is almost exactly the same as ⁠1/ 11 ⁠ syntonic comma meantone tuning (1.955 cents vs. 1.95512).

## Notable meantone temperaments

Quarter-comma meantone, which tempers each of the twelve perfect fifths by ⁠ 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 C G D A E) tempered by ⁠ 1 / 4 ⁠ comma (and lowered by two octaves) produce a just major third (C E) (with ratio 5 : 4), which is one syntonic comma (or about 22 cents) narrower than the Pythagorean third that would result from four perfect fifths.
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, C) and goes up by six successive fifths (always adjusting by dividing by powers of  2  to remain within the octave above the fundamental), and similarly down, by six successive fifths (adjusting back to the octave by multiplying by powers of 2 ). However, instead of using the ⁠ 3 / 2 ⁠ ratio, which gives perfect fifths, this must be divided by the fourth root of ⁠ 81 / 80 ⁠, which is the syntonic comma: the ratio of the Pythagorean third ⁠ 81 / 64 ⁠ to the just major third ⁠ 5 / 4 ⁠. Equivalently, one can use $^4$√5  instead of ⁠ 3 / 2 ⁠, which produces the same slightly reduced fifths. This results in the interval C E being a just major third ⁠ 5 / 4 ⁠, and the intermediate seconds (C D, D E) dividing C E uniformly, so D C and E D are equal ratios, whose square is ⁠ 5 / 4 ⁠. The same is true of the major second sequences F G A and G A B.
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 F♯ and G♭ if the starting point is chosen as C, which, adjusted for the octave, are in the ratio of ⁠ 125 / 128 ⁠ or -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 ⁠ 1 / 3 ⁠ of a syntonic comma. It follows that three descending fifths (such as A D G C) produce a just minor third (A C) of ratio ⁠ 6 / 5 ⁠, which is nearly 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 the whole tone (as C D) is reached after two fifths (as C G D) (lowered by an octave), while the major third is reached after four fifths (C G D A E) (lowered by two octaves). It follows that in ⁠ 1 / 4 ⁠ comma meantine the whole tone is exactly half of the just major third (in cents) or, equivalently, the square root of the frequency ratio of ⁠ 5 / 4 ⁠.
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: $\ {\sqrt {{\tfrac {10}{\ 9\ }}\cdot {\tfrac {\ 9\ }{8}}\ }}={\sqrt {{\tfrac {\ 5\ }{4}}\ }}=1.1180340$, 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.

## 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). Pietro Aron (Venice, 1523) 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 ⁠ 1 / 4 ⁠ comma, ⁠ 1 / 3 ⁠ comma, and ⁠ 2 / 7 ⁠ comma 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 (31 TET) as an excellent approximation for the ⁠ 1 / 4 ⁠ comma meantone system, mentioning prior writings of Zarlino and Salinas, and dissenting from the negative opinion of Mersenne (1639). He made a detailed comparison of the frequency ratios in the 31 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 ⁠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 (~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 ⁠5/ 4 ⁠, by eliminating the beats.
For 12 tone equally-tempered tuning, the fifths have to be tempered by considerably less than a ⁠1/4⁠ comma (very close to a ⁠1/11⁠ syntonic comma, or a ⁠1/12⁠ Pythagorean comma), since they must form a perfect cycle, with no gap at the end, whereas ⁠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.
Although meantone is best known as a tuning system associated with earlier music of the Renaissance and Baroque, there is evidence of its continuous usage as a keyboard temperament well into the 19th century.
    "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." — G. Grove (1890)
It has had a considerable revival for early music performance in the late 20th century and in newly composed works specifically demanding meantone by some composers, such as Adams, Ligeti, and Leedy.

## Meantone temperaments

A meantone temperament is a regular temperament, distinguished by the fact that the correction factor to the Pythagorean perfect fifths, given usually as a specific fraction of the syntonic comma, is chosen to make the whole tone intervals equal, as closely as possible, to the geometric mean of the major tone and the minor tone. Historically, commonly used meantone temperaments, discussed below, occupy a narrow portion of this tuning continuum, with fifths ranging from approximately 695 to 699 cents.
Meantone temperaments can be specified in various ways: By what fraction of a syntonic comma the fifth is being flattened (as above), the width of the tempered perfect fifth in cents, or the ratio of the whole tone (in cents) to the diatonic semitone. This last ratio was termed "R" by American composer, pianist and theoretician Easley Blackwood. If R happens to be a rational number R = ${\scriptstyle {\frac {N}{D}}}$, then $2^{\frac {3R+1}{5R+2}}$ is the closest approximation to the corresponding meantone tempered fifth within the equitempered division of the octave into $5N+2D$ 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 systonic 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 = $\scriptstyle {\frac {N}{D}}$, and the fifth is the number $5N+2D$ of equitempered (ET ) microtones in an octave.
Meantone vs. Equitempered tunings :
| Meantone fraction of (syntonic) comma | 5-limit rational intervals | Size of ET fifths as fractions of an octave | Error between meantone fifths and ET fifths (in cents) | Blackwood’s ratio R =$\scriptstyle {\frac {N}{D}}$ | Number of ET microtones $5N+2D$ |
| ⁠1/ 315 ⁠ (very nearly Pythagorean tuning) | For all practical purposes, the fifth is a "perfect" ⁠ 3 / 2 ⁠ . | ⁠ 31 / 53 ⁠ | +0.000066 (+6.55227×10$^{−5}$) | ⁠ 9 / 4 ⁠ = 2.25 | 53 |
| ⁠1/ 11 ⁠ ( or ⁠1/ 12 ⁠ Pythagorean comma) | ⁠ 16384 / 10935 ⁠ =⁠ 2$^{14}$ / 3$^7$ × 5 ⁠ ( Kirnberger fifth: a just fifth flattened by a schisma. Equals meantone to 6 significant figures.) | ⁠ 7 / 12⁠ | +0.000116 (+1.16371×10$^{−4}$) | ⁠ 2 / 1 ⁠ = 2.00 | 12 |
| ⁠ 1 / 6 ⁠ | ⁠ 45 / 32 ⁠ and ⁠ 64 / 45 ⁠ (tritones) | ⁠ 32 / 55 ⁠ | −0.188801 | ⁠ 9 / 5 ⁠ = 1.80 | 55 |
| ⁠ 1 / 5 ⁠ | ⁠16/ 15 ⁠ and ⁠15/ 8 ⁠ (diatonic semitone and major seventh) | ⁠ 25 / 43 ⁠ | +0.0206757 | ⁠ 7 / 4 ⁠ = 1.75 | 43 |
| ⁠ 1 / 4 ⁠ | ⁠ 5 / 4 ⁠ and ⁠ 8 / 5 ⁠ (just major third and minor sixth) | ⁠ 18 / 31 ⁠ | +0.195765 | ⁠ 5 / 3 ⁠ = 1.66 | 31 |
| ⁠ 2 / 7 ⁠ | ⁠ 25 / 24 ⁠ and ⁠ 48 / 25 ⁠ (chromatic semitone and major seventh ) | ⁠ 29 / 50 ⁠ | +0.189653 | ⁠ 8 / 5 ⁠ = 1.60 | 50 |
| ⁠ 1 / 3 ⁠ | ⁠6/ 5 ⁠ and ⁠5/ 3 ⁠ (just minor third and major sixth) | ⁠ 11 / 19 ⁠ | −0.0493956 | ⁠ 3 / 2 ⁠ = 1.50 | 19 |
| ⁠ 2 / 5 ⁠ | ⁠ 27/ 25 ⁠ (large limma) | ⁠ 26 / 45 ⁠ | +0.0958 | ⁠ 7 / 5 ⁠ = 1.40 | 45 |
| ⁠ 1 / 2 ⁠ | ⁠10/ 9 ⁠ and ⁠9/ 5 ⁠ (just minor tone and diminished seventh) | ⁠ 19 / 33 ⁠ | −0.292765 | ⁠ 5/ 4 ⁠ = 1.25 | 33 |

### 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 ( "N TET" ), in which the octave is divided into some number (N) of equally wide intervals.
Equal temperaments that are useful as approximations to meantone tunings include (in order of increasing generator width) 19 TET (⁠~ + 1 / 3 ⁠ comma), 50 TET (⁠~ + 2 / 7 ⁠ comma), 31 TET (⁠~ + 1 / 4 ⁠ comma), 43 TET (⁠~ + 1 / 5 ⁠ comma), and 55 TET (⁠~ + 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 tempering the partials to match the tuning, which is possible, however, only on electronic synthesizers.
* 12-ET
* 19-ET
* 31-ET
* 43-ET
* 50-ET
* 55-ET

## Wolf intervals

A whole number of just perfect fifths will never add up to a whole number of octaves, because log$_2$ 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 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 G♯ and E♭). Likewise, 11 of the 12 perfect fourths are also in tune, but the remaining fourth is 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. This can be shown most easily using an isomorphic keyboard, 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 E♯. The note that's a perfect fifth higher than E♯ is B♯, which is not included on the keyboard shown (although it could be included in a larger keyboard, placed just to the right of A♯, hence maintaining the keyboard's consistent note-pattern). Because there is no B♯ button, when playing an E♯ power chord (open fifth chord), one must choose some other note, such as C, to play instead of the missing B♯.
Even edge conditions produce wolf intervals only if the isomorphic keyboard has fewer buttons per octave than the tuning has enharmonically-distinct notes (Milne 2007 harvnb error: no target: CITEREFMilne2007 (help)). For example, the isomorphic keyboard in figure 2 has 19 buttons per octave, so the above-cited edge-condition, from E♯ to C, is not a wolf interval in 12 tone equal temperament (TET), 17 TET, or 19 TET; however, it is a wolf interval in 26 TET, 31 TET, 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 (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 (Milne 2007 harvnb error: no target: CITEREFMilne2007 (help)). One-dimensional N key keyboards (where N is some number) can expose accurately the invariant properties of only a single one-dimensional tuning in N TET; hence, the one-dimensional piano-style keyboard, with 12 keys per octave, can expose the invariant properties of only one tuning: 12 TET.
When the perfect fifth is exactly 700 cents wide (that is, tempered by approximately ⁠1/11⁠ of a syntonic comma, or exactly ⁠1/12⁠ of a Pythagorean comma) then the tuning is identical to the familiar 12 tone equal temperament. This appears in the table above when R = 2:1 .
Because of the compromises (and wolf intervals) forced on meantone tunings by the one-dimensional 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, B♭ to C♯), and four "major thirds" are actually diminished fourths (for example, B to E♭). Several triads (like B E♭ F♯ and B♭ C♯ F) 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 (F♯ to B♯), seven flat notes (B♭ to F♭) (which is the limit of the orchestral harp, which allows 21 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, 19, or 31), this infinity of notes still exists, although some notes will be equivalent. For example, in 19 ET, E♯ and F♭ are the same pitch; and in just intonation for C major, C♯ D are within 8.1 ¢, and so can be tempered to be identical.
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.
On the other hand, the piano keyboard has only twelve physical note-controlling devices per octave, making it poorly suited to any tunings other than 12 ET. 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.
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 G♯ and E♭).
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 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 notes per octave, including Vincento's Archicembalo, Mersenne's 19 ET harpsichord, Colonna's 31 ET sambuca, and Huygens's 31 ET harpsichord. Other instruments extended the keyboard by only a few notes. Some period harpsichords and organs have split D♯ / E♭ keys, such that both E major / C♯ minor (4 sharps) and E♭ major / C minor (3 flats) can be played with no wolf fifths. Many of those instruments also have split G♯ / A♭ keys, and a few have all the five accidental keys split.
All of these alternative instruments were "complicated" and "cumbersome" (Isacoff 2009), 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 (Plamondon 2009 harvnb error: no target: CITEREFPlamondon2009 (help)). 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.