# Meantone temperament

Comparison between Pythagorean tuning (blue), equal-tempered (black), 1/4-comma meantone (red) and 1/3-comma meantone (green). For each, the common origin is arbitrarily chosen as C. The degrees are arranged in the order of the cycle of fifths; as in each of these tunings all fifths are of the same size, the tunings appear as straight lines, the slope indicating the relative tempering with respect to Pythagorean, which has pure fifths (3:2, 702 cents). The Pythagorean A (at the left) is at 792 cents, G (at the right) at 816 cents; the difference is the Pythagorean comma. Equal temperament by definition is such that A and G are at the same level. 1/4 comma meantone produces the "just" major third (5:4, 386 cents, a syntonic comma lower than the Pythagorean one of 408 cents). 1/3 comma meantone produces the "just" minor third (6:5, 316 cents, a syntonic comma higher than the Pythagorean one of 294 cents). In both these meantone temperaments, the enharmony, here the difference between A and G, is much larger than in Pythagorean, and with the flat degree higher than the sharp one.

Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 used in 12 equal temperament. The meantone temperament:

• generates all non-octave intervals from a stack of tempered perfect fifths; and
• by choosing an appropriate size for major and minor thirds, tempers the syntonic comma to unison.

Quarter-comma meantone is the best known type of meantone temperament, and the term meantone temperament is often used to refer to it specifically.

## Meantone temperaments

Figure 1: The linear temperament continuum (Milne 2007). It covers the range of linear temperaments that can also be used as generators for Regular diatonic tunings. In other words these are linear temperaments that include subsets with the familiar pattern of intervals of the diatonic scale. Notable meantone tunings are labeled, at center-left, with names of the form "n/d-comma".

"Meantone" can receive the following equivalent definitions:

• The meantone is the mean between the major whole tone (9/8 in just intonation) and the minor whole tone (10/9 in just intonation), i.e., the geometric mean of 9/8 and 10/9.
• The meantone is the mean of the just major third, i.e. the square root of 5/4.

The family of meantone temperaments share the common characteristic that they form a stack of identical fifths, the tone being the result of two fifths minus one octave, the major third of four fifths minus two octaves. Meantone temperaments are often described by the fraction of the syntonic comma by which the fifths are tempered: quarter-comma meantone, the most common type, tempers the fifths by 1/4 syntonic comma, with the result that four fifths produce a just major third, a syntonic comma lower than a Pythagorean major third; third-comma meantone tempers by 1/3 syntonic comma, three fifths producing a just major sixth, a syntonic comma lower than a Pythagorean one.

All meantone temperaments are linear temperament's,[1] distinguished by the width of its generator in cents, as shown in the central column of Figure 1. Historically notable meantone temperaments, discussed below, occupy a narrow portion of the this tuning continuum, ranging from approximately 695 to 699 cents.

While the term meantone temperament refers primarily to the tempering of 5-limit musical intervals, optimum values for the 5-limit also work well for the 7-limit, defining septimal meantone temperament. In Figure 1, the valid tuning ranges of 5-limit, 7-limit, and 11-limit tunings are shown, and can be seen to include many notable meantone tunings.

Meantone temperaments can be specified in various ways: by what fraction (logarithmically) of a syntonic comma the fifth is being flattened (as above), what equal temperament has the meantone fifth in question, the width of the tempered perfect fifth in cents, or the ratio of the whole tone to the diatonic semitone. This last ratio was termed "R" by American composer, pianist and theoretician Easley Blackwood, but in effect has been in use for much longer than that. It is useful because it gives us an idea of the melodic qualities of the tuning, and because if R is a rational number N/D, so is (3R+1)/(5R+2) or (3N+D)/(5N+2D), which is the size of fifth in terms of logarithms base 2, and which immediately tells us what division of the octave we will have. If we multiply by 1200, we have the size of fifth in cents.

In these terms, some historically notable meantone tunings are listed below. The relationship between the first two columns is exact, while that between them and the third is closely approximate.

Meantone tunings
R Size of the fifth in octaves Fraction of a (syntonic) comma
9/4 31/53 1/315 (nearly Pythagorean Tuning)
2/1 7/12 1/11 (1/12 Pythagorean comma)
9/5 32/55 1/6
7/4 25/43 1/5
5/3 18/31 1/4
8/5 29/50 2/7
3/2 11/19 1/3

### Equal temperaments

Neither the just fifth nor the quarter-comma meantone fifth is 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-ET"), in which the octave is divided into some number (N) of equally wide intervals.

Equal temperaments useful as meantone tunings include (in order of increasing generator width) 19-ET, 50-ET, 31-ET, 43-ET, and 55-ET. The farther the tuning gets away from quarter-comma meantone, however, the less related[2] the tuning is to harmonic timbres, which can be overcome by tempering the timbre to match the tuning.[3]

## Wolf intervals

A whole number of just perfect fifths will never add up to a whole number of octaves, because they are incommensurable (see Fundamental theorem of arithmetic). If a stacked-up whole number of perfect fifths is to close with the octave, then one of the fifths must have a different width than all of the others. For example, to make the 12-note chromatic scale in Pythagorean tuning close at the octave, one fifth must be out of tune by the Pythagorean comma; this altered fifth is called a wolf fifth.

Wolf intervals are an artifact of keyboard design.[4] This can be shown most easily using an isomorphic keyboard, such as that shown in Figure 2.

Fig. 2: The Wicki isomorphic keyboard, invented by Kaspar Wicki 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 2, from any given note, the note that's a perfect fifth higher is always up-and-rightwardly 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, 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). 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-ET, 17-ET, or 19-ET; however, it is a wolf interval 26-ET, 31-ET, 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.[5]

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). One-dimensional N-key keyboards can expose accurately the invariant properties of only a single one-dimensional N-ET tuning; hence, the one-dimensional piano-style keyboard, with 12 keys per octave, can expose the invariant properties of only one tuning: 12-ET.

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 BEF and BCF) contain both these intervals and have normal fifths.

## Extended meantones

Figure 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, have a conceptually infinite number of notes in each octave, that is, seven natural notes, seven sharp notes (F to B), seven flat notes (B to F), 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/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 enharmonic. For example, in 19-ET, E and F are the same pitch.

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/A and D/E).

Throughout the Renaissance and Enlightenment, theorists as varied as Nicola Vicentino, Francisco de Salinas, Fabio Colonna, Marin Mersenne, Constantijn Huygens, and Isaac Newton advocated the use of meantone tunings that were extended beyond the keyboard's twelve notes,[6][7][8] and hence have come to be called "extended" meantone tunings. These efforts required a concomitant extension of keyboard instruments to offer means of controlling more than 12 notes per octave, including Vincento's Archicembalo (shown in Figure 3), Mersenne's 19-ET harpsichord, Colonna's 31-ET sambuca, and Huygens' 31-ET harpsichord.[9] 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 without 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, 2003), due to (a) not being isomorphic, and (b) not having the ability to transpose electronically, which can significantly reduce the number of note-controlling buttons needed on an isomorphic keyboard (Plamondon, 2009). Both of these criticisms could be addressed by electronic isomorphic keyboard instruments (such as the open source jammer keyboard), which could be simpler, less cumbersome, and more expressive than existing keyboard instruments.[10]

## Use of meantone temperament

References to tuning systems that could possibly refer to meantone were published as early as 1496 (Gafori) and Aron (1523) is unmistakably referring to meantone. However, the first mathematically precise Meantone tuning descriptions are found in late 16th century treatises by Francisco de Salinas and Gioseffo Zarlino. Salinas (in De musica libra septum) describes three different mean tone temperaments: the 1/3 comma system, the 2/7 comma system, and the 1/4 comma system. He is the likely inventor of the 1/3 system, while he and Zarlino both wrote on the 2/7 system, apparently independently. Lodovico Fogliano mentions the 1/4 comma system, but offers no discussion of it.

In the past, meantone temperaments were sometimes used or referred to under other names or descriptions. For example, in 1691 Christiaan Huygens wrote his "Lettre touchant le cycle harmonique" ("Letter concerning the harmonic cycle") with the purpose of introducing what he believed to be a new division of the octave. In this letter Huygens referred several times, in a comparative way, to a conventional tuning arrangement, which he indicated variously as "temperament ordinaire", or "the one that everyone uses". But Huygens' description of this conventional arrangement was quite precise, and is clearly identifiable with what is now classified as (quarter-comma) meantone temperament.[11]

Although meantone is best known as a tuning environment associated with earlier music of the Renaissance and Baroque, there is evidence of continuous usage of meantone as a keyboard temperament well into the middle of the 19th century. Meantone temperament has had considerable revival for early music performance in the late 20th century and in newly composed works specifically demanding meantone by composers including John Adams, György Ligeti and Douglas Leedy.

## New uses of meantone tunings

Meantone is one of many possible tuning effects found in Dynamic Tonality (Plamondon, 2009).

## References

1. Milne, Andrew; Sethares, W.A., Plamondon, J. (December 2007). "Invariant Fingerings Across a Tuning Continuum". Computer Music Journal. 31 (4): 15–32. doi:10.1162/comj.2007.31.4.15. Retrieved 2009-09-20. Cite uses deprecated parameter `|coauthors=` (help)
2. Sethares, William (September 1993). "Local consonance and the relationship between timbre and scale". Journal of the Acoustical Society of America. 94 (3): 1218–1228. doi:10.1121/1.408175.
3. Sethares, William; Milne, A.; Tiedje, S.; Prechtl, A.; Plamondon, J. (2009). "Spectral Tools for Dynamic Tonality and Audio Morphing". Computer Music Journal. 33 (2): 71–84. doi:10.1162/comj.2009.33.2.71. Retrieved 2009-09-20.
4. Milne, Andrew; Sethares, W.A.; Plamondon, J. (March 2008). "Tuning Continua and Keyboard Layouts". Journal of Mathematics and Music. 2 (1): 1–19. doi:10.1080/17459730701828677. Retrieved 2009-09-20.
5. Plamondon, Jim; Milne, A.; Sethares, W.A. (2009). "Dynamic Tonality: Extending the Framework of Tonality into the 21st Century" (PDF). Proceedings of the Annual Conference of the South Central Chapter of the College Music Society.
6. Barbour, J.M., 2004, Tuning and Temperament: A Historical Survey.
7. Duffin, R.W., 2006, How Equal Temperament Ruined Harmony (and Why You Should Care).
8. Isacoff, Stuart, 2003, Temperament: How Music Became a Battleground for the Great Minds of Western Civilization
9. Stembridge, Christopher (1993). "The Cimbalo Cromatico and Other Italian Keyboard Instruments with Nineteen or More Divisions to the Octave". Performance Practice Review. vi (1): 33–59.
10. Paine, G.; Stevenson, I.; Pearce, A. (2007). "The Thummer Mapping Project (ThuMP)" (PDF). Proceedings of the 7th international conference on New Interfaces for Musical Expression (NIME07): 70–77.
11. (See references cited in article 'Temperament Ordinaire'.)