Meantone temperaments aremusical temperaments;[1] that is, a variety oftuning systems constructed, similarly toPythagorean tuning, as a sequence of equal fifths, both rising and descending, scaled to remain within the same octave. But rather than usingperfect fifths, consisting of frequency ratios of value, these aretempered by a suitable factor that narrows them to ratios that are slightly less than, in order to bring the major or minor thirds closer tothe just intonation ratio of or, respectively. Among temperaments constructed as a sequence of fifths, aregular temperament is one in which all the fifths are chosen to be of the same size.
Twelve-toneequal temperament (12TET) is obtained by making all semitones the same size, with each equal to one-twelfth of an octave; i.e. with ratios12√2 : 1. Relative toPythagorean tuning, it narrows theperfect fifths by about 2 cents or1/ 12 of aPythagorean comma to give a frequency ratio of. This produces major thirds that are wide by about 13 cents, or1/ 8 of a semitone. Twelve-tone equal temperament is almost exactly the same as1/ 11 syntonic comma meantone tuning (1.955cents vs. 1.95512).
Quarter-comma meantone, which tempers each of the twelveperfect fifths by 1 / 4 of asyntonic comma, is the best known type of meantone temperament, and the termmeantone temperament is often used to refer to it specifically. Four ascending fifths (asC G D A E) tempered by 1 / 4 comma (and then lowered by two octaves) produce a justmajor third (C E) (with ratio5 : 4), which is one syntonic comma (or about 22 cents) narrower than the Pythagorean third that would result from fourperfect fifths.
It was commonly used from the early 16th century till the early 18th, after which twelve-tone equal temperament eventually came into general use. For church organs and some other keyboard purposes, it continued to be used well into the 19th century, and is sometimes revived in early music performances today.Quarter-comma meantone can be well approximated by a division of the octave into31 equal steps.
It proceeds in the same way asPythagorean 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 bypowers of2 ). However, instead of using the 3 / 2 ratio, which givesperfect fifths, this must be divided by the fourth root of 81 / 80, which is thesyntonic comma: the ratio of the Pythagorean third 81 / 64 to the just major third 5 / 4. Equivalently, one can use4√5 instead of 3 / 2, which produces the same slightly reduced fifths. This results in the intervalC E being ajust major third 5 / 4, and the intermediate seconds (C D,D E) dividingC E uniformly, soD C andE D are equal ratios, whose square is 5 / 4. The same is true of the major second sequencesF G A andG 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. betweenF♯ andG♭ if the starting point is chosen asC, 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.
Inthird-comma meantone, the fifths are tempered by 1 / 3 of a syntonic comma. It follows that three descending fifths (such asA D G C) produce ajust minor third (A C) of ratio 6 / 5, which is one syntonic comma wider than the minor third resulting from Pythagorean tuning of threeperfect fifths. Third-comma meantone can be very well approximated by a division of the octave into19 equal steps.
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 themajor and minor tones (9:8 and 10:9 respectively) ofjust intonation, which differ from each other by asyntonic comma. In any regular system[1] the whole tone (asC D) is reached after two fifths (asC 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 meantone 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 thegeometric mean of the major tone and the minor tone: equivalent to 193.157 cents: the quarter-comma whole-tone size. However,any intermediate tone qualifies as a "mean" in the sense of being intermediate, and hence as a valid choice for some meantone system.
In the case of quarter-comma meantone, where the major third is made narrower by a syntonic comma, the whole tone is made half a comma narrower than the major tone of just intonation (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 (incents) between its possible extremes.[1]
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 byZarlino[5] andde Salinas.[6] 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 onmusic 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 1691Huygens[8] advocated the use of the 31 tone equitempered system(31TET) as an excellent approximation for the 1 / 4 comma meantone system, mentioning prior writings ofZarlino andSalinas, and dissenting from the negative opinion ofMersenne (1639). He made a detailed comparison of the frequency ratios in the31TET system and thequarter-comma meantone temperament, which he referred to variously astemperament ordinaire, or "the one that everyone uses". (See references cited in the articleTemperament 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 thejust3/ 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, nearmiddleC (~264 Hz), is about onehertz, 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 tojust major thirds, of ratio5/ 4 , by eliminating the beats.
For 12 toneequally-tempered tuning, the fifths have to be tempered by considerably less than a1/4 comma (very close to a1/11syntonic comma, or a1/12Pythagorean comma), since they must form a perfect cycle, with no gap at the end ("circle of fifths"). For1/4 comma meantone tuning, if one artificially stops after filling the octave with only 12 pitches, one has aresidual gap between sharps and their enharmonic flats that is slightly smaller than thePythagorean one, in the opposite direction. Both quarter-comma meantone and the Pythagorean system do not have acircle but rather aspiral of fifths, which continues indefinitely. Slightly tempered versions of the two systems thatdo close into a much larger circle of fifths are31TET for meantone, and53TET forPythagorean.
Although meantone is best known as a tuning system associated with earlier music of the Renaissance and Baroque, there is evidence of its continuous use as a keyboard temperament well into the 19th century.
It has had a considerable revival for early music performance in the late 20th century and in newly composed works specifically demanding meantone by some composers, such asAdams,Ligeti, andLeedy.
A meantone temperament is aregular 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 diatonicsemitone. This last ratio was termed"R" by American composer, pianist and theoreticianEasley Blackwood. IfR happens to be a rational number then is the closest approximation to the corresponding meantone tempered fifth within the equitempered division of the octave into equal parts. Such divisions of the octave into a number of small parts greater than 12 are sometimes refererred to asmicrotonality, and the smallest intervals calledmicrotones.
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 lists5 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, incents. The fifth is the corresponding value of the fraction and the fifth is the number of equitempered (ET ) microtones in an octave.
| Meantone fraction of (syntonic) comma | 5-limit rational intervals | Size ofET fifths as fractions of an octave | Error between meantone fifths andET fifths (in cents) | Blackwood’s ratio R = | Number ofET microtones |
|---|---|---|---|---|---|
1/ 315 (very nearly | 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 ( or1/ 12 Pythagorean comma) | 16384 / 10935 = 214 / 37 × 5 ( Kirnberger fifth: a just fifth flattened by aschisma. | 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 and15/ 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 and5/ 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 and9/ 5 (just minor tone and diminished seventh) | 19 / 33 | −0.292765 | 5/ 4 = 1.25 | 33 |
In neither the twelve tone equitemperament nor the quarter-comma meantone is the fifth arational fraction of the octave, but several tunings exist which approximate the fifth by such an interval; these are a subset of theequal temperaments( "NTET" ), 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 increasinggenerator width)19TET(~+ 1 / 3 comma),50TET(~+ 2 / 7 comma),31TET(~+ 1 / 4 comma),43TET(~+ 1 / 5 comma),55TET(~+ 1 / 6 comma),12TET(=+ 1 / 11 comma), and53TET(~+ 1 / 315 comma).53TET almost perfectly fits bothPythagorean tuning and 5 limitjust 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 bytempering the partials to match the tuning, which is possible, however, only on electronic synthesizers.[10] The following table gives various meantone temperaments
| 12ET | 19ET | 31ET | 43ET | 50ET | 53ET | 55ET |
|---|---|---|---|---|---|---|
| =+ 1 / 11 comma | ~+ 1 / 3 comma | ~+ 1 / 4 comma | ~+ 1 / 5 comma | ~+ 2 / 7 comma | ~+ 1 / 315 comma | ~+ 1 / 6 comma |
 |  |  |  |  |  |  |
A whole number of just perfect fifths will never add up to a whole number of octaves, because log2 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 note chromatic scale inPythagorean tuning close at the octave, one of the fifth intervals must be lowered ("out-of-tune") by thePythagorean 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 isactually a diminished sixth (e.g. betweenG♯ andE♭). Likewise, 11 of the 12 perfect fourths are also in tune, but the remaining fourth isactually 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]
The issue can be most easily shown by using anisomorphic keyboard, with many more than just 12 keys per octave, such as that shown inFigure 2 (on anisomorphic keyboard, any given musical interval has the same shape wherever it appears, except at the edges). Here's an example: On the keyboard shown inFigure 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 noteE♯. The note that's a perfect fifth higher thanE♯ isB♯, which is not included on the keyboard shown (although it could be included in a larger keyboard, placed just to the right ofA♯, hence maintaining the keyboard's consistent note-pattern). Because there is noB♯ button, when playing anE♯power chord (open fifth chord), one must choose some other note, such asC, to play instead of the missingB♯.
Even edge conditions produce wolf intervals only if the isomorphic keyboard has fewer buttons per octave than the tuning hasenharmonically-distinct notes.[12] For example, the isomorphic keyboard in Figure 2 has 19 buttons per octave, so the above-cited edge-condition, fromE♯ toC, isnot a wolf interval in12 tone equal temperament (TET), 17 TET, or 19 TET; however, itis 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.[13]
Isomorphic keyboards expose the invariant properties of the meantone tunings of thesyntonic 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.[14] One-dimensionalN key keyboards (whereN is some number) can expose accurately the invariant properties of only a single one-dimensional tuning inNTET; 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 almost exactly1/11 of asyntonic comma, or exactly1/12 of aPythagorean comma) then the tuning is identical to the familiar 12 toneequal temperament. This appears in the table above whenR = 2:1 .
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 oneaugmented seventh; seven octaves are equal to eleven fifths plus onediminished sixth. Given this, three "minor thirds" are actuallyaugmented seconds (for example,B♭ toC♯), and four "major thirds" are actuallydiminished fourths (for example,B toE♭). Several triads (likeBE♭F♯ andB♭C♯F) contain both these intervals and have normal fifths.
All meantone tunings fall into the valid tuning range of thesyntonic temperament, so all meantone tunings are syntonic tunings. All syntonic tunings, including the meantones and the variousjust intonations, conceivably have an infinite number of notes in each octave, that is, seven natural notes, seven sharp notes (F♯ toB♯), seven flat notes (B♭ toF♭) (which is the limit of theorchestral harp, which allows 21 distinct pitches per octave); then double sharp notes (F
toB), double flat notes (F
toB), 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 as12,19, or31ET), this extended set of notes still exists, but isnot infinite, since some notes will be equivalent. For example, in19ET,E♯ andF♭ are the same pitch; in31ET,C♯ andE are identical, as areE andG; and injust intonation forC major,C♯D 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 thehuman voice, thetrombone, unfretted strings such as theviolin family andfretless guitars, andlutes 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 than12ET orwell 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 ofwell temperament, instrumental music generally stayed in a number of "safe" tonalities that did not involve the "wolf fifth" (which was generally put betweenG♯ andE♭).
Throughout the Renaissance and Enlightenment, theorists as varied asNicola Vicentino,Francisco de Salinas,Fabio Colonna,Marin Mersenne,Christiaan Huygens, andIsaac Newton advocated the use of meantone tunings that were extended beyond the keyboard's twelve notes,[1][15][16] 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 includeVincento'sarchicembalo,Mersenne's19ETharpsichord,Colonna's31ETsambuca rota, andHuygens's31ET harpsichord.[17]
Other instruments extended the keyboard by only a few notes. Some periodharpsichords andorgans have splitD♯ /E♭ keys, such that bothE major /C♯ minor (4 sharps) andE♭ major /C minor (3 flats) can be played with no wolf fifths. Many of those instruments also have splitG♯ /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
which can significantly reduce the number of note-controlling buttons needed on anisomorphic keyboard (Plamondon 2009 harvnb error: no target: CITEREFPlamondon2009 (help)). Both of these criticisms could be addressed by electronic isomorphic keyboard instruments (such as theopen-source hardware jammer keyboard), which could be simpler, less cumbersome, and more expressive than existing keyboard instruments.[18]
{{cite book}}:ISBN / Date incompatibility (help)Translation to English by Roger E. Chapman (The Hague, 1957)
