Meantone temperament is a
musical temperamentIn musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...
, which is a system of
musical tuningIn music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
. In general, a meantone is constructed the same way as
Pythagorean tuningPythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
, as a stack of
perfect fifthIn classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
s, but in meantone, each fifth is
narrow compared to the ratio 2
7/12:1 in 12 equal temperament, the opposite of Pythagorean. The meantone temperament:
- generates all non-octave intervals from a stack of tempered
In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...
perfect fifths; and
- by choosing an appropriate size for major and minor thirds, tempers the syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
to unison.
Quarter-comma meantoneQuarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning...
is the best known type of meantone temperament, and the term
meantone temperament is often used to refer to it specifically.
Meantone temperaments
Though quarter-comma meantone is the most common type, other systems that flatten the fifth by some amount, but that still equate the major whole tone (9/8 in just intonation) with the minor whole tone (10/9 in just intonation), are also called meantone systems. Since (9/8) / (10/9) = (81/80)—the
syntonic commaIn music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
—the fundamental characteristics of meantone systems are that all intervals are generated from fifths, and the syntonic comma is tempered to a unison.
All meantone temperaments fall on the
syntonic temperamentThe syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect fifth, with the width of the tempered major third being equal to four tempered perfect fifths minus two octaves and the width of...
's tuning continuum, and as such are "syntonic tunings." The distinguishing feature of each unique syntonic tuning is the width of its generator in
centsThe cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
, as shown in the central column of Figure 1. Historically notable meantone temperaments, discussed below, occupy a narrow portion of the syntonic temperament's tuning continuum, ranging from approximately 695 to 699 cents. The criteria which define the limits (if any) of the meantone range of tunings within the syntonic temperament's tuning continuum are not yet well-defined.
While the term
meantone temperament refers primarily to the tempering of
5-limitIn music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...
musical intervals, optimum values for the 5-limit also work well for the 7-limit, defining
septimal meantone temperamentIn music, septimal meantone temperament, also called standard septimal meantone or simply septimal meantone, refers to the tempering of 7-limit musical intervals by a meantone temperament tuning in the range from fifths flattened by the amount of fifths for 12 equal temperament to those as flat as...
. In Figure 1, the valid tuning ranges of 5-limit, 7-limit, and 11-limit syntonic 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 temperamentAn equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
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
semitoneA semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....
. 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 numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer 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
logarithmThe logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
s 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 In music, 53 equal temperament, called 53-TET, 53-EDO, or 53-ET, is the tempered scale derived by dividing the octave into 53 equal steps . Each step represents a frequency ratio of 21/53, or 22.6415 cents , an interval sometimes called the Holdrian comma.- History :Theoretical interest in this...
|
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 In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...
|
1/4 |
| 8/5 |
29/50 |
2/7 |
| 3/2 |
11/19 In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...
|
1/3 |
Equal temperaments
Neither the just fifth nor the quarter-comma meantone fifth is a
rationalIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
fraction of the octave, but several tunings exist which approximate the fifth by such an interval; these are a subset of the
equal temperamentAn equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
s ("
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-ETIn music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...
, 50-ET,
31-ETIn music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...
, 43-ET, and 55-ET. The farther the tuning gets away from quarter-comma meantone, however, the less related the tuning is to harmonic timbres, which can be overcome by
tempering the timbreDynamic tonality is tonal music which uses real-time changes in tuning and timbre to perform new musical effects such as polyphonic tuning bends, new chord progressions, and temperament modulations, with the option of consonance. The performance of dynamic tonality requires an isomorphic keyboard...
to match the tuning.
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 arithmeticIn number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...
). 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 tuningPythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
close at the octave, one fifth must be out of tune by the
Pythagorean commaIn musical tuning, the Pythagorean comma , named after the ancient mathematician and philosopher Pythagoras, is the small interval existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B , or D and C...
; this altered fifth is called a wolf fifth.
Wolf intervals are an artifact of keyboard design. 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 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 chordIn music, a power chord is a chord consisting of only the root note of the chord and the fifth interval, usually played on electric guitar, and typically through an amplification process that imparts distortion...
, 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
enharmonicallyIn modern musical notation and tuning, an enharmonic equivalent is a note , interval , or key signature which is equivalent to some other note, interval, or key signature, but "spelled", or named, differently...
-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.
Isomorphic keyboards expose the invariant properties of the meantone tunings of the
syntonic temperamentThe syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect fifth, with the width of the tempered major third being equal to four tempered perfect fifths minus two octaves and the width of...
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-2In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the...
) 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
centsThe cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
wide (that is, tempered by approximately 1/11 of a syntonic comma) then the tuning is identical to the familiar 12-tone
equal temperamentAn equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
. 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 temperamentWell temperament is a type of tempered tuning described in 20th-century music theory. The term is modelled on the German word wohltemperiert which appears in the title of J.S. Bach's famous composition, The Well-Tempered Clavier...
s and eventually equal temperament became more popular.
Using standard interval names, twelve fifths equal six octaves plus one
augmented seventhIn classical music from Western culture, an augmented seventh is an interval produced by widening a major seventh by a chromatic semitone. For instance, the interval from C to B is a major seventh, eleven semitones wide, and both the intervals from C to B, and from C to B are augmented sevenths,...
; seven octaves are equal to eleven fifths plus one
diminished sixthIn classical music from Western culture, a diminished sixth is an interval produced by narrowing a minor sixth by a chromatic semitone. For example, the interval from A to F is a minor sixth, eight semitones wide, and both the intervals from A to F, and from A to F are diminished sixths, spanning...
. Given this, three "minor thirds" are actually
augmented secondIn classical music from Western culture, an augmented second is an interval produced by widening a major second by a chromatic semitone. For instance, the interval from C to D is a major second, two semitones wide, and both the intervals from C to D, and from C to D are augmented seconds, spanning...
s (for example, B to C), and four "major thirds" are actually
diminished fourthIn classical music from Western culture, a diminished fourth is an interval produced by narrowing a perfect fourth by a chromatic semitone. For example, the interval from C to F is a perfect fourth, five semitones wide, and both the intervals from C to F, and from C to F are diminished fourths,...
s (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 temperamentThe syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect fifth, with the width of the tempered major third being equal to four tempered perfect fifths minus two octaves and the width of...
, so all meantone tunings are syntonic tunings. All syntonic tunings, including the meantones, 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), double sharp notes, double flat notes, triple sharps and flats, and so on. In reality, double sharps/flats are uncommon, but still needed; triple sharps/flats are never seen. In any syntonic tuning that happens to divide the octave into a small number of equally wide smallest intervals (such as 12,
19In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...
, or
31In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...
), 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 able to divide the octave into a nearly infinite number of notes, 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 12 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 temperamentWell temperament is a type of tempered tuning described in 20th-century music theory. The term is modelled on the German word wohltemperiert which appears in the title of J.S. Bach's famous composition, The Well-Tempered Clavier...
, 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 VicentinoNicola Vicentino was an Italian music theorist and composer of the Renaissance. He was one of the most visionary musicians of the age, inventing, among other things, a microtonal keyboard, and devising a practical system of chromatic writing two hundred years before the rise of equal...
,
Francisco de SalinasFrancisco de Salinas was a Spanish music theorist and organist, noted as among the first to describe meantone temperament in mathematically precise terms, and one of the first to describe, in effect, 19 equal temperament. In his De musica libri septem of 1577 he discusses 1/3-, 1/4- and 2/7-comma...
, Fabio Colonna,
Marin MersenneMarin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the "father of acoustics"...
,
Constantijn HuygensConstantijn Huygens , was a Dutch Golden Age poet and composer. He was secretary to two Princes of Orange: Frederick Henry and William II, and the father of the scientist Christiaan Huygens.-Biography:...
, and
Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
advocated the use of meantone tunings that were extended beyond the piano's twelve notes, and hence have come to be called "extended" meantone tunings. These efforts required a concomitant extension of keyboard instruments to offer means of contolling more than 12 notes per octave, including Vincento's Archicembalo (shown in Figure 3), Mersenne's 19-ET harpsihord, Colonna's 31-ET sambuca, and Huygens' 31-ET harpsichord. Other instruments extended the piano keyboard by only a few notes. Some period harpsichords and organs have split D/E keys, such that both
E majorE major is a major scale based on E, with the pitches E, F, G, A, B, C, and D. Its key signature has four sharps .Its relative minor is C-sharp minor, and its parallel minor is E minor....
/C
Meantone temperament is a
musical temperamentIn musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...
, which is a system of
musical tuningIn music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
. In general, a meantone is constructed the same way as
Pythagorean tuningPythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
, as a stack of
perfect fifthIn classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
s, but in meantone, each fifth is
narrow compared to the ratio 2
7/12:1 in 12 equal temperament, the opposite of Pythagorean. The meantone temperament:
- generates all non-octave intervals from a stack of tempered
In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...
perfect fifths; and
- by choosing an appropriate size for major and minor thirds, tempers the syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
to unison.
Quarter-comma meantoneQuarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning...
is the best known type of meantone temperament, and the term
meantone temperament is often used to refer to it specifically.
Meantone temperaments
Though quarter-comma meantone is the most common type, other systems that flatten the fifth by some amount, but that still equate the major whole tone (9/8 in just intonation) with the minor whole tone (10/9 in just intonation), are also called meantone systems. Since (9/8) / (10/9) = (81/80)—the
syntonic commaIn music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
—the fundamental characteristics of meantone systems are that all intervals are generated from fifths, and the syntonic comma is tempered to a unison.
All meantone temperaments fall on the
syntonic temperamentThe syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect fifth, with the width of the tempered major third being equal to four tempered perfect fifths minus two octaves and the width of...
's tuning continuum, and as such are "syntonic tunings." The distinguishing feature of each unique syntonic tuning is the width of its generator in
centsThe cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
, as shown in the central column of Figure 1. Historically notable meantone temperaments, discussed below, occupy a narrow portion of the syntonic temperament's tuning continuum, ranging from approximately 695 to 699 cents. The criteria which define the limits (if any) of the meantone range of tunings within the syntonic temperament's tuning continuum are not yet well-defined.
While the term
meantone temperament refers primarily to the tempering of
5-limitIn music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...
musical intervals, optimum values for the 5-limit also work well for the 7-limit, defining
septimal meantone temperamentIn music, septimal meantone temperament, also called standard septimal meantone or simply septimal meantone, refers to the tempering of 7-limit musical intervals by a meantone temperament tuning in the range from fifths flattened by the amount of fifths for 12 equal temperament to those as flat as...
. In Figure 1, the valid tuning ranges of 5-limit, 7-limit, and 11-limit syntonic 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 temperamentAn equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
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
semitoneA semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....
. 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 numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer 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
logarithmThe logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
s 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 In music, 53 equal temperament, called 53-TET, 53-EDO, or 53-ET, is the tempered scale derived by dividing the octave into 53 equal steps . Each step represents a frequency ratio of 21/53, or 22.6415 cents , an interval sometimes called the Holdrian comma.- History :Theoretical interest in this...
|
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 In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...
|
1/4 |
| 8/5 |
29/50 |
2/7 |
| 3/2 |
11/19 In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...
|
1/3 |
Equal temperaments
Neither the just fifth nor the quarter-comma meantone fifth is a
rationalIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
fraction of the octave, but several tunings exist which approximate the fifth by such an interval; these are a subset of the
equal temperamentAn equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
s ("
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-ETIn music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...
, 50-ET,
31-ETIn music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...
, 43-ET, and 55-ET. The farther the tuning gets away from quarter-comma meantone, however, the less related the tuning is to harmonic timbres, which can be overcome by
tempering the timbreDynamic tonality is tonal music which uses real-time changes in tuning and timbre to perform new musical effects such as polyphonic tuning bends, new chord progressions, and temperament modulations, with the option of consonance. The performance of dynamic tonality requires an isomorphic keyboard...
to match the tuning.
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 arithmeticIn number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...
). 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 tuningPythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
close at the octave, one fifth must be out of tune by the
Pythagorean commaIn musical tuning, the Pythagorean comma , named after the ancient mathematician and philosopher Pythagoras, is the small interval existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B , or D and C...
; this altered fifth is called a wolf fifth.
Wolf intervals are an artifact of keyboard design. 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 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{{music|sharp}}. The note that's a perfect fifth higher than E{{music|sharp}} is B{{music|sharp}}, which is not included on the keyboard shown (although it could be included in a larger keyboard, placed just to the right of A{{music|sharp}}, hence maintaining the keyboard's consistent note-pattern). Because there is no B{{music|sharp}} button, when playing an E{{music|sharp}}
power chordIn music, a power chord is a chord consisting of only the root note of the chord and the fifth interval, usually played on electric guitar, and typically through an amplification process that imparts distortion...
, one must choose some other note, such as C, to play instead of the missing B{{music|sharp}}.
Even edge conditions produce wolf intervals only if the isomorphic keyboard has fewer buttons per octave than the tuning has
enharmonicallyIn modern musical notation and tuning, an enharmonic equivalent is a note , interval , or key signature which is equivalent to some other note, interval, or key signature, but "spelled", or named, differently...
-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{{music|sharp}} 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.
Isomorphic keyboards expose the invariant properties of the meantone tunings of the
syntonic temperamentThe syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect fifth, with the width of the tempered major third being equal to four tempered perfect fifths minus two octaves and the width of...
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-2In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the...
) 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
centsThe cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
wide (that is, tempered by approximately 1/11 of a syntonic comma) then the tuning is identical to the familiar 12-tone
equal temperamentAn equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
. 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 temperamentWell temperament is a type of tempered tuning described in 20th-century music theory. The term is modelled on the German word wohltemperiert which appears in the title of J.S. Bach's famous composition, The Well-Tempered Clavier...
s and eventually equal temperament became more popular.
Using standard interval names, twelve fifths equal six octaves plus one
augmented seventhIn classical music from Western culture, an augmented seventh is an interval produced by widening a major seventh by a chromatic semitone. For instance, the interval from C to B is a major seventh, eleven semitones wide, and both the intervals from C to B, and from C to B are augmented sevenths,...
; seven octaves are equal to eleven fifths plus one
diminished sixthIn classical music from Western culture, a diminished sixth is an interval produced by narrowing a minor sixth by a chromatic semitone. For example, the interval from A to F is a minor sixth, eight semitones wide, and both the intervals from A to F, and from A to F are diminished sixths, spanning...
. Given this, three "minor thirds" are actually
augmented secondIn classical music from Western culture, an augmented second is an interval produced by widening a major second by a chromatic semitone. For instance, the interval from C to D is a major second, two semitones wide, and both the intervals from C to D, and from C to D are augmented seconds, spanning...
s (for example, B{{music|flat}} to C{{music|sharp}}), and four "major thirds" are actually
diminished fourthIn classical music from Western culture, a diminished fourth is an interval produced by narrowing a perfect fourth by a chromatic semitone. For example, the interval from C to F is a perfect fourth, five semitones wide, and both the intervals from C to F, and from C to F are diminished fourths,...
s (for example, B to E{{music|flat}}). Several triads (like B–E{{music|flat}}–F{{music|sharp}} and B{{music|flat}}–C{{music|sharp}}–F) contain both these intervals and have normal fifths.
Extended meantones
All meantone tunings fall into the valid tuning range of the
syntonic temperamentThe syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect fifth, with the width of the tempered major third being equal to four tempered perfect fifths minus two octaves and the width of...
, so all meantone tunings are syntonic tunings. All syntonic tunings, including the meantones, have an infinite number of notes in each octave, that is, seven natural notes, seven sharp notes (F{{music|#}} to B{{music|#}}), seven flat notes (B{{music|b}} to F{{music|b}}), double sharp notes, double flat notes, triple sharps and flats, and so on. In reality, double sharps/flats are uncommon, but still needed; triple sharps/flats are never seen. In any syntonic tuning that happens to divide the octave into a small number of equally wide smallest intervals (such as 12,
19In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...
, or
31In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...
), this infinity of notes still exists, although some notes will be enharmonic. For example, in 19-ET, E{{music|#}} and F{{music|b}} are the same pitch.
Many musical instruments are able to divide the octave into a nearly infinite number of notes, 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 12 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 temperamentWell temperament is a type of tempered tuning described in 20th-century music theory. The term is modelled on the German word wohltemperiert which appears in the title of J.S. Bach's famous composition, The Well-Tempered Clavier...
, instrumental music generally stayed in a number of "safe" tonalities that did not involve the "wolf fifth" (which was generally put between G{{music|#}}/A{{music|b}} and D{{music|#}}/E{{music|b}}).
Throughout the Renaissance and Enlightenment, theorists as varied as
Nicola VicentinoNicola Vicentino was an Italian music theorist and composer of the Renaissance. He was one of the most visionary musicians of the age, inventing, among other things, a microtonal keyboard, and devising a practical system of chromatic writing two hundred years before the rise of equal...
,
Francisco de SalinasFrancisco de Salinas was a Spanish music theorist and organist, noted as among the first to describe meantone temperament in mathematically precise terms, and one of the first to describe, in effect, 19 equal temperament. In his De musica libri septem of 1577 he discusses 1/3-, 1/4- and 2/7-comma...
, Fabio Colonna,
Marin MersenneMarin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the "father of acoustics"...
,
Constantijn HuygensConstantijn Huygens , was a Dutch Golden Age poet and composer. He was secretary to two Princes of Orange: Frederick Henry and William II, and the father of the scientist Christiaan Huygens.-Biography:...
, and
Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
advocated the use of meantone tunings that were extended beyond the piano's twelve notes, and hence have come to be called "extended" meantone tunings. These efforts required a concomitant extension of keyboard instruments to offer means of contolling more than 12 notes per octave, including Vincento's Archicembalo (shown in Figure 3), Mersenne's 19-ET harpsihord, Colonna's 31-ET sambuca, and Huygens' 31-ET harpsichord. Other instruments extended the piano keyboard by only a few notes. Some period harpsichords and organs have split D{{music|#}}/E{{music|b}} keys, such that both
E majorE major is a major scale based on E, with the pitches E, F, G, A, B, C, and D. Its key signature has four sharps .Its relative minor is C-sharp minor, and its parallel minor is E minor....
/C
Meantone temperament is a
musical temperamentIn musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...
, which is a system of
musical tuningIn music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
. In general, a meantone is constructed the same way as
Pythagorean tuningPythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
, as a stack of
perfect fifthIn classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
s, but in meantone, each fifth is
narrow compared to the ratio 2
7/12:1 in 12 equal temperament, the opposite of Pythagorean. The meantone temperament:
- generates all non-octave intervals from a stack of tempered
In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...
perfect fifths; and
- by choosing an appropriate size for major and minor thirds, tempers the syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
to unison.
Quarter-comma meantoneQuarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning...
is the best known type of meantone temperament, and the term
meantone temperament is often used to refer to it specifically.
Meantone temperaments
Though quarter-comma meantone is the most common type, other systems that flatten the fifth by some amount, but that still equate the major whole tone (9/8 in just intonation) with the minor whole tone (10/9 in just intonation), are also called meantone systems. Since (9/8) / (10/9) = (81/80)—the
syntonic commaIn music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
—the fundamental characteristics of meantone systems are that all intervals are generated from fifths, and the syntonic comma is tempered to a unison.
All meantone temperaments fall on the
syntonic temperamentThe syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect fifth, with the width of the tempered major third being equal to four tempered perfect fifths minus two octaves and the width of...
's tuning continuum, and as such are "syntonic tunings." The distinguishing feature of each unique syntonic tuning is the width of its generator in
centsThe cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
, as shown in the central column of Figure 1. Historically notable meantone temperaments, discussed below, occupy a narrow portion of the syntonic temperament's tuning continuum, ranging from approximately 695 to 699 cents. The criteria which define the limits (if any) of the meantone range of tunings within the syntonic temperament's tuning continuum are not yet well-defined.
While the term
meantone temperament refers primarily to the tempering of
5-limitIn music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...
musical intervals, optimum values for the 5-limit also work well for the 7-limit, defining
septimal meantone temperamentIn music, septimal meantone temperament, also called standard septimal meantone or simply septimal meantone, refers to the tempering of 7-limit musical intervals by a meantone temperament tuning in the range from fifths flattened by the amount of fifths for 12 equal temperament to those as flat as...
. In Figure 1, the valid tuning ranges of 5-limit, 7-limit, and 11-limit syntonic 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 temperamentAn equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
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
semitoneA semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....
. 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 numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer 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
logarithmThe logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
s 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 In music, 53 equal temperament, called 53-TET, 53-EDO, or 53-ET, is the tempered scale derived by dividing the octave into 53 equal steps . Each step represents a frequency ratio of 21/53, or 22.6415 cents , an interval sometimes called the Holdrian comma.- History :Theoretical interest in this...
|
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 In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...
|
1/4 |
| 8/5 |
29/50 |
2/7 |
| 3/2 |
11/19 In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...
|
1/3 |
Equal temperaments
Neither the just fifth nor the quarter-comma meantone fifth is a
rationalIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
fraction of the octave, but several tunings exist which approximate the fifth by such an interval; these are a subset of the
equal temperamentAn equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
s ("
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-ETIn music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...
, 50-ET,
31-ETIn music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...
, 43-ET, and 55-ET. The farther the tuning gets away from quarter-comma meantone, however, the less related the tuning is to harmonic timbres, which can be overcome by
tempering the timbreDynamic tonality is tonal music which uses real-time changes in tuning and timbre to perform new musical effects such as polyphonic tuning bends, new chord progressions, and temperament modulations, with the option of consonance. The performance of dynamic tonality requires an isomorphic keyboard...
to match the tuning.
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 arithmeticIn number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...
). 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 tuningPythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
close at the octave, one fifth must be out of tune by the
Pythagorean commaIn musical tuning, the Pythagorean comma , named after the ancient mathematician and philosopher Pythagoras, is the small interval existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B , or D and C...
; this altered fifth is called a wolf fifth.
Wolf intervals are an artifact of keyboard design. 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 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{{music|sharp}}. The note that's a perfect fifth higher than E{{music|sharp}} is B{{music|sharp}}, which is not included on the keyboard shown (although it could be included in a larger keyboard, placed just to the right of A{{music|sharp}}, hence maintaining the keyboard's consistent note-pattern). Because there is no B{{music|sharp}} button, when playing an E{{music|sharp}}
power chordIn music, a power chord is a chord consisting of only the root note of the chord and the fifth interval, usually played on electric guitar, and typically through an amplification process that imparts distortion...
, one must choose some other note, such as C, to play instead of the missing B{{music|sharp}}.
Even edge conditions produce wolf intervals only if the isomorphic keyboard has fewer buttons per octave than the tuning has
enharmonicallyIn modern musical notation and tuning, an enharmonic equivalent is a note , interval , or key signature which is equivalent to some other note, interval, or key signature, but "spelled", or named, differently...
-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{{music|sharp}} 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.
Isomorphic keyboards expose the invariant properties of the meantone tunings of the
syntonic temperamentThe syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect fifth, with the width of the tempered major third being equal to four tempered perfect fifths minus two octaves and the width of...
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-2In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the...
) 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
centsThe cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
wide (that is, tempered by approximately 1/11 of a syntonic comma) then the tuning is identical to the familiar 12-tone
equal temperamentAn equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
. 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 temperamentWell temperament is a type of tempered tuning described in 20th-century music theory. The term is modelled on the German word wohltemperiert which appears in the title of J.S. Bach's famous composition, The Well-Tempered Clavier...
s and eventually equal temperament became more popular.
Using standard interval names, twelve fifths equal six octaves plus one
augmented seventhIn classical music from Western culture, an augmented seventh is an interval produced by widening a major seventh by a chromatic semitone. For instance, the interval from C to B is a major seventh, eleven semitones wide, and both the intervals from C to B, and from C to B are augmented sevenths,...
; seven octaves are equal to eleven fifths plus one
diminished sixthIn classical music from Western culture, a diminished sixth is an interval produced by narrowing a minor sixth by a chromatic semitone. For example, the interval from A to F is a minor sixth, eight semitones wide, and both the intervals from A to F, and from A to F are diminished sixths, spanning...
. Given this, three "minor thirds" are actually
augmented secondIn classical music from Western culture, an augmented second is an interval produced by widening a major second by a chromatic semitone. For instance, the interval from C to D is a major second, two semitones wide, and both the intervals from C to D, and from C to D are augmented seconds, spanning...
s (for example, B{{music|flat}} to C{{music|sharp}}), and four "major thirds" are actually
diminished fourthIn classical music from Western culture, a diminished fourth is an interval produced by narrowing a perfect fourth by a chromatic semitone. For example, the interval from C to F is a perfect fourth, five semitones wide, and both the intervals from C to F, and from C to F are diminished fourths,...
s (for example, B to E{{music|flat}}). Several triads (like B–E{{music|flat}}–F{{music|sharp}} and B{{music|flat}}–C{{music|sharp}}–F) contain both these intervals and have normal fifths.
Extended meantones
All meantone tunings fall into the valid tuning range of the
syntonic temperamentThe syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect fifth, with the width of the tempered major third being equal to four tempered perfect fifths minus two octaves and the width of...
, so all meantone tunings are syntonic tunings. All syntonic tunings, including the meantones, have an infinite number of notes in each octave, that is, seven natural notes, seven sharp notes (F{{music|#}} to B{{music|#}}), seven flat notes (B{{music|b}} to F{{music|b}}), double sharp notes, double flat notes, triple sharps and flats, and so on. In reality, double sharps/flats are uncommon, but still needed; triple sharps/flats are never seen. In any syntonic tuning that happens to divide the octave into a small number of equally wide smallest intervals (such as 12,
19In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...
, or
31In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...
), this infinity of notes still exists, although some notes will be enharmonic. For example, in 19-ET, E{{music|#}} and F{{music|b}} are the same pitch.
Many musical instruments are able to divide the octave into a nearly infinite number of notes, 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 12 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 temperamentWell temperament is a type of tempered tuning described in 20th-century music theory. The term is modelled on the German word wohltemperiert which appears in the title of J.S. Bach's famous composition, The Well-Tempered Clavier...
, instrumental music generally stayed in a number of "safe" tonalities that did not involve the "wolf fifth" (which was generally put between G{{music|#}}/A{{music|b}} and D{{music|#}}/E{{music|b}}).
Throughout the Renaissance and Enlightenment, theorists as varied as
Nicola VicentinoNicola Vicentino was an Italian music theorist and composer of the Renaissance. He was one of the most visionary musicians of the age, inventing, among other things, a microtonal keyboard, and devising a practical system of chromatic writing two hundred years before the rise of equal...
,
Francisco de SalinasFrancisco de Salinas was a Spanish music theorist and organist, noted as among the first to describe meantone temperament in mathematically precise terms, and one of the first to describe, in effect, 19 equal temperament. In his De musica libri septem of 1577 he discusses 1/3-, 1/4- and 2/7-comma...
, Fabio Colonna,
Marin MersenneMarin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the "father of acoustics"...
,
Constantijn HuygensConstantijn Huygens , was a Dutch Golden Age poet and composer. He was secretary to two Princes of Orange: Frederick Henry and William II, and the father of the scientist Christiaan Huygens.-Biography:...
, and
Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
advocated the use of meantone tunings that were extended beyond the piano's twelve notes, and hence have come to be called "extended" meantone tunings. These efforts required a concomitant extension of keyboard instruments to offer means of contolling more than 12 notes per octave, including Vincento's Archicembalo (shown in Figure 3), Mersenne's 19-ET harpsihord, Colonna's 31-ET sambuca, and Huygens' 31-ET harpsichord. Other instruments extended the piano keyboard by only a few notes. Some period harpsichords and organs have split D{{music|#}}/E{{music|b}} keys, such that both
E majorE major is a major scale based on E, with the pitches E, F, G, A, B, C, and D. Its key signature has four sharps .Its relative minor is C-sharp minor, and its parallel minor is E minor....
/C{{music (4 sharps) and E{{music/
C minorC minor is a minor scale based on C, consisting of the pitches C, D, E, F, G, A, and B. The harmonic minor raises the B to B. Changes needed for the melodic and harmonic versions of the scale are written in with naturals and accidentals as necessary.Its key signature consists of three flats...
(3 flats) can be played without wolf fifths. Many of those instruments also have split G{{music|#}}/A{{music|b}} keys, and a few have all the 5 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 keyboardAn isomorphic keyboard is a musical input device consisting of a two-dimensional array of note-controlling elements on which any given sequence and/or combination of musical intervals has the “same shape” on the keyboard wherever it occurs – within a key, across keys, across octaves, and across...
instruments (such as the
open sourceThe term open source describes practices in production and development that promote access to the end product's source materials. Some consider open source a philosophy, others consider it a pragmatic methodology...
Thummer, shown in Figure 4), which could be simpler, less cumbersome, and more expressive than existing keyboard instruments.
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 SalinasFrancisco de Salinas was a Spanish music theorist and organist, noted as among the first to describe meantone temperament in mathematically precise terms, and one of the first to describe, in effect, 19 equal temperament. In his De musica libri septem of 1577 he discusses 1/3-, 1/4- and 2/7-comma...
and
Gioseffo ZarlinoGioseffo Zarlino was an Italian music theorist and composer of the Renaissance. He was possibly the most famous music theorist between Aristoxenus and Rameau, and made a large contribution to the theory of counterpoint as well as to musical tuning.-Life:Zarlino was born in Chioggia, near Venice...
. 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)Quarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning...
meantone temperament.
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
György LigetiGyörgy Sándor Ligeti was a composer of contemporary classical music. Born in a Hungarian Jewish family in Transylvania, Romania, he briefly lived in Hungary before becoming an Austrian citizen.-Early life:...
and
Douglas LeedyDouglas Leedy is an American composer, performer and music scholar.-Biography:Born in Portland, Oregon, Leedy studied with Karl Kohn at Pomona College and at the University of California, Berkeley, where he was in a composition seminar with membership including La Monte Young and Terry Riley...
.
New uses of meantone tunings
Meantone tunings are particularly well-suited for use with an
isomorphic keyboardAn isomorphic keyboard is a musical input device consisting of a two-dimensional array of note-controlling elements on which any given sequence and/or combination of musical intervals has the “same shape” on the keyboard wherever it occurs – within a key, across keys, across octaves, and across...
, because such keyboards offer transpositional invariance and tuning invariance (Milne, 2007; Milne, 2008; Sethares, 2009) across the
syntonic temperamentThe syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect fifth, with the width of the tempered major third being equal to four tempered perfect fifths minus two octaves and the width of...
's tuning continuum (shown in Figure 1 above), which includes the entire range of extended meantone tunings. Tuning invariance also enables a suite of new musical effects called
Dynamic TonalityDynamic tonality is tonal music which uses real-time changes in tuning and timbre to perform new musical effects such as polyphonic tuning bends, new chord progressions, and temperament modulations, with the option of consonance. The performance of dynamic tonality requires an isomorphic keyboard...
(Plamondon, 2009).
See also
- Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
- Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
- Interval
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
- Mathematics of musical scales
- Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
- Semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....
- Well temperament
Well temperament is a type of tempered tuning described in 20th-century music theory. The term is modelled on the German word wohltemperiert which appears in the title of J.S. Bach's famous composition, The Well-Tempered Clavier...
- Regular temperament
Regular temperament is any tempered system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios...
- List of meantone intervals
External links
{{Musical tuning}}