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Encyclopedia > Mean tone

Meantone temperament is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a chain of perfect fifths, but in a meantone, each fifth is narrowed by the same amount (or equivalently, each fourth widened) in order to make the other intervals like the major third closer to their ideal just ratios. This page is about musical systems of tuning, for the musical process of tuning see tuning. ... Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. ... The perfect fifth or diapente is one of three musical intervals that span five diatonic scale degrees; the others being the diminished fifth, which is one semitone smaller, and the augmented fifth, which is one semitone larger. ... The perfect fourth or diatessaron, abbreviated P4, is one of two musical intervals that span four diatonic scale degrees; the other being the augmented fourth, which is one semitone larger. ... A major third is the larger of two commonly occuring musical intervals that span three diatonic scale degrees. ... In music, Just intonation, also called rational intonation, is any musical tuning in which the frequencies of notes are related by whole number ratios; that is, by positive rational numbers. ...

The term meantone temperament is sometimes used to refer specifically to quarter-comma meantone. However, systems which flatten the fifth by differing amounts but which still equate the major whole tone, which in just intonation is 9/8, with the minor whole tone, tuned justly to 10/9, are also called meantone systems. Since (9/8) / (10/9) = (81/80), the syntonic comma, the fundamental character of a meantone tuning is that all intervals are generated from fifths, and the syntonic comma is tempered to a unison. While the term meantone temperament refers primarily to the tempering of 5-limit musical intervals, optimum values for the 5-limit also work well for the 7-limit, defining septimal meantone temperament. Quarter-comma meantone was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. ... The syntonic comma, also known as the comma of Didymus or Ptolemaic comma, is a small interval between two musical notes, equal to the frequency ratio 81:80, or around 21. ... Just intonation tunings and scales can be described by giving an upper bound on the complexity of the harmonies admitted by the tuning or scale. ... In 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 19 equal...

Meantones can be specified in various ways. We can, as above, specify what fraction (logarithmically) of a syntonic comma the fifth is being flattened by, what equal temperament has the meantone fifth in question, or what the ratio of the whole tone to the diatonic semitone is. This ratio was termed "R" by American composer, pianist and theoretician Easley Blackwood, but in effect has been in use for much longer than that. It is useful because it gives us an idea of the melodic qualities of the tuning, and because if R is a rational number, so is (3R+1)/(5R+2), which is the size of fifth in terms of logarithms base 2, and which immediately tells us what division of the octave we will have. If we multiply by 1200, we have the size of fifth in cents. Equal temperament is a scheme of musical tuning in which the octave is divided into a series of equal steps (equal frequency ratios). ... Easley Blackwood {April 21, 1933-) is a professor of music, a composer of music using unusual tunings, and the author of books on music theory. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

In these terms, some historically important 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 comma
2 7/12 1/11
9/5 32/55 1/6
7/4 25/43 1/5
5/3 18/31 7/29
33/20 119/205 1/4
8/5 29/50 2/7
3/2 11/19 1/3

## Wolf intervals and extended meantones

A whole number of just perfect fifths will never add up to a whole number of octaves, because they are incommensurable (see Fundamental theorem of arithmetic). Therefore, a chromatic scale in Pythagorean tuning must have one fifth that is out of tune by the Pythagorean comma, called a wolf fifth. Most meantone temperaments share this problem, except for the case where the fifth is exactly 700 cents (tempered by approximately 1/11 of a syntonic comma) and the meantone becomes the familiar 12-tone equal temperament. This appears in the table above when R=2. In mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer greater than 1 is either a prime number or can be written as a product of prime numbers. ... Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. ... When one ascends by a cycle of justly tuned perfect fifths (ratio 3:2), leapfrogging 12 times, one eventually reaches a note around seven octaves above the note one started on, which, when lowered to the same octave as the starting point, is 23. ... When the twelve notes within the octave are tuned using meantone temperament, one of the fifths will be much sharper than the rest. ... The cent is a logarithmic unit of measure used for musical intervals. ... Equal temperament is a scheme of musical tuning in which the octave is divided into a series of equal steps (equal frequency ratios). ...

Because of this wolf fifth which arises when twelve notes to the octave are tuned to a meantone with fifths significantly flatter than the 1/11-comma of equal temperament, well temperaments and eventually equal temperament (a special case of the former) became more popular. Equal temperament is a scheme of musical tuning in which the octave is divided into a series of equal steps (equal frequency ratios). ... Well temperament is a 20th-century term in music theory denoting a type of tuning. ...

Another way to solve the problem of the wolf fifth is to forsake enharmonic equivalence (so, for example, G♯ and A♭ are actually different pitches) and use a temperament with more than 12 pitches to the octave. This is known as extended meantone. Its advantage is the ability to modulate into arbitrarily distant keys without wolf fifths, but an obvious disadvantage is the necessity of using instruments capable of playing more than twelve pitches in an octave, such as fretless string instruments or modified keyboard instruments with extra keys, like the archicembalo. In music, an enharmonic is a note which is the equivalent of some other note, but spelled differently. ... In music, modulation is most commonly the act or process of changing from one key (tonic, or tonal center) to another. ... The archicembalo of Nicola Vicentino is a kind of harpsichord of 36 notes to the octave which Vicentino constructed in 1555. ...

The existence of the "wolf fifth" is one of the reasons why, before the introduction of well temperament, instrumental music generally stayed in a number of "safe" tonalities that did not involve the "wolf fifth" (which was generally put between G♯/A♭ and D♯/E♭). Some period harpsichords and organs have split D♯/E♭ keys, such that both Emajor/C♯minor (4 sharps) and E♭major/Cminor (3 flats) can be played without wolf fifths. Well temperament is a 20th-century term in music theory denoting a type of tuning. ...

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