12edo: Difference between revisions
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== Theory == | == Theory == | ||
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament, tempering out [[81/80]], equating four [[3/2 | 12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament, tempering out [[81/80]], equating four [[3/2]] perfect fifths with the [[5/1|5th harmonic]]. | ||
It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has | It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has an approximate [[3/2]] perfect fifth which is quite accurate at 700 cents, two cents flat of [[just intonation]]. It has a [[5/4]] major third which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5]] minor third is slightly less accurate than that, being 15.6 cents flat of just. | ||
Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}} | Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}} | ||
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12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. | 12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. | ||
[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. | [[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Various [[12th-octave temperaments]] that augment 12edo exist, most prominent examples being [[compton]] and [[catler]]. | ||
== Intervals == | == Intervals == | ||
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== Scales == | == Scales == | ||
{{ | {{See also| List of MOS scales in 12edo }} | ||
The two most common 12edo | The two most common 12edo MOS scales are meantone[5] and meantone[7]. | ||
* Diatonic | * Diatonic: [[5L 2s]] – 2221221 (generator = 7\12) | ||
* Pentatonic | * Pentatonic: [[2L 3s]] – 22323 (generator = 7\12) | ||
The diminished and augmented scales are also MOS scales. | |||
* Diminished: [[4L 4s]] – 12121212 (generator = 1\12, period = 3\12) | |||
* Augmented: [[3L 3s]] – 131313 (generator = 1\12, period = 4\12) | |||
* Harmonic | Other widely used scales include: | ||
* | * Melodic minor – 2122221 | ||
* Harmonic minor – 2122131 | |||
* Harmonic major – 2212131 | |||
* Hungarian minor – 2131131 | * Hungarian minor – 2131131 | ||
* Maqam hijaz / double harmonic major – 1312131 | * Maqam hijaz / double harmonic major – 1312131 | ||
== Well temperaments == | == Well temperaments == | ||