3/2: Difference between revisions

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'''3/2''', the '''just perfect fifth''', is the largest [[superparticular]] [[interval]], spanning the distance between the 2nd and 3rd harmonics. It is an interval with low [[harmonic entropy]], and therefore high consonance.

'''3/2''', the '''just perfect fifth''', is the largest [[superparticular]] [[interval]], spanning the distance between the 2nd and 3rd harmonics. It is an interval with low [[harmonic entropy]], and therefore high consonance. In composition, the presence of perfect fifths can provide a "ground" upon which unusual intervals may be placed while still sounding structurally coherent. Systems excluding perfect fifths can thus sound more "xenharmonic".

Variations of the [[Perfect_fifth|fifth]] (whether just or not) appear in most music of the world. On a harmonic instrument, the third harmonic is usually the loudest one that is not an octave double of the fundamental. Treatment of the perfect fifth as consonant historically precedes treatment of the major third- specifically [[5/4]]- as consonant. 3/2 is the simple JI interval best approximated by [[12edo]], after the [[octave]].

~~Variations~~ of ~~the~~ [[~~Perfect_fifth|fifth~~]] ~~(whether just or~~ not~~) appear in most music of the world. On~~ a ~~harmonic instrument~~, ~~the third harmonic~~ is ~~usually~~ the ~~loudest one~~ that ~~is not an octave double of~~ the ~~fundamental~~. ~~Treatment of~~ the perfect fifth ~~as consonant historically precedes treatment of~~ the major third (see [[~~5/4|5:4~~]]) ~~as consonant~~. ~~3:2 is~~ the ~~simple JI interval best~~ approximated by [[~~12edo~~]], ~~after~~ the [[~~octave~~]].

Producing a chain of just perfect fifths yields [[Pythagorean tuning]]. Such a chain does not close at a circle, but continues infinitely. [[12edo]] is a system which flattens the perfect fifth by about 2 cents so that the circle close at 12 tones. Meanwhile, [[meantone]] temperaments are systems which flatten the perfect fifth such that the major third generated by four fifths upward be closer to 5/4 – or, in the case of [[quarter-comma meantone]] (see [[31edo]]), identical. Then there's the possibility of [[schismatic]] temperaments, which flatten the perfect fifth such that an approximated 5/4 is generated by stacking eight fifths downwards; however, without a notation system that properly accounts for the [[syntonic comma]] (such as [[Syntonic-Rastmic Subchroma Notation]]), the 5/4 will be invariably classified as a diminished fourth due to being enharmonic with [[8192/6561]].

Producing a chain of just perfect fifths yields Pythagorean tuning. Such a chain does not close at a circle, but continues infinitely. [[12edo]] is a system which flattens the perfect fifth by about 2 cents so that the circle close at 12 tones. Meanwhile, [[meantone]] temperaments are systems which flatten the perfect fifth such that the major third generated by four fifths upward be closer to 5:4 – or, in the case of [[quarter-comma meantone]] (see [[31edo]]), identical.

Some better (compared to 12edo) approximations of the perfect fifth are [[29edo]], [[41edo]], and [[53edo]]. The latter is particularly noteworthy in regards to [[telicity]] as while the 12edo is a 2-strong 3-2 telic system, 53edo is a 3-strong 3-2 telic system.

In composition, the presence of perfect fifths can provide a "ground" upon which unusual intervals may be placed while still sounding structurally coherent. Systems excluding perfect fifths can thus sound more "xenharmonic".

Some better (compared to 12edo) approximations of the perfect fifth are [[29edo]], [[41edo]], and [[53edo]].

== Approximations by EDOs ==

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-'''3/2''', the '''just perfect fifth''', is the largest [[superparticular]] [[interval]], spanning the distance between the 2nd and 3rd harmonics. It is an interval with low [[harmonic entropy]], and therefore high consonance.
+'''3/2''', the '''just perfect fifth''', is the largest [[superparticular]] [[interval]], spanning the distance between the 2nd and 3rd harmonics. It is an interval with low [[harmonic entropy]], and therefore high consonance. In composition, the presence of perfect fifths can provide a "ground" upon which unusual intervals may be placed while still sounding structurally coherent. Systems excluding perfect fifths can thus sound more "xenharmonic".
+Variations of the [[Perfect_fifth|fifth]] (whether just or not) appear in most music of the world. On a harmonic instrument, the third harmonic is usually the loudest one that is not an octave double of the fundamental. Treatment of the perfect fifth as consonant historically precedes treatment of the major third- specifically [[5/4]]- as consonant. 3/2 is the simple JI interval best approximated by [[12edo]], after the [[octave]].
-Variations of the [[Perfect_fifth|fifth]] (whether just or not) appear in most music of the world. On a harmonic instrument, the third harmonic is usually the loudest one that is not an octave double of the fundamental. Treatment of the perfect fifth as consonant historically precedes treatment of the major third (see [[5/4|5:4]]) as consonant. 3:2 is the simple JI interval best approximated by [[12edo]], after the [[octave]].
+Producing a chain of just perfect fifths yields [[Pythagorean tuning]]. Such a chain does not close at a circle, but continues infinitely. [[12edo]] is a system which flattens the perfect fifth by about 2 cents so that the circle close at 12 tones. Meanwhile, [[meantone]] temperaments are systems which flatten the perfect fifth such that the major third generated by four fifths upward be closer to 5/4 – or, in the case of [[quarter-comma meantone]] (see [[31edo]]), identical. Then there's the possibility of [[schismatic]] temperaments, which flatten the perfect fifth such that an approximated 5/4 is generated by stacking eight fifths downwards; however, without a notation system that properly accounts for the [[syntonic comma]] (such as [[Syntonic-Rastmic Subchroma Notation]]), the 5/4 will be invariably classified as a diminished fourth due to being enharmonic with [[8192/6561]].
-Producing a chain of just perfect fifths yields Pythagorean tuning. Such a chain does not close at a circle, but continues infinitely. [[12edo]] is a system which flattens the perfect fifth by about 2 cents so that the circle close at 12 tones. Meanwhile, [[meantone]] temperaments are systems which flatten the perfect fifth such that the major third generated by four fifths upward be closer to 5:4 – or, in the case of [[quarter-comma meantone]] (see [[31edo]]), identical.
+Some better (compared to 12edo) approximations of the perfect fifth are [[29edo]], [[41edo]], and [[53edo]].  The latter is particularly noteworthy in regards to [[telicity]] as while the 12edo is a 2-strong 3-2 telic system, 53edo is a 3-strong 3-2 telic system.
-In composition, the presence of perfect fifths can provide a "ground" upon which unusual intervals may be placed while still sounding structurally coherent. Systems excluding perfect fifths can thus sound more "xenharmonic".
-Some better (compared to 12edo) approximations of the perfect fifth are [[29edo]], [[41edo]], and [[53edo]].
 == Approximations by EDOs ==