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. In composition, the presence of perfect fifths can provide a "ground" upon which unusual intervals may be placed while still sounding structurally coherent. Not only that, but there are many other uses of 3/2, and thus, systems excluding perfect fifths can thus sound more "xenharmonic". Variations of the perfect 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 [[just intonation]] 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. 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. In such systems, and in common practice theory, the perfect fifth consists of four diatonic semitones and three chromatic semitones. In [[12edo]], and hence in most discussions these days, this is simplified to seven semitones, which is fitting seeing as 12edo is a system which flattens the perfect fifth by about 2 cents so that the circle close at 12 tones. On the other hand, in just intonation the perfect fifth consists of four just diatonic semitones of [[16/15]], three just chromatic semitones of [[25/24]], and two syntonic commas of [[81/80]], and is the just perfect fifth of 3/2.

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]].

Some better (compared to 12edo) approximations of the perfect fifth are [[29edo]], [[41edo]], and [[53edo]]. Of the aforementioned systems, 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 [[meantone]], and in common practice theory, the perfect fifth consists of four diatonic semitones and three chromatic semitones. In 12edo, and hence in most discussions these days, that is simplified to seven semitones. On the other hand, in just intonation it consists of four just diatonic semitones of [[16/15]], three just chromatic semitones of [[25/24]], and two syntonic commas of [[81/80]], and is the just perfect fifth of 3/2.

== 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. In composition, the presence of perfect fifths can provide a "ground" upon which unusual intervals may be placed while still sounding structurally coherent. Not only that, but there are many other uses of 3/2, and thus, systems excluding perfect fifths can thus sound more "xenharmonic". Variations of the perfect 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 [[just intonation]] 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. 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. In such systems, and in common practice theory, the perfect fifth consists of four diatonic semitones and three chromatic semitones. In [[12edo]], and hence in most discussions these days, this is simplified to seven semitones, which is fitting seeing as 12edo is a system which flattens the perfect fifth by about 2 cents so that the circle close at 12 tones. On the other hand, in just intonation the perfect fifth consists of four just diatonic semitones of [[16/15]], three just chromatic semitones of [[25/24]], and two syntonic commas of [[81/80]], and is the just perfect fifth of 3/2.
+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]].
 Some better (compared to 12edo) approximations of the perfect fifth are [[29edo]], [[41edo]], and [[53edo]].  Of the aforementioned systems, 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 [[meantone]], and in common practice theory, the perfect fifth consists of four diatonic semitones and three chromatic semitones. In 12edo, and hence in most discussions these days, that is simplified to seven semitones. On the other hand, in just intonation it consists of four just diatonic semitones of [[16/15]], three just chromatic semitones of [[25/24]], and two syntonic commas of [[81/80]], and is the just perfect fifth of 3/2.
 == Approximations by EDOs ==