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 [[Wikipedia: Perfect fifth|corresponding Wikipedia article]] attests there are many other uses of 3/2, and thus, 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]].

'''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 as the [[Wikipedia: Perfect fifth|corresponding Wikipedia article]] attests, there are many other uses of 3/2, and thus, 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]].

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]].

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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 [[Wikipedia: Perfect fifth|corresponding Wikipedia article]] attests there are many other uses of 3/2, and thus, 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]].
+'''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 as the [[Wikipedia: Perfect fifth|corresponding Wikipedia article]] attests, there are many other uses of 3/2, and thus, 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]].
 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]].