3/2: Difference between revisions

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'''3/2''', the '''just perfect fifth''', is the second 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. There are many other uses of 3/2, and thus, systems excluding perfect fifths can thus sound more "xenharmonic". On a harmonic instrument, the third harmonic is usually the loudest one that is not an octave double of the fundamental, with 3/2 itself being the [[octave reduced]] form of this interval. Variations of the perfect fifth (whether just or not) appear in most music of the world. Treatment of the perfect fifth as consonant historically precedes treatment of the major third – specifically [[5/4]] – as consonant. 3/2 is the simplest [[just intonation]] interval to be very well approximated by [[12edo]], after the [[octave]].

'''3/2''', the '''just perfect fifth''', is the second 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. There are many other uses of 3/2, and thus, systems excluding perfect fifths can thus sound more "xenharmonic". On a harmonic instrument, the third harmonic is usually the loudest one that is not an octave double of the fundamental, with 3/2 itself being the [[octave reduced]] form of this interval. Variations of the perfect fifth (whether just or not) appear in most music of the world. Treatment of the perfect fifth as consonant historically precedes treatment of the major third—specifically [[5/4]]—as consonant. 3/2 is the simplest [[just intonation]] interval to be very well approximated by [[12edo]], after the [[octave]].

Producing a chain of just perfect fifths yields [[Pythagorean tuning]]. Since log<sub>2</sub>(3) is an irrational number, a chain of just fifths continues indefinitely and will never returns to the starting note in either direction. Nevertheless, even in xenharmonic circles, the common label "perfect" for this interval retains value in at least some of the [[moment of symmetry]] scales created by this tuning – specifically in the [[TAMNAMS]] system – due to it being an interval that can be thought of as a multiple of the period plus or minus 0 or 1 generators. An example of such a scale is the familiar [[Wikipedia:Diatonic scale #Iteration of the fifth|Pythagorean diatonic scale]].

Producing a chain of just perfect fifths yields [[Pythagorean tuning]]. Since log<sub>2</sub>(3) is an irrational number, a chain of just fifths continues indefinitely and will never returns to the starting note in either direction. Nevertheless, even in xenharmonic circles, the common label "perfect" for this interval retains value in at least some of the [[moment of symmetry]] scales created by this tuning—specifically in the [[TAMNAMS]] system – due to it being an interval that can be thought of as a multiple of the period plus or minus 0 or 1 generators. An example of such a scale is the familiar [[Wikipedia:Diatonic scale #Iteration of the fifth|Pythagorean diatonic scale]].

Meanwhile, [[meantone]] temperaments 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 also [[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 5-limit just intonation, the just 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]].

Meanwhile, [[meantone]] temperaments 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 also [[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 5-limit just intonation, the just 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]].

There are also [[superpyth]] (or "superpythagorean") temperaments, which ''sharpen'' the fifth from just so that the interval generated by four fifths upwards is closer to 9/7 and the interval generated by three fifths downnward is closer to 7/6.

There are also [[superpyth]] (or "superpythagorean") temperaments, which ''sharpen'' the fifth from just so that the interval generated by four fifths upwards is closer to 9/7 and the interval generated by three fifths downnward is closer to 7/6. This also means that intervals such as A–G or C–B♭ approximate 7/4 instead of 9/5.

Then there is 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 [[ups and downs notation]] or [[Syntonic-Rastmic Subchroma notation]]), the 5/4 will be invariably classified as a diminished fourth due to being enharmonic with [[8192/6561]], and this in turn results in common chords such as conventional [[Wikipedia: Major chord|Major]] and [[Wikipedia: Minor chord|Minor]] triads being awkward to notate.

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 {{Wikipedia|Perfect fifth}}
-'''3/2''', the '''just perfect fifth''', is the second 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. There are many other uses of 3/2, and thus, systems excluding perfect fifths can thus sound more "xenharmonic". On a harmonic instrument, the third harmonic is usually the loudest one that is not an octave double of the fundamental, with 3/2 itself being the [[octave reduced]] form of this interval. Variations of the perfect fifth (whether just or not) appear in most music of the world. Treatment of the perfect fifth as consonant historically precedes treatment of the major third – specifically [[5/4]] – as consonant. 3/2 is the simplest [[just intonation]] interval to be very well approximated by [[12edo]], after the [[octave]].
+'''3/2''', the '''just perfect fifth''', is the second 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. There are many other uses of 3/2, and thus, systems excluding perfect fifths can thus sound more "xenharmonic". On a harmonic instrument, the third harmonic is usually the loudest one that is not an octave double of the fundamental, with 3/2 itself being the [[octave reduced]] form of this interval. Variations of the perfect fifth (whether just or not) appear in most music of the world. Treatment of the perfect fifth as consonant historically precedes treatment of the major third&mdash;specifically [[5/4]]&mdash;as consonant. 3/2 is the simplest [[just intonation]] interval to be very well approximated by [[12edo]], after the [[octave]].
-Producing a chain of just perfect fifths yields [[Pythagorean tuning]]. Since log<sub>2</sub>(3) is an irrational number, a chain of just fifths continues indefinitely and will never returns to the starting note in either direction. Nevertheless, even in xenharmonic circles, the common label "perfect" for this interval retains value in at least some of the [[moment of symmetry]] scales created by this tuning – specifically in the [[TAMNAMS]] system – due to it being an interval that can be thought of as a multiple of the period plus or minus 0 or 1 generators. An example of such a scale is the familiar [[Wikipedia:Diatonic scale #Iteration of the fifth|Pythagorean diatonic scale]].
+Producing a chain of just perfect fifths yields [[Pythagorean tuning]]. Since log<sub>2</sub>(3) is an irrational number, a chain of just fifths continues indefinitely and will never returns to the starting note in either direction. Nevertheless, even in xenharmonic circles, the common label "perfect" for this interval retains value in at least some of the [[moment of symmetry]] scales created by this tuning&mdash;specifically in the [[TAMNAMS]] system – due to it being an interval that can be thought of as a multiple of the period plus or minus 0 or 1 generators. An example of such a scale is the familiar [[Wikipedia:Diatonic scale #Iteration of the fifth|Pythagorean diatonic scale]].
-Meanwhile, [[meantone]] temperaments 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 also [[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 5-limit just intonation, the just 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]].
+Meanwhile, [[meantone]] temperaments flatten the perfect fifth such that the major third generated by four fifths upward be closer to 5/4&mdash;or, in the case of [[quarter-comma meantone]] (see also [[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 5-limit just intonation, the just 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]].
-There are also [[superpyth]] (or "superpythagorean") temperaments, which ''sharpen'' the fifth from just so that the interval generated by four fifths upwards is closer to 9/7 and the interval generated by three fifths downnward is closer to 7/6.
+There are also [[superpyth]] (or "superpythagorean") temperaments, which ''sharpen'' the fifth from just so that the interval generated by four fifths upwards is closer to 9/7 and the interval generated by three fifths downnward is closer to 7/6. This also means that intervals such as A&ndash;G or C&ndash;B&#x266D; approximate 7/4 instead of 9/5.
 Then there is 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 [[ups and downs notation]] or [[Syntonic-Rastmic Subchroma notation]]), the 5/4 will be invariably classified as a diminished fourth due to being enharmonic with [[8192/6561]], and this in turn results in common chords such as conventional [[Wikipedia: Major chord|Major]] and [[Wikipedia: Minor chord|Minor]] triads being awkward to notate.