3/2: Difference between revisions

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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]], and this in turn results in common chords such as conventional [[Wikipedia: Major and Minor|~~Major and~~ Minor]] triads being awkward to notate.

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]], 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.

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 12edo is a 2-strong 3-2 telic system, 53edo is a 3-strong 3-2 telic system.

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 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]], and this in turn results in common chords such as conventional [[Wikipedia: Major and Minor|Major and Minor]] triads being awkward to notate.
+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]], 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.
 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 12edo is a 2-strong 3-2 telic system, 53edo is a 3-strong 3-2 telic system.