3/2: Difference between revisions

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'''3/2''', the '''just perfect fifth''', is a very [[consonance|consonant]] interval, due to the numerator and denominator of its ratio being very small numbers, with only the [[~~2/1|~~octave]] and the [[3/1|tritave]] having smaller numbers. As such, it is very important in western music and many musical traditions, and approximating it is key in systems like [[12edo]] and other [[edo]]s.

'''3/2''', the '''just perfect fifth''', is a very [[consonance|consonant]] interval, due to the numerator and denominator of its ratio being very small numbers, with only the [[octave]] and the [[3/1|tritave]] having smaller numbers. As such, it is very important in western music and many musical traditions, and approximating it is key in systems like [[12edo]] and other [[edo]]s.

For harmonic [[timbre]]s, the loudest harmonics are usually the second and third ones (2/1 and 3/1). 3/2 is the interval between these two harmonics (which incidentally makes 3/2 [[superparticular]]). Thus 3/2 is easy to tune by ear, and it is easy to hear if it is mistuned.

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Because 3/2 is a very simple and concordant interval, it is still recognizable even when heavily tempered. Often it is tempered so that an octave-reduced stack of fourths or fifths approximates some other interval. Some examples:

[[Meantone]] temperament flattens the fifth from just (to around 695–700 cents) such that the major third generated by stacking four fifths is closer to (or even identical to) 5/4. The minor ~~3rd~~ generated by stacking three fourths is closer to 6/5.

[[Meantone]] temperament flattens the fifth from just (to around 695–700 cents) such that the major third generated by stacking four fifths is closer to (or even identical to) 5/4. The minor third generated by stacking three fourths is closer to 6/5.

[[Superpyth]] temperaments ''sharpen'' the fifth from just so that the major third is closer to 9/7 and the minor third is closer to 7/6. Thus the minor seventh 16/9 approximates 7/4 instead of 9/5.

* One may choose to prioritize the accurate tuning of either the thirds or the harmonic seventh, leading to a ~710{{c}} tuning when prioritizing the thirds, or a ~715{{c}} tuning when prioritizing 7/4.

[[Schismic]] temperament adjusts the fifth such that the ''diminished fourth'' generated by stacking eight fourths approximates 5/4. As this is already a close approximation, the tuning of the fifth can be varied around its just tuning, but is most accurately flattened by a tiny amount. Thus a triad with 5/4 is written as {{nowrap|{{dash|C, F♭, G}}}} (unless the notation has accidentals for [[81/80]], e.g. {{nowrap|{{dash|C, vE, G}}}}).

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 {{Wikipedia|Perfect fifth}}
-'''3/2''', the '''just perfect fifth''', is a very [[consonance|consonant]] interval, due to the numerator and denominator of its ratio being very small numbers, with only the [[2/1|octave]] and the [[3/1|tritave]] having smaller numbers. As such, it is very important in western music and many musical traditions, and approximating it is key in systems like [[12edo]] and other [[edo]]s.
+'''3/2''', the '''just perfect fifth''', is a very [[consonance|consonant]] interval, due to the numerator and denominator of its ratio being very small numbers, with only the [[octave]] and the [[3/1|tritave]] having smaller numbers. As such, it is very important in western music and many musical traditions, and approximating it is key in systems like [[12edo]] and other [[edo]]s.
 For harmonic [[timbre]]s, the loudest harmonics are usually the second and third ones (2/1 and 3/1). 3/2 is the interval between these two harmonics (which incidentally makes 3/2 [[superparticular]]). Thus 3/2 is easy to tune by ear, and it is easy to hear if it is mistuned.
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 Because 3/2 is a very simple and concordant interval, it is still recognizable even when heavily tempered. Often it is tempered so that an octave-reduced stack of fourths or fifths approximates some other interval. Some examples:
-[[Meantone]] temperament flattens the fifth from just (to around 695–700 cents) such that the major third generated by stacking four fifths is closer to (or even identical to) 5/4. The minor 3rd generated by stacking three fourths is closer to 6/5.
+[[Meantone]] temperament flattens the fifth from just (to around 695–700 cents) such that the major third generated by stacking four fifths is closer to (or even identical to) 5/4. The minor third generated by stacking three fourths is closer to 6/5.
 [[Superpyth]] temperaments ''sharpen'' the fifth from just so that the major third is closer to 9/7 and the minor third is closer to 7/6. Thus the minor seventh 16/9 approximates 7/4 instead of 9/5.
-* One may choose to prioritize the accurate tuning of either the thirds or the harmonic seventh, leading to a ~710{{c}} tuning when prioritizing the thirds, or a ~715{{c}} tuning when prioritizing 7/4.
 [[Schismic]] temperament adjusts the fifth such that the ''diminished fourth'' generated by stacking eight fourths approximates 5/4. As this is already a close approximation, the tuning of the fifth can be varied around its just tuning, but is most accurately flattened by a tiny amount. Thus a triad with 5/4 is written as {{nowrap|{{dash|C, F♭, G}}}} (unless the notation has accidentals for [[81/80]], e.g. {{nowrap|{{dash|C, vE, G}}}}).