3/2: Difference between revisions

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~~{{main|Just~~ perfect fifth}}

'''3/2''', the '''just perfect fifth''', is the largest [[superparticular]] [[Gallery_of_Just_Intervals|interval]], spanning the distance between the 2nd and 3rd harmonics. It is an interval with low [[harmonic entropy]], and therefore high consonance.

~~'''~~3/2~~'''~~ is the [[~~frequency ratio~~]] of the [[just perfect ~~fifth]]~~. ~~What tunes it well~~, ~~is one of variants of~~ [[12edo]] or [[~~17edo~~]] (such as [[~~24edo~~]], [[~~34edo~~]] ~~and 36edo~~). ~~Other edos tune it well too~~ (~~5, 7, 29, 41, 53, 200~~)~~. But not all edos~~ are ~~like this. 35edo is great for 2~~, 5, ~~7, 9, 11~~ and ~~17 but fails on 3~~.

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 (see [[5/4|5:4]]) as consonant. 3:2 is the simple JI interval best approximated by [[12edo|12edo]], after the [[Octave|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.

In composition, the presence of perfect fifths can provide a "ground" upon which unusual intervals may be placed while still sounding structurally coherent. Systems excluding perfect fifths can thus sound more "xenharmonic".

Some better (compared to 12edo) approximations of the perfect fifth are [[29edo]], [[41edo]], and [[53edo]].

== Approximations by EDOs ==

~~Following~~ [[EDO]]s (up to 200) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 3/2. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (↑) or flat (↓).

The following [[EDO]]s (up to 200) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 3/2. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (↑) or flat (↓).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"

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See a list of EDOs with increasingly better approximations of 3:2 (and by extension 4:3) at {{OEIS|A060528}}. Also relevant are the {{OEIS|A005664|denominators of the convergents to log<sub>2</sub>(3)}}

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 }}
-{{main|Just perfect fifth}}
+'''3/2''', the '''just perfect fifth''', is the largest [[superparticular]] [[Gallery_of_Just_Intervals|interval]], spanning the distance between the 2nd and 3rd harmonics. It is an interval with low [[harmonic entropy]], and therefore high consonance.
-'''3/2''' is the [[frequency ratio]] of the [[just perfect fifth]]. What tunes it well, is one of variants of [[12edo]] or [[17edo]] (such as [[24edo]], [[34edo]] and 36edo). Other edos tune it well too (5, 7, 29, 41, 53, 200). But not all edos are like this. 35edo is great for 2, 5, 7, 9, 11 and 17 but fails on 3.
+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 (see [[5/4|5:4]]) as consonant. 3:2 is the simple JI interval best approximated by [[12edo|12edo]], after the [[Octave|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.
+In composition, the presence of perfect fifths can provide a "ground" upon which unusual intervals may be placed while still sounding structurally coherent. Systems excluding perfect fifths can thus sound more "xenharmonic".
+Some better (compared to 12edo) approximations of the perfect fifth are [[29edo]], [[41edo]], and [[53edo]].
 == Approximations by EDOs ==
-Following [[EDO]]s (up to 200) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 3/2. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&uarr;) or flat (&darr;).
+The following [[EDO]]s (up to 200) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 3/2. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&uarr;) or flat (&darr;).
 {| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
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 |-
 |}
+See a list of EDOs with increasingly better approximations of 3:2 (and by extension 4:3) at {{OEIS|A060528}}.  Also relevant are the {{OEIS|A005664|denominators of the convergents to log<sub>2</sub>(3)}}
 <references/>