Perfect fifth: Difference between revisions
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{{About|the [[interval region]]|the just perfect fifth|3/2}} | {{About|the [[interval region]]|the just perfect fifth|3/2}} | ||
A '''perfect fifth (P5)''' is an interval that is near 700 [[cent]]s in size, distinct from | A '''perfect fifth (P5)''' is an interval that spans four steps of the [[5L 2s|diatonic]] scale with a perfect quality, i.e. the quality that exists in all but one modes. Depending on the specific tuning, it ranges from 686 to 720 [[cent]]s ([[7edo|4\7]] to [[5edo|3\5]]). | ||
In [[just intonation]], the just perfect fifth is [[3/2]]. Other intervals are also classified as perfect fifths, sometimes called '''wolf fifths''' or '''imperfect fifths''', if they are reasonably mapped to 4\7 and [[24edo|14\24]] (precisely four steps of the diatonic scale and seven steps of the chromatic scale). The use of 24edo's 14\24 as the mapping criteria here rather than [[12edo]]'s 7\12 better captures the characteristics of many intervals in the [[11-limit|11-]] and [[13-limit]]. | |||
As a concrete [[interval region]], it is typically near 700 [[cent]]s in size, distinct from semidiminished fifths of rougly 650 cents. A rough tuning range for the perfect fifth is about 670 to 730 [[cents]] according to [[Margo Schulter]]'s theory of interval regions. Another common range is the stricter range from 686 to 720 cents, which generates a diatonic scale. | |||
== In just intonation == | == In just intonation == | ||
The only "perfect" fourth in JI is the '''Pythagorean perfect fifth''' of [[3/2]], about 702 cents in size, which corresponds to the | The only "perfect" fourth in JI is the '''Pythagorean perfect fifth''' of [[3/2]], about 702 cents in size, which corresponds to the mos-based interval category of the diatonic perfect fifth, and is the generator for Pythagorean tuning and the diatonic scale. However, various "out of tune" fifths exist, such as the '''Pythagorean wolf fifth [[262144/177147]]''', which is flat of 3/2 by one [[Pythagorean comma]], and is about 678 cents in size. | ||
Other "out of tune" fifths in higher [[ | Other "out of tune" fifths in higher [[prime limit|limits]] include: | ||
*The 5-limit '''grave fifth''' is a ratio of 40/27, and is about 680 cents | * The 5-limit '''grave fifth''' is a ratio of 40/27, and is about 680 cents | ||
*The 7-limit '''superfifth''' is a ratio of 32/21, and is about 729 cents. | * The 7-limit '''superfifth''' is a ratio of 32/21, and is about 729 cents. | ||
*The 11-limit '''diminished fifth''' is a ratio of 22/15, and is about 663 cents. | * The 11-limit '''diminished fifth''' is a ratio of 22/15, and is about 663 cents. | ||
**There is also an 11-limit '''acute fifth,''' which is a ratio of 50/33, and is about 720 cents. | ** There is also an 11-limit '''acute fifth,''' which is a ratio of 50/33, and is about 720 cents. | ||
*The 13-limit '''ultrafifth''' is a ratio of 20/13, and is about 746 cents, but it might be better analyzed as an [[ | * The 13-limit '''ultrafifth''' is a ratio of 20/13, and is about 746 cents, but it might be better analyzed as an [[minor sixth|inframinor sixth]]. Despite that, it is also here for completeness. | ||
== In | == In edos == | ||
The following table lists the best tuning of 3/2, as well as other fifths if present, in various significant [[ | The following table lists the best tuning of 3/2, as well as other fifths if present, in various significant [[edos]]. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
! | !Edo | ||
!3/2 | !3/2 | ||
!Other fifths | !Other fifths | ||
| Line 92: | Line 96: | ||
The simplest perfect 5th ratio is 3/2. The following notable temperaments are generated by it: | The simplest perfect 5th ratio is 3/2. The following notable temperaments are generated by it: | ||
===Temperaments that use 3/2 as a generator=== | === Temperaments that use 3/2 as a generator === | ||
* [[Meantone]], the temperament flattening 3/2 such that four 3/2s stack to [[5/4]] | |||
*[[Meantone]], the temperament flattening 3/2 such that four 3/2s stack to [[5/4]] | * [[Schismatic]], the temperament slightly sharpening 3/2 such that nine 3/2s stack to [[6/5]] | ||
*[[Schismatic]], the temperament slightly sharpening 3/2 such that nine 3/2s stack to [[6/5]] | * [[Superpyth]], the temperament sharpening 3/2 such that four 3/2s stack to [[9/7]] | ||
*[[Superpyth]], the temperament sharpening 3/2 such that four 3/2s stack to [[9/7]] | * [[Compton]], the temperament of the Pythagorean comma, equivalent to 12edo | ||
*[[Compton]], the temperament of the Pythagorean comma, equivalent to 12edo | * [[Mavila]], the temperament flattening 3/2 such that four 3/2s stack to [[6/5]] | ||
*[[Mavila]], the temperament flattening 3/2 such that four 3/2s stack to [[6/5]] | * Various historical [[well temperament]]s generated by tempered 4/3s or 3/2s, equivalent to 12edo as compton and meantone | ||
*Various historical [[ | |||
{{Navbox intervals}} | {{Navbox intervals}} | ||