Perfect fifth: Difference between revisions
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''This page is about the interval region. For the just perfect fifth, see [[3/2]].'' | |||
A perfect fifth is an interval that is near 700 [[Cent|cents]] in size, distinct from diminished fifths (a type of [[tritone]], about 600 cents). A rough tuning range for the perfect fifth is about 650 to 750 [[cents]], though this is extremely wide; some might prefer to restrict it to around 670-730 cents. | |||
"Perfect fifth" may also refer to the [[diatonic perfect fifth]], which is a tempered fifth used to generate the diatonic scale, and is not the subject of this article. | |||
==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 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 [[Prime limit|limits]] include: | |||
*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 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. | |||
*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 EDOs== | |||
The following table lists the best tuning of 3/2, as well as other fifths if present, in various significant [[EDOs]]. | |||
{| class="wikitable" | |||
|+ | |||
!EDO | |||
!3/2 | |||
!Other fifths | |||
|- | |||
|5 | |||
|720c | |||
| | |||
|- | |||
|7 | |||
|686c | |||
| | |||
|- | |||
|12 | |||
|700c | |||
| | |||
|- | |||
|15 | |||
|720c | |||
| | |||
|- | |||
|16 | |||
|675c | |||
|750c ≈ 20/13 | |||
|- | |||
|17 | |||
|706c | |||
| | |||
|- | |||
|19 | |||
|694c | |||
| | |||
|- | |||
|22 | |||
|709c | |||
|654c ≈ 22/15 | |||
|- | |||
|24 | |||
|700c | |||
|750c ≈ 20/13, 650c ≈ 22/15 | |||
|- | |||
|25 | |||
|720c | |||
|672c ≈ 40/27 | |||
|- | |||
|26 | |||
|692c | |||
|738c ≈ 32/21, 20/13 | |||
|- | |||
|27 | |||
|711c | |||
|666c ≈ 22/15 | |||
|- | |||
|29 | |||
|704c | |||
|745c ≈ 20/13, 663c ≈ 22/15 | |||
|- | |||
|31 | |||
|697c | |||
|736c ≈ 32/21, 659c ≈ 22/15 | |||
|- | |||
|34 | |||
|706c | |||
|742c ≈ 20/13, 671c ≈ 40/27, 22/15 | |||
|- | |||
|41 | |||
|702c | |||
|732c ≈ 32/21, 674c ≈ 40/27 | |||
|- | |||
|53 | |||
|702c | |||
|748c ≈ 20/13, 724c ≈ 32/21, 679c ≈ 40/27, 657c ≈ 22/15 | |||
|} | |||
==In temperaments== | |||
The simplest perfect 5th ratio is 3/2. The following notable temperaments are generated by it: | |||
===Temperaments that use 3/2 as a generator=== | |||
*[[Compton]], the temperament of the Pythagorean comma, equivalent to 12edo | |||
**The 3-limit [[Circular temperament|circular temperaments]] in general | |||
*[[Archy]], the temperament sharpening 3/2 such that four 3/2s stack to [[9/7]] | |||
*[[Meantone]], the temperament flattening 3/2 such that four 3/2s stack to [[5/4]] | |||
*[[Mavila]], the temperament flattening 3/2 such that four 3/2s stack to [[6/5]] | |||
*Various historical [[Well temperament|well temperaments]] generated by tempered 4/3s or 3/2s, equivalent to 12edo as compton and meantone | |||
{{Navbox intervals}} | |||