Editing Wolf interval (section)

=== Five-limit tuning ===
[[Five-limit tuning]] was designed to maximize the number of pure intervals, but even in this system several intervals are markedly impure. 5-limit tuning yields a much larger number of wolf intervals with respect to [[Pythagorean tuning]], which can be considered a 3-limit just intonation tuning. Namely, while Pythagorean tuning determines only 2&nbsp;wolf intervals (a fifth and a fourth), the 5-limit symmetric scales produce 12 of them, and the asymmetric scale&nbsp;14. It is also important to note that the two fifths, three minor thirds, and three major sixths marked in orange in the tables (ratio {{nobr|{{math|40:27}},}} {{nobr|{{math|32:27}},}} and {{math|27:16}}; or G↓, E{{music|flat}}↓, and A↑), even though they do not completely meet the conditions to be wolf intervals, deviate from the corresponding pure ratio by an amount (1&nbsp;[[syntonic comma]], i.e., {{nobr|{{math|81:80}},}} or about {{nobr|{{math|21.5}} cents}}) large enough to be clearly perceived as [[consonance and dissonance|dissonant]].

[[Five-limit tuning]] determines one diminished sixth of size {{math|1024:675}} (about {{nobr|{{math|722}} cents,}} i.e. {{nobr|{{math|20}} cents}} sharper than the {{math|3:2}} Pythagorean perfect fifth). Whether this interval should be considered dissonant enough to be called a wolf fifth is a controversial matter.

Five-limit tuning also creates two ''impure'' perfect fifths of size {{math|40:27}}.
Five-limit fifths are about {{nobr|{{math|680}} cents;}} less ''pure'' than the {{math|3:2}} Pythagorean and/or [[just intonation|just]] {{nobr|{{math|701.95500 cent}} perfect fifth .
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They are not diminished sixths, but relative to the Pythagorean perfect fifth they are less consonant (about {{nobr|{{math|20}} cents}} flatter) and hence, they might be considered to be wolf fifths. The corresponding [[Inversion (interval)|inversion]] is an ''impure'' perfect fourth(also called Acute Fourth<ref>https://www.huygens-fokker.org/docs/intervals.html</ref>) of size {{math|27:20}} (about {{nobr|{{math|520}} cents}}). For instance, in the [[major scale|C&nbsp;major]] [[diatonic scale]], an impure perfect fifth arises between D and A, and its inversion arises between A and D.

Since in this context the term ''perfect'' is interpreted to mean 'perfectly consonant',<ref>{{cite book |first=Godfrey |last=Weber |year=1841 |section=Definition of ''perfect consonance'' |title=General Music Teacher |quote=perfect concord. |via=Internet Archive (archive.org)
|url=https://archive.org/details/bub_gb_20oBAAAAYAAJ}}</ref> the impure perfect fourth and perfect fifth are sometimes simply called ''the imperfect'' fourth and fifth.<ref name=Baker/> However, the widely adopted standard naming convention for [[Interval (music)|musical intervals]] classifies them as ''perfect'' intervals, together with the [[octave]] and [[unison]]. This is also true for any perfect fourth or perfect fifth which slightly deviates from the perfectly consonant {{math|4:3}} or {{math|3:2}} ratios (for instance, those tuned using [[12-tone equal temperament|12&nbsp;tone equal]] or [[quarter-comma meantone]] temperament). Conversely, the expressions ''imperfect fourth'' and ''imperfect fifth'' do not conflict with the standard naming convention when they refer to a dissonant [[augmented third]] or [[diminished sixth]] (e.g. the wolf fourth and fifth in Pythagorean tuning).