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Wolf interval
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== Temperament and the wolf == The reason for "wolf" tones in meantone tunings is the bad practice of performers pressing the key for an [[enharmonic equivalence|enharmonic note]] as a substitute for a note that has not been tuned on the keyboard; e.g. pressing the black key tuned to G{{music|#}} when the music calls for A{{music|b}}. In all [[meantone tuning]] systems, sharps and flats are '''''not''''' equivalent; a relic of which, that persists in modern musical practice, is to fastidiously distinguish the [[musical notation]] for two notes which are the same pitch in [[equal temperament]] ([[enharmonic equivalence|"enharmonic"]]) and played with the same key on an equal tempered keyboard (such as C{{music|#}} and D{{music|b}}, or E{{music|#}} and F{{music|n}}), despite the fact that they are the same in all but [[musical theory|theory]]. In order to close the [[circle of fifths]] in 12 note scales, twelve [[perfect fifth|fifths]] must average out to {{nobr|{{math|700}} [[cent (music)|cents]]}}.{{efn| No such {{nobr|{{math|700}} [[cent (music)|cents]]}} exact average for fifth inervals exists meantone systems: Their fifths – and all repeated intervals – form a [[helix]], not a circle. }} Each of the first eleven fifths (starting with the fifth ''below'' the [[tonic (music)|tonic]], the [[subdominant]]: F in the key of C, when each black key is tuned to a meantone sharp / no flats) has a value of {{nobr|{{math|700 − ''ε''}} cents}}, where {{mvar|ε}} is some small number of cents that all fifths are detuned by.{{efn| The size of {{mvar|ε}} is around 1–4 cents, and is different for each particular meantone system used. As a technicality, [[equal temperament]] happens to be a [[meantone temperament]] for which the value of {{mvar|ε}} is zero. }} In [[meantone temperament]] tuning systems, the twelfth and last fifth does not exist in the 12 note octave on the keyboard. The actual note available is really a [[diminished sixth]]: The interval is {{nobr|{{math|700 + 11 ''ε''}} cents}}, and is not a correct meantone fifth, which would be {{nobr|{{math|700 − ''ε''}} cents.}} The difference of {{nobr|{{math|12 ''ε''}} cents}} between the available pitch and the intended pitch is the source of the "wolf". The "wolf" effect is particularly grating for values of {{nobr|{{math|12 ''ε''}} cents}} that approach {{nobr|{{math|20~25}} cents}}.{{efn| {{nobr|{{math|20~25}} cents}}, or a quarter-sharp / quarter flat, is the typical size of the several discrepant musical intervals called [[comma (music)|"commas"]]. Note that a quarter-comma is a different interval than a quarter-sharp. }} A simplistic reaction to the problem is: ''"Of {{underline|course}} it sounds awful: You're playing the wrong note!"'' With only 12 notes available in a conventional keyboard's octave, in meantone tunings there must always be omitted notes. For example, one choice for tuning an instrument in meantone, to play music in the [[key of C]]{{music|n}}, would be :{| style="text-align:center;" |-" | '''A'''  || {{grey|[no A{{music|#}}]}}  || '''B{{music|b}}'''  || '''B'''  || {{grey|[no B{{music|#}} and no C{{music|b}}]}}  || '''C'''  || '''C{{music|#}}'''  || {{grey|[no D{{music|b}}]}}  ||  || |- | '''D'''  || {{grey|[no D{{music|#}}]}}  || '''E{{music|b}}'''  || '''E'''  || {{grey|[no E{{music|#}} and no F{{music|b}}]}}  || '''F'''  || '''F{{music|#}}'''  || {{grey|[no G{{music|b}}]}}  || '''G'''  || '''G{{music|#}}'''   {{grey|[choose one of either G{{music|#}} or A{{music|b}}]}} |} with this set of chosen notes in bold face, and some of the omitted notes shown in grey.{{efn| Of course, [[double sharp]]s and [[double flat]]s are infeasible for the key of C major / A minor. }} This limitation on the set meantone notes and their sharps and flats that can be tuned on a keyboard at any one time, was the main reason that [[Baroque (music)|Baroque period]] keyboard and [[orchestral harp]] performers were obliged to retune their instruments in mid-performance breaks, in order to make available all the [[accidental (music)|accidentals]] called for by the next piece of music.{{efn| If a performer could get the use of an extra instrument, an alternative to retuning is to switch seats to a spare instrument already tuned for the upcoming piece. }}{{efn| Note that [[wind instrument]]s, [[bowed stringed instrument]]s, and [[singing|singers]] have no such need for a retuning session, since players always microtune every note they produce "on the fly". On the other hand, players of stringed instruments with movable [[fret]]s, such as the [[oud]] face a similar problem; performers on fixed-fret instruments likewise are limited to the keys which are compatible with the positions of the frets, although it is possible to microtune by tugging on a string using the finger that presses it down. }} Some music that modulates too far between keys cannot be played on a single keyboard or single harp, no matter how it is tuned: In the example tuning above, music that modulates from [[C major]] into both [[A major]] (which needs G{{music|#}} for the [[major seventh|seventh note]]) and [[C minor]] (which needs A{{music|b}} for its [[minor sixth|sixth note]]) is not possible, since each of the two meantone notes, G{{music|#}} and A{{music|b}}, both require the same string in each octave on the instrument to be tuned to their different pitches. For expediency, keyboard players substitute the wrong diminished sixth interval for a genuine meantone fifth (or neglect retuning their instrument). Though not available, a genuine meantone fifth ''would'' be consonant, but in meantone tuning systems (where {{big|{{mvar|ε}}}} isn't zero) the sharp of any note is ''always'' different from the flat of the note above it. A meantone keyboard that allowed unlimited modulation theoretically would require an infinite number of separate sharp and flat keys, and then double sharps and double flats, and so on: There must inevitably be missing pitches on a standard keyboard with only 12 notes in an octave. The value of {{mvar|ε}} changes depending on the tuning system. In other tuning systems (such as [[Pythagorean tuning]] and twelfth-comma meantone), each of the eleven fifths may have a size of {{nobr|{{math|700 + ''ε''}} cents,}} thus the diminished sixth is {{nobr|{{math|700 − 11 ''ε''}} cents.}} If their difference {{nobr| {{math|12 ''ε''}} ,}} is very large, as in the quarter-comma meantone tuning system, the diminished sixth is used as a substitute for a fifth, it is called a "wolf fifth". In terms of [[frequency]] [[ratio]]s, in order to close the [[circle of fifths]], the [[product (mathematics)|product]] of the fifths' ratios must be {{math|128}} (since the twelve fifths, if closed in a circle, span seven octaves exactly; an octave is {{nobr|{{math|2:1}},}} and {{nobr|{{math|2{{sup|7}} {{=}} 128}}),}} and if {{mvar|f}} is the size of a fifth, {{nobr|{{math|128 : ''f''{{sup| 11}}}},}} or {{nobr|{{math|''f''{{sup| 11}} : 128}},}} will be the size of the wolf. We likewise find varied tunings for the thirds: [[Major third]]s must average {{nobr|{{math|400}} cents,}} and to each pair of thirds of size {{nobr|{{math|400 ∓ 4 ''ε''}} cents}} we have a third (or diminished fourth) of {{nobr|{{math|400 ± 8 ''ε''}} cents,}} leading to eight thirds {{nobr|{{math|4 ''ε''}} cents}} narrower or wider, and four diminished fourths {{nobr|{{math|8 ''ε''}} cents}} wider or narrower than average. Three of these diminished fourths form major [[triad (music)|triad]]s with perfect fifths, but one of them forms a major triad substituting the diminished sixth for a real fifth. If the diminished sixth is a wolf interval, this triad is called the '''wolf major triad'''. Similarly, we obtain nine [[minor third]]s of {{nobr|{{math|300 ± 3 ''ε''}} cents}} and three minor thirds (or augmented seconds) of {{nobr|{{math|300 ∓ 9 ''ε''}} cents.}} === Quarter comma meantone === In [[quarter-comma meantone]], the frequency ratio for the fifth is {{math|{{radic| 5 | 4 }} }}, which is about {{nobr|{{math|3.42157}} cents}} flatter than an equal tempered {{nobr|{{math|700}} cents,}} (or exactly one twelfth of a [[diesis]]) and so the wolf is about {{nobr|{{math|737.637}} cents,}} or {{nobr|{{math|35.682}} cents}} sharper than a [[perfect fifth]] of ratio exactly {{nobr|{{math|3:2}},}} and this is the original "howling" wolf fifth. The flat minor thirds are only about {{nobr|{{math|2.335}} cents}} sharper than a [[Septimal minor third|subminor third]] of ratio {{math|7:6}}, and the sharp major thirds, of ratio exactly {{nobr|{{math|32:25}},}} are about {{nobr|{{math|7.712}} cents}} flatter than the [[Septimal major third|supermajor third]] of {{nobr|{{math|9:7}} .}} Meantone tunings with slightly flatter fifths produce even closer approximations to the subminor and supermajor thirds and corresponding triads. These thirds therefore hardly deserve the appellation of wolf, and in fact historically have not been given that name. The wolf fifth of quarter-comma meantone can be approximated by the 7-limit [[just intonation|just]] interval {{math|49:32}}, which has a size of {{nobr|{{math|737.652}} cents.}} === Pythagorean tuning === In [[Pythagorean tuning]], there are eleven [[just intonation|justly tuned]] fifths sharper than {{nobr|{{math|700}} cents}} by about {{nobr|{{math|1.955}} cents}} (or exactly one twelfth of a [[Pythagorean comma]]), and hence one fifth will be flatter by twelve times that, which is {{nobr|{{math|23.460}} cents}} (one Pythagorean comma) flatter than a just fifth. A fifth this flat can also be regarded as "howling like a wolf." There are also now eight sharp and four flat major thirds. === 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 wolf intervals (a fifth and a fourth), the 5-limit symmetric scales produce 12 of them, and the asymmetric scale 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 [[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 . }} 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 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 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).
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