Editing Wolf interval (section)

=== Keep the two-dimensional tuning system ===
[[Image:Isomorphic Note Layout.jpg|thumb|top|550px|Figure 1: The Wicki isomorphic keyboard, invented by Kaspar Wicki in 1896.<ref name=Gaskins2003>
{{cite web
 | first1 = Robert  | last1 = Gaskins
 | date = September 2003
 | title = The Wicki system – an 1896 precursor of the Hayden system
 | website = Concertina Library: Digital Reference Collection for Concertinas
 | url = http://www.concertina.com/gaskins/wicki/
 | access-date = 2013-07-11
}}
</ref>]]
[[Image:Rank-2 temperaments with the generator close to a fifth and period an octave.jpg|right|250px|thumb|Figure 2: The [[syntonic temperament]]’s tuning continuum.<ref name=Milne2007 />]]
A lesser-known alternative method that allows the use of multi-dimensional temperaments without wolf intervals is to use a two-dimensional keyboard that is "[[isomorphic]]" with that temperament. A keyboard and temperament are isomorphic if they are generated by the same intervals. For example, the Wicki keyboard shown in Figure 1 is generated by the same musical intervals as the [[syntonic temperament]]—that is, by the [[octave]] and tempered [[perfect fifth]]—so they are isomorphic.

On an [[isomorphic keyboard]], any given musical interval has the same shape wherever it appears—in any octave, key, and tuning—except at the edges. For example, on Wicki's keyboard, from any given note, the note that is a tempered perfect fifth higher is always up-and-rightwardly adjacent to the given note. There are no wolf intervals within the note-span of this keyboard. The only problem is at the edge, on the note E{{music|sharp}}. The note that is a tempered perfect fifth higher than E{{music|sharp}} is B{{music|sharp}}, which is not included on the keyboard shown (although it could be included in a larger keyboard, placed just to the right of A{{music|sharp}}, hence maintaining the keyboard's consistent note-pattern). Because there is no B{{music|sharp}} button, when playing an E{{music|sharp}} [[power chord]], one must choose some other note that is close in pitch to B{{music|sharp}}, such as C, to play instead of the missing B{{music|sharp}}. That is, the interval from E{{music|sharp}} to C would be a "wolf interval" on this keyboard. In [[19 equal temperament|19-TET]], the interval from E{{music|sharp}} to C{{music|flat}} would be (enharmonic to) a perfect fifth.

However, such edge conditions produce wolf intervals only if the isomorphic keyboard has fewer buttons per octave than the tuning has [[enharmonic]]ally distinct notes.<ref name=Milne2007/> For example, the isomorphic keyboard in Figure 2 has 19 buttons per octave, so the above-cited edge condition, from E{{music|sharp}} to C, is ''not'' a wolf interval in [[Equal temperament|12-TET]], [[17 equal temperament|17-TET]], or [[19 equal temperament|19-TET]]; however, it ''is'' a wolf interval in 26-TET, [[31 equal temperament|31-TET]], and [[53 equal temperament|53-TET]]. In these latter tunings, using electronic transposition could keep the current key's notes centered on the isomorphic keyboard, in which case these wolf intervals would very rarely be encountered in tonal music, despite modulation to exotic keys.<ref name=Plamondon2009>
{{cite conference
 | first1 = J.   | last1 = Plamondon
 | first2 = A.   | last2 = Milne
 | first3 = W.A. | last3 = Sethares 
 | year = 2009
 | title =  Dynamic tonality: Extending the framework of tonality into the 21st&nbsp;century
 | conference = Annual Conference of the South Central Chapter of the College Music Society
 | book-title = Proceedings of the Annual Conference of the South Central Chapter of the College Music Society
 | url = http://sethares.engr.wisc.edu/paperspdf/CMS2009.pdf
 | via = sethares.engr.wisc.edu
}}</ref>

A keyboard that is isomorphic with the syntonic temperament, such as Wicki's keyboard above, retains its isomorphism in any tuning within the tuning continuum of the syntonic temperament, even when changing tuning dynamically among such tunings.<ref name=Plamondon2009/> Plamondon, Milne & Sethares (2009),<ref name=Plamondon2009/> Figure&nbsp;2, shows the valid tuning range of the syntonic temperament.