Editing Time signature (section)

== Irrational meters ==
<!-- This section is linked from [[Stereolab]] -->
{{Image frame
|width=386
|align=right
|content=
<score>
{
\time 4/3
\times 2/3 {c''2 d'' e'' f''}
\time 4/2
c'' d'' e'' f''
}
</score>
|caption=Example of an irrational {{music|time|4|3}} time signature: here there are four (4) third notes (3) per measure. A "third note" would be one third of a whole note, and thus is a half-note triplet. The second measure of {{music|time|4|2}} presents the same notes, so the {{music|time|4|3}} time signature serves to indicate the precise speed relationship between the notes in the two measures.
|alt=Printed music staff with treble-clef symbol. The first bar has a {{music|time|4|3}} time signature, and 4 half-note triplets: treble C, D, E, and F. The second bar is in {{music|time|4|2}} and has 4 half notes: treble C, D, E, and F.
}}
Irrational time signatures (rarely, "non-dyadic time signatures") are used for so-called ''irrational bar lengths'',<ref name="Sospeso">[https://web.archive.org/web/20110721014850/http://www.sospeso.com/contents/articles/ferneyhough_p1.html "Brian Ferneyhough"], ''The Ensemble Sospeso''</ref> that have a [[denominator]] that is not a [[power of two]] (1, 2, 4, 8, 16, 32, etc.). These are based on beats expressed in terms of fractions of full beats in the prevailing tempo—for example {{music|time|3|10}} or {{music|time|5|24}}.<ref name="Sospeso"/> For example, where {{music|time|4|4}} implies a bar construction of four quarter-parts of a whole note (i.e., four quarter notes), {{music|time|4|3}} implies a bar construction of four third-parts of it. These signatures are of utility only when juxtaposed with other signatures with varying denominators; a piece written entirely in {{music|time|4|3}}, say, could be more legibly written out in {{music|time|4|4}}.

{{Image frame
|width=386
|align=right
|content=
<score>
{
\time 4/2
c''2 d'' e'' f'' |
c''^\markup {
  \note {1.} #1
  =
  \note {1} #1
} d'' e'' f''
}
</score>
|caption=The same example written using metric modulation instead of irrational time signatures. Three half notes in the first measure (making up a dotted whole note) are equal in duration to two half notes in the second (making up a whole note).|alt=Printed music staff with treble-clef symbol. Each bar has a {{music|time|4|2}} time signature, and 4 half notes: treble C, D, E, and F.
}}
{{Image frame
|width=386
|align=right
|content=
<score>
{
\time 4/2
c''2 d'' e'' f'' |
\time 12/4
c''2. d'' e'' f''
}
</score>
|caption=The same example written using a change in time signature.|alt=Printed music staff with treble-clef symbol. The first bar has a {{music|time|4|2}} time signature, and 4 half notes: treble C, D, E, and F. The second bar has a {{music|time|12|4}} time signature, and 4 dotted half notes: treble C, D, E, and F.
}}
According to [[Brian Ferneyhough]], [[metric modulation]] is "a somewhat distant analogy" to his own use of "irrational time signatures" as a sort of rhythmic dissonance.<ref name="Sospeso"/> It is disputed whether the use of these signatures makes metric relationships clearer or more obscure to the musician; it is always possible to write a passage using non-irrational signatures by specifying a relationship between some note length in the previous bar and some other in the succeeding one. Sometimes, successive metric relationships between bars are so convoluted that the pure use of irrational signatures would quickly render the notation extremely hard to penetrate. Good examples, written entirely in conventional signatures with the aid of between-bar specified metric relationships, occur a number of times in [[John Adams (composer)|John Adams]]' opera ''[[Nixon in China]]'' (1987), where the sole use of irrational signatures would quickly produce massive numerators and denominators.{{Citation needed|date=March 2010}}

Historically, this device has been prefigured wherever composers wrote [[tuplet]]s. For example, a {{music|time|2|4}} bar of 3 triplet quarter notes could be written as a bar of {{music|time|3|6}}. [[Henry Cowell]]'s piano piece ''[[Fabric (Cowell)|Fabric]]'' (1920) employs separate divisions of the bar (1 to 9) for the three [[contrapuntal]] parts, using a scheme of shaped [[notehead]]s to visually clarify the differences, but the pioneering of these signatures is largely due to [[Brian Ferneyhough]], who says that he finds that "such 'irrational' measures serve as a useful buffer between local changes of event density and actual changes of base [[tempo]]".<ref name="Sospeso"/> [[Thomas Adès]] has also used them extensively—for example in ''[[Traced Overhead]]'' (1996), the second movement of which contains, among more conventional meters, bars in such signatures as {{music|time|2|6}}, {{music|time|9|14}} and {{music|time|5|24}}.

A gradual process of diffusion into less rarefied musical circles seems underway.{{Citation needed|date=April 2015}}<!--A single quirky example hardly consistutes a trend. Who thinks there is one?--> For example, [[John Pickard (composer)|John Pickard]]'s ''Eden'', commissioned for the 2005 finals of the [[National Brass Band Championships of Great Britain]], contains bars of {{music|time|3|10}} and {{music|time|7|12}}.<ref>John Pickard: Eden, full score, Kirklees Music, 2005.</ref>

Notationally, rather than using Cowell's elaborate series of notehead shapes, the same convention has been invoked as when normal tuplets are written; for example, one beat in {{music|time|4|5}} is written as a normal quarter note, four quarter notes complete the bar, but the whole bar lasts only {{frac|4|5}} of a reference [[whole note]], and a beat {{frac|1|5}} of one (or {{frac|4|5}} of a normal quarter note). This is notated in exactly the same way that one would write if one were writing the first four quarter notes of five quintuplet quarter notes.

'''Some video samples are shown below.'''

These video samples show two time signatures combined to make a [[polymeter]], since {{music|time|4|3}}, say, in isolation, is identical to {{music|time|4|4}}.

{| class="wikitable"
|- valign="top"
| [[File:Polymeter-4o4c4o3.theora.ogv|thumb|center|Polymeter {{music|time|4|4}} and {{music|time|4|3}} played together has three beats of {{music|time|4|3}} to four beats of {{music|time|4|4}}]]
| [[File:Polymeter-2o6c3o4.theora.ogv|thumb|center|Polymeter {{music|time|2|6}} and {{music|time|3|4}} played together has six beats of {{music|time|2|6}} to four beats of {{music|time|3|4}}]]
| [[File:Polymeter-2o5c2o3.theora.ogv|thumb|center|Polymeter {{music|time|2|5}} and {{music|time|2|3}} played together has five beats of {{music|time|2|5}} to three beats of {{music|time|2|3}}. The displayed numbers count the underlying [[polyrhythm]], which is 5:3]]
|}