Editing Polyrhythm (section)

== Examples ==
The following is an example of a 3 against 2 polyrhythm, given in [[time unit box system]] (TUBS) notation; each box represents a fixed unit of time; time progresses from the left of the diagram to the right. It is in bad form to teach a student to play 3:2 polyrhythms as simply quarter note, eighth note, eighth note, quarter note. The proper way is to establish sound bases for both the quarter-notes, and the triplet-quarters, and then to layer them upon each other, forming multiple rhythms. Beats are indicated with an X; rests are indicated with a blank.
:{| class="wikitable"

|+ 3 against 2 polyrhythm
|- style="text-align:center;"
| 3-beat rhythm
| style="width:20px; background:silver;" | X
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | X
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | X
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px;" | X
| style="width:20px;" | &nbsp;
| style="width:20px;" | X
| style="width:20px;" | &nbsp;
| style="width:20px;" | X
| style="width:20px;" | &nbsp;
| style="width:20px; background:silver;" | X
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | X
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | X
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px;" | X
| style="width:20px;" | &nbsp;
| style="width:20px;" | X
| style="width:20px;" | &nbsp;
| style="width:20px;" | X
| style="width:20px;" | &nbsp;
|- style="text-align:center;"
| 2-beat rhythm
| style="width:20px; background:silver;" | X
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | X
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px;" | X
| style="width:20px;" | &nbsp;
| style="width:20px;" | &nbsp;
| style="width:20px;" | X
| style="width:20px;" | &nbsp;
| style="width:20px;" | &nbsp;
| style="width:20px; background:silver;" | X
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | X
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px;" | X
| style="width:20px;" | &nbsp;
| style="width:20px;" | &nbsp;
| style="width:20px;" | X
| style="width:20px;" | &nbsp;
| style="width:20px;" | &nbsp;
|}

A common memory aid to help with the 3 against 2 polyrhythm is that it has the same rhythm as the phrase "not difficult"; the simultaneous beats occur on the word "not"; the second and third of the triple beat land on "dif" and "cult", respectively. The second 2-beat lands on the "fi" in "difficult". Try saying "not difficult" over and over in time with the sound file above. This will emphasize the "3 side" of the 3 against 2 feel. Now try saying the phrase "not a problem", stressing the syllables "not" and "prob-". This will emphasize the "2 side" of the 3 against 2 feel. More phrases with the same rhythm are "cold cup of tea", "four funny frogs", "come, if you please", and "[[Carol of the Bells|ring, Christmas bells]]".

:[[File:Music cross-rhythm, cold cup of tea.PNG|230x230px]]

Similar phrases for the 4 against 3 polyrhythm are "pass the golden butter"<ref name="Fingertips"/> or "pass the goddamn butter"<ref>Sep 2007. ''Guitar World'', p.102. Vol. 28, No. 9. ISSN 1045-6295.</ref> and "what atrocious weather" (or "what a load of rubbish" in British English); the 4 against 3 polyrhythm is shown below.
:{| class="wikitable"

|+ 4 against 3 polyrhythm
|- style="text-align:center;"
| 4-beat rhythm
| style="width:20px; background:silver;" | X
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | X
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | X
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | X
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px;" | X
| style="width:20px;" | &nbsp;
| style="width:20px;" | &nbsp;
| style="width:20px;" | X
| style="width:20px;" | &nbsp;
| style="width:20px;" | &nbsp;
| style="width:20px;" | X
| style="width:20px;" | &nbsp;
| style="width:20px;" | &nbsp;
| style="width:20px;" | X
| style="width:20px;" | &nbsp;
| style="width:20px;" | &nbsp;
|- style="text-align:center;"
| 3-beat rhythm
| style="width:20px; background:silver;" | X
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | X
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | X
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px; background:silver;" | &nbsp;
| style="width:20px;" | X
| style="width:20px;" | &nbsp;
| style="width:20px;" | &nbsp;
| style="width:20px;" | &nbsp;
| style="width:20px;" | X
| style="width:20px;" | &nbsp;
| style="width:20px;" | &nbsp;
| style="width:20px;" | &nbsp;
| style="width:20px;" | X
| style="width:20px;" | &nbsp;
| style="width:20px;" | &nbsp;
| style="width:20px;" | &nbsp;
|}

{| class="wikitable"
|-
| [[File:Music cross-rhythm, what atrocious weather.PNG|center|alt=|230x230px]]
{{Listen|filename=Polyrhythm 3 against 4.ogg|title=A 3 against 4 polyrhythm}}
| [[File:Polyrhythm4c3.theora.ogv|thumb|right|Polyrhythm {{music|time|4|4}} with {{music|time|3|4}} simultaneously (cross rhythm) as bounce inside oval]]
|}

As can be seen from above, the counting for polyrhythms is determined by the [[Least common multiple|lowest common multiple]], so if one wishes to count 2 against 3, one needs to count a total of 6 beats, as lcm(2,3) = 6 (<samp>'''''1'''''2'''''3'''''4'''''5'''''6</samp> and <samp>'''''1'''''23'''''4'''''56</samp>). However this is only useful for very simple polyrhythms, or for getting a feel for more complex ones, as the total number of beats rises quickly. To count 4 against 5, for example, requires a total of 20 beats, and counting thus slows the tempo considerably. However some players, such as [[Indian Classical Music|classical Indian musicians]], can intuitively play high polyrhythms such as 7 against 8.{{citation needed|date=May 2012}}

Polyrhythms are quite common in late [[Romantic Music]] and [[20th-century classical music]]. Works for keyboard often set odd rhythms against one another in separate hands. A good example is in the soloist's cadenza in [[Grieg]]'s [[Piano Concerto (Grieg)|Concerto in A Minor]]; the left hand plays [[arpeggios]] of seven notes to a beat; the right hand plays an ostinato of eight notes per beat while also playing the melody in octaves, which uses whole notes, dotted eighth notes, and triplets.
Other instances occur often in [[Rachmaninoff]]'s [[Piano Concerto No. 2 (Rachmaninoff)|Piano Concerto No. 2]].{{citation needed|date=April 2011}} The piano arpeggios that constitute much of the soloist's material in the first movement often have anywhere from four to eleven notes per beat. In the last movement, the piano's opening run, marked 'quasi [[glissando]]', fits 52 notes into the space of one [[Bar (music)|measure]], making for a glissando-like effect while keeping the mood of the music. Other instances in this movement include a scale that juxtaposes ten notes in the right hand against four in the left, and one of the main themes in the piano, which imposes an eighth-note melody on a triplet harmony.

=== List of basic polyrhythms ===
{| class="wikitable"
|+
!First rhythm
!Second rhythm
!Least common multiple
|-
|2
|3
|6
|-
|2
|5
|10
|-
|3
|4
|12
|-
|2
|7
|14
|-
|3
|5
|15
|-
|2
|9
|18
|-
|4
|5
|20
|-
|3
|7
|21
|-
|3
|8
|24
|-
|4
|6
|12
|-
|4
|7
|28
|-
|5
|6
|30
|-
|5
|7
|35
|-
|4
|9
|36
|-
|5
|8
|40
|-
|6
|7
|42
|-
|5
|9
|45
|-
|6
|8
|24
|-
|6
|9
|18
|-
|7
|8
|56
|-
|7
|9
|63
|-
|7
|11
|77
|-
|8
|9
|72
|}