Hemiola

Beginning in the 15th century, both words were also used to describe rhythmic relationships, specifically the substitution (usually through the use of coloration—red notes in place of black ones, or black in place of "white", hollow noteheads) of three imperfect notes (divided into two parts) for two perfect ones (divided into three parts) in tempus perfectum or in prolatio maior.

[7] Grove's Dictionary, on the other hand, has maintained from the first edition of 1880 down to the most recent edition of 2001 that the Greek and Latin terms are equivalent and interchangeable, both in the realms of pitch and rhythm,[8][3] although David Hiley, E. Thomas Stanford, and Paul R. Laird hold that, though similar in effect, hemiola properly applies to a momentary occurrence of three duple values in place of two triple ones, whereas sesquialtera represents a proportional metric change between successive sections.

Novotney observes: "The 3:2 relationship (and [its] permutations) is the foundation of most typical polyrhythmic textures found in West African musics.

"[10] Agawu states: "[The] resultant [3:2] rhythm holds the key to understanding ... there is no independence here, because 2 and 3 belong to a single Gestalt.

John Daverio says that the movement's "fanciful hemiolas... serve to legitimize the dance-like material as a vehicle for symphonic elaboration.

"[16] At the beginning of the second movement, Assez vif – très rythmé, of his String Quartet (1903), Ravel "uses the pizzicato as a vehicle for rhythmic interplay between 68 and 34.

"[17] Peter Manuel, in the context of an analysis of the flamenco soleá song form, refers to the following figure as a horizontal hemiola or "sesquialtera" (which mistranslates as: "six that alters").

It is "a cliché of various Spanish and Latin American musics ... well established in Spain since the sixteenth century", a twelve-beat scheme with internal accents, consisting of a 68 bar followed by one in 34, for a 3 + 3 + 2 + 2 + 2 pattern.

This interpretational switch has been exploited, for example, by Leonard Bernstein, in the song "America" from West Side Story, as can be heard in the prominent motif (suggesting a duple beat scheme, followed by a triple beat scheme): Hemiola can be used to describe the ratio of the lengths of two strings as three-to-two (3:2), that together sound a perfect fifth.

The 3:2 just perfect fifth arises in the justly tuned C major scale between C and G.[20] Later Greek authors such as Aristoxenus and Ptolemy use the word to describe smaller intervals as well, such as the hemiolic chromatic pyknon, which is one-and-a-half times the size of the semitone comprising the enharmonic pyknon.