Hemiola

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In modern musical parlance, a hemiola is a metrical pattern in which two bars in simple triple time (3/2 or 3/4 for example) are articulated as if they were three bars in simple duple time (2/2 or 2/4).

The word hemiola derives from the Greek hemiolios, meaning "one and a half". (The term hemiola or "one and a half" was also used by the Greeks to refer to a galley powered by one and a half banks of oars). It was originally used in music to refer to the frequency ratio 3:2; that is, the interval of a justly tuned perfect fifth.

Later, from around the 15th century, the word came to mean the use of three breves in a bar when the prevailing metrical scheme had two dotted breves in each bar.^[1] This usage was later extended to its modern sense of two bars in simple triple time articulated or phrased as if they were three bars in simple duple time. (The pulse stays constant, and the duration of the beat changes.) An example can be found in measures 64 and 65 of this excerpt from the first movement of Wolfgang Amadeus Mozart's Piano Sonata, K. 332:

The effect can clearly be seen in the bottom staff, played by the left hand: the accented beats are those with two notes; hearing this passage one gets a sensation of "1 2 3, 1 2 3, 1 2, 1 2, 1 2".

Hemiola is found in many Renaissance pieces at areas of cadential repose such as the compositions of Josquin des Prez and Jacob Obrecht.

Perhaps one of the most obvious examples is the Ukrainian Bell Carol, Carol of the Bells.

Hemiolas (in the modern sense) often occur in certain dances, particularly the courante. Composers of classical music who have used the device particularly extensively include Arcangelo Corelli, George Friedrich Handel and most famously in the music of Johannes Brahms (e.g. the opening of Symphony no 3).

Musicians' common speech has extended the definition of "hemiola" to include any occasion of a "three-against-two" metrical feel --- including some mixed meters and polyrhythms --- contrary to the word's original meaning. For example, "America" from Leonard Bernstein's West Side Story is often said to contain good examples of hemiola. However, though "America" does alternate between 6/8 time and 3/4 time, this is not strictly hemiola; hemiola is specifically the regrouping of notes in simple triple meter into groups of two beats rather than three.

Likewise, three-against-two polyrhythms are not hemiola, since 1) they may or may not occur over two bars of triple meter, and 2) in hemiola, the triple-meter feel is altogether absent from the two bars in question.

Were the metrical impulse to be not a three beat pattern changing to a two beat one (as in the Mozart example above), but one where a two beat impulse changes to a two beat one, the pattern of 2:3 would be known as sesquialtera. (Note, this does not specifically refer to the "sesquialtera" organ stop.)