Гемиола: Музыкальное отношение 3:2

Musical ratio of 3:2
In music, hemiola (also hemiolia) is the ratio 3:2. The equivalent Latin term is sesquialtera. In rhythm, hemiola refers to three beats of equal value in the time normally occupied by two beats. In pitch, hemiola refers to the interval of a perfect fifth.

The word hemiola comes from the Greek adjective ἡμιόλιος, hemiolios, meaning "containing one and a half," "half as much again," "in the ratio of one and a half to one (3:2), as in musical sounds." The words "hemiola" and "sesquialtera" both signify the ratio 3:2, and in music were first used to describe relations of pitch. Dividing the string of a monochord in this ratio produces the interval of a perfect fifth. 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.

In rhythm, hemiola refers to three beats of equal value in the time normally occupied by two beats.

The Oxford Dictionary of Music illustrates hemiola with a superimposition of three notes in the time of two and vice versa. One textbook states that, although the word "hemiola" is commonly used for both simultaneous and successive durational values, describing a simultaneous combination of three against two is less accurate than for successive values and the "preferred term for a vertical two against three is sesquialtera." The New Harvard Dictionary of Music states that in some contexts, a sesquialtera is equivalent to a hemiola. 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,

A repeating vertical hemiola is known as polyrhythm, or more specifically, cross rhythm. The most basic rhythmic cell of sub Saharan Africa is the 3:2 cross rhythm. Novotney observes: "The 3:2 relationship (and [its] permutations) is the foundation of most typical polyrhythmic textures found in West African musics." 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." In the following example, a Ghanaian gyil plays a hemiola as the basis of an ostinato melody. The left hand (lower notes) sounds the two main beats, while the right hand (upper notes) sounds the three cross beats.

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 bar followed by one in , for a 3 + 3 + 2 + 2 + 2 pattern. This figure is a common African bell pattern, used by the Hausa people of Nigeria, in Haitian Vodou drumming, Cuban palo, and many other drumming systems. The horizontal hemiola suggests metric modulation ( changing to ). 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.

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.