harmonic division


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harmonic division

[här′män·ik di′vizh·ən]
(mathematics)
The division of a line segment externally and internally in the same ratio; that is, the division of a line segment by the harmonic conjugates of its end points.
References in periodicals archive ?
Five panels more than six feet square (with a gold-silver impasto on the edge of the frame) and three six-by-three-foot panels enshroud the viewer's body in a visual rhythm, guided by the tonal modulations and the harmonic division of the space, the perfect proportions of which are an interpretation of Renaissance equilibrium.
In 1931 he designed an original instrument constructed by Lev Termen--the Rhythmicon--used in his work Rhythmicana to generate low frequencies according to the harmonic division of a duration of a second as a reference unit (Slonimsky 1988, 151).
This group works by marked points within the twinned totals of 108 that represent -- but again proportionally -- perfect musical intervals seen either as "a group of poems, differentiated at each end by some structural mark (the ends of a sequence, say, or the ends of a series of poems in the same stanzaic pattern)" (71) along a monochord or, inversely, as harmonic divisions of the whole such as "an octave that could be divided by similar means into tetrachords, or tones, by counting equal divisions of the whole -- under this scheme, the fifth would occur seven-twelfths of the way through the group" (71).
After a brief survey of the use of alternative tuning systems in twentieth-century music, Chalmers begins his elucidation and expansion of tetrachordal theories with the arithmetic approach of Pythagoras and Ptolemy, in which intervals are expressed as numerical ratios, or as harmonic divisions - 1/2, 1/3, 1/4, and so on - of a string, and the complementary approach of Aristoxenus and his followers, who thought of musical intervals as spatial distances divisible into parts in the same way that a line can be divided with a ruler (and much as on a piano keyboard, where equal intervals span constant distances).