Since the time of the ancient Greeks, we have known that two tones whose frequencies were related by a simple ratio like 2:1 (an octave) or 3:2 (a perfect fifth) produce the most pleasing, or consonant, musical intervals
This not only facilitated the mathematical examination and manipulation of intervals but also enabled theorists to depict musical intervals visually in equal terms.
This includes experiments using sounding glasses and sounding wheels, attempts to measure the absolute frequency of particular pitches and determine ratios associated with particular musical intervals.
More than 2000 years ago, Pythagoras reportedly discovered that pleasing musical intervals
could be described using simple ratios.
In the sixteenth century, the plays' topics were often religious, derived from biblical, hagiographic, and devotional sources; musical intervals
also became an integral part of the staging.
An order of consonance was assigned on the musical intervals
, which was shown mathematically sound and consistent with previous studies.
The researchers then compared these musical intervals
with those between important tonal frequencies in spoken vowelsMovie Camera uttered by American English speakers in either excited or subdued voices.
The work's second movement, with instruments entering in pairs at different musical intervals
, is not unlike the way that jazz band leaders used to introduce players - "And now on trumpet .
Oxford brilliantly inaugurates its new series, Studies in Music Theory, with a collection of essays that gives a more complete sense of Lewin's achievement than if we had been left with only the two extraordinary but highly technical volumes published during his lifetime, Generalized Musical Intervals
and Transformations (New Haven: Yale University Press, 1987) and Musical Form and Transformation (New Haven: Yale University Press, 1993).
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).
They found that the sound spectra of the speech tones could be sorted the same way as the music, with excited speech exhibiting more major musical intervals
and subdued speech more minor ones.
David Lewin's analytical essays are an extension of the theory of transformational networks he set forth in his earlier book Generalized Musical Intervals
and Transformations (New Haven: Yale University Press, 1987).
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).
Perhaps most unsatisfactory of all is the author's presentation of one of the cosmic harmony's crucial features, the relationship between number and musical intervals
For example, West provides an introduction to the diatonic scale, ratios, the establishment of the whole-tone interval, and the representation of musical intervals