Examples
Example 1: Take the 2-dimensional lattice of perfect fifths (ratio 3/2) and just major thirds (ratio 5/4). Choose the commas 128/125 (the diesis, the distance by which three just major thirds fall short of an octave, about 41 cents) and 81/80 (the syntonic comma, the difference between four perfect fifths and a just major third, about 21.5 cents). The result is a block of twelve, showing how twelve-tone equal temperament approximates the ratios of the 5-limit.
Example 2: However, if we were to reject the diesis as a unison vector and instead choose the difference between five major thirds (minus an octave) and a fourth, 3125/3072 (about 30 cents), the result is a block of 19, showing how 19-TET approximates ratios of the 5-limit.
Example 3: In the 3-dimensional lattice of perfect fifths, just major thirds, and just minor sevenths (ratio 7/4), the identification of the syntonic comma, the septimal kleisma (225/224, about 8 cents), and the ratio 1029/1024 (the difference between three septimal whole tones and a perfect fifth, about 8.4 cents) results in a block of 31, showing how 31-TET approximates ratios of the 7-limit.
Read more about this topic: Fokker Periodicity Blocks
Famous quotes containing the word examples:
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)