Types of Quarter Tones
The term quarter tone can refer to a number of different intervals, all very close in size. For example, some 17th- and 18th-century theorists used the term to describe the distance between a sharp and enharmonically distinct flat in mean-tone temperaments (e.g., D♯–E♭). In the quarter tone scale, also called 24 tone equal temperament (24-TET), the quarter tone is 50 cents, or a frequency ratio of 21/24 or approximately 1.0293, and divides the octave into 24 equal steps (equal temperament). In this scale the quarter tone is the smallest step. A semitone is thus made of two steps, and three steps make a three-quarter tone play or neutral second, half of a minor third.
In just intonation the quarter tone can be represented as 36:35 or 33:32, approximately half the semitone of 16:15 or 25:24. The ratio of 36:35 is only 1.23 cents narrower than a 24-TET quarter tone. This just ratio is also the difference between a minor third (6:5) and septimal minor third (7:6).
Quarter tones and intervals close to them also occur in a number of other equally tempered tuning systems. 22-TET contains an interval of 54.55 cents, slightly wider than a quarter-tone, whereas 53-TET has an interval of 45.28 cents, slightly smaller. 72-TET also has equally-tempered quarter-tones, and indeed contains 3 quarter tone scales, since 72 is divisible by 24.
Composer Ben Johnston, to accommodate the just septimal quarter tone, uses a small "7" as an accidental to indicate a note is lowered 49 cents, or an upside down "∠" to indicate a note is raised 49 cents, or a ratio of 36/35. Johnston uses an upward and downward arrow to indicate a note is raised or lowered by a ratio of 33/32, or 53 cents.
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