The one other tuning system that does<\/i> satisfy the comma difference between sharps and flats with a nine-comma whole tone is Just Intonation. David Boyden, in fact, wrote an article proposing that Just intonation was the tuning system actually intended by these 18th-century theorists (JAMS<\/i>, 1951). However, in Just intonation, the nine-comma whole tone is a large, Pythagorean 9:8 tone of 204c, based on a comma of 22.7c. This comma is too large to meet the 55 commas to the octave criterion: 55 x 22.7c = 1246.2c\u2014much wider than an octave. Besides, two different whole tone sizes are needed in various contexts in Just intonation (9:8 and 10:9), and two whole tone sizes do not satisfy the accepted identity of the whole tone as a single interval consisting of nine commas. Besides, the necessity of two different whole tones requires too much tuning flexibility to make Just intonation usable in situations where a fixed-pitch instrument like a keyboard takes part, especially where the music includes chromaticism and complex harmonies.<\/p>\n","protected":false},"excerpt":{"rendered":"
The one other tuning system that does<\/i> satisfy the comma difference between sharps and flats with a nine-comma whole tone is Just Intonation. David Boyden, in fact, wrote an article proposing that Just intonation was the tuning system actually intended by these 18th-century theorists (JAMS<\/i>, 1951). However, in Just intonation, the nine-comma whole tone is a large, Pythagorean 9:8 tone of 204c, based on a comma of 22.7c. This comma is too large to meet the 55 commas to the octave criterion: 55 x 22.7c = 1246.2c\u2014much wider than an octave. Besides, two different whole tone sizes are needed in various contexts in Just intonation (9:8 and 10:9),<\/p>\n