Odd limit: Difference between revisions

This conjecture has two implications. First, a given JI chord has an ideal voicing. This voicing may be rather far-flung, and a more compact voicing may be almost as consonant. For example, 1:3:5:7 has a large gap between the two lowest voices, and 2:3:5:7 is more practical. Second, a voicing can imply a tuning. For example, if a piece has a minor chord with the 3rd voiced as a 10th, 7/3 may be preferred over 12/5 for the 3rd. If it's voiced as a 10th plus an octave, either 14/3 or 19/4 may be preferred to 24/5.

The concept of odd limit can be generalized to prime three in a [[Nonoctave|non-octave]] ("no-twos") tritave-equivalent context such as [[Bohlen-Pierce]]. Just as the words even and odd refer to divisibility by two, mathematicians use the words '''threeven''' and '''throdd''' for divisibility by three. The '''throdd limit''' of a ratio is found by repeatedly dividing the numerator or denominator by three, and selecting the larger of the two numbers. Example: the throdd limit of 15/7 is 7. Other limits can be generalized too. The '''double throdd limit''' of 15/7 is (7,5). Its '''all-throdd voicing''' is 7/5. The 1/1 - 9/7 - 9/5 - 3/1 chord has extended ratio 35:45:63:105. Its '''intervallic throdd limit''' is 7, and its '''otonal throdd limit''' is 35.