Odd limit: Difference between revisions

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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.

==== ~~Nonoctave equaves~~ ====

==== Non-octave settings ====

~~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 1–9/7–9/5–3/1 chord has extended ratio 35:45:63:105. Its '''intervallic throdd limit''' is 7, and its '''otonal throdd limit~~''' is 35.~~

~~==== Equave~~ limit ~~====~~

The concept of odd limit can be generalized to [[nonoctave|non-octave]] contexts such as 3/1-equivalent [[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 1–9/7–9/5–3/1 chord has extended ratio 35:45:63:105. Its ''intervallic throdd limit'' is 7, and its ''otonal throdd limit'' is 35.

~~See~~ [[~~Equave limit~~]].

== See also ==

@@ Line 59: / Line 59: @@
 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.
-==== Nonoctave equaves ====
+==== Non-octave settings ====
-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 1–9/7–9/5–3/1 chord has extended ratio 35:45:63:105. Its '''intervallic throdd limit''' is 7, and its '''otonal throdd limit''' is 35.
+{{Main| Equave limit }}
-==== Equave limit ====
+The concept of odd limit can be generalized to [[nonoctave|non-octave]] contexts such as 3/1-equivalent [[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 1–9/7–9/5–3/1 chord has extended ratio 35:45:63:105. Its ''intervallic throdd limit'' is 7, and its ''otonal throdd limit'' is 35.
-See [[Equave limit]].
 == See also ==