1789edo: Difference between revisions

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A "proper" jacobin scale in 1789edo, the maximum evenness scale that uses 822 as a generator, contains only 37 notes. The step sizes are 48 and 49, making them indistinguishable to human ear at this scale.

Using the maximal evenness method of finding rank two temperaments, we get a 37 & 1789 temperament with the comma set 6656/6655 and {{Monzo|-74, -51, 0, 52}}.

Using the maximal evenness method of finding rank two temperaments, we get a 37 & 1789 temperament with the comma set 6656/6655 and {{Monzo|-74, -51, 0, 52}}. The first is the well known jacobin comma, and it means that a stack of three 11/8 is equated with 13/10. Second comma represents a stack of 52 13/5s with a 5/1 closing at 74 octaves, or when octave-rediced, a stack of 52 13/10s being equal to 8/5.

The monotonicity can be fixed by using divisors of 822 as a generaator, for example 137\1789 "6th root of 11/8" temperament having 222 notes. In addition, this can be re-interpreted by using 13/10 as a generator instead which produces a more vibrant 1205 out of 1789, and partitioning the resulting 13/5s in three around the octave. Addition of 29 and 31 harmonic intervals may also be suitable to spice up an otherwise monotonous scale.

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 A "proper" jacobin scale in 1789edo, the maximum evenness scale that uses 822 as a generator, contains only 37 notes. The step sizes are 48 and 49, making them indistinguishable to human ear at this scale.
-Using the maximal evenness method of finding rank two temperaments, we get a 37 & 1789 temperament with the comma set 6656/6655 and {{Monzo|-74, -51, 0, 52}}.
+Using the maximal evenness method of finding rank two temperaments, we get a 37 & 1789 temperament with the comma set 6656/6655 and {{Monzo|-74, -51, 0, 52}}. The first is the well known jacobin comma, and it means that a stack of three 11/8 is equated with 13/10. Second comma represents a stack of 52 13/5s with a 5/1 closing at 74 octaves, or when octave-rediced, a stack of 52 13/10s being equal to 8/5.
 The monotonicity can be fixed by using divisors of 822 as a generaator, for example 137\1789 "6th root of 11/8" temperament having 222 notes. In addition, this can be re-interpreted by using 13/10 as a generator instead which produces a more vibrant 1205 out of 1789, and partitioning the resulting 13/5s in three around the octave. Addition of 29 and 31 harmonic intervals may also be suitable to spice up an otherwise monotonous scale.