5-odd-limit

Odd limit

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5-odd-limit intervals within an octave

The 5-odd-limit is the set of all rational intervals which can be written as 2^k(a/b) where a, b ≤ 5 and k is an integer. To the 3-odd-limit, it adds 2 pairs of octave-reduced intervals involving 5.

Below is a list of all octave-reduced intervals in the 5-odd-limit.

• 1/1
• 6/5, 5/3
• 5/4, 8/5
• 4/3, 3/2

Ratio	Size (¢)	Color name		Name
6/5	315.641	g3	gu 3rd	minor third
5/4	386.314	y3	yo 3rd	major third
8/5	813.686	g6	gu 6th	minor sixth
5/3	884.359	y6	yo 6th	major sixth

The smallest equal division of the octave which is consistent in the 5-odd-limit is 3edo.

The one which is distinctly consistent in the same is 9edo.

The density of edos consistent in the 5-odd-limit is 3/4. See below for proof.

Proof for the density of edos consistent in the 5-odd-limit

Let the error of harmonic 3 be A. Let the error of harmonic 5 be B. By the equidistribution theorem, the relative error of any individual interval is equidistributed from −50% to +50%, so we have the probability density functions f_A(x) = 1 if −1/2 ≤ x ≤ +1/2 and 0 otherwise, f_B(y) = 1 if −1/2 ≤ y ≤ +1/2 and 0 otherwise.

An edo is consistent in the 5-odd-limit if the error C = B − A of the only compound interval – 5/3 – falls into the range from −50% to +50%.

It is easy to show the probability density function f_C(z) is the triangular distribution 1 − |z| if −1 ≤ z ≤ +1 and 0 otherwise.

The density of edos consistent in the 5-odd-limit therefore equals (∫ +1/2
−1/2  f_C(z)dz)/(∫ +∞
−∞  f_C(z)dz), which evaluates to 3/4. [math]\displaystyle{ \square }[/math]

See also

• 5-limit (prime limit)
• Diamond5 – as a scale