Consistent circle

Intuitively, a circle^{[idiosyncratic term]} of an interval a/b in an edo describes a case where an interval so accurate that you can rely on navigating with it being consistent with respect to the "circle of notes" defined. It is closely related to the concept of telicity, except that a circle can be of any JI interval, as long as it is consistently mapped.

Motivation

The circle of fifths/fourths can be confusing to navigate in edos which are not telic in that a circle of 3/2's / 4/3's fails to "close", for example in 31edo where the difference between 31 fifths and 18 octaves is 415% of a 31edostep. Usually, in such edos, there is present other intervals, such as 5/4 and 7/4 in the case of 31edo, which are far more accurate and therefore far more reliable for navigation. In the case of 31edo, 5/4 and 7/4 are in fact so accurate that stacking either of them 31 times (and in fact, any combination of them or their octave complements 31 times, as long as there is 31 intervals in total) will keep the result off by less than a 31edostep (meaning they form weak circles), even if the result isn't guaranteed to be consistent.

Definition

We define a circle^{[idiosyncratic term]} of some interval a/b as an interval with such extremely low relative error (using the direct approximation) with respect to N-edo that when we stack it m > 0 times, where m is the minimal amount required to reach a whole number of octaves, the combined interval is consistent with its actual size in JI, which is to say it is off by less than 0.5\N = 1200 ¢ / N / 2 (a.k.a. 50% relative error). Note that this definition implies that the circle need not reach all notes of the EDO if the circle occurs in a subset EDO, but that the circle must have low enough error that within the full EDO it is still consistent.

Note that when a/b does generate all notes of the edo (meaning N = m), then that means that (a/b)^{m = N} reaches m = N octaves. This will always be true in a prime edo, such as 31edo, meaning we can easily deduce that stacking 35/32 31 times gets us at 4 octaves, because 35/32's direct mapping is 4\31. This same reasoning can be applied in general if you think instead in terms of the subset edo generated.

Closing error

The closing error of a/b is defined as follows: If k is the number of steps that a/b is consistently mapped to (even if one of a and b are themselves inconsistent), meaning that:

k = round(N log₂(a/b))

...then the closing error of a/b in N-edo is:

N² log₂(a/b) / GCD(k,N)

Therefore, N-edo is a circle of a/b's iff:

|N² log₂(a/b) / GCD(k,N)| < 1/2

And is a weak circle if it is < 1 instead.

Weak circle

A weak circle is a circle with closing error of less than an EDOstep, so that going half of the way around the circle in either direction is consistent. This is a much more common type of circle, hence useful to distinguish.

Examples

80edo is a circle of 12/11's because (12/11)⁸ / 2 = ~5.1 ¢ < 0.5\80 = 7.5 ¢. ~5.1 ¢ (the size of the undecimal octatonic comma) is thus the closing error of the circle of 12/11's in any multiple of 8edo where 12/11 is mapped consistently to 1\8. 80edo is not a proper circle of 13/10's, even though 8edo is a circle of 13/10's (which requires that 13/10 is mapped consistently by the val chosen), because the accrued error exceeds 0.5\80 = 7.5 ¢.

Another example from before is that 31edo is a weak circle of 5/4's and 7/4's, but note that 31edo is a circle of (5/4)/(8/7) = 5/4 * 7/4 / 2 = 35/32's (meaning that 31edo is a (strong) circle of septimal neutral seconds).