Consistent circle

Intuitively, a consistent circle of ~a/b's in an edo describes a case where a/b is so accurately approximated 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 to the unison when octave-reduced. (It also admits generalization to any equave.) Note that the precise term consistent circle of ~a/b's may be shortened to circle of a/b's⁠ ⁠^{[idiosyncratic term ]} in writing for the sake of brevity wherever it's unambiguous (or in informal writing).

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 just fifths and 18 octaves is 415% of a 31edostep. Usually, in such edos, there are 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 isn't more than 31 intervals in total) will keep the result off by less than a 31edostep (meaning they form weakly consistent circles), even if the result isn't guaranteed to be consistent beyond floor(31/2) = 15 moves. (Alternatively, stated, this means the result is guaranteed to be consistent if you stack at most floor(31/2) = 15 of them w.r.t. a starting note.)

Definitions

We define a consistent circle (abbreviatable to just circle⁠ ⁠^{[idiosyncratic term ]}) of some (usually JI) interval a/b as: an interval with such extremely low relative error with respect to N-edo that when we stack its best approximation m > 0 times, where m is the minimum required to reach a whole number of octaves, the combined interval is consistent with its actual (untempered) size, 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\N is the best approximation of a/b, 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)

Circle

Therefore, N-edo is a circle of ~a/b's iff the closing error c = |N² log₂(a/b) / GCD(k,N)| < 1/2,

and is a weak circle if c < 1 instead, or a strong circle if c < 1/4.

Weakly consistent circle

A weakly consistent 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 that is still generally reliable for most purposes, hence useful to distinguish.

When a circle satisfies this consistency but no stronger, we say its consistency is weak. If it fails to satisfy this bound but generates a subset, it might qualify for #having a sub-weak circle.

Strongly consistent circle

A strongly consistent circle is a circle with closing error of less than a quarter of an edostep, so that it is in some sense as reliable as you could possibly ask of it.

This is useful because it means that when you tune perfectly, it will have significantly more plausible "closure" than a non-strong circle.

For example, 12edo, 53edo and 665edo are the first three strong circles of ~3/2, given that the remnant is less than a quarter of an edostep in each case.

When a circle satisfies this consistency but no stronger (which is guaranteed if it generates the whole edo), we say its consistency is strong.

Is vs. has

An important distinction must be made (when N is not prime) in the case where a/b does not generate all notes of N-edo but only a subset of the notes.

In such a case, we say that N-edo "has a circle of ~a/b's"; it is incorrect to say that N-edo "is a circle of ~a/b's" because that would imply all notes are reached by repeatedly stacking a/b.

Having a sub-weak circle

A "sub-weak circle of ~a/b's" in N-edo describes a case where the subset edo generated by a/b qualifies as a weak circle of a/b's.

This can be a useful property to distinguish; for example 80edo has a sub-weak circle of ~10/9's, because (10/9)²⁰ / 8 (~48.1 ¢) is smaller in JI than 1\20 = 60 ¢.

This means that if one is satisfied with the circle in the subset edo, one may find it to be sufficiently accurate for navigation with in the larger edo, because of familiarity with it the subset edo.

When a circle satisfies this consistency but no stronger (which is only possible if the circle does not generate the full edo), we say its consistency is sub-weak.

Having a super-strong circle

A "super-strong circle of ~a/b's" in N-edo describes a case where a/b generates a subset of N-edo but is accurate enough that you can stack floor(N/2)-many a/b's and still have it be consistent w.r.t. the superset edo.

In other words, if N-edo has a super-strong circle of ~a/b's, that means that were GCD(k, N) = 1, it would still qualify as a weak circle.

(We use this weaker/more generous bound rather than the default bound for closing error because such a circle is already going "above-and-beyond" in terms of what's necessary to produce a consistent circle.)

When a circle satisfies this strongest sense of consistency (which is only possible if the circle does not generate the full edo), we say its consistency is super-strong.

Examples

80edo has 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 (assuming 12/11 is mapped to 1\8). 80edo does not have a circle of ~13/10's, even though 8edo is a circle of 13/10's (which requires that 13/10 is mapped to 3\8 consistently by the val chosen), because the accrued error exceeds 0.5\80 = 7.5 ¢. (This does however mean, because 8edo is a circle of ~13/10's, that 80edo has an sub-weak circle of 13/10's. If 8edo had only a weak circle of ~13/10's, this would still be true.)

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 ~35/32's (meaning that 31edo is a (strong) circle of septimal neutral seconds), where 35/32 = (5/4)/(8/7) = 5/4 * 7/4 / 2.

Comparing with telicity

At first glance, it would appear that the concept of telicity and having a circle are identical, however they are not upon closer inspection of their definitions: a circle concerns any rational interval with respect to closing at some equave, while telicity (usually) concerns primes. (The case where telicity does not refer to primes is dealt with in #Vs. subgroup telicity.) This means that "closure" is usually concerning being closed w.r.t. a psychoacoustic equave — by default the octave — while telicity allows closing w.r.t. any prime up to octave-reduction, so is conceptualized differently, because the target at which the circle is closed is no longer a specific equave. In other words, consistent circles concern closure of some rational w.r.t. the equave while telicity concerns reliability of connection between generators.

Another key difference is that being telic is often a more strict requirement than having a consistent circle of some kind; an edo can have a sub-weak circle without qualifying for even 0.5-strong 2-a/b telicity (which would usually not be considered as being telic anyways), or it can having a weak circle without qualifying for 1-strong 2-a/b telicity (again not usually considered as being telic), because these do not require reliability of the full circle, but rather a weaker sense of reliability that is sufficient for many of its practical/musical applications.

Furthermore, #having a super-strong circle is arbitrarily stricter than 2-a/b telicity, which means that in general, it corresponds to s-strong 2-a/b telicity, with s = GCD(N, round(N log₂(a/b))) / 2.

Vs. subgroup telicity

There is an application of telicity that concerns rationals in general — subgroup telicity — which can be seen as roughly equivalent to using one of the generators as an equave and the other as the interval a/b, in which case a consistent circle refers to 1-strong pairwise telicity iff the circle generates all notes w.r.t. equave-reduction, but notice this "iff". Similarly, iff the circle generates all notes as mentioned prior, then qualifying for 2-strong pairwise telicity is equivalent to having a strongly consistent circle of a/b's w.r.t. the equave, and qualifying for 0.5-strong pairwise telicity to having a weakly consistent circle of a/b's w.r.t. the equave. But notice how wordy and technical it sounds in the case where the concepts overlap - hence using the concept of "circle" might be preferred as more easily understood in the case where it's applicable, and needs to be used in inapplicable cases where the definition of telicity is too strict or otherwise too niche.

Consider, for example, that "the only type of telicity available to the 3-prime is 3-2 telicity" - this is in a sense true for consistent circles too, iff you assume your equave is the octave, but otherwise not. In other words, consistent circles refer to less strict types of a/b-2 telicity in the case where the equave is the octave, and to less strict types of a/b-E telicity if E is some alternative equave.