Consistent circle: Difference between revisions
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Intuitively, a [[circle]] of ~a/b's | 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 [[consistency#mathematical definition|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}} in writing for the sake of brevity wherever it's unambiguous (or in informal writing). | ||
== Motivation == | == 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 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 complement]]s 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 [[#Weak circle|'' | 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 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 complement]]s 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 [[#Weak circle|''weakly consistent circles'']]), even if the result isn't guaranteed to be [[consistent]] beyond floor(31/2) 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 == | == Definitions == | ||
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and is a ''weak circle'' if ''c'' < 1 instead, or a ''strong circle'' if ''c'' < 1/4. | and is a ''weak circle'' if ''c'' < 1 instead, or a ''strong circle'' if ''c'' < 1/4. | ||
=== | === Weakly consistent circle === | ||
A | 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. | 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 a [[having a sub-weak circle]]. | ||
A | |||
=== 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. | 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. | 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 [[Having a super-strong circle|if]] it generates the whole edo), we say its consistency is ''strong''. | |||
=== Is vs. has === | === Is vs. has === | ||
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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''. | 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- | === Having a sub-weak circle === | ||
A "'''sub'''-weak circle of ~a/b's" in ''N''-edo describes a case where the '''sub'''set edo generated by a/b qualifies as a weak circle of a/b's. | A "'''sub'''-weak circle of ~a/b's" in ''N''-edo describes a case where the '''sub'''set edo generated by a/b qualifies as a weak circle of a/b's. | ||