Consistent circle: Difference between revisions

Line 2:

== 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|''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.)

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 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) = 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 ==

@@ Line 2: / Line 2: @@
 == 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|''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.)
+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 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) = 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 ==