Godtone

:In terms of circles of intervals, my current favourite EDO is [[80edo|80 EDO]] which has a lot of amazingly strong circles that close at the unison when octave-reduced, and where some of these generate all of 80 EDO while others generate sub-EDOs of 80 EDO, although that's just one reason I like 80 EDO. The intervals of significance that generate the entirety of 80 EDO - with less than half a step of error left over - are [[11/10]], [[39/38]], [[17/16]] and [[9/7]] (in order of increasing error). (116/115 is a very good and consistent approximation of 1\80, but it accrues a little too much error to be included in that restriction.) Remarkable commas tempered involving these intervals are (9/7)^3/(17/8) and (9/7)/(11/10)^2/(17/16), with 39/38 instead being linked to the 10 EDO subset being a circle of [[16/13]]'s through (39/38)(17/16)^2/(16/13) and providing a high accuracy "skeleton" for the 19-prime-limit.  As you seem to be interested in [[[159edo|159 EDO]], I did notice that it is almost exactly half of that, due to 3\80 being very close in size to 2\53 to the extent that you can use 80 ED8 as an alternative tuning of 53 ED4, with both representing the 2.3.5.13.19 subgroup.<br/>

:I will also mention that [[87edo|87 EDO]] is very related to 80 EDO, but emphasizes accuracy in the 5- and 13-prime-limit as opposed to the 19-prime-limit of 80 EDO (and I'd argue 80 EDO deals generally well with the 29- (or at least 23-)prime-limit for its size), as both are tunings of the [[Tolermic family]] and its extensions up to the 17-prime-limit, and it may be interesting to you too as it has a [[29edo|29 EDO]] circle of fifths, but all primes up to and including 13 are one step flat of the nearest 29 EDO note, creating a very simple and elegant model of connectivity. 87 is (IMO) very recommendable if you want approximations of the 13-limit but still want all of the intervals to be musically meaningful to distinguish in the senses of [[User:Godtone#Colourful_EDOs|colour]] and melody.

--[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 08:45, 22 January 2021 (UTC)

:: I'm afraid your understanding of the concept of telicity is an oversimplification.  While the concept of telicity does in fact include the idea of a "circle of n'ths" where "n" is some interval of interest, incomplete circles are still counted, hence the term "chains", and while the concept of telicity not only involves connectivity between multiple chains- specifically of primes- and the the patent val for an EDO agreeing with the connection, the fact remains that the [[direct mapping]] for every interval in both circles up to the point of connection must also agree with the connection.  Stated more mathematically, where "N" is the number of steps in a given EDO, "r" is the ratio of an interval in one of the two circles, and "M" is the monzo of "r", the equation {N, round(log2(3)*N), round(log2(5)*N), round(log2(7)*N), round(log2(11)*N), ...}.{M} = round(log2(r)*N) ''must'' hold true along ''both'' prime chains up until the point of connection.