Godtone
Joined 17 December 2020
→My Python 3 code: add important derived 23-limit EDOs |
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=== Important 23-limit EDOs === | === Important 23-limit EDOs === | ||
Produced with the help of my above code and corrected with love. Below are theoretically important [[23-limit]] [[EDO]]s in that they do an increasingly better job at representing the [[27-odd-limit]] minus zero or one of composite odds 25 and 27. The exception is that if an EDO (such as 94) achieves less than 2 inconsistencies in the [[25-odd-limit]] or no-25's [[27-odd-limit]] then we don't consider a record to be set to give a chance to larger EDOs that are often interesting for having better approximations (of the intervals mentioned or of other intervals in higher limits). The logical stopping point is [[311edo]] due to its extreme efficiency and elegance as a [[23-limit]] and [[41-limit]] temperament (plus some optional large primes). Please note that there may be one or two important 23-limit EDOs missing here, because I've only checked patent val and occasionally a non-patent-val will perform better for consistency, however, one can argue that a patent val representation should be a requirement for any serious 23-limit EDO. | Produced with the help of my above code and corrected with love. Below are theoretically important [[23-limit]] [[EDO]]s in that they do an increasingly better job at representing the [[27-odd-limit]] minus zero or one of composite odds 25 and 27. The exception is that if an EDO (such as 94) achieves less than 2 inconsistencies in the [[25-odd-limit]] or no-25's [[27-odd-limit]] then we don't consider a record to be set to give a chance to larger EDOs that are often interesting for having better approximations (of the intervals mentioned or of other intervals in higher limits). The logical stopping point is [[311edo]] due to its extreme efficiency and elegance as a [[23-limit]] and [[41-limit]] temperament (plus some optional large primes). Please note that there may be one or two important 23-limit EDOs missing here, because I've only checked patent val and occasionally a non-patent-val will perform better for consistency, however, one can argue that a patent val representation should be a requirement for any serious 23-limit EDO. | ||
It should hopefully go without saying that by "serious 23-limit EDO" I mean in terms of archetypal and structural importance; to me it's obvious, with the exception of maybe EDOs 46 and 53 (corresponding to [[Eros]] (rank 3) and maybe also [[Amity]] (rank 2)), that any 23-limit EDO must have at least 77 notes, to allow for the fitting of a submajor third ([[21/17]] plus various higher-complexity submajor thirds), a superneutral third ([[16/13]]), a subneutral third ([[11/9]]) and a supraminor third (most notably [[17/14]] and [[23/19]]; equated in any "small" system) between 5/4 and 6/5, corresponding to splitting 25/24 into 5 equal parts. If you do this with one part equated to [[81/80]] you get [[absurdity]], explaining its non-absurdity and, to the contrary, ''intuitiveness'', as a [[29-limit]] temperament (given that prime 29 is essentially free for multiples of 7 EDO). If you instead choose to exaggerate 81/80 to 2 parts you get an extension of 5-limit [[artoneutral]] (I specify 5-limit as [[80edo]], one logical tuning for it, uses a different mapping for 7). | |||
[[10edo]] gets 13 wrong in the no-25's 27-odd-limit (and 14 wrong in the 25-odd-limit (noticed the code produced a mistake here)) | [[10edo]] gets 13 wrong in the no-25's 27-odd-limit (and 14 wrong in the 25-odd-limit (noticed the code produced a mistake here)) | ||