Diaschismic family: Difference between revisions

Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the 22 & 58 temperament. [[58edo]] or [[80edo]] make for good tunings, or their vals can be added to {{val| 138 219 321 388 }} (138cde).

Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 896/891 or 540/539 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-odd-limit diamond to within about six cents of error, within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note mos much more. Better yet, it is related to three important 11-limit edos: 22edo, a trivial tuning, is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with [[hedgehog]]; [[58edo]] is the smallest tuning that is distinctly consistent in the 11-odd-limit and [[80edo]] is the third smallest distinctly consistent in the 11-odd-limit and arguably about as accurate as echidna can be, although it has a significantly sharp 7/4 (which some prefer).  

In the 11-limit the generator of echidna can be interpreted as 11/10, the period complement of 9/7, as a stack of 11/10 and 9/7 makes [[99/70]] which is extremely close to 600{{cent}} and is equal to it if we temper [[9801/9800|S99]]. Three 11/10's then make a 4/3 (tempering [[4000/3993|S10/S11]] thus making 10/9 and 12/11 equidistant from 11/10), implying a flat 4/3.

A surprisingly natural extension to the 13-limit is possible by observing that since we temper [[176/175]], tempering [[351/350]] and [[352/351]] (which sum to 176/175) is very elegant. This is notable as this mapping of 13 is supported by patent val by the three main echidna EDOs of 80, 58 and 22 (the trivial tuning), of which all except 22 are consistent in the [[17-odd-limit]]; see the 17-limit for more details.

In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with [[srutal archagall]], leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). [[58edo]] and [[80edo]] are both interesting tunings with different advantages and both are consistent in the 17-limit.