62edo: Difference between revisions
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== Theory == | == Theory == | ||
62edo's [[patent val]] of 62edo is a contorted [[31edo]] through the 11-limit, but it makes for a good tuning in the higher limits. In the 13-limit it tempers out [[169/168]], [[1188/1183]], [[847/845]] and [[676/675]]; in the 17-limit [[221/220]], [[273/272]], and [[289/288]]; in the 19-limit [[153/152]], [[171/170]], [[209/208]], [[286/285]], and [[361/360]]. Unlike 31edo, which has a sharp profile for primes [[13/1|13]], [[17/1|17]], [[19/1|19]] and [[23/1|23]], 62edo has a flat profile for these, as it removes the distinction of otonal and utonal [[superparticular]] pairs of the primes (e.g. 13/12 vs 14/13 for prime 13) by tempering out the corresponding [[square-particular]]s. This flat tendency extends to higher primes too, as the first prime harmonic that is tuned sharper than its [[5/4]] is its [[59/32]]. Interestingly, the relative size differences between consecutive | 62edo's [[patent val]] of 62edo is a contorted [[31edo]] through the 11-limit, but it makes for a good tuning in the higher limits. In the 13-limit it tempers out [[169/168]], [[1188/1183]], [[847/845]] and [[676/675]]; in the 17-limit [[221/220]], [[273/272]], and [[289/288]]; in the 19-limit [[153/152]], [[171/170]], [[209/208]], [[286/285]], and [[361/360]]. Unlike 31edo, which has a sharp profile for primes [[13/1|13]], [[17/1|17]], [[19/1|19]] and [[23/1|23]], 62edo has a flat profile for these, as it removes the distinction of otonal and utonal [[superparticular]] pairs of the primes (e.g. 13/12 vs 14/13 for prime 13) by tempering out the corresponding [[square-particular]]s. This flat tendency extends to higher primes too, as the first prime harmonic that is tuned sharper than its [[5/4]] is its [[59/32]]. Interestingly, the relative size differences between consecutive harmonics are well preserved for all first 24 harmonics, and 62edo is one of the few [[meantone]] edos that achieve this, great for those who seek higher-limit meantone harmony. | ||
It provides the [[optimal patent val]] for [[31st-octave temperaments#Gallium|gallium]], [[Starling temperaments #Valentine|semivalentine]] and [[Meantone family #Hemimeantone|hemimeantone]] temperaments. | It provides the [[optimal patent val]] for [[31st-octave temperaments#Gallium|gallium]], [[Starling temperaments #Valentine|semivalentine]] and [[Meantone family #Hemimeantone|hemimeantone]] temperaments. | ||
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<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||