62edo: Difference between revisions

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== Theory ==

62 = 2 × 31 and the [[patent val]] 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. Interestingly, the relative size differences of 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.

62 = 2 × 31 and the [[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. Interestingly, the relative size differences of 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.

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=== Odd harmonics ===

=== Subsets and supersets ===

Since 62 factors into {{factorization|62}}, 62edo does not contain nontrivial subset edos other than [[2edo]] and 31edo. [[186edo]] and [[248edo]] are notable supersets.

=== Miscellaneous properties ===

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 == Theory ==
-= 2 × 31 and the [[patent val]] 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. Interestingly, the relative size differences of 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.
+= 2 × 31 and the [[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. Interestingly, the relative size differences of 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.
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 === Odd harmonics ===
 {{Harmonics in equal|62}}
+=== Subsets and supersets ===
+Since 62 factors into {{factorization|62}}, 62edo does not contain nontrivial subset edos other than [[2edo]] and 31edo. [[186edo]] and [[248edo]] are notable supersets.
 === Miscellaneous properties ===