62edo: Difference between revisions

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== Theory ==

62 = 2 × 31 and the [[patent val]] is a contorted [[31edo]] through the 11-limit; in the 13-limit it tempers out [[169/168]], [[1188/1183]], [[847/845]] and [[676/675]]. It provides the [[optimal patent val]] for [[31 comma temperaments #Gallium|gallium]], [[Starling temperaments #Valentine|semivalentine]] and [[~~Meantone_family~~#Hemimeantone|hemimeantone]] temperaments.

62 = 2 × 31 and the [[patent val]] is a contorted [[31edo]] through the 11-limit; in the 13-limit it tempers out [[169/168]], [[1188/1183]], [[847/845]] and [[676/675]]. It provides the [[optimal patent val]] for [[31 comma temperaments #Gallium|gallium]], [[Starling temperaments #Valentine|semivalentine]] and [[Meantone family #Hemimeantone|hemimeantone]] temperaments.

Using the 35\62 generator, which leads to the {{val| 62 97 143 173 }} val, 62edo is also an excellent tuning for septimal mavila temperament; alternatively {{val| 62 97 143 172 }} [[support]]s hornbostel.

Using the 35\62 generator, which leads to the {{val| 62 97 143 173 }} val, 62edo is also an excellent tuning for septimal [[mavila]] temperament; alternatively {{val| 62 97 143 172 }} [[support]]s [[hornbostel]].

=== ~~Relation to a calendar reform~~ ===

=== Odd harmonics ===

=== Miscellaneous properties ===

62 years is the amount of years in a leap week calendar cycle which corresponds to a year of 365 days 5 hours 48 minutes 23 seconds, meaning it is both a simple cycle for a calendar, and 62 being a multiple of 31 makes it a harmonically useful and playable cycle. The corresponding maximal evenness scales are 15 & 62 and 11 & 62.

11 & 62 ~~is best interpreted~~ in the 2.9.7 subgroup~~, where it~~ tempers out 44957696/43046721, and the three generators of 17\62 correspond to [[16/9]]. It's possible to extend this to the 11-limit with comma basis {896/891, 1331/1296}, where 17\62 is mapped to [[11/9]] and two of them make a [[16/11]]. In addition, three generators make the patent val 9/8, which is also created by combining the flat patent val fifth from 31edo with the sharp 37\62 fifth.

The 11 & 62 temperament in the 2.9.7 subgroup tempers out 44957696/43046721, and the three generators of 17\62 correspond to [[16/9]]. It is possible to extend this to the 11-limit with comma basis {896/891, 1331/1296}, where 17\62 is mapped to [[11/9]] and two of them make [[16/11]]. In addition, three generators make the patent val 9/8, which is also created by combining the flat patent val fifth from 31edo with the sharp 37\62 fifth.

The 15 & 62 temperament, corresponding to the leap day cycle, is ~~just contorted~~ [[valentine]]~~, order 2~~.

The 15 & 62 temperament, corresponding to the leap day cycle, is an unnamed extension to [[valentine]] in the 13-limit.

~~=== Odd harmonics ===~~

~~{{harmonics in equal|62}}~~

== Intervals ==

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| [[Gallium]]

|}

~~[[Category:Equal divisions of the octave|##]] ~~

@@ Line 3: / Line 3: @@
 == Theory ==
-= 2 × 31 and the [[patent val]] is a contorted [[31edo]] through the 11-limit; in the 13-limit it tempers out [[169/168]], [[1188/1183]], [[847/845]] and [[676/675]]. It provides the [[optimal patent val]] for [[31 comma temperaments #Gallium|gallium]], [[Starling temperaments #Valentine|semivalentine]] and [[Meantone_family#Hemimeantone|hemimeantone]] temperaments.
+= 2 × 31 and the [[patent val]] is a contorted [[31edo]] through the 11-limit; in the 13-limit it tempers out [[169/168]], [[1188/1183]], [[847/845]] and [[676/675]]. It provides the [[optimal patent val]] for [[31 comma temperaments #Gallium|gallium]], [[Starling temperaments #Valentine|semivalentine]] and [[Meantone family #Hemimeantone|hemimeantone]] temperaments.
-Using the 35\62 generator, which leads to the {{val| 62 97 143 173 }} val, 62edo is also an excellent tuning for septimal mavila temperament; alternatively {{val| 62 97 143 172 }} [[support]]s hornbostel.
+Using the 35\62 generator, which leads to the {{val| 62 97 143 173 }} val, 62edo is also an excellent tuning for septimal [[mavila]] temperament; alternatively {{val| 62 97 143 172 }} [[support]]s [[hornbostel]].
-=== Relation to a calendar reform ===
+=== Odd harmonics ===
+{{Harmonics in equal|62}}
+=== Miscellaneous properties ===
 years is the amount of years in a leap week calendar cycle which corresponds to a year of 365 days 5 hours 48 minutes 23 seconds, meaning it is both a simple cycle for a calendar, and 62 being a multiple of 31 makes it a harmonically useful and playable cycle. The corresponding maximal evenness scales are 15 & 62 and 11 & 62.
-& 62 is best interpreted in the 2.9.7 subgroup, where it tempers out 44957696/43046721, and the three generators of 17\62 correspond to [[16/9]]. It's possible to extend this to the 11-limit with comma basis {896/891, 1331/1296}, where 17\62 is mapped to [[11/9]] and two of them make a [[16/11]]. In addition, three generators make the patent val 9/8, which is also created by combining the flat patent val fifth from 31edo with the sharp 37\62 fifth.
+The 11 & 62 temperament in the 2.9.7 subgroup tempers out 44957696/43046721, and the three generators of 17\62 correspond to [[16/9]]. It is possible to extend this to the 11-limit with comma basis {896/891, 1331/1296}, where 17\62 is mapped to [[11/9]] and two of them make [[16/11]]. In addition, three generators make the patent val 9/8, which is also created by combining the flat patent val fifth from 31edo with the sharp 37\62 fifth.
-The 15 & 62 temperament, corresponding to the leap day cycle, is just contorted [[valentine]], order 2.
+The 15 & 62 temperament, corresponding to the leap day cycle, is an unnamed extension to [[valentine]] in the 13-limit.
-=== Odd harmonics ===
-{{harmonics in equal|62}}
 == Intervals ==
@@ Line 390: / Line 390: @@
 | [[Gallium]]
 |}
-[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->