52edo: Difference between revisions

Line 1:

~~'''52edo''' divides the [[octave]] into~~ 52 ~~equal parts of about 23.1 [[cent]]s each. It has 26edo's very flat meantone fifth and a very sharp fifth close to 1/2 [[64/63|septimal comma]] superpyth.~~

== Theory ==

52edo has 26edo's very flat meantone fifth and a very sharp fifth close to 1/2 [[64/63|septimal comma]] superpyth. The patent val has the same mapping for 3, 7, 11 and 13 as 26 does, but its 5 is sharp rather than flat. From this it tempers out [[648/625]] rather than [[81/80]] in the 5-limit, and [[225/224]] and [[1029/1024]] in the 7-limit, showing it [[support|supports]] miracle, albeit badly, and may be defined by the tempering out of both 648/625 and miracle. In the 11-limit it tempers out [[99/98]] and [[176/175]] and in the 13-limit [[78/77]], [[144/143]] and [[169/168]]. It supplies the optimal patent val for then 12&40 temperament of the diminished family in the 7- and 11-limits, and also in the 13-limit where it can be defined as tempering out 78/77, 99/98, 176/175, 567/550 rather than by two patent vals. It also gives the 13-limit patent val for the 21&52 variant of miracle.

~~{{primes in edo~~|~~52}}~~

~~The 52 equal division divides the octave into 52 equal parts of 23.077 cents each~~. The patent val has the same mapping for 3, 7, 11 and 13 as 26 does, but its 5 is sharp rather than flat. From this it tempers out [[648/625]] rather than [[81/80]] in the 5-limit, and [[225/224]] and [[1029/1024]] in the 7-limit, showing it [[support|supports]] miracle, albeit badly, and may be defined by the tempering out of both 648/625 and miracle. In the 11-limit it tempers out [[99/98]] and [[176/175]] and in the 13-limit [[78/77]], [[144/143]] and [[169/168]]. It supplies the optimal patent val for then 12&40 temperament of the diminished family in the 7- and 11-limits, and also in the 13-limit where it can be defined as tempering out 78/77, 99/98, 176/175, 567/550 rather than by two patent vals. It also gives the 13-limit patent val for the 21&52 variant of miracle.

Using the sharp fifth rather than the flat fifth (that is, using the 52b val), it contains a version of [[Porcupine|porcupine]] temperament, and combining 30\52 with 31\52 leads to a whole tone of 9\52, or 208 cents, which can be used inconsistently.

Line 13:

Line 9:

The 11\52 (253.846¢) [[semifourth]] is a very accurate [[22/19]], with an error of only +0.041¢ and a closing error of only 9.3%.

=== Odd harmonics ===

== Intervals ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
+{{EDO intro|52}}
-'''52edo''' divides the [[octave]] into 52 equal parts of about 23.1 [[cent]]s each. It has 26edo's very flat meantone fifth and a very sharp fifth close to 1/2 [[64/63|septimal comma]] superpyth.
 == Theory ==
+edo has 26edo's very flat meantone fifth and a very sharp fifth close to 1/2 [[64/63|septimal comma]] superpyth. The patent val has the same mapping for 3, 7, 11 and 13 as 26 does, but its 5 is sharp rather than flat. From this it tempers out [[648/625]] rather than [[81/80]] in the 5-limit, and [[225/224]] and [[1029/1024]] in the 7-limit, showing it [[support|supports]] miracle, albeit badly, and may be defined by the tempering out of both 648/625 and miracle. In the 11-limit it tempers out [[99/98]] and [[176/175]] and in the 13-limit [[78/77]], [[144/143]] and [[169/168]]. It supplies the optimal patent val for then 12&amp;40 temperament of the diminished family in the 7- and 11-limits, and also in the 13-limit where it can be defined as tempering out 78/77, 99/98, 176/175, 567/550 rather than by two patent vals. It also gives the 13-limit patent val for the 21&amp;52 variant of miracle.
-{{primes in edo|52}}
-The 52 equal division divides the octave into 52 equal parts of 23.077 cents each. The patent val has the same mapping for 3, 7, 11 and 13 as 26 does, but its 5 is sharp rather than flat. From this it tempers out [[648/625]] rather than [[81/80]] in the 5-limit, and [[225/224]] and [[1029/1024]] in the 7-limit, showing it [[support|supports]] miracle, albeit badly, and may be defined by the tempering out of both 648/625 and miracle. In the 11-limit it tempers out [[99/98]] and [[176/175]] and in the 13-limit [[78/77]], [[144/143]] and [[169/168]]. It supplies the optimal patent val for then 12&amp;40 temperament of the diminished family in the 7- and 11-limits, and also in the 13-limit where it can be defined as tempering out 78/77, 99/98, 176/175, 567/550 rather than by two patent vals. It also gives the 13-limit patent val for the 21&amp;52 variant of miracle.
 Using the sharp fifth rather than the flat fifth (that is, using the 52b val), it contains a version of [[Porcupine|porcupine]] temperament, and combining 30\52 with 31\52 leads to a whole tone of 9\52, or 208 cents, which can be used inconsistently.
@@ Line 13: / Line 9: @@
 The 11\52 (253.846¢) [[semifourth]] is a very accurate [[22/19]], with an error of only +0.041¢ and a closing error of only 9.3%.
+=== Odd harmonics ===
+{{harmonics in equal|52}}
 == Intervals ==