52edo: Difference between revisions

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== 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.

52edo has [[26edo]]'s very flat [[meantone]] [[perfect|fifth]] and a very sharp fifth close to 1/2-[[64/63|septimal-comma]] [[superpyth]]. The [[patent val]] has the same mapping for [[3/1|3]], [[7/1|7]], [[11/1|11]] and [[13/1|13]] as 26 does, but its [[5/1|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]]s [[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-limit, 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.

Using the sharp fifth rather than the flat fifth (that is, using the 52b val), it contains a version of [[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.

The 5\52 interval approximates [[31/29]] well, and when used as a generator produces [[tricesimoprimal miracloid]] temperament. The relationship is also preserved exactly in the period-52 [[french deck]] temperament.

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 ==

{| class="wikitable center-all right-2 left-3"

|-

! Degrees

! [[~~Cents|~~Cents]]

! [[Cents]]s

! colspan="3" | [[Ups and Downs Notation]]

|-

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 {{Infobox ET}}
 {{EDO intro|52}}
 == 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.
+edo has [[26edo]]'s very flat [[meantone]] [[perfect|fifth]] and a very sharp fifth close to 1/2-[[64/63|septimal-comma]] [[superpyth]]. The [[patent val]] has the same mapping for [[3/1|3]], [[7/1|7]], [[11/1|11]] and [[13/1|13]] as 26 does, but its [[5/1|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]]s [[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-limit, 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.
+Using the sharp fifth rather than the flat fifth (that is, using the 52b val), it contains a version of [[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.
 The 5\52 interval approximates [[31/29]] well, and when used as a generator produces [[tricesimoprimal miracloid]] temperament. The relationship is also preserved exactly in the period-52 [[french deck]] temperament.
 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}}
+{{Harmonics in equal|52}}
 == Intervals ==
 {| class="wikitable center-all right-2 left-3"
 |-
 ! Degrees
-! [[Cents|Cents]]
+! [[Cents]]s
 ! colspan="3" | [[Ups and Downs Notation]]
 |-