Luna and hemithirds: Difference between revisions

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The [[7-limit]] '''hemithirds''' temperament functions as a strong extension of [[didacus]], the 2.5.7 subgroup temperament, ~~in the range between [[25edo]] and [[31edo]] tuning,~~ defined by tempering out [[3136/3125]] such that two of its generators (hemithird, [[~]][[28/25]], around 193.2 [[cent]]s) reach ~[[5/4]], three reach ~[[7/5]], and therefore five reach ~[[7/4]]. Hemithirds extends didacus by tempering out [[1029/1024]], such that three intervals of ~[[8/7]] reach ~[[3/2]], therefore finding ~[[4/3]] after fifteen generators in total. The canonical extension to the [[13-limit]] tempers out [[385/384]] and [[441/440]] to reach ~[[55/32]] at four ~8/7s and therefore ~[[11/8]] at 22 generators down, and then [[196/195]] (along with [[352/351]], [[625/624]], and [[1001/1000]]) to interpret the generator as ~[[143/128]] and find ~[[13/8]] at 23 generators up.

The [[7-limit]] '''hemithirds''' temperament functions as a strong extension of [[didacus]], the 2.5.7 subgroup temperament, which is defined by tempering out [[3136/3125]] such that two of its generators (hemithird, [[~]][[28/25]], around 193.2 [[cent]]s) reach ~[[5/4]], three reach ~[[7/5]], and therefore five reach ~[[7/4]]. Hemithirds extends didacus in the range between [[25edo]] and [[31edo]] tuning, by tempering out [[1029/1024]], such that three intervals of ~[[8/7]] reach ~[[3/2]], therefore finding ~[[4/3]] after fifteen generators in total. The canonical extension to the [[13-limit]] tempers out [[385/384]] and [[441/440]] to reach ~[[55/32]] at four ~8/7s and therefore ~[[11/8]] at 22 generators down, and then [[196/195]] (along with [[352/351]], [[625/624]], and [[1001/1000]]) to interpret the generator as ~[[143/128]] and find ~[[13/8]] at 23 generators up.

'''Luna''' is a restriction of hemithirds to the [[5-limit]] that is a [[microtemperament]], supported by such high-precision tuning systems as [[323edo]] and [[441edo]]; another notable tuning of luna is [[1000edo]]. It can further be re-extended to the 7-limit in the form of [[lunatic]] by adding [[4375/4374]] to the comma list, but that extension is extremely complex (finding the 7th harmonic at 113 generators down).

See [[Hemimean clan #Hemithirds]] and [[Luna family #Luna]] for more information.

== Intervals ==

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-The [[7-limit]] '''hemithirds''' temperament functions as a strong extension of [[didacus]], the 2.5.7 subgroup temperament, in the range between [[25edo]] and [[31edo]] tuning, defined by tempering out [[3136/3125]] such that two of its generators (hemithird, [[~]][[28/25]], around 193.2 [[cent]]s) reach ~[[5/4]], three reach ~[[7/5]], and therefore five reach ~[[7/4]]. Hemithirds extends didacus by tempering out [[1029/1024]], such that three intervals of ~[[8/7]] reach ~[[3/2]], therefore finding ~[[4/3]] after fifteen generators in total. The canonical extension to the [[13-limit]] tempers out [[385/384]] and [[441/440]] to reach ~[[55/32]] at four ~8/7s and therefore ~[[11/8]] at 22 generators down, and then [[196/195]] (along with [[352/351]], [[625/624]], and [[1001/1000]]) to interpret the generator as ~[[143/128]] and find ~[[13/8]] at 23 generators up.
+The [[7-limit]] '''hemithirds''' temperament functions as a strong extension of [[didacus]], the 2.5.7 subgroup temperament, which is defined by tempering out [[3136/3125]] such that two of its generators (hemithird, [[~]][[28/25]], around 193.2 [[cent]]s) reach ~[[5/4]], three reach ~[[7/5]], and therefore five reach ~[[7/4]]. Hemithirds extends didacus in the range between [[25edo]] and [[31edo]] tuning, by tempering out [[1029/1024]], such that three intervals of ~[[8/7]] reach ~[[3/2]], therefore finding ~[[4/3]] after fifteen generators in total. The canonical extension to the [[13-limit]] tempers out [[385/384]] and [[441/440]] to reach ~[[55/32]] at four ~8/7s and therefore ~[[11/8]] at 22 generators down, and then [[196/195]] (along with [[352/351]], [[625/624]], and [[1001/1000]]) to interpret the generator as ~[[143/128]] and find ~[[13/8]] at 23 generators up.
 '''Luna''' is a restriction of hemithirds to the [[5-limit]] that is a [[microtemperament]], supported by such high-precision tuning systems as [[323edo]] and [[441edo]]; another notable tuning of luna is [[1000edo]]. It can further be re-extended to the 7-limit in the form of [[lunatic]] by adding [[4375/4374]] to the comma list, but that extension is extremely complex (finding the 7th harmonic at 113 generators down).
 See [[Hemimean clan #Hemithirds]] and [[Luna family #Luna]] for more information.
 == Intervals ==