Luna and hemithirds: Difference between revisions

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{{Infobox Regtemp

| Title = Hemithirds

| Subgroups = 2.3.5.7

| Comma basis = [[1029/1024]], [[3136/3125]] (2.3.5.7); <br> [[176/175]], [[1375/1372]] (2.5.7.11)

| Edo join 1 = 25 | Edo join 2 = 31

| Generator = 28/25 | Generator tuning = 193.239 | Optimization method = CWE

| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]], [[25L 6s]]

| Mapping = 1; -15 2 5

| Odd limit 1 = 7 | Mistuning 1 = ??? | Complexity 1 = ???

| Odd limit 2 = 11 | Mistuning 2 = ??? | Complexity 2 = ???

}}

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 [[1001/1000]] to interpret the generator as ~[[143/128]] and find ~[[13/8]] at 23 generators up.

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 {{URWTC}}
+{{Infobox Regtemp
+| Title = Hemithirds
+| Subgroups = 2.3.5.7
+| Comma basis = [[1029/1024]], [[3136/3125]] (2.3.5.7); <br> [[176/175]], [[1375/1372]] (2.5.7.11)
+| Edo join 1 = 25 | Edo join 2 = 31
+| Generator = 28/25 | Generator tuning = 193.239 | Optimization method = CWE
+| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]], [[25L 6s]]
+| Mapping = 1; -15 2 5
+| Odd limit 1 = 7 | Mistuning 1 = ??? | Complexity 1 = ???
+| Odd limit 2 = 11 | Mistuning 2 = ??? | Complexity 2 = ???
+}}
 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 [[1001/1000]] to interpret the generator as ~[[143/128]] and find ~[[13/8]] at 23 generators up.