3136/3125: Difference between revisions
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Tempering out this comma in the full [[7-limit]] leads to the rank-3 [[hemimean family]] of temperaments, which splits the [[81/80|syntonic comma]] into two equal parts, each representing [[126/125]]~[[225/224]]. Note that if we temper both of those commas individually we get [[septimal meantone]]. | Tempering out this comma in the full [[7-limit]] leads to the rank-3 [[hemimean family]] of temperaments, which splits the [[81/80|syntonic comma]] into two equal parts, each representing [[126/125]]~[[225/224]]. Note that if we temper both of those commas individually we get [[septimal meantone]]. | ||
==== Hemimean orion ==== | ==== [[Hemimean family#Hemimean orion|Hemimean orion]] ==== | ||
As tempering either [[256/255|S16]]/[[324/323|S18]] = [[1216/1215]] or [[324/323|S18]]/[[400/399|S20]] = [[1701/1700]] implies the other in the context of orion with the effect of extending to include prime 3 in the subgroup and as this therefore gives us both S16 = S18 = S20 and S17 = S19, it can be considered natural to add these commas, because {S16/S18, S17/S19, S18/S20} implies all the aforementioned commas of orion. However, this is an extension of hemimean because the ~17/16 generator of orion is no longer present and instead we have a ~3/2 generator. Orion is described next. | As tempering either [[256/255|S16]]/[[324/323|S18]] = [[1216/1215]] or [[324/323|S18]]/[[400/399|S20]] = [[1701/1700]] implies the other in the context of orion with the effect of extending to include prime 3 in the subgroup and as this therefore gives us both S16 = S18 = S20 and S17 = S19, it can be considered natural to add these commas, because {S16/S18, S17/S19, S18/S20} implies all the aforementioned commas of orion. However, this is an extension of hemimean because the ~17/16 generator of orion is no longer present and instead we have a ~3/2 generator. Orion is described next. | ||