3136/3125: Difference between revisions
The mathematical facts aren't in causal relationship |
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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 ==== | ||
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 | 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 a strong extension of hemimean as we have a ~3/2 generator slicing the second generator of orion into five. (The temperament orion is described next on this page.) | ||
See [[Hemimean family #Hemimean orion]]. | |||
=== Orion === | === Orion === | ||
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[[Badness]]: 0.0150 | [[Badness]]: 0.0150 | ||
=== Semiorion === | ==== Semiorion ==== | ||
As [[1445/1444]] = [[289/288|S17]]/[[361/360|S19]] we can extend orion to include prime 3 in its subgroup by tempering both [[289/288|S17]] and [[361/360|S19]]. However, note that (because of tempering [[289/288|S17]]) this splits the period in half, representing a [[17/12]]~[[24/17]] half-octave. This has the consequence that the [[17/16]] generator can be described as a [[3/2]] because [[17/16]] up from [[24/17]] is [[3/2]]. As a result, this equates the generators of hemimean orion and orion up to period-equivalence and is a weak extension of both and neither. | As [[1445/1444]] = [[289/288|S17]]/[[361/360|S19]] we can extend orion to include prime 3 in its subgroup by tempering both [[289/288|S17]] and [[361/360|S19]]. However, note that (because of tempering [[289/288|S17]]) this splits the period in half, representing a [[17/12]]~[[24/17]] half-octave. This has the consequence that the [[17/16]] generator can be described as a [[3/2]] because [[17/16]] up from [[24/17]] is [[3/2]]. As a result, this equates the generators of hemimean orion and orion up to period-equivalence and is a weak extension of both and neither. | ||
See [[Hemimean family #Semiorion]]. | |||
[[Category:Hemimean]] | [[Category:Hemimean]] | ||