Squares: Difference between revisions

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At its most basic level, '''~~skwares~~''' can be thought of as a [[2.3.7 subgroup]] temperament, generated by a flat [[~]][[9/7]] such that four of them stack to the perfect eleventh, [[8/3]], therefore tempering out the comma [[19683/19208]]. However, it is more natural to think of the temperament first as [[2.3.7.11 subgroup]], tempering out [[99/98]] so as to identify the generator with [[14/11]] in addition to 9/7 and so that two generators stack to the undecimal neutral sixth, [[18/11]], two of which are then identified with 8/3 due to tempering out [[243/242]]. This can also be thought of as an octavization of the [[3/1|tritave]]-equivalent 3.7.11 [[Mintaka]] temperament by identifying 2/1 with a false octave corresponding to {{nowrap|243/121 ~ 99/49}}, in a manner similar to [[sensi]]'s relation to [[BPS]].

At its most basic level, '''squares''' can be thought of as a [[2.3.7 subgroup]] temperament (sometimes called ''skwares''), generated by a flat [[~]][[9/7]] such that four of them stack to the perfect eleventh, [[8/3]], therefore tempering out the comma [[19683/19208]]. However, it is more natural to think of the temperament first as [[2.3.7.11 subgroup]], tempering out [[99/98]] so as to identify the generator with [[14/11]] in addition to 9/7 and so that two generators stack to the undecimal neutral sixth, [[18/11]], two of which are then identified with 8/3 due to tempering out [[243/242]]. This can also be thought of as an octavization of the [[3/1|tritave]]-equivalent 3.7.11 [[Mintaka]] temperament by identifying 2/1 with a false octave corresponding to {{nowrap|243/121 ~ 99/49}}, in a manner similar to [[sensi]]'s relation to [[BPS]].

However, since the fifth in skwares is tuned flat, it is very natural to combine the temperament with [[meantone]] to create full [[11-limit]] '''squares''', which additionally can be restricted to the [[7-limit]] as the temperament with comma basis [[81/80]] and [[2401/2400]]. The 11-limit temperament is considered below.

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	At its most basic level, '''~~skwares~~''' can be thought of as a [[2.3.7 subgroup]] temperament, generated by a flat [[~]][[9/7]] such that four of them stack to the perfect eleventh, [[8/3]], therefore tempering out the comma [[19683/19208]]. However, it is more natural to think of the temperament first as [[2.3.7.11 subgroup]], tempering out [[99/98]] so as to identify the generator with [[14/11]] in addition to 9/7 and so that two generators stack to the undecimal neutral sixth, [[18/11]], two of which are then identified with 8/3 due to tempering out [[243/242]]. This can also be thought of as an octavization of the [[3/1\|tritave]]-equivalent 3.7.11 [[Mintaka]] temperament by identifying 2/1 with a false octave corresponding to {{nowrap\|243/121 ~ 99/49}}, in a manner similar to [[sensi]]'s relation to [[BPS]].		At its most basic level, '''squares''' can be thought of as a [[2.3.7 subgroup]] temperament (sometimes called ''skwares''), generated by a flat [[~]][[9/7]] such that four of them stack to the perfect eleventh, [[8/3]], therefore tempering out the comma [[19683/19208]]. However, it is more natural to think of the temperament first as [[2.3.7.11 subgroup]], tempering out [[99/98]] so as to identify the generator with [[14/11]] in addition to 9/7 and so that two generators stack to the undecimal neutral sixth, [[18/11]], two of which are then identified with 8/3 due to tempering out [[243/242]]. This can also be thought of as an octavization of the [[3/1\|tritave]]-equivalent 3.7.11 [[Mintaka]] temperament by identifying 2/1 with a false octave corresponding to {{nowrap\|243/121 ~ 99/49}}, in a manner similar to [[sensi]]'s relation to [[BPS]].

	However, since the fifth in skwares is tuned flat, it is very natural to combine the temperament with [[meantone]] to create full [[11-limit]] '''squares''', which additionally can be restricted to the [[7-limit]] as the temperament with comma basis [[81/80]] and [[2401/2400]]. The 11-limit temperament is considered below.		However, since the fifth in skwares is tuned flat, it is very natural to combine the temperament with [[meantone]] to create full [[11-limit]] '''squares''', which additionally can be restricted to the [[7-limit]] as the temperament with comma basis [[81/80]] and [[2401/2400]]. The 11-limit temperament is considered below.