Squares: Difference between revisions

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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]].

At its most basic level, '''squares''' can be thought of as a [[2.3.7 subgroup|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 3.7.11-subgroup [[mintaka]] temperament by identifying [[2/1]] with a false octave corresponding to 99/49~243/121, 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]]. This 11-limit temperament is considered below.

There is also a natural extension adding prime 23 by equating the generator to [[23/18]], and so finding 23 itself seven generators down, tempering out [[162/161]].

There is also a natural extension adding [[prime interval|prime]] [[23/1|23]] by equating the generator to [[23/18]], and so finding 23 itself seven generators down, tempering out [[162/161]].

As for prime 13, the way to map it is less clear. The canonical squares mapping tempers out [[144/143]] in order to equate the tridecimal neutral sixth, [[13/8]], with 18/11, finding 13 two generators up, while '''agora''' tempers out [[105/104]] to equate [[8/7]] with [[15/13]], finding the 13th harmonic 29 generators down. These two mappings are enharmonically equivalent in [[31edo]]. Finally, '''squad''' tempers out [[351/343]] (which is the same as 3.7.11.13 [[~~Mintaka#Extensions of Mintaka|Minalzidar~~]]'s tempering of that prime) so that 13 is equated with (7/3)<sup>3</sup>, and found 15 generators down.

As for prime [[13/1|13]], the way to map it is less clear. The canonical squares mapping tempers out [[144/143]] in order to equate the tridecimal neutral sixth, [[13/8]], with 18/11, finding 13 two generators up, while '''agora''' tempers out [[105/104]] to equate [[8/7]] with [[15/13]], finding the 13th harmonic 29 generators down. These two mappings are enharmonically equivalent in [[31edo]]. Finally, '''squad''' tempers out [[351/343]] (which is the same as 3.7.11.13 [[minalzidar]]'s tempering of that prime) so that 13 is equated with (7/3)<sup>3</sup>, and found 15 generators down.

See [[Meantone family #Squares]] and [[No-fives subgroup temperaments #Skwares]] for more technical data.

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{| class="wikitable center-1 right-2"

|-

! rowspan="3" | &#~~35;~~

! rowspan="3" | #

! rowspan="3" | Cents*

! colspan="4" | Approximate ~~Ratios~~

! colspan="4" | Approximate ratios

|-

! rowspan="2" | 11-limit

! colspan="3" | 13-limit ~~Extension~~

! colspan="3" | 13-limit extensions

|-

! Squares

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-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]].
+At its most basic level, '''squares''' can be thought of as a [[2.3.7 subgroup|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 3.7.11-subgroup [[mintaka]] temperament by identifying [[2/1]] with a false octave corresponding to 99/49~243/121, 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]]. This 11-limit temperament is considered below.
-There is also a natural extension adding prime 23 by equating the generator to [[23/18]], and so finding 23 itself seven generators down, tempering out [[162/161]].
+There is also a natural extension adding [[prime interval|prime]] [[23/1|23]] by equating the generator to [[23/18]], and so finding 23 itself seven generators down, tempering out [[162/161]].
-As for prime 13, the way to map it is less clear. The canonical squares mapping tempers out [[144/143]] in order to equate the tridecimal neutral sixth, [[13/8]], with 18/11, finding 13 two generators up, while '''agora''' tempers out [[105/104]] to equate [[8/7]] with [[15/13]], finding the 13th harmonic 29 generators down. These two mappings are enharmonically equivalent in [[31edo]]. Finally, '''squad''' tempers out [[351/343]] (which is the same as 3.7.11.13 [[Mintaka#Extensions of Mintaka|Minalzidar]]'s tempering of that prime) so that 13 is equated with (7/3)<sup>3</sup>, and found 15 generators down.
+As for prime [[13/1|13]], the way to map it is less clear. The canonical squares mapping tempers out [[144/143]] in order to equate the tridecimal neutral sixth, [[13/8]], with 18/11, finding 13 two generators up, while '''agora''' tempers out [[105/104]] to equate [[8/7]] with [[15/13]], finding the 13th harmonic 29 generators down. These two mappings are enharmonically equivalent in [[31edo]]. Finally, '''squad''' tempers out [[351/343]] (which is the same as 3.7.11.13 [[minalzidar]]'s tempering of that prime) so that 13 is equated with (7/3)<sup>3</sup>, and found 15 generators down.
 See [[Meantone family #Squares]] and [[No-fives subgroup temperaments #Skwares]] for more technical data.
@@ Line 14: / Line 14: @@
 {| class="wikitable center-1 right-2"
 |-
-! rowspan="3" | &#35;
+! rowspan="3" | #
 ! rowspan="3" | Cents*
-! colspan="4" | Approximate Ratios
+! colspan="4" | Approximate ratios
 |-
 ! rowspan="2" | 11-limit
-! colspan="3" | 13-limit Extension
+! colspan="3" | 13-limit extensions
 |-
 ! Squares