Kleismic family: Difference between revisions

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=== 2.3.5.13 subgroup (cata) ===

The ~~kleisma has~~ the [[S-expression]] ~~S252<~~/~~sup>⋅S26~~, ~~which explains how the tone is trisected, and thus tempering out its constituents~~ [[625/624]] ~~({{S|25}}~~) ~~and~~ [[676/675]] ({{S|26}}~~) is a readily natural subgroup extension. As a result of coincidental S-expression equivalences,~~ [[~~325~~/~~324~~]] (~~S25⋅S26) is also tempered out, and two generator steps stand in for ~~~[[13/9]]. ~~Finally~~, ~~since this step~~ is ~~one third of a perfect twelfth,~~ the ~~comma~~ [[~~2197~~/~~2187~~]] ~~(S25⋅S26~~2~~) is tempered~~ out~~. The third~~ tone is ~~now~~ 25~~/24~~~26~~/25~~~27~~/26~~, ~~which means~~ 27/25 ~~and~~ [[13/12]] ~~are made equal~~.

The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap| {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]],) [[676/675|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] {{nowrap| S13/S15 {{=}} ([[4/3|16/12]])/([[15/13]])2. }} From here, we notice that {{nowrap| (26/15)/(13/9) {{=}} 6/5,}} so all that remains is dividing 13/9 into two 6/5's via the semiparticular {{nowrap| S10/S12 {{=}} (13/9)/([[6/5|12/10]])2, }} hence explaining the mapping of the entire 2.3.5.13 subgroup, and dividing 3/1 into three 13/9's via [[2197/2187]]. (Note that relatedly, because of tempering out S25 and S26, the tone is trisected via {{nowrap| 24:25:26:27, }} also implying {{nowrap| [[27/25]][[~]][[13/12]].) }}

The accuracy of cata as providing a slightly flat 5/4 in ideal tunings lends a possible (but complex) extension for prime 41 via [[32/25]][[~]][[41/32]], tempering out [[1025/1024]].

For a version of cata with a 1\5 period, see [[Thunderclysmic]].

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Comma list: 325/324, 625/624

~~Subgroup-val mapping:~~ {{~~mapping~~| 1 0 1 0 | 0 6 5 14 }}

Optimal tunings:

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Comma list: 169/168, 225/224, 325/324

~~Subgroup-val mapping:~~ {{~~mapping~~| 1 0 1 -3 0 | 0 6 5 22 14 }}

Optimal tunings:

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 === 2.3.5.13 subgroup (cata) ===
-The kleisma has the [[S-expression]] S25<sup>2</sup>⋅S26, which explains how the tone is trisected, and thus tempering out its constituents [[625/624]] ({{S|25}}) and [[676/675]] ({{S|26}}) is a readily natural subgroup extension. As a result of coincidental S-expression equivalences, [[325/324]] (S25⋅S26) is also tempered out, and two generator steps stand in for ~[[13/9]]. Finally, since this step is one third of a perfect twelfth, the comma [[2197/2187]] (S25⋅S26<sup>2</sup>) is tempered out. The third tone is now 25/24~26/25~27/26, which means 27/25 and [[13/12]] are made equal.
+The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap| {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]],) [[676/675|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] {{nowrap| S13/S15 {{=}} ([[4/3|16/12]])/([[15/13]])<sup>2</sup>. }} From here, we notice that {{nowrap| (26/15)/(13/9) {{=}} 6/5,}} so all that remains is dividing 13/9 into two 6/5's via the semiparticular {{nowrap| S10/S12 {{=}} (13/9)/([[6/5|12/10]])<sup>2</sup>, }} hence explaining the mapping of the entire 2.3.5.13 subgroup, and dividing 3/1 into three 13/9's via [[2197/2187]]. (Note that relatedly, because of tempering out S25 and S26, the tone is trisected via {{nowrap| 24:25:26:27, }} also implying {{nowrap| [[27/25]][[~]][[13/12]].) }}
+The accuracy of cata as providing a slightly flat 5/4 in ideal tunings lends a possible (but complex) extension for prime 41 via [[32/25]][[~]][[41/32]], tempering out [[1025/1024]].
 For a version of cata with a 1\5 period, see [[Thunderclysmic]].
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 Comma list: 325/324, 625/624
-Subgroup-val mapping: {{mapping| 1 0 1 0 | 0 6 5 14 }}
+{{Mapping|legend=2| 1 0 1 0 | 0 6 5 14 }}
 Optimal tunings:
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 Comma list: 169/168, 225/224, 325/324
-Subgroup-val mapping: {{mapping| 1 0 1 -3 0 | 0 6 5 22 14 }}
+{{Mapping|legend=2| 1 0 1 -3 0 | 0 6 5 22 14 }}
 Optimal tunings: