Kleismic family: Difference between revisions

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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 it 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]].

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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 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 it 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]].
+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]].