Schismic–commatic equivalence continuum: Difference between revisions

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* [[Quindromeda]] ({{nowrap|''n'' {{=}} 5}}) does not split the octave but splits the fourth in five, as 5 is coprime with 12.

Alternatively, because the [[duodene|5-limit otonal detemper]] of 12edo is a 4x3 rectangle (known as the [[duodene]]), we may be interested in expressing the continuum in terms of the boundary commas of this detemper, that is, as ([[81/80]])<sup>''k''</sup> ~ ([[128/125]]). This corresponds to these commas' structural significance via 128/125 being entirely in the 2.5 subgroup while 81/80 explains 5 in the simplest way relative to the 3-limit. This choice of coordinates has the advantage of finding all temperaments discussed in a relatively intuitive way and is noted for its nontrivial relation to the other better-motivated coordinates discussed; specifically, it's related to the inverted continuum by a translation followed by a flip. Its JIP is at 1.90915584... which is approximated closely by the microtemperament atomic at 21/11 = 1.90909... so that the main continuum can be seen as taking successive mediants towards 2/1 (schismic) starting from 1/1 (diaschismic).

Alternatively, because the [[duodene|5-limit otonal detemper]] of 12edo is a 4x3 rectangle (known as the [[duodene]]), we may be interested in expressing the continuum in terms of the boundary commas of this detemper, that is, as {{nowrap| ([[81/80]])<sup>''k''</sup> ~ ([[128/125]]) }}. This corresponds to these commas' structural significance via 128/125 being entirely in the 2.5 subgroup while 81/80 explains 5 in the simplest way relative to the 3-limit. This choice of coordinates has the advantage of finding all temperaments discussed in a relatively intuitive way and is noted for its nontrivial relation to the other better-motivated coordinates discussed; specifically, it's related to the inverted continuum by a translation followed by a flip. Its JIP is at 1.90915584... which is approximated closely by the microtemperament atomic at 21/11 = 1.90909... so that the main continuum can be seen as taking successive mediants towards 2/1 (schismic) starting from 1/1 (diaschismic).

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 * [[Quindromeda]]  ({{nowrap|''n'' {{=}} 5}}) does not split the octave but splits the fourth in five, as 5 is coprime with 12.
-Alternatively, because the [[duodene|5-limit otonal detemper]] of 12edo is a 4x3 rectangle (known as the [[duodene]]), we may be interested in expressing the continuum in terms of the boundary commas of this detemper, that is, as ([[81/80]])<sup>''k''</sup> ~ ([[128/125]]). This corresponds to these commas' structural significance via 128/125 being entirely in the 2.5 subgroup while 81/80 explains 5 in the simplest way relative to the 3-limit. This choice of coordinates has the advantage of finding all temperaments discussed in a relatively intuitive way and is noted for its nontrivial relation to the other better-motivated coordinates discussed; specifically, it's related to the inverted continuum by a translation followed by a flip. Its JIP is at 1.90915584... which is approximated closely by the microtemperament atomic at 21/11 = 1.90909... so that the main continuum can be seen as taking successive mediants towards 2/1 (schismic) starting from 1/1 (diaschismic).
+Alternatively, because the [[duodene|5-limit otonal detemper]] of 12edo is a 4x3 rectangle (known as the [[duodene]]), we may be interested in expressing the continuum in terms of the boundary commas of this detemper, that is, as {{nowrap| ([[81/80]])<sup>''k''</sup> ~ ([[128/125]]) }}. This corresponds to these commas' structural significance via 128/125 being entirely in the 2.5 subgroup while 81/80 explains 5 in the simplest way relative to the 3-limit. This choice of coordinates has the advantage of finding all temperaments discussed in a relatively intuitive way and is noted for its nontrivial relation to the other better-motivated coordinates discussed; specifically, it's related to the inverted continuum by a translation followed by a flip. Its JIP is at 1.90915584... which is approximated closely by the microtemperament atomic at 21/11 = 1.90909... so that the main continuum can be seen as taking successive mediants towards 2/1 (schismic) starting from 1/1 (diaschismic).
 {| class="wikitable center-1 center-2"