Schismic–Pythagorean 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. | * [[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 {{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 | 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 and simple way so that less accurate but structurally simpler temperaments are found at integer points while microtemperaments converging to atomic are found as successive mediants towards the JIP. Specifically, its JIP is at 1.90915584... which is approximated very 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). It is noted for its nontrivial relation to the other better-motivated (in terms of mapping) coordinates discussed. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||