User:Godtone/diaschismic-tetracot equivalence continuum: Difference between revisions
mNo edit summary |
m just value of m & exact forms for just values |
||
| Line 1: | Line 1: | ||
The '''diaschismic-tetracot equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] that describes the set of all [[5-limit]] temperaments supported by [[34edo]]. It is equivalent to the ''diaschismic-gothic equivalence continuum'', where the [[gothic comma]] is found as ([[2048/2025]])<sup>2</sup> * [[20000/19683]], at ''n'' = -2. | The '''diaschismic-tetracot equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] that describes the set of all [[5-limit]] temperaments supported by [[34edo]]. It is equivalent to the ''diaschismic-gothic equivalence continuum'', where the [[gothic comma]] is found as ([[2048/2025]])<sup>2</sup> * [[20000/19683]], at ''n'' = -2. | ||
Each ''n'' on the continuum is defined by equating ([[2048/2025]])<sup>''n''</sup> with [[20000/19683]]. The just value of ''n'' is 1.41464…, and temperaments near this tend to be the most accurate. However, due to this continuum being defined through two reasonably-accurate temperaments and due to the strength of 34edo as a 5-limit temperament (supporting many notable tempered equivalences), simple fractional values of ''n'' in the general proximity of the just value are also often notable. | Each ''n'' on the continuum is defined by equating ([[2048/2025]])<sup>''n''</sup> with [[20000/19683]]. The just value of ''n'' is 1.41464… = log<sub>2</sub>([[20000/19683]])/log<sub>2</sub>([[2048/2025]]), and temperaments near this tend to be the most accurate. However, due to this continuum being defined through two reasonably-accurate temperaments and due to the strength of 34edo as a 5-limit temperament (supporting many notable tempered equivalences), simple fractional values of ''n'' in the general proximity of the just value are also often notable. | ||
: A reasonable way of defining this continuum equates a number of [[2048/2025|diaschismas (2048/2025)]] with the [[393216/390625|Würschmidt comma (393216/390625)]], so that (2048/2025)<sup>''n''</sup> ~ 393216/390625. As a result, this may also be called the ''wurschmidt-diaschismic equivalence continuum'', or the ''diaschismic-gothic equivalence continuum'', which is more or less the same thing. The just value of ''n'' is 0.5853…, and temperaments near this tend to be the most accurate. The [[17-comma|gothic comma]] (134217728/129140163) is the characteristic [[3-limit]] comma tempered out in 34edo, and it has a value of ''n'' = 4. Therefore, one can additionally define ''k'' = 4 - ''n'', which has notable advantages - in particular, due to being determined in terms of the 3-limit comma and the comma with the next lowest power of 5, (twice the numerator of) the value of ''k'' represents the number of generator steps required to reach the 3rd harmonic. | : A reasonable way of defining this continuum equates a number of [[2048/2025|diaschismas (2048/2025)]] with the [[393216/390625|Würschmidt comma (393216/390625)]], so that (2048/2025)<sup>''n''</sup> ~ 393216/390625. As a result, this may also be called the ''wurschmidt-diaschismic equivalence continuum'', or the ''diaschismic-gothic equivalence continuum'', which is more or less the same thing. The just value of ''n'' is 0.5853…, and temperaments near this tend to be the most accurate. The [[17-comma|gothic comma]] (134217728/129140163) is the characteristic [[3-limit]] comma tempered out in 34edo, and it has a value of ''n'' = 4. Therefore, one can additionally define ''k'' = 4 - ''n'', which has notable advantages - in particular, due to being determined in terms of the 3-limit comma and the comma with the next lowest power of 5, (twice the numerator of) the value of ''k'' represents the number of generator steps required to reach the 3rd harmonic. | ||
| Line 94: | Line 94: | ||
: <nowiki>*</nowiki> in projective tuning space, ∞ = -∞. | : <nowiki>*</nowiki> in projective tuning space, ∞ = -∞. | ||
We may invert the continuum by setting '' | We may invert the continuum by setting ''m'' such that 1/''n'' - 1/''m'' = 1. The just value of ''m'' is 3.41173… = log<sub>2</sub>([[20000/19683]])/log<sub>2</sub>([[15625/15552]]), and temperaments near this tend to be the most accurate ones. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||