Diaschismic–gothmic equivalence continuum: Difference between revisions

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Note that all of these correspond to half-integer points of either ''n'' or ''k'' (define below), hence part of the usefulness of the inversion discussed in the [[#kleismic-tetracot continuum]] subsection. They are presented here for potential insight into meanings of their fractional values of ''n'' and ''m'' as they relate to the pergen structures of the temperaments.

== ~~significance~~ of tetracot ==

== Significance of tetracot ==

Tetracot appears as the unique (simplest) minimal positive integer ''n'' which achieves:

Tetracot appears as the unique simplest minimal positive integer ''n'' which achieves:

1. The simplest comma (compare the monzos, ratios or expressions of ~~gothmic~~ (''n''=0) and immunity (''n''=1)).

1. The simplest comma (compare the monzos, ratios or expressions of gothic (''n'' = 0) and immunity (''n'' = 1)).

2. The simplest temperament mapping (compare the mappings of ~~gothmic~~ (which has a whopping ''17'' periods per octave, but lacks the accuracy of something like [[chlorine]]) and immunity which takes slightly more generators to reach the same intervals of tetracot, so initially seems comparable, but whose generator's 5-limit interpretation is questionably damaged and complex compared to tetracot).

2. The simplest temperament mapping (compare the mappings of gothic (which has a whopping ''17'' periods per octave, but lacks the accuracy of something like [[chlorine]]) and immunity which takes slightly more generators to reach the same intervals of tetracot, so initially seems comparable, but whose generator's 5-limit interpretation is questionably damaged and complex compared to tetracot).

3. A characteristic damage of [[34edo|34et]] which is not trivial; ~~gothmic~~ is trivial in that it's just a subgroup restriction, and immunity, though not trivial and comparable in complexity to tetracot, is not characteristic, because it asks for a flat tuning of 5, so that it is arguably more or equally as characteristic of [[63edo]]'s or [[97edo]]'s representation of the 5-limit, but the fact that it has so many 3's in it when 34edo does not do so well in consistency of 3's to long distance should be a hint that we can do better. Compare with tetracot, which admits comparatively more and lower damage patent tunings and is clearly a type of imperfect simplification corresponding to a structural awkwardness in [[5-limit]] [[JI]] – the [[20000/19683|minimal diesis]] – so it has characteristic damage on 10/9 (flatwards) and 9/8 (sharpwards) by exaggerating the difference; this is to say, if you look at 34et's tuning of the 5-limit, its damage is strongly characteristic of tetracot. (Exaggerating this difference in this way ''is'' also characteristic of immunity, but its general tunings are at odds with those of tetracot's so that they only merge in 34edo, which is arguably a more unusual tuning for immunity than it is for tetracot, where it is clearly characteristic.)

3. A characteristic damage of [[34edo|34et]] which is not trivial; gothic is trivial in that it is just a subgroup restriction, and immunity, though not trivial and comparable in complexity to tetracot, is not characteristic, because it asks for a flat tuning of 5, so that it is arguably more or equally as characteristic of [[63edo]]'s or [[97edo]]'s representation of the 5-limit, but the fact that it has so many 3's in it when 34edo does not do so well in consistency of 3's to long distance should be a hint that we can do better. Compare with tetracot, which admits comparatively more and lower damage patent tunings and is clearly a type of imperfect simplification corresponding to a structural awkwardness in [[5-limit]] [[JI]] – the [[20000/19683|minimal diesis]] – so it has characteristic damage on 10/9 (flatwards) and 9/8 (sharpwards) by exaggerating the difference; this is to say, if you look at 34et's tuning of the 5-limit, its damage is strongly characteristic of tetracot. Exaggerating this difference in this way ''is'' also characteristic of immunity, but its general tunings are at odds with those of tetracot's so that they only merge in 34edo, which is arguably a more unusual tuning for immunity than it is for tetracot, where it is clearly characteristic.

4. As aforementioned, a convenient point to invert the scale to define the '''kleismic-tetracot continuum''' nicely, discussed below.

== ~~kleismic~~-tetracot continuum ==

== Kleismic-tetracot continuum ==

We may also describe the set of all [[5-limit]] [[regular temperament|temperaments]] supported by [[34edo|34et]] by expressing the continuum (15625/15552)''k'' ~ 20000/19683, for a value of ''k'' defined such that 1/''r'' + 1/''k'' = 1 – corresponding to an inversion of the diaschismic-tetracot continuum with respect to tetracot. Varying ''k'' (for number of kleismas) results in different temperaments listed in the table below. It converges to [[hanson]] as ''k'' approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas as discovered by [[User:Lériendil|Lériendil]]. The just value of ''k'' is 3.4117…, and temperaments near this tend to be the most accurate. This also suggests that the kleisma is (loosely speaking) a type of "super-comma" or "meta-comma" for the 5-limit, in its ability to equate so many commas simultaneously into a general purpose comma.

We may also describe the set of all [[5-limit]] [[regular temperament|temperaments]] supported by [[34edo|34et]] by expressing the continuum (15625/15552)''k'' ~ 20000/19683, for a value of ''k'' defined such that 1/''r'' + 1/''k'' = 1 – corresponding to an inversion of the diaschismic-tetracot continuum with respect to tetracot. Varying ''k'' (for number of kleismas) results in different temperaments listed in the table below. It converges to [[hanson]] as ''k'' approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas as discovered by [[User:Lériendil|Lériendil]]. The just value of ''k'' is 3.4117…, and temperaments near this tend to be the most accurate. This also suggests that the kleisma is, loosely speaking, a type of "super-comma" or "meta-comma" for the 5-limit, in its ability to equate so many commas simultaneously into a general purpose comma.

{| class="wikitable center-1 center-2"

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 Note that all of these correspond to half-integer points of either ''n'' or ''k'' (define below), hence part of the usefulness of the inversion discussed in the [[#kleismic-tetracot continuum]] subsection. They are presented here for potential insight into meanings of their fractional values of ''n'' and ''m'' as they relate to the pergen structures of the temperaments.
-== significance of tetracot ==
+== Significance of tetracot ==
-Tetracot appears as the unique (simplest) minimal positive integer ''n'' which achieves:
+Tetracot appears as the unique simplest minimal positive integer ''n'' which achieves:
-. The simplest comma (compare the monzos, ratios or expressions of gothmic (''n''=0) and immunity (''n''=1)).
+. The simplest comma (compare the monzos, ratios or expressions of gothic (''n'' = 0) and immunity (''n'' = 1)).
-. The simplest temperament mapping (compare the mappings of gothmic (which has a whopping ''17'' periods per octave, but lacks the accuracy of something like [[chlorine]]) and immunity which takes slightly more generators to reach the same intervals of tetracot, so initially seems comparable, but whose generator's 5-limit interpretation is questionably damaged and complex compared to tetracot).
+. The simplest temperament mapping (compare the mappings of gothic (which has a whopping ''17'' periods per octave, but lacks the accuracy of something like [[chlorine]]) and immunity which takes slightly more generators to reach the same intervals of tetracot, so initially seems comparable, but whose generator's 5-limit interpretation is questionably damaged and complex compared to tetracot).
-. A characteristic damage of [[34edo|34et]] which is not trivial; gothmic is trivial in that it's just a subgroup restriction, and immunity, though not trivial and comparable in complexity to tetracot, is not characteristic, because it asks for a flat tuning of 5, so that it is arguably more or equally as characteristic of [[63edo]]'s or [[97edo]]'s representation of the 5-limit, but the fact that it has so many 3's in it when 34edo does not do so well in consistency of 3's to long distance should be a hint that we can do better. Compare with tetracot, which admits comparatively more and lower damage patent tunings and is clearly a type of imperfect simplification corresponding to a structural awkwardness in [[5-limit]] [[JI]] – the [[20000/19683|minimal diesis]] – so it has characteristic damage on 10/9 (flatwards) and 9/8 (sharpwards) by exaggerating the difference; this is to say, if you look at 34et's tuning of the 5-limit, its damage is strongly characteristic of tetracot. (Exaggerating this difference in this way ''is'' also characteristic of immunity, but its general tunings are at odds with those of tetracot's so that they only merge in 34edo, which is arguably a more unusual tuning for immunity than it is for tetracot, where it is clearly characteristic.)
+. A characteristic damage of [[34edo|34et]] which is not trivial; gothic is trivial in that it is just a subgroup restriction, and immunity, though not trivial and comparable in complexity to tetracot, is not characteristic, because it asks for a flat tuning of 5, so that it is arguably more or equally as characteristic of [[63edo]]'s or [[97edo]]'s representation of the 5-limit, but the fact that it has so many 3's in it when 34edo does not do so well in consistency of 3's to long distance should be a hint that we can do better. Compare with tetracot, which admits comparatively more and lower damage patent tunings and is clearly a type of imperfect simplification corresponding to a structural awkwardness in [[5-limit]] [[JI]] – the [[20000/19683|minimal diesis]] – so it has characteristic damage on 10/9 (flatwards) and 9/8 (sharpwards) by exaggerating the difference; this is to say, if you look at 34et's tuning of the 5-limit, its damage is strongly characteristic of tetracot. Exaggerating this difference in this way ''is'' also characteristic of immunity, but its general tunings are at odds with those of tetracot's so that they only merge in 34edo, which is arguably a more unusual tuning for immunity than it is for tetracot, where it is clearly characteristic.
 . As aforementioned, a convenient point to invert the scale to define the '''kleismic-tetracot continuum''' nicely, discussed below.
-== kleismic-tetracot continuum ==
+== Kleismic-tetracot continuum ==
-We may also describe the set of all [[5-limit]] [[regular temperament|temperaments]] supported by [[34edo|34et]] by expressing the continuum (15625/15552)<sup>''k''</sup> ~ 20000/19683, for a value of ''k'' defined such that 1/''r'' + 1/''k'' = 1 – corresponding to an inversion of the diaschismic-tetracot continuum with respect to tetracot. Varying ''k'' (for number of <u>k</u>leismas) results in different temperaments listed in the table below. It converges to [[hanson]] as ''k'' approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas as discovered by [[User:Lériendil|Lériendil]]. The just value of ''k'' is 3.4117…, and temperaments near this tend to be the most accurate. This also suggests that the kleisma is (loosely speaking) a type of "super-comma" or "meta-comma" for the 5-limit, in its ability to equate so many commas simultaneously into a general purpose comma.
+We may also describe the set of all [[5-limit]] [[regular temperament|temperaments]] supported by [[34edo|34et]] by expressing the continuum (15625/15552)<sup>''k''</sup> ~ 20000/19683, for a value of ''k'' defined such that 1/''r'' + 1/''k'' = 1 – corresponding to an inversion of the diaschismic-tetracot continuum with respect to tetracot. Varying ''k'' (for number of <u>k</u>leismas) results in different temperaments listed in the table below. It converges to [[hanson]] as ''k'' approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas as discovered by [[User:Lériendil|Lériendil]]. The just value of ''k'' is 3.4117…, and temperaments near this tend to be the most accurate. This also suggests that the kleisma is, loosely speaking, a type of "super-comma" or "meta-comma" for the 5-limit, in its ability to equate so many commas simultaneously into a general purpose comma.
 {| class="wikitable center-1 center-2"