Diaschismic–gothmic equivalence continuum: Difference between revisions

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We may invert the continuum by setting ''m'' such that 1/''n'' + 1/''m'' = 1. The just value of ''m'' is 1.41414…, and temperaments near this tend to be the most accurate ones. The resulting continuum equates a number of [[immunity comma]]s to the [[gothic comma]], but as the immunity comma is both larger and more complex than the diaschisma, this continuum does not contain as many useful temperaments at simple points which aren't already found by (half-)integer points on the diaschismic-gothmic and kleismic-tetracot continua.

It is worth briefly noting that on this continuum: ''m'' = 0 yields [[gothic]], ''m'' = 1 yields [[diaschismic]] (a.k.a. srutal), ''m'' = 2 yields [[tetracot]] and the simplest non-integer convergent, ''m'' = 3/2, yields [[Hanson|kleismic]]~~, with~~ ''m'' = 1/2 yielding the 34 & 29c temperament which may also be described as the 34 & 107 temperament, which is essentially complementary to the simpler [[immunity]].

It is worth briefly noting that on this continuum: ''m'' = 0 yields [[gothic]], ''m'' = 1 yields [[diaschismic]] (a.k.a. srutal), ''m'' = 2 yields [[tetracot]], ''m'' = 3 yields the 34 & 36c temperament occurring at ''n'' = -1/2, and the simplest non-integer convergent that approximates the [[JIP]], ''m'' = 3/2, yields [[Hanson|kleismic]]. A unique (but not very good) temperament in this continuum is ''m'' = 1/2, yielding the 34 & 29c temperament which may also be described as the 34 & 107 temperament, which is essentially complementary (w.r.t. [[34edo|34et]]) to the simpler [[immunity]].

We may also examine temperaments that are structurally nontrivial in that they correspond to non-half-integer fractional ''n'' and ''m'', 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.

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 We may invert the continuum by setting ''m'' such that 1/''n'' + 1/''m'' = 1. The just value of ''m'' is 1.41414…, and temperaments near this tend to be the most accurate ones. The resulting continuum equates a number of [[immunity comma]]s to the [[gothic comma]], but as the immunity comma is both larger and more complex than the diaschisma, this continuum does not contain as many useful temperaments at simple points which aren't already found by (half-)integer points on the diaschismic-gothmic and kleismic-tetracot continua.
-It is worth briefly noting that on this continuum: ''m'' = 0 yields [[gothic]], ''m'' = 1 yields [[diaschismic]] (a.k.a. srutal), ''m'' = 2 yields [[tetracot]] and the simplest non-integer convergent, ''m'' = 3/2, yields [[Hanson|kleismic]], with ''m'' = 1/2 yielding the 34 & 29c temperament which may also be described as the 34 & 107 temperament, which is essentially complementary to the simpler [[immunity]].
+It is worth briefly noting that on this continuum: ''m'' = 0 yields [[gothic]], ''m'' = 1 yields [[diaschismic]] (a.k.a. srutal), ''m'' = 2 yields [[tetracot]], ''m'' = 3 yields the 34 & 36c temperament occurring at ''n'' = -1/2, and the simplest non-integer convergent that approximates the [[JIP]], ''m'' = 3/2, yields [[Hanson|kleismic]]. A unique (but not very good) temperament in this continuum is ''m'' = 1/2, yielding the 34 & 29c temperament which may also be described as the 34 & 107 temperament, which is essentially complementary (w.r.t. [[34edo|34et]]) to the simpler [[immunity]].
 We may also examine temperaments that are structurally nontrivial in that they correspond to non-half-integer fractional ''n'' and ''m'', 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.