Diaschismic–gothmic equivalence continuum: Difference between revisions

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The '''diaschismic-gothic 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]].

The '''diaschismic-gothic equivalence continuum''', also referrable as the ''diaschismic-würschmidt equivalence continuum'', ''diaschismic-kleismic equivalence continuum'', or ''kleismic-würschmidt 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]].

All temperaments in the continuum satisfy (2048/2025)''n'' ~ {{monzo| 27 -17 }}, equating a number of [[2048/2025|diaschismas (2048/2025)]] with the [[gothic comma]]. Varying ''n'' results in different temperaments listed in the table below. It converges to [[diaschismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 34edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 3.41464…, and temperaments having ''n'' near this value tend to be the most accurate ones.

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: <nowiki>*</nowiki> in projective tuning space, ∞ = -∞.

~~{| class="wikitable"~~

All temperaments in the continuum also satisfy (15625/15552)''s'' ~ 393216/390625, for a value of ''s'' defined such that 1/''k'' - 1/''s'' = 1; equivalently, we can offset ''s'' by 1, and equate a number of [[15625/15552|kleismas (15625/15552)]] with the diaschisma, giving rise to the name ''diaschismic-kleismic equivalence continuum''. Varying ''s'' results in different temperaments listed in the table below. It converges to [[hanson]] as ''s'' approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas.

~~|+ Temperaments with fractional ''n''~~

|-

~~! ''n'' !! ''k'' !! Temperament !! Comma~~

|-

~~| 5/2 = 2.5 || 3/2 = 1.5 || [[Fifive]] || {{monzo| -1 -14 10 }}~~

|-

~~| 13/4 = 3.25 || 3/4 = 0.75 || [[Quatracot]] || {{monzo| -33 -16 26 }}~~

|-

~~| 10/3 = 3.{{overline|3}} || 2/3 = 0.{{overline|6}} || [[Gammic]] || {{monzo| -29 -11 20 }}~~

|-

~~| 17/5 = 3.4 || 3/5 = 0.6 || [[Chlorine]] || {{monzo| -52 -17 34 }}~~

|-

~~| 7/2 = 3.5 || 1/2 = 0.5 || [[Vishnu]] || {{monzo| 23 6 -14 }}~~

|-

~~| 11/3 = 3.{{overline|6}} || 1/3 = 0.{{overline|3}} || [[Majvam]] || {{monzo| 40 7 -22 }}~~

|-

| 9/2 = 4.5 || -1/2 = -0.5 || [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.17&reduce=on&weights=tenney&target=&edos=&commas=35184372088832%2F34332275390625%2C+289%2F288%2C+2197%2F2187&submit_comma=submit 34 & 142] || {{monzo| 45 -2 -18 }}

|}

All temperaments in the continuum also satisfy (15625/15552)''m'' ~ 393216/390625, for a value of ''m'' defined such that 1/''k'' - 1/''m'' = 1; equivalently, we can offset ''m'' by 1, and equate a number of [[15625/15552|kleismas (15625/15552)]] with the diaschisma, giving rise to the name ''diaschismic-kleismic equivalence continuum''. Varying ''m'' results in different temperaments listed in the table below. It converges to [[hanson]] as ''m'' approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas.

{| class="wikitable center-1"

|+ Temperaments with integer ''m'' in the kleismic-würschmidt continuum

|+ Temperaments with integer ''s'' in the kleismic-würschmidt continuum

|-

! rowspan="2" | ''m''

! rowspan="2" | ''s''

! rowspan="2" | Temperament

! colspan="2" | Comma

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{| class="wikitable"

|+ Temperaments with fractional ''k'' and ''m''

|+ Temperaments with fractional ''k'' and ''s''

|-

! ''m'' !! ''k'' !! Temperament !! Comma

! ''s'' !! ''k'' !! ''n'' !! Temperament !! Comma

|-

| 1/2 = 0.5 || 1/3 = 0.{{overline|3}} || [[Majvam]] || {{monzo| 40 7 -22 }}

| 1/2 = 0.5 || 1/3 = 0.{{overline|3}} || 11/3 = 3.{{overline|6}} || [[Majvam]] || {{monzo| 40 7 -22 }}

|-

| 3/2 = 1.5 || 3/5 = 0.6 || [[Chlorine]] || {{monzo| -52 -17 34 }}

| 3/2 = 1.5 || 3/5 = 0.6 || 17/5 = 3.4 || [[Chlorine]] || {{monzo| -52 -17 34 }}

|-

| -1/3 = -0.{{overline|3}} || -1/2 = -0.5 || [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.17&reduce=on&weights=tenney&target=&edos=&commas=35184372088832%2F34332275390625%2C+289%2F288%2C+2197%2F2187&submit_comma=submit 34 & 142] || {{monzo| 45 -2 -18 }}

| -1/3 = -0.{{overline|3}} || -1/2 = -0.5 || 9/2 = 4.5 || [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.17&reduce=on&weights=tenney&target=&edos=&commas=35184372088832%2F34332275390625%2C+289%2F288%2C+2197%2F2187&submit_comma=submit 34 & 142] || {{monzo| 45 -2 -18 }}

|}

[[Category:34edo]]

[[Category:Equivalence continua]]

@@ Line 1: / Line 1: @@
-The '''diaschismic-gothic 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]].
+The '''diaschismic-gothic equivalence continuum''', also referrable as the ''diaschismic-würschmidt equivalence continuum'', ''diaschismic-kleismic equivalence continuum'', or ''kleismic-würschmidt 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]].
 All temperaments in the continuum satisfy (2048/2025)<sup>''n''</sup> ~ {{monzo| 27 -17 }}, equating a number of [[2048/2025|diaschismas (2048/2025)]] with the [[gothic comma]]. Varying ''n'' results in different temperaments listed in the table below. It converges to [[diaschismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 34edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 3.41464…, and temperaments having ''n'' near this value tend to be the most accurate ones.
@@ Line 74: / Line 74: @@
 : <nowiki>*</nowiki> in projective tuning space, ∞ = -∞.
-{| class="wikitable"
+All temperaments in the continuum also satisfy (15625/15552)<sup>''s''</sup> ~ 393216/390625, for a value of ''s'' defined such that 1/''k'' - 1/''s'' = 1; equivalently, we can offset ''s'' by 1, and equate a number of [[15625/15552|kleismas (15625/15552)]] with the diaschisma, giving rise to the name ''diaschismic-kleismic equivalence continuum''. Varying ''s'' results in different temperaments listed in the table below. It converges to [[hanson]] as ''s'' approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas.
-|+ Temperaments with fractional ''n''
-|-
-! ''n'' !! ''k'' !! Temperament !! Comma
-|-
-| 5/2 = 2.5 || 3/2 = 1.5 || [[Fifive]] || {{monzo| -1 -14 10 }}
-|-
-| 13/4 = 3.25 || 3/4 = 0.75 || [[Quatracot]] || {{monzo| -33 -16 26 }}
-|-
-| 10/3 = 3.{{overline|3}} || 2/3 = 0.{{overline|6}} || [[Gammic]] || {{monzo| -29 -11 20 }}
-|-
-| 17/5 = 3.4 || 3/5 = 0.6 || [[Chlorine]] || {{monzo| -52 -17 34 }}
-|-
-| 7/2 = 3.5 || 1/2 = 0.5 || [[Vishnu]] || {{monzo| 23 6 -14 }}
-|-
-| 11/3 = 3.{{overline|6}} || 1/3 = 0.{{overline|3}} || [[Majvam]] || {{monzo| 40 7 -22 }}
-|-
-| 9/2 = 4.5 || -1/2 = -0.5 || [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.17&reduce=on&weights=tenney&target=&edos=&commas=35184372088832%2F34332275390625%2C+289%2F288%2C+2197%2F2187&submit_comma=submit 34 & 142] || {{monzo| 45 -2 -18 }}
-|}
-All temperaments in the continuum also satisfy (15625/15552)<sup>''m''</sup> ~ 393216/390625, for a value of ''m'' defined such that 1/''k'' - 1/''m'' = 1; equivalently, we can offset ''m'' by 1, and equate a number of [[15625/15552|kleismas (15625/15552)]] with the diaschisma, giving rise to the name ''diaschismic-kleismic equivalence continuum''. Varying ''m'' results in different temperaments listed in the table below. It converges to [[hanson]] as ''m'' approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas.
 {| class="wikitable center-1"
-|+ Temperaments with integer ''m'' in the kleismic-würschmidt continuum
+|+ Temperaments with integer ''s'' in the kleismic-würschmidt continuum
 |-
-! rowspan="2" | ''m''
+! rowspan="2" | ''s''
 ! rowspan="2" | Temperament
 ! colspan="2" | Comma
@@ Line 158: / Line 138: @@
 {| class="wikitable"
-|+ Temperaments with fractional ''k'' and ''m''
+|+ Temperaments with fractional ''k'' and ''s''
 |-
-! ''m'' !! ''k'' !! Temperament !! Comma
+! ''s'' !! ''k'' !! ''n'' !! Temperament !! Comma
 |-
-| 1/2 = 0.5 || 1/3 = 0.{{overline|3}} || [[Majvam]] || {{monzo| 40 7 -22 }}
+| 1/2 = 0.5 || 1/3 = 0.{{overline|3}} || 11/3 = 3.{{overline|6}} || [[Majvam]] || {{monzo| 40 7 -22 }}
 |-
-| 3/2 = 1.5 || 3/5 = 0.6 || [[Chlorine]] || {{monzo| -52 -17 34 }}
+| 3/2 = 1.5 || 3/5 = 0.6 || 17/5 = 3.4 || [[Chlorine]] || {{monzo| -52 -17 34 }}
 |-
-| -1/3 = -0.{{overline|3}} || -1/2 = -0.5 || [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.17&reduce=on&weights=tenney&target=&edos=&commas=35184372088832%2F34332275390625%2C+289%2F288%2C+2197%2F2187&submit_comma=submit 34 & 142] || {{monzo| 45 -2 -18 }}
+| -1/3 = -0.{{overline|3}} || -1/2 = -0.5 || 9/2 = 4.5 || [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.17&reduce=on&weights=tenney&target=&edos=&commas=35184372088832%2F34332275390625%2C+289%2F288%2C+2197%2F2187&submit_comma=submit 34 & 142] || {{monzo| 45 -2 -18 }}
 |}
 [[Category:34edo]]
 [[Category:Equivalence continua]]