Kleismic family: Difference between revisions

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The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma, which is the amount by which a stack of six [[6/5|classical minor third]]s falls short of the [[3/1|3rd~~]] [[~~harmonic]], or equivalently the amount by which a stack of three [[25/24]] overshoots [[9/8]]. As a result, all kleismic temperaments trisect 9/8, for which 1 third tone is 25/24 and two ~~are~~ 27/25.

The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma, which is the amount by which a stack of six [[6/5|classical minor third]]s falls short of the [[3/1|3rd harmonic]], or equivalently the amount by which a stack of three [[25/24]] overshoots the [[9/8]] whole tone. As a result, all kleismic temperaments trisect 9/8, for which one third-tone represents [[25/24]] and two represents [[27/25]].

== Kleismic a.k.a. hanson ==

The [[generator]] of kleismic is a [[6/5|classical minor third]], and to get to the interval class of [[5/4|major thirds]] requires five of these, and so to get to [[3/2|fifths]] requires six. In fact, (6/5)5 = (5/2)⋅(15625/15552). This 5-limit temperament (virtually a [[microtemperament]]) is sometimes called ''hanson'', and [[53edo|14\53]] is about perfect as a generator, though [[34edo|9\34]] also makes sense, and [[19edo|5\19]] and [[15edo|4\15]] are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].

The [[generator]] of kleismic is a [[6/5|classical minor third]], and to get to the interval class of [[5/4|major thirds]] requires five of these, and so to get to [[3/2|perfect fifths]] requires six. In fact, (6/5)5 = (5/2)⋅(15625/15552). This 5-limit temperament (virtually a [[microtemperament]]) is sometimes called ''hanson'', and [[53edo|14\53]] is about perfect as a generator, though [[34edo|9\34]] also makes sense, and [[19edo|5\19]] and [[15edo|4\15]] are possible. Other edo tunings include [[72edo]], [[87edo]] and [[140edo]].

[[Subgroup]]: 2.3.5

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=== Overview to extensions ===

==== Full 7-limit extensions ====

The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. [[4375/4374]], the ragisma, gives catakleismic. [[875/864]], the keemic comma, gives keemun. [[5120/5103]], ~~hemifamity~~, gives countercata. [[179200/177147]], the tolerant comma, gives metakleismic. [[64/63]], the archytas comma, gives catalan. Catakleismic, keemun, countercata, metakleismic, and catalan all have octave period and use the minor third as a generator; catakleismic, countercata, and metakleismic define the 7/4 more complexly but more accurately than keemun and catalan.

The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. [[4375/4374]], the ragisma, gives catakleismic. [[875/864]], the keemic comma, gives keemun. [[5120/5103]], the aberschisma, gives countercata. [[179200/177147]], the tolerant comma, gives metakleismic. [[64/63]], the archytas comma, gives catalan. Catakleismic, keemun, countercata, metakleismic, and catalan all have octave period and use the minor third as a generator; catakleismic, countercata, and metakleismic define the 7/4 more complexly but more accurately than keemun and catalan.

[[6144/6125]], the porwell comma, gives [[#Hemikleismic|hemikleismic]]. [[245/243]], sensamagic, gives [[#Clyde|clyde]]. [[1029/1024]], the gamelisma, gives [[#Tritikleismic|tritikleismic]]. [[10976/10935]], hemimage, gives [[#Marfifths|marfifths]]. [[1728/1715]], the orwellismia, gives [[#Kleiboh|kleiboh]]. [[2401/2400]], the breedsma, gives [[#Quadritikleismic|quadritikleismic]]. [[2460375/2458624]], the breeze comma, gives [[#Marthirds|marthirds]]. Hemikleismic splits the 6/5 in half to get a neutral second generator of ~35/32, and clyde similarly splits the 5/3 in half to get a ~9/7 generator. Marfifths splits the 12/5 into three. Kleiboh splits the 24/5 into three. Marthirds splits the 12/5 into four. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator.

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=== 2.3.5.13 subgroup (cata) ===

The kleisma has the [[S-expression]] ~~S25²*S26~~ which explains how the tone is trisected, thus tempering out its ~~constituyents S25 =~~ [[625/624]] and ~~S26 =~~ [[676/675]] is a readily natural subgroup extension. As a result of coincidental S-expression equivalences, [[325/324]] ~~= S25*S26 it~~ is also tempered out~~. Cata finds the third-tritave [[³√3]] to be~~ [[13/9]] ~~as its~~ comma [[2197/2187]] ~~also has the expression S25*S26², and the half-tritave [[√3]] to be [[26~~/~~15]], whose difference~~ is ~~the basis S26~~.

The kleisma has the [[S-expression]] S252⋅S26, which explains how the tone is trisected, and thus tempering out its constituents [[625/624]] ({{S|25}}) and [[676/675]] ({{S|26}}) is a readily natural subgroup extension. As a result of coincidental S-expression equivalences, [[325/324]] (S25⋅S26) is also tempered out, and two generator steps stand in for ~[[13/9]]. Finally, since this step is one third of a perfect twelfth, the comma [[2197/2187]] (S25⋅S262) is tempered out. The third tone is now 25/24~26/25~27/26, which means 27/25 and [[13/12]] are made equal.

The third tone is now 25/24 ~ 26/25 ~ 27/26, which means 27/25 ~ [[13/12]] are made equal.

For a version of cata with a 1\5 period, see [[Thunderclysmic]].

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Comma list: 325/324, 625/624

~~Subgroup-val mapping:~~ {{~~mapping~~| 1 0 1 0 | 0 6 5 14 }}

Optimal tunings:

* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}[[error map]]: {{val|+0.121 +0.690 -0.655 -1.022}}

* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}[[error map]]: {{val|+0.000 +0.597 -0.854 -1.240}}

: error map: {{val| +0.121 +0.690 -0.655 -1.022 }}

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}

: error map: {{val| 0.000 +0.597 -0.854 -1.240 }}

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Catakleismic tempers out 225/224, the [[marvel comma]], and 4375/4374, the [[ragisma]], and may be described as the {{nowrap| 53 & 72 }} temperament. [[125edo]] and especially [[197edo]] make for excellent tunings. In here, the JI intervals 25/~~24 - 27~~/~~25 - 9~~/~~8 - 7~~/6 are made equidistant.

Catakleismic tempers out 225/224, the [[marvel comma]], and 4375/4374, the [[ragisma]], and may be described as the {{nowrap| 53 & 72 }} temperament. [[125edo]] and especially [[197edo]] make for excellent tunings. In here, the JI intervals 25/24–27/25–9/8–7/6 are made equidistant.

=== 7-limit ===

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==== 2.3.5.7.13 subgroup ====

Catakleismic extends easily with [[prime interval|prime]] [[13/1|13]] as seen before with cata, which has the S-expression list {S25, S26}. The marvel comma has an equivalent S-expression ~~S25*S26*S27~~ and the ragisma S25/S27, thus ~~S27 =~~ [[729/728]] is another constituent that can be naturally tempered out. As a result, the third tone becomes 25/24~26/25~27/26~28/27, and the equivalence 27/25 ~ 13/12 is added.

Catakleismic extends easily with [[prime interval|prime]] [[13/1|13]] as seen before with cata, which has the S-expression list {S25, S26}. The marvel comma has an equivalent S-expression S25⋅S26⋅S27 and the ragisma S25/S27, thus [[729/728]] ({{S|27}}) is another constituent that can be naturally tempered out. As a result, the third tone becomes 25/24~26/25~27/26~28/27, and the equivalence 27/25~13/12 is added.

Add-13 catakleismic is less accurate than cata or catakleismic alone as the error of 7 and 13 go in separate directions - cata wants a sharper 6/5 but catakleismic wants a flatter one. Nontheless, its structural properties make it quite notable.

Add-13 catakleismic is less accurate than cata or catakleismic alone as the error of 7 and 13 go in separate directions – cata wants a sharper 6/5 but catakleismic wants a flatter one. Nontheless, its structural properties make it quite notable.

Subgroup: 2.3.5.7.13

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Comma list: 169/168, 225/224, 325/324

~~Subgroup-val mapping:~~ {{~~mapping~~| 1 0 1 -3 0 | 0 6 5 22 14 }}

Optimal tunings:

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma, which is the amount by which a stack of six [[6/5|classical minor third]]s falls short of the [[3/1|3rd]] [[harmonic]], or equivalently the amount by which a stack of three [[25/24]] overshoots [[9/8]]. As a result, all kleismic temperaments trisect 9/8, for which 1 third tone is 25/24 and two are 27/25.
+The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma, which is the amount by which a stack of six [[6/5|classical minor third]]s falls short of the [[3/1|3rd harmonic]], or equivalently the amount by which a stack of three [[25/24]] overshoots the [[9/8]] whole tone. As a result, all kleismic temperaments trisect 9/8, for which one third-tone represents [[25/24]] and two represents [[27/25]].
 == Kleismic a.k.a. hanson ==
 {{Main| Kleismic }}
-The [[generator]] of kleismic is a [[6/5|classical minor third]], and to get to the interval class of [[5/4|major thirds]] requires five of these, and so to get to [[3/2|fifths]] requires six. In fact, (6/5)<sup>5</sup> = (5/2)⋅(15625/15552). This 5-limit temperament (virtually a [[microtemperament]]) is sometimes called ''hanson'', and [[53edo|14\53]] is about perfect as a generator, though [[34edo|9\34]] also makes sense, and [[19edo|5\19]] and [[15edo|4\15]] are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].
+The [[generator]] of kleismic is a [[6/5|classical minor third]], and to get to the interval class of [[5/4|major thirds]] requires five of these, and so to get to [[3/2|perfect fifths]] requires six. In fact, (6/5)<sup>5</sup> = (5/2)⋅(15625/15552). This 5-limit temperament (virtually a [[microtemperament]]) is sometimes called ''hanson'', and [[53edo|14\53]] is about perfect as a generator, though [[34edo|9\34]] also makes sense, and [[19edo|5\19]] and [[15edo|4\15]] are possible. Other edo tunings include [[72edo]], [[87edo]] and [[140edo]].
 [[Subgroup]]: 2.3.5
@@ Line 30: / Line 30: @@
 === Overview to extensions ===
 ==== Full 7-limit extensions ====
-The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. [[4375/4374]], the ragisma, gives catakleismic. [[875/864]], the keemic comma, gives keemun. [[5120/5103]], hemifamity, gives countercata. [[179200/177147]], the tolerant comma, gives metakleismic. [[64/63]], the archytas comma, gives catalan. Catakleismic, keemun, countercata, metakleismic, and catalan all have octave period and use the minor third as a generator; catakleismic, countercata, and metakleismic define the 7/4 more complexly but more accurately than keemun and catalan.
+The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. [[4375/4374]], the ragisma, gives catakleismic. [[875/864]], the keemic comma, gives keemun. [[5120/5103]], the aberschisma, gives countercata. [[179200/177147]], the tolerant comma, gives metakleismic. [[64/63]], the archytas comma, gives catalan. Catakleismic, keemun, countercata, metakleismic, and catalan all have octave period and use the minor third as a generator; catakleismic, countercata, and metakleismic define the 7/4 more complexly but more accurately than keemun and catalan.
 [[6144/6125]], the porwell comma, gives [[#Hemikleismic|hemikleismic]]. [[245/243]], sensamagic, gives [[#Clyde|clyde]]. [[1029/1024]], the gamelisma, gives [[#Tritikleismic|tritikleismic]]. [[10976/10935]], hemimage, gives [[#Marfifths|marfifths]]. [[1728/1715]], the orwellismia, gives [[#Kleiboh|kleiboh]]. [[2401/2400]], the breedsma, gives [[#Quadritikleismic|quadritikleismic]]. [[2460375/2458624]], the breeze comma, gives [[#Marthirds|marthirds]]. Hemikleismic splits the 6/5 in half to get a neutral second generator of ~35/32, and clyde similarly splits the 5/3 in half to get a ~9/7 generator. Marfifths splits the 12/5 into three. Kleiboh splits the 24/5 into three. Marthirds splits the 12/5 into four. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator.
@@ Line 40: / Line 40: @@
 === 2.3.5.13 subgroup (cata) ===
-The kleisma has the [[S-expression]] S25²*S26 which explains how the tone is trisected, thus tempering out its constituyents S25 = [[625/624]] and S26 = [[676/675]] is a readily natural subgroup extension. As a result of coincidental S-expression equivalences, [[325/324]] = S25*S26 it is also tempered out. Cata finds the third-tritave [[³√3]] to be [[13/9]] as its comma [[2197/2187]] also has the expression S25*S26², and the half-tritave [[√3]] to be [[26/15]], whose difference is the basis S26.
+The kleisma has the [[S-expression]] S25<sup>2</sup>⋅S26, which explains how the tone is trisected, and thus tempering out its constituents [[625/624]] ({{S|25}}) and [[676/675]] ({{S|26}}) is a readily natural subgroup extension. As a result of coincidental S-expression equivalences, [[325/324]] (S25⋅S26) is also tempered out, and two generator steps stand in for ~[[13/9]]. Finally, since this step is one third of a perfect twelfth, the comma [[2197/2187]] (S25⋅S26<sup>2</sup>) is tempered out. The third tone is now 25/24~26/25~27/26, which means 27/25 and [[13/12]] are made equal.
-The third tone is now 25/24 ~ 26/25 ~ 27/26, which means 27/25 ~ [[13/12]] are made equal.
 For a version of cata with a 1\5 period, see [[Thunderclysmic]].
@@ Line 50: / Line 48: @@
 Comma list: 325/324, 625/624
-Subgroup-val mapping: {{mapping| 1 0 1 0 | 0 6 5 14 }}
+{{Mapping|legend=2| 1 0 1 0 | 0 6 5 14 }}
 Optimal tunings:
-* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}[[error map]]: {{val|+0.121 +0.690 -0.655 -1.022}}
+* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}[[error map]]: {{val|+0.000 +0.597 -0.854 -1.240}}
+: error map: {{val| +0.121 +0.690 -0.655 -1.022 }}
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}
+: error map: {{val| 0.000 +0.597 -0.854 -1.240 }}
 {{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246 }}
@@ Line 63: / Line 63: @@
 {{Main| Catakleismic }}
-Catakleismic tempers out 225/224, the [[marvel comma]], and 4375/4374, the [[ragisma]], and may be described as the {{nowrap| 53 & 72 }} temperament. [[125edo]] and especially [[197edo]] make for excellent tunings. In here, the JI intervals 25/24 - 27/25 - 9/8 - 7/6 are made equidistant.
+Catakleismic tempers out 225/224, the [[marvel comma]], and 4375/4374, the [[ragisma]], and may be described as the {{nowrap| 53 & 72 }} temperament. [[125edo]] and especially [[197edo]] make for excellent tunings. In here, the JI intervals 25/24–27/25–9/8–7/6 are made equidistant.
 === 7-limit ===
@@ Line 87: / Line 87: @@
 ==== 2.3.5.7.13 subgroup ====
-Catakleismic extends easily with [[prime interval|prime]] [[13/1|13]] as seen before with cata, which has the S-expression list {S25, S26}. The marvel comma has an equivalent S-expression S25*S26*S27 and the ragisma S25/S27, thus S27 = [[729/728]] is another constituent that can be naturally tempered out. As a result, the third tone becomes 25/24~26/25~27/26~28/27, and the equivalence 27/25 ~ 13/12 is added.
+Catakleismic extends easily with [[prime interval|prime]] [[13/1|13]] as seen before with cata, which has the S-expression list {S25, S26}. The marvel comma has an equivalent S-expression S25⋅S26⋅S27 and the ragisma S25/S27, thus [[729/728]] ({{S|27}}) is another constituent that can be naturally tempered out. As a result, the third tone becomes 25/24~26/25~27/26~28/27, and the equivalence 27/25~13/12 is added.
-Add-13 catakleismic is less accurate than cata or catakleismic alone as the error of 7 and 13 go in separate directions - cata wants a sharper 6/5 but catakleismic wants a flatter one. Nontheless, its structural properties make it quite notable.
+Add-13 catakleismic is less accurate than cata or catakleismic alone as the error of 7 and 13 go in separate directions – cata wants a sharper 6/5 but catakleismic wants a flatter one. Nontheless, its structural properties make it quite notable.
 Subgroup: 2.3.5.7.13
@@ Line 95: / Line 95: @@
 Comma list: 169/168, 225/224, 325/324
-Subgroup-val mapping: {{mapping| 1 0 1 -3 0 | 0 6 5 22 14 }}
+{{Mapping|legend=2| 1 0 1 -3 0 | 0 6 5 22 14 }}
 Optimal tunings: