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~~]].

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 ~~structure of~~ the ~~temperament as dividing 3/1 into 6 equal parts can be deduced completely from its~~ [[S-expression]]~~-based comma list of {{nowrap| {[[325~~/~~324|S10/S12 = S25⋅S26]]~~, ([[625/624~~|S25~~]],) [[676/675~~|S13/S15 = S26~~]]} }}. ~~Specifically, dividing 3/1 into two halves~~ of ~~~26/15 is equivalent to dividing 4/3 into two halves of ~15/13~~, ~~hence the~~ [[~~semiparticular~~]] ~~S13/S15 =~~ ([[4/~~3|16/12~~]]~~)/(~~[[15/13]])2. ~~From here, we notice that (~~26/~~15)~~/~~(13/9) = 6/5~~, ~~so all that remains is dividing 13/9 into two 6~~/~~5's via the semiparticular S10/S12 = (13/9)/(~~[[6/5|12~~/10~~]]~~)2, hence explaining the mapping of the entire 2.3.5.13 subgroup~~.

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 accuracy of it as providing a slightly flat 5/4 in ideal tunings lends a possible (but complex) extension for prime 41 via [[32/25]][[~]][[41/32]], tempering out [[1025/1024]].~~

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

Subgroup: 2.3.5.13

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Optimal tunings:

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

: 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.

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 extends easily with [[prime interval|prime]] [[13/1|13]]. The [[S-expression]]-based comma list of this extension is {[[169/168|S13]]~~, ~~[[225~~/~~224|S15 = S25⋅S26⋅S27]], [[325~~/~~324|S10~~/~~S12 = S25⋅S26]], ([[625/624|S25]], [[676/675|S26 = S13/S15]], [[729~~/~~728|S27]])}~~.

=== 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 [[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.

Subgroup: 2.3.5.7.13

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Badness (Sintel): 1.61

[[Category:Kleismic family| ]]

[[Category:Kleismic]]

[[Category:Temperament families]]

[[Category:Catalogs of rank-2 temperaments]]

@@ 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]].
+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 structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap| {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]],) [[676/675|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] S13/S15 = ([[4/3|16/12]])/([[15/13]])<sup>2</sup>. From here, we notice that (26/15)/(13/9) = 6/5, so all that remains is dividing 13/9 into two 6/5's via the semiparticular S10/S12 = (13/9)/([[6/5|12/10]])<sup>2</sup>, hence explaining the mapping of the entire 2.3.5.13 subgroup.
+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 accuracy of it as providing a slightly flat 5/4 in ideal tunings lends a possible (but complex) extension for prime 41 via [[32/25]][[~]][[41/32]], tempering out [[1025/1024]].
+For a version of cata with a 1\5 period, see [[Thunderclysmic]].
-For a version of cata with a 1\5 period, see [[Thunderclysmic]].
 Subgroup: 2.3.5.13
@@ Line 54: / Line 52: @@
 Optimal tunings:
 * WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}
+: 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.
+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 extends easily with [[prime interval|prime]] [[13/1|13]]. The [[S-expression]]-based comma list of this extension is {[[169/168|S13]], [[225/224|S15 = S25⋅S26⋅S27]], [[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]], [[676/675|S26 = S13/S15]], [[729/728|S27]])}.
 === 7-limit ===
@@ Line 89: / 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 [[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.
 Subgroup: 2.3.5.7.13
@@ Line 1,249: / Line 1,251: @@
 Badness (Sintel): 1.61
+[[Category:Kleismic family| ]] <!-- main article -->
+[[Category:Kleismic]]
+[[Category:Temperament families]]
+[[Category:Catalogs of rank-2 temperaments]]