Hemimean family: Difference between revisions

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The '''hemimean family''' of ~~temperaments~~ are rank-3 ~~temperaments tempering~~ out [[3136/3125]].

The '''hemimean family''' of [[temperament]]s are [[rank-3 temperament]]s which [[temper out]] [[3136/3125]].

The hemimean comma, 3136/3125, is the ratio between the ~~diesis~~ and the ~~tritonic diesis~~, ~~or jubilisma; that is~~, (128/125)/(50/49).

The hemimean comma, 3136/3125, is the ratio between the [[126/125|septimal semicomma (126/125)]] and the [[225/224|septimal kleisma (225/224)]]. This fact alone makes hemimean a very notable rank-3 temperament, as any non-meantone tuning of hemimean will split the [[81/80|syntonic comma (81/80)]] into two equal parts, each representing 126/125~225/224.

Other equivalences characteristic to hemimean are [[128/125]]~[[50/49]] and [[49/45]]~([[25/24]])2.

== Hemimean ==

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[[Comma list]]: 3136/3125 (hemimean)

~~[[Mapping]]: [~~{{~~val~~| 1 0 0 -3 ~~}}, {{val~~| 0 1 0 0 ~~}}, {{val~~| 0 0 2 5 }}]

~~Mapping~~ generators: ~2, ~3, ~56/25

: mapping generators: ~2, ~3, ~56/25

[[Mapping to lattice]]: [{{~~val~~| 0 0 2 5 ~~}}, {{val~~| 0 1 0 0 }}]

[[Mapping to lattice]]: {{mapping| 0 0 2 5 | 0 1 0 0 }}

Lattice basis:

: 28/25 length = 0.5055, 3/2 length = 1.5849

: Angle (28/25, 3/2) = 90 degrees

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9550, ~28/25 = 193.6499

[[Minimax tuning]]:

* 7- and [[9-odd-limit]]

* [[7-odd-limit|7-]] and [[9-odd-limit]]

: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 6/5 0 0 2/5 }}, {{monzo| 0 0 0 1 }}]

: [[~~Eigenmonzo~~ basis]]: 2.3.7

: [[eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.7

{{~~Val list~~|legend=1| 12, 19, 31, 68, 80, 87, 99, 217, 229, 328, 347, 446~~, 545c~~, 675c }}

{{Optimal ET sequence|legend=1| 12, 19, 31, 68, 80, 87, 99, 217, 229, 328, 347, 446, 675c }}

[[Badness]]: 0.160 × 10-3

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[[Projection pair]]s: 5 3136/625 7 68841472/9765625 to 2.3.25/7

==== Hemimean orion ====

=== Hemimean orion ===

As tempering ~~either~~ [[~~256~~/~~255|S16~~]]/[[~~324~~/~~323|S18~~]] = [[~~1216~~/~~1215~~]] or [[~~324~~/~~323|S18~~]]/[[~~400~~/~~399|S20~~]] = [[1701/1700]] ~~implies~~ the ~~other in~~ the ~~context of orion~~ with the ~~effect of extending to include prime~~ 3 ~~in the~~ subgroup ~~and as this therefore gives us both~~ S16 = S18 ~~= S20 and~~ S17 = S19, ~~it can be considered natural to add these commas~~, ~~because {~~S16/S18, S17/S19, S18/S20} ~~implies all the aforementioned commas of orion~~. ~~However~~, ~~this is an extension of hemimean because the~~ ~~~17/16 generator of orion is no longer present and instead we have a~~ ~3/2 ~~generator~~. ~~Orion is described next~~.

As the second generator of hemimean, [[28/25]], is close to [[19/17]], and as the latter is the [[mediant]] of [[10/9]] and [[9/8]], it is natural to extend hemimean to the 2.3.5.7.17.19 subgroup by tempering out ([[28/25]])/([[19/17]]) = [[476/475]], or equivalently stated, the [[semiparticular]] (5/4)/(19/17)2 = [[1445/1444]]. Notice 3136/3125 = (476/475)([[2128/2125]]) and that 2128/2125 = ([[1216/1215]])([[1701/1700]]), so it makes sense to temper out 1216/1215 and/or 1701/1700 as well. An interesting tuning not in the optimal ET sequence is [[111edo]]. This temperament finds the harmonic 17 and 19 at (+5, +1) and (+5, +2), respectively, with virtually no additional error.

The [[S-expression]]-based comma list for the 2.3.5.7.17.19 subgroup extension is {[[1216/1215|S16/S18]], [[1445/1444|S17/S19]], [[1701/1700|S18/S20]](, ([[136/135|S16*S17]])/([[190/189|S19*S20]]) = [[476/475|S16/S18 * S17/S19 * S18/S20]])}.

Subgroup: 2.3.5.7.17

Comma list: 1701/1700, 3136/3125

Sval mapping: {{mapping| 1 0 0 -3 -5 | 0 1 0 0 5 | 0 0 2 5 1 }}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.1960, ~28/25 = 193.6548

{{Optimal ET sequence|legend=1| 12, 19g, 31g, …, 87, 99, 217, 229, 316, 328h, 446, 545c, 873cg }}

Badness: 0.573

==== 2.3.5.7.17.19 subgroup ====

Subgroup: 2.3.5.7.17.19

Comma list: 476/475, 1216/1215, 1445/1444

~~Mapping~~: [{{~~val~~| 1 0 0 -3 -5 -6 ~~}}, {{val~~| 0 1 0 0 5 5 ~~}}, {{val~~| 0 0 2 5 1 2 }}]

Sval mapping: {{mapping| 1 0 0 -3 -5 -6 | 0 1 0 0 5 5 | 0 0 2 5 1 2 }}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~28/25 = 193.647

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~19/17 = 193.647

Optimal ~~GPV~~ sequence~~: {{Val list~~| 12, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}

{{Optimal ET sequence|legend=1| 12, 19gh, 31gh, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}

Badness: 0.456

=== Semiorion ===

Semiorion is an alternative subgroup extension of lower complexity, which splits the octave into two. The [[S-expression]]-based comma list for the 2.3.5.7.17.19 subgroup extension is {[[289/288|S17]], [[361/360|S19]], [[1216/1215|S16/S18]](, [[1701/1700|S18/S20]], [[476/475]] = [[2128/2125|S16/S20]] * [[1445/1444|S17/S19]])}.

Subgroup: 2.3.5.7.17

Comma list: 289/288, 3136/3125

Sval mapping: {{mapping| 2 0 0 -6 5 | 0 1 0 0 1 | 0 0 2 5 0 }}

: sval mapping generators: ~17/12, ~3, ~56/25

Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.3471, ~28/25 = 193.6499

{{Optimal ET sequence|legend=1| 12, 30d, 38d, 50, 62, 68, 106d, 118, 248g, 316g }}

Badness: 1.095

==== 2.3.5.7.17.19 subgroup ====

Subgroup: 2.3.5.7.17.19

Comma list: 289/288, 361/360, 476/475

Mapping: {{mapping| 2 0 0 -6 5 3 | 0 1 0 0 1 1 | 0 0 2 5 0 1 }}

Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.509, ~28/25 = 193.669

{{Optimal ET sequence|legend=1| 12, …, 50, 68, 106d, 118, 248g, 316g }}

Badness: 0.569

== Belobog ==

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[[Comma list]]: 441/440, 3136/3125

~~[[Mapping]]: [~~{{~~val~~| 1 0 0 -3 -9 ~~}}, {{val~~| 0 1 0 0 2 ~~}}, {{val~~| 0 0 2 5 8 }}]

~~Mapping~~ generators: ~2, ~3, ~56/25

: mapping generators: ~2, ~3, ~56/25

Mapping to lattice: [{{~~val~~| 0 -2 2 5 4 ~~}}, {{val~~| 0 -1 0 0 -2 }}]

Mapping to lattice: {{mapping| 0 -2 2 5 4 | 0 -1 0 0 -2 }}

Lattice basis:

: 28/25 length = 0.3829, 16/15 length = 1.1705

: Angle (28/25, 16/15) = 93.2696

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7205, ~28/25 = 193.5545

[[Minimax tuning]]:

* [[11-odd-limit]]

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 27/22 6/11 -5/22 -3/11 5/22 }}, {{monzo| 24/11 -4/11 -2/11 2/11 2/11 }}, {{monzo| 27/11 -10/11 -5/11 5/11 5/11 }}, {{monzo| 24/11 -4/11 -13/11 2/11 13/11 }}]

: [[Eigenmonzo basis]]: 2.9/7.11/5

: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.9/7.11/5

{{~~Val list~~|legend=1| 12, 19e, 31, 87, 99e, 118, 130, 217, 248~~, 378, 626, 961cd~~ }}

{{Optimal ET sequence|legend=1| 12, 19e, 31, 68e, 87, 99e, 118, 130, 217, 248 }}

[[Badness]]: 0.609 × 10-3

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Comma list: 441/440, 1001/1000, 3136/3125

Mapping: [{{~~val~~| 1 0 0 -3 -9 15 ~~}}, {{val~~| 0 1 0 0 2 -2 ~~}}, {{val~~| 0 0 2 5 8 -7 }}]

Mapping: {{mapping| 1 0 0 -3 -9 15 | 0 1 0 0 2 -2 | 0 0 2 5 8 -7 }}

Optimal ~~GPV sequence~~: {{~~Val list~~| 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e }}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.8219, ~28/25 = 193.5816

{{Optimal ET sequence|legend=1| 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e }}

Badness: 1.11 × 10-3

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Comma list: 196/195, 352/351, 625/624

Mapping: [{{~~val~~| 0 0 -3 -9 -4 ~~}}, {{val~~| 0 1 0 0 2 -1 ~~}}, {{val~~| 0 0 2 5 8 8 }}]

Mapping: {{mapping| 1 0 0 -3 -9 -4 | 0 1 0 0 2 -1 | 0 0 2 5 8 8 }}

Optimal ~~GPV sequence~~: {{~~Val list~~| 12f, 19e, 31, 56, 68e, 87, 118, 205d~~, 263f, 304f, 391df, 509df~~ }}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.5667, ~28/25 = 193.2493

{{Optimal ET sequence|legend=1| 12f, 19e, 31, 56, 68e, 87, 118, 186ef, 205d }}

Badness: 1.26 × 10-3

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[[Comma list]]: 540/539, 3136/3125

~~[[Mapping]]: [~~{{~~val~~| 1 0 0 -3 8 ~~}}, {{val~~| 0 1 0 0 3 ~~}}, {{val~~| 0 0 2 5 -8 }}]

~~Mapping~~ generators: ~2, ~3, ~~~768~~/~~343~~

: mapping generators: ~2, ~3, ~56/25

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.1636, ~28/25 = 193.8645

[[Minimax tuning]]:

* [[11-odd-limit]]

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 8/5 3/5 1/5 0 -1/5 }}, {{monzo| 1 3/2 1/2 0 -1/2 }}, {{monzo| 8/5 3/5 -4/5 0 4/5 }}]

: [[Eigenmonzo basis]]: 2.3.11/10

: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.11/5

{{~~Val list~~|legend=1| 12e, 19, 31, 68e, 80, 99e, 130, 241, 340ce, 371ce, ~~470cde~~, ~~711cde~~ }}

{{Optimal ET sequence|legend=1| 12e, 18e, 19, 31, 68e, 80, 99e, 130, 210e, 241, 340ce, 371ce, 470cdee, 501cde, 581cdee, 711ccdee }}

[[Badness]]: 0.870 × 10-3

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[[Comma list]]: 3025/3024, 3136/3125

~~[[Mapping]]: [~~{{~~val~~| 1 0 2 2 1 ~~}}, {{val~~| 0 1 2 5 2 ~~}}, {{val~~| 0 0 -4 -10 -1 }}]

: mapping generators: ~2, ~3, ~18/11

~~Mapping generators~~: ~2, ~3, ~18/11

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.2875, ~18/11 = 854.3132

{{~~Val list~~|legend=1| 31, 80, 87, 111, 118, 198, 316, 545c~~, 861ce~~ }}

{{Optimal ET sequence|legend=1| 24d, 31, 80, 87, 111, 118, 198, 316, 514c, 545c }}

[[Badness]]: 0.819 × 10-3

[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Hemimean family| ]]

[[Category:Hemimean]]

[[Category:Rank 3]]

@@ Line 1: / Line 1: @@
-The '''hemimean family''' of temperaments are rank-3 temperaments tempering out [[3136/3125]].
+{{Technical data page}}
+The '''hemimean family''' of [[temperament]]s are [[rank-3 temperament]]s which [[temper out]] [[3136/3125]].
-The hemimean comma, 3136/3125, is the ratio between the diesis and the tritonic diesis, or jubilisma; that is, (128/125)/(50/49).
+The hemimean comma, 3136/3125, is the ratio between the [[126/125|septimal semicomma (126/125)]] and the [[225/224|septimal kleisma (225/224)]]. This fact alone makes hemimean a very notable rank-3 temperament, as any non-meantone tuning of hemimean will split the [[81/80|syntonic comma (81/80)]] into two equal parts, each representing 126/125~225/224.
+Other equivalences characteristic to hemimean are [[128/125]]~[[50/49]] and [[49/45]]~([[25/24]])<sup>2</sup>.
 == Hemimean ==
@@ Line 8: / Line 11: @@
 [[Comma list]]: 3136/3125 (hemimean)
-[[Mapping]]: [{{val| 1 0 0 -3 }}, {{val| 0 1 0 0 }}, {{val| 0 0 2 5 }}]
+{{Mapping|legend=1| 1 0 0 -3 | 0 1 0 0 | 0 0 2 5 }}
-Mapping generators: ~2, ~3, ~56/25
+: mapping generators: ~2, ~3, ~56/25
-[[Mapping to lattice]]: [{{val| 0 0 2 5 }}, {{val| 0 1 0 0 }}]
+[[Mapping to lattice]]: {{mapping| 0 0 2 5 | 0 1 0 0 }}
 Lattice basis:
 : 28/25 length = 0.5055, 3/2 length = 1.5849
 : Angle (28/25, 3/2) = 90 degrees
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9550, ~28/25 = 193.6499
 [[Minimax tuning]]:
-* 7- and [[9-odd-limit]]
+* [[7-odd-limit|7-]] and [[9-odd-limit]]
 : [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 6/5 0 0 2/5 }}, {{monzo| 0 0 0 1 }}]
-: [[Eigenmonzo basis]]: 2.3.7
+: [[eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.7
-{{Val list|legend=1| 12, 19, 31, 68, 80, 87, 99, 217, 229, 328, 347, 446, 545c, 675c }}
+{{Optimal ET sequence|legend=1| 12, 19, 31, 68, 80, 87, 99, 217, 229, 328, 347, 446, 675c }}
 [[Badness]]: 0.160 × 10<sup>-3</sup>
@@ Line 31: / Line 36: @@
 [[Projection pair]]s: 5 3136/625 7 68841472/9765625 to 2.3.25/7
-==== Hemimean orion ====
+=== Hemimean orion ===
-As tempering either [[256/255|S16]]/[[324/323|S18]] = [[1216/1215]] or [[324/323|S18]]/[[400/399|S20]] = [[1701/1700]] implies the other in the context of orion with the effect of extending to include prime 3 in the subgroup and as this therefore gives us both S16 = S18 = S20 and S17 = S19, it can be considered natural to add these commas, because {S16/S18, S17/S19, S18/S20} implies all the aforementioned commas of orion. However, this is an extension of hemimean because the ~17/16 generator of orion is no longer present and instead we have a ~3/2 generator. Orion is described next.
+As the second generator of hemimean, [[28/25]], is close to [[19/17]], and as the latter is the [[mediant]] of [[10/9]] and [[9/8]], it is natural to extend hemimean to the 2.3.5.7.17.19 subgroup by tempering out ([[28/25]])/([[19/17]]) = [[476/475]], or equivalently stated, the [[semiparticular]] (5/4)/(19/17)<sup>2</sup> = [[1445/1444]]. Notice 3136/3125 = (476/475)([[2128/2125]]) and that 2128/2125 = ([[1216/1215]])([[1701/1700]]), so it makes sense to temper out 1216/1215 and/or 1701/1700 as well. An interesting tuning not in the optimal ET sequence is [[111edo]]. This temperament finds the harmonic 17 and 19 at (+5, +1) and (+5, +2), respectively, with virtually no additional error.
+The [[S-expression]]-based comma list for the 2.3.5.7.17.19 subgroup extension is {[[1216/1215|S16/S18]], [[1445/1444|S17/S19]], [[1701/1700|S18/S20]](, ([[136/135|S16*S17]])/([[190/189|S19*S20]]) = [[476/475|S16/S18 * S17/S19 * S18/S20]])}.
+Subgroup: 2.3.5.7.17
+Comma list: 1701/1700, 3136/3125
+Sval mapping: {{mapping| 1 0 0 -3 -5 | 0 1 0 0 5 | 0 0 2 5 1 }}
+Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.1960, ~28/25 = 193.6548
+{{Optimal ET sequence|legend=1| 12, 19g, 31g, …, 87, 99, 217, 229, 316, 328h, 446, 545c, 873cg }}
+Badness: 0.573
+==== 2.3.5.7.17.19 subgroup ====
 Subgroup: 2.3.5.7.17.19
 Comma list: 476/475, 1216/1215, 1445/1444
-Mapping: [{{val| 1 0 0 -3 -5 -6 }}, {{val| 0 1 0 0 5 5 }}, {{val| 0 0 2 5 1 2 }}]
+Sval mapping: {{mapping| 1 0 0 -3 -5 -6 | 0 1 0 0 5 5 | 0 0 2 5 1 2 }}
-Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~28/25 = 193.647
+Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~19/17 = 193.647
-Optimal GPV sequence: {{Val list| 12, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}
+{{Optimal ET sequence|legend=1| 12, 19gh, 31gh, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}
 Badness: 0.456
+=== Semiorion ===
+Semiorion is an alternative subgroup extension of lower complexity, which splits the octave into two. The [[S-expression]]-based comma list for the 2.3.5.7.17.19 subgroup extension is {[[289/288|S17]], [[361/360|S19]], [[1216/1215|S16/S18]](, [[1701/1700|S18/S20]], [[476/475]] = [[2128/2125|S16/S20]] * [[1445/1444|S17/S19]])}.
+Subgroup: 2.3.5.7.17
+Comma list: 289/288, 3136/3125
+Sval mapping: {{mapping| 2 0 0 -6 5 | 0 1 0 0 1 | 0 0 2 5 0 }}
+: sval mapping generators: ~17/12, ~3, ~56/25
+Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.3471, ~28/25 = 193.6499
+{{Optimal ET sequence|legend=1| 12, 30d, 38d, 50, 62, 68, 106d, 118, 248g, 316g }}
+Badness: 1.095
+==== 2.3.5.7.17.19 subgroup ====
+Subgroup: 2.3.5.7.17.19
+Comma list: 289/288, 361/360, 476/475
+Mapping: {{mapping| 2 0 0 -6 5 3 | 0 1 0 0 1 1 | 0 0 2 5 0 1 }}
+Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.509, ~28/25 = 193.669
+{{Optimal ET sequence|legend=1| 12, …, 50, 68, 106d, 118, 248g, 316g }}
+Badness: 0.569
 == Belobog ==
@@ Line 51: / Line 101: @@
 [[Comma list]]: 441/440, 3136/3125
-[[Mapping]]: [{{val| 1 0 0 -3 -9 }}, {{val| 0 1 0 0 2 }}, {{val| 0 0 2 5 8 }}]
+{{Mapping|legend=1| 1 0 0 -3 -9 | 0 1 0 0 2 | 0 0 2 5 8 }}
-Mapping generators: ~2, ~3, ~56/25
+: mapping generators: ~2, ~3, ~56/25
-Mapping to lattice: [{{val| 0 -2 2 5 4 }}, {{val| 0 -1 0 0 -2 }}]
+Mapping to lattice: {{mapping| 0 -2 2 5 4 | 0 -1 0 0 -2 }}
 Lattice basis:
 : 28/25 length = 0.3829, 16/15 length = 1.1705
 : Angle (28/25, 16/15) = 93.2696
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.7205, ~28/25 = 193.5545
 [[Minimax tuning]]:
 * [[11-odd-limit]]
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 27/22 6/11 -5/22 -3/11 5/22 }}, {{monzo| 24/11 -4/11 -2/11 2/11 2/11 }}, {{monzo| 27/11 -10/11 -5/11 5/11 5/11 }}, {{monzo| 24/11 -4/11 -13/11 2/11 13/11 }}]
-: [[Eigenmonzo basis]]: 2.9/7.11/5
+: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.9/7.11/5
-{{Val list|legend=1| 12, 19e, 31, 87, 99e, 118, 130, 217, 248, 378, 626, 961cd }}
+{{Optimal ET sequence|legend=1| 12, 19e, 31, 68e, 87, 99e, 118, 130, 217, 248 }}
 [[Badness]]: 0.609 × 10<sup>-3</sup>
@@ Line 79: / Line 131: @@
 Comma list: 441/440, 1001/1000, 3136/3125
-Mapping: [{{val| 1 0 0 -3 -9 15 }}, {{val| 0 1 0 0 2 -2 }}, {{val| 0 0 2 5 8 -7 }}]
+Mapping: {{mapping| 1 0 0 -3 -9 15 | 0 1 0 0 2 -2 | 0 0 2 5 8 -7 }}
-Optimal GPV sequence: {{Val list| 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e }}
+Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.8219, ~28/25 = 193.5816
+{{Optimal ET sequence|legend=1| 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e }}
 Badness: 1.11 × 10<sup>-3</sup>
@@ Line 90: / Line 144: @@
 Comma list: 196/195, 352/351, 625/624
-Mapping: [{{val| 0 0 -3 -9 -4 }}, {{val| 0 1 0 0 2 -1 }}, {{val| 0 0 2 5 8 8 }}]
+Mapping: {{mapping| 1 0 0 -3 -9 -4 | 0 1 0 0 2 -1 | 0 0 2 5 8 8 }}
-Optimal GPV sequence: {{Val list| 12f, 19e, 31, 56, 68e, 87, 118, 205d, 263f, 304f, 391df, 509df }}
+Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.5667, ~28/25 = 193.2493
+{{Optimal ET sequence|legend=1| 12f, 19e, 31, 56, 68e, 87, 118, 186ef, 205d }}
 Badness: 1.26 × 10<sup>-3</sup>
@@ Line 101: / Line 157: @@
 [[Comma list]]: 540/539, 3136/3125
-[[Mapping]]: [{{val| 1 0 0 -3 8 }}, {{val| 0 1 0 0 3 }}, {{val| 0 0 2 5 -8 }}]
+{{Mapping|legend=1| 1 0 0 -3 8 | 0 1 0 0 3 | 0 0 2 5 -8 }}
-Mapping generators: ~2, ~3, ~768/343
+: mapping generators: ~2, ~3, ~56/25
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.1636, ~28/25 = 193.8645
 [[Minimax tuning]]:
 * [[11-odd-limit]]
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 8/5 3/5 1/5 0 -1/5 }}, {{monzo| 1 3/2 1/2 0 -1/2 }}, {{monzo| 8/5 3/5 -4/5 0 4/5 }}]
-: [[Eigenmonzo basis]]: 2.3.11/10
+: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.11/5
-{{Val list|legend=1| 12e, 19, 31, 68e, 80, 99e, 130, 241, 340ce, 371ce, 470cde, 711cde }}
+{{Optimal ET sequence|legend=1| 12e, 18e, 19, 31, 68e, 80, 99e, 130, 210e, 241, 340ce, 371ce, 470cdee, 501cde, 581cdee, 711ccdee }}
 [[Badness]]: 0.870 × 10<sup>-3</sup>
@@ Line 119: / Line 177: @@
 [[Comma list]]: 3025/3024, 3136/3125
-[[Mapping]]: [{{val| 1 0 2 2 1 }}, {{val| 0 1 2 5 2 }}, {{val| 0 0 -4 -10 -1 }}]
+{{Mapping|legend=1| 1 0 2 2 1 | 0 1 2 5 2 | 0 0 -4 -10 -1 }}
+: mapping generators: ~2, ~3, ~18/11
-Mapping generators: ~2, ~3, ~18/11
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.2875, ~18/11 = 854.3132
-{{Val list|legend=1| 31, 80, 87, 111, 118, 198, 316, 545c, 861ce }}
+{{Optimal ET sequence|legend=1| 24d, 31, 80, 87, 111, 118, 198, 316, 514c, 545c }}
 [[Badness]]: 0.819 × 10<sup>-3</sup>
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Hemimean family| ]] <!-- main article -->
 [[Category:Hemimean]]
 [[Category:Rank 3]]