Trisedodge family: Difference between revisions

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== Trisedodge ==

The generator of trisedodge is ~864/625 at around 554 cents, which in all 11-limit extensions is used to represent [[11/8]], and three of them and a period is equal to the harmonic 3. This generator, when reduced to the minimal size, represents 25/24. However, another possible generator is ~6/5, reached by a period plus 25/24, that is, (144/125)(25/24) = 6/5.

In the 11-limit the generator can be taken to be ~11/10, reached as a period minus 25/24, that is, (55/48)/(25/24) = 11/10. Therefore, since a period plus a gen is 6/5 and a period minus a gen is 11/10, we reach 12/11 at 2 gens.

[[Subgroup]]: 2.3.5

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: mapping generators: ~144/125, ~864/625

[[Optimal tuning]] ([[POTE]]): ~144/125 = 1\5, ~25/24 = 74.0077

[[Optimal tuning]]s:

* [[CTE]]: ~144/125 = 1\5, ~864/625 = 553.8249 (~25/24 = 73.8249)

* [[POTE]]: ~144/125 = 1\5, ~864/625 = 554.0077 (~25/24 = 74.0077)

{{Optimal ET sequence|legend=1| 15, 50, 65, 340c, 405c, 470c, 535c, 600c }}

{{Optimal ET sequence|legend=1| 15, 35, 50, 65, 340c, 405c, 470c, 535c, 600c }}

[[Badness]]: 0.252724

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Sval mapping: {{mapping| 5 1 7 15 | 0 3 2 1 }}

Optimal tuning (POTE): ~55/48 = 1\5, ~25/24 = 74.1247

[[Optimal tuning]]s:

* CTE: ~55/48 = 1\5, ~11/8 = 553.7951 (~25/24 = 73.7951)

* POTE: ~55/48 = 1\5, ~11/8 = 554.1247 (~25/24 = 74.1247)

{{Optimal ET sequence|legend=1| 15, 35, 50, 65, 210e, 275e, 340ce }}

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== Septimal trisedodge ==

The generator of this temperament, when reduced to be as small as possible (by using period-reduction and period-complement-reduction) is what is shown in the optimal tunings, however note that the generator is also interpretable as ~6/5 (reached by a period plus 25/24, that is, 144/125 * 25/24 = 6/5). Other possible generators are described below in the limits in which they appear.

[[Subgroup]]: 2.3.5.7

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[[Optimal tuning]]s:

* [[CTE]]: ~144/125 = 1\5, ~25/24 = 74.~~515~~

* [[CTE]]: ~144/125 = 1\5, ~175/128 = 554.5146 (~25/24 = 74.5146)

* [[POTE]]: ~144/125 = 1\5, ~25/24 = 74.9480

* [[POTE]]: ~144/125 = 1\5, ~175/128 = 554.9480 (~25/24 = 74.9480)

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=== 11-limit ===

In the 11-limit the generator can be taken to be ~11/10 (reached as a period minus 25/24, that is, (55/48)/(25/24) = 11/10). Therefore, because a period plus a gen is 6/5 and a period minus a gen is 11/10, we reach 12/11 at 2 gens.

Subgroup: 2.3.5.7.11

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Optimal tunings:

* CTE: ~55/48 = 1\5, ~25/24 = 74.~~466~~

* CTE: ~55/48 = 1\5, ~11/8 = 554.4664 (~25/24 = 74.4664)

* POTE: ~55/48 = 1\5, ~25/24 = 74.9401

* POTE: ~55/48 = 1\5, ~11/8 = 554.9401 (~25/24 = 74.9401)

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Optimal tunings:

* CTE: ~55/48 = 1\5, ~25/24 = 74.~~741~~

* CTE: ~55/48 = 1\5, ~11/8 = 554.7405 (~25/24 = 74.7405)

* POTE: ~55/48 = 1\5, ~25/24 = 75.261

* POTE: ~55/48 = 1\5, ~11/8 = 555.261 (~25/24 = 75.261)

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== Coblack ==

In addition to 126/125, the coblack temperament tempers out the [[cloudy comma]], 16807/16384, which is the amount by which five septimal supermajor seconds ([[8/7]]) fall short of an octave.

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[[Optimal tuning]] ([[POTE]]): ~8/7 = 1\5, ~21/20 = 73.044

[[Optimal tuning]]s:

* [[CTE]]: ~8/7 = 1\5, ~48/35 = 553.8429 (~21/20 = 73.8429)

* [[POTE]]: ~8/7 = 1\5, ~48/35 = 553.044 (~21/20 = 73.044)

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Mapping: {{mapping| 5 1 7 14 15 | 0 3 2 0 1 }}

Optimal ~~tuning~~ (POTE): ~8/7 = 1\5, ~21/20 = 73.264

Optimal tunings:

* CTE: ~8/7 = 1\5, ~11/8 = 553.7951 (~21/20 = 73.7951)

* POTE: ~8/7 = 1\5, ~11/8 = 553.264 (~21/20 = 73.264)

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 == Trisedodge ==
+The generator of trisedodge is ~864/625 at around 554 cents, which in all 11-limit extensions is used to represent [[11/8]], and three of them and a period is equal to the harmonic 3. This generator, when reduced to the minimal size, represents 25/24. However, another possible generator is ~6/5, reached by a period plus 25/24, that is, (144/125)(25/24) = 6/5.
+In the 11-limit the generator can be taken to be ~11/10, reached as a period minus 25/24, that is, (55/48)/(25/24) = 11/10. Therefore, since a period plus a gen is 6/5 and a period minus a gen is 11/10, we reach 12/11 at 2 gens.
 [[Subgroup]]: 2.3.5
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 : mapping generators: ~144/125, ~864/625
-[[Optimal tuning]] ([[POTE]]): ~144/125 = 1\5, ~25/24 = 74.0077
+[[Optimal tuning]]s:
+* [[CTE]]: ~144/125 = 1\5, ~864/625 = 553.8249 (~25/24 = 73.8249)
+* [[POTE]]: ~144/125 = 1\5, ~864/625 = 554.0077 (~25/24 = 74.0077)
-{{Optimal ET sequence|legend=1| 15, 50, 65, 340c, 405c, 470c, 535c, 600c }}
+{{Optimal ET sequence|legend=1| 15, 35, 50, 65, 340c, 405c, 470c, 535c, 600c }}
 [[Badness]]: 0.252724
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 Sval mapping: {{mapping| 5 1 7 15 | 0 3 2 1 }}
-Optimal tuning (POTE): ~55/48 = 1\5, ~25/24 = 74.1247
+[[Optimal tuning]]s:
+* CTE: ~55/48 = 1\5, ~11/8 = 553.7951 (~25/24 = 73.7951)
+* POTE: ~55/48 = 1\5, ~11/8 = 554.1247 (~25/24 = 74.1247)
 {{Optimal ET sequence|legend=1| 15, 35, 50, 65, 210e, 275e, 340ce }}
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 == Septimal trisedodge ==
-The generator of this temperament, when reduced to be as small as possible (by using period-reduction and period-complement-reduction) is what is shown in the optimal tunings, however note that the generator is also interpretable as ~6/5 (reached by a period plus 25/24, that is, 144/125 * 25/24 = 6/5). Other possible generators are described below in the limits in which they appear.
 [[Subgroup]]: 2.3.5.7
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 [[Optimal tuning]]s:
-* [[CTE]]: ~144/125 = 1\5, ~25/24 = 74.515
+* [[CTE]]: ~144/125 = 1\5, ~175/128 = 554.5146 (~25/24 = 74.5146)
-* [[POTE]]: ~144/125 = 1\5, ~25/24 = 74.9480
+* [[POTE]]: ~144/125 = 1\5, ~175/128 = 554.9480 (~25/24 = 74.9480)
 {{Optimal ET sequence|legend=1| 15, 50d, 65d, 80 }}
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 === 11-limit ===
-In the 11-limit the generator can be taken to be ~11/10 (reached as a period minus 25/24, that is, (55/48)/(25/24) = 11/10). Therefore, because a period plus a gen is 6/5 and a period minus a gen is 11/10, we reach 12/11 at 2 gens.
 Subgroup: 2.3.5.7.11
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 Optimal tunings:
-* CTE: ~55/48 = 1\5, ~25/24 = 74.466
+* CTE: ~55/48 = 1\5, ~11/8 = 554.4664 (~25/24 = 74.4664)
-* POTE: ~55/48 = 1\5, ~25/24 = 74.9401
+* POTE: ~55/48 = 1\5, ~11/8 = 554.9401 (~25/24 = 74.9401)
 {{Optimal ET sequence|legend=1| 15, 50d, 65d, 80 }}
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 Optimal tunings:
-* CTE: ~55/48 = 1\5, ~25/24 = 74.741
+* CTE: ~55/48 = 1\5, ~11/8 = 554.7405 (~25/24 = 74.7405)
-* POTE: ~55/48 = 1\5, ~25/24 = 75.261
+* POTE: ~55/48 = 1\5, ~11/8 = 555.261 (~25/24 = 75.261)
 {{Optimal ET sequence|legend=1| 15, 80, 175bcde, 255bcdde }}
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 == Coblack ==
-{{see also| Cloudy clan #Coblack }}
+{{See also| Cloudy clan #Coblack }}
 In addition to 126/125, the coblack temperament tempers out the [[cloudy comma]], 16807/16384, which is the amount by which five septimal supermajor seconds ([[8/7]]) fall short of an octave.
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 {{Mapping|legend=1| 5 1 7 14 | 0 3 2 0 }}
-[[Optimal tuning]] ([[POTE]]): ~8/7 = 1\5, ~21/20 = 73.044
+[[Optimal tuning]]s:
+* [[CTE]]: ~8/7 = 1\5, ~48/35 = 553.8429 (~21/20 = 73.8429)
+* [[POTE]]: ~8/7 = 1\5, ~48/35 = 553.044 (~21/20 = 73.044)
 {{Optimal ET sequence|legend=1| 15, 35, 50, 65, 115d }}
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 Mapping: {{mapping| 5 1 7 14 15 | 0 3 2 0 1 }}
-Optimal tuning (POTE): ~8/7 = 1\5, ~21/20 = 73.264
+Optimal tunings:
+* CTE: ~8/7 = 1\5, ~11/8 = 553.7951 (~21/20 = 73.7951)
+* POTE: ~8/7 = 1\5, ~11/8 = 553.264 (~21/20 = 73.264)
 {{Optimal ET sequence|legend=1| 15, 35, 50, 65, 115d }}