Sensamagic family: Difference between revisions

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Sensamagic is generated by a perfect fifth and a wide supermajor third of ~[[9/7]], two of which make ~[[5/3]].

Sensamagic is generated by a perfect fifth and a wide supermajor third of ~[[9/7]], two of which make ~[[5/3]]. Among the good edo tunings are [[87edo]] and [[128edo]], as well as the [[optimal patent val]] [[283edo]].

A notable tuning ~~of sensamagic~~ is given by [[TE]], [[CTE]] and [[POTE]], all coinciding at 703.7424{{c}}, 440.9020{{c}} with pure octaves since prime 2 is not involved in the comma to begin with.

Another notable tuning is given by [[TE]], [[CTE]] and [[POTE]], all coinciding at 703.7424{{c}}, 440.9020{{c}} with pure octaves since prime 2 is not involved in the comma to begin with, though its difference from [[CWE]] is practically unnoticeable.

[[Subgroup]]: 2.3.5.7

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=== Overview to extensions ===

Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]). ~~Considered~~ below ~~are undecimal sensamagic, sensawer, octarod, shrusus, bisector and sensigh~~.

The second comma in the comma list defines which [[11-limit]] family member we are looking at. Undecimal sensamagic adds [[385/384]], sensawer adds [[441/440]], octarod adds [[100/99]], shrusus adds [[176/175]]. These temperaments use the same generators as sensamagic. Bisector adds [[121/120]] with a half-octave period.

Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]) and [[sensigh]] (→ [[Sengic family #Sensigh|Sengic family]]). The rest are considered below.

== Undecimal sensamagic ==

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== Octarod ==

Octarod tempers out 100/99 and the interval class of [[11/1|11]] is found as a stack of four ~9/7's. The name ''octarod'' was the former name of the sensamagic comma before being ~~used~~ for this 11-limit extension, and comes from [[octacot]] and [[rodan]]; it should be noted however that rodan does not temper out 100/99 and therefore does not support this temperament.

Octarod tempers out [[100/99]] and the interval class of [[11/1|11]] is found as a stack of four ~9/7's. The name ''octarod'' was the former name of the sensamagic comma before being reused for this 11-limit extension, and comes from [[octacot]] and [[rodan]]; it should be noted however that rodan does not temper out 100/99 and therefore does not support this temperament.

[[Subgroup]]: 2.3.5.7.11

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: error map: {{val| 0.000 +2.570 -3.230 +0.944 +5.801 }}

{{Optimal ET sequence|legend=1| 14c, 19, 22, 27e, 41, 90e, 131e }} *

{{Optimal ET sequence|legend=1| 14c, 19, 22, 27e, 41, 90e, 131e}}*

<nowiki/>[[~~optimal~~ patent val]]: [[104edo|104]]

<nowiki/>*[[Optimal patent val]]: [[104edo|104]]

[[Badness]] (Sintel): 0.698

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[[Badness]] (Sintel): 1.31

~~== Sensigh ==~~

~~Sensigh uses the same mapping as 7-limit [[sensi]] with an independent generator for prime 11.~~

~~[[Subgroup]]: 2.3.5.7.11~~

~~[[Comma list]]: 126/125, 245/243~~

~~{{Mapping|legend=1| 1 -1 -1 -2 3 | 0 7 9 13 0 | 0 0 0 0 1 }}~~

~~: mapping generators: ~2, ~9/7, ~11~~

~~[[Optimal tuning]]s:~~

* [[WE]]: ~2 = 1199.7081{{c}}, ~9/7 = 443.2748{{c}}, ~11/8 = 552.1736{{c}}

~~: [[error map]]: {{val| -0.2919 +1.2608 +3.4518 -5.6691 -0.0202 }}~~

* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3493{{c}}, ~11/8 = 551.8069{{c}}

~~: error map: {{val| 0.0000 +1.4899 +3.8297 -5.2854 +0.4890 }}~~

~~{{Optimal ET sequence|legend=1| 27e, 38df, 46, 111d }}~~

~~[[Badness]] (Sintel): 1.48~~

~~=== 13-limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

~~Comma list: 91/90, 126/125, 169/168~~

~~Mapping: {{mapping| 1 -1 -1 -2 3 0 | 0 7 9 13 0 10 | 0 0 0 0 1 0 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.0000{{c}}, ~9/7 = 443.4379{{c}}, ~11/8 = 550.3462{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}, ~11/8 = 550.7092{{c}}

~~{{Optimal ET sequence|legend=0| 27e, 38df, 46, 111df }}~~

~~Badness (Sintel): 0.878~~

~~=== 17-limit ===~~

~~Subgroup: 2.3.5.7.11.13.17~~

~~Comma list: 91/90, 126/125, 154/153, 169/168~~

~~Mapping: {{mapping| 1 -1 -1 -2 3 0 4 | 0 7 9 13 0 10 -1 | 0 0 0 0 1 0 1 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.2286{{c}}, ~9/7 = 443.4291{{c}}, ~11/8 = 549.2790{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3707{{c}}, ~11/8 = 549.5775{{c}}

~~{{Optimal ET sequence|legend=0| 27eg, 38df, 46 }}~~

~~Badness (Sintel): 0.917~~

[[Category:Temperament families]]

~~[[Category:Pages with mostly numerical content]]~~

[[Category:Sensamagic family| ]]

~~[[Category:Sensamagic| ]] ~~

[[Category:Rank 3]]

@@ Line 7: / Line 7: @@
 {{Main| Sensamagic }}
-Sensamagic is generated by a perfect fifth and a wide supermajor third of ~[[9/7]], two of which make ~[[5/3]].
+Sensamagic is generated by a perfect fifth and a wide supermajor third of ~[[9/7]], two of which make ~[[5/3]]. Among the good edo tunings are [[87edo]] and [[128edo]], as well as the [[optimal patent val]] [[283edo]].
-A notable tuning of sensamagic is given by [[TE]], [[CTE]] and [[POTE]], all coinciding at 703.7424{{c}}, 440.9020{{c}} with pure octaves since prime 2 is not involved in the comma to begin with.
+Another notable tuning is given by [[TE]], [[CTE]] and [[POTE]], all coinciding at 703.7424{{c}}, 440.9020{{c}} with pure octaves since prime 2 is not involved in the comma to begin with, though its difference from [[CWE]] is practically unnoticeable.
 [[Subgroup]]: 2.3.5.7
@@ Line 54: / Line 54: @@
 === Overview to extensions ===
-Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]). Considered below are undecimal sensamagic, sensawer, octarod, shrusus, bisector and sensigh.
+The second comma in the comma list defines which [[11-limit]] family member we are looking at. Undecimal sensamagic adds [[385/384]], sensawer adds [[441/440]], octarod adds [[100/99]], shrusus adds [[176/175]]. These temperaments use the same generators as sensamagic. Bisector adds [[121/120]] with a half-octave period.
+Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]) and [[sensigh]] (→ [[Sengic family #Sensigh|Sengic family]]). The rest are considered below.
 == Undecimal sensamagic ==
@@ Line 132: / Line 134: @@
 == Octarod ==
-Octarod tempers out 100/99 and the interval class of [[11/1|11]] is found as a stack of four ~9/7's. The name ''octarod'' was the former name of the sensamagic comma before being used for this 11-limit extension, and comes from [[octacot]] and [[rodan]]; it should be noted however that rodan does not temper out 100/99 and therefore does not support this temperament.
+Octarod tempers out [[100/99]] and the interval class of [[11/1|11]] is found as a stack of four ~9/7's. The name ''octarod'' was the former name of the sensamagic comma before being reused for this 11-limit extension, and comes from [[octacot]] and [[rodan]]; it should be noted however that rodan does not temper out 100/99 and therefore does not support this temperament.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 146: / Line 148: @@
 : error map: {{val| 0.000 +2.570 -3.230 +0.944 +5.801 }}
-{{Optimal ET sequence|legend=1| 14c, 19, 22, 27e, 41, 90e, 131e }} *
+{{Optimal ET sequence|legend=1| 14c, 19, 22, 27e, 41, 90e, 131e}}*
-<nowiki/>[[optimal patent val]]: [[104edo|104]]
+<nowiki/>*[[Optimal patent val]]: [[104edo|104]]
 [[Badness]] (Sintel): 0.698
@@ Line 203: / Line 205: @@
 [[Badness]] (Sintel): 1.31
-== Sensigh ==
-Sensigh uses the same mapping as 7-limit [[sensi]] with an independent generator for prime 11.
-[[Subgroup]]: 2.3.5.7.11
-[[Comma list]]: 126/125, 245/243
-{{Mapping|legend=1| 1 -1 -1 -2 3 | 0 7 9 13 0 | 0 0 0 0 1 }}
-: mapping generators: ~2, ~9/7, ~11
-[[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.7081{{c}}, ~9/7 = 443.2748{{c}}, ~11/8 = 552.1736{{c}}
-: [[error map]]: {{val| -0.2919 +1.2608 +3.4518 -5.6691 -0.0202 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3493{{c}}, ~11/8 = 551.8069{{c}}
-: error map: {{val| 0.0000 +1.4899 +3.8297 -5.2854 +0.4890 }}
-{{Optimal ET sequence|legend=1| 27e, 38df, 46, 111d }}
-[[Badness]] (Sintel): 1.48
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 91/90, 126/125, 169/168
-Mapping: {{mapping| 1 -1 -1 -2 3 0 | 0 7 9 13 0 10 | 0 0 0 0 1 0 }}
-Optimal tunings:
-* WE: ~2 = 1200.0000{{c}}, ~9/7 = 443.4379{{c}}, ~11/8 = 550.3462{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}, ~11/8 = 550.7092{{c}}
-{{Optimal ET sequence|legend=0| 27e, 38df, 46, 111df }}
-Badness (Sintel): 0.878
-=== 17-limit ===
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 91/90, 126/125, 154/153, 169/168
-Mapping: {{mapping| 1 -1 -1 -2 3 0 4 | 0 7 9 13 0 10 -1 | 0 0 0 0 1 0 1 }}
-Optimal tunings:
-* WE: ~2 = 1200.2286{{c}}, ~9/7 = 443.4291{{c}}, ~11/8 = 549.2790{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3707{{c}}, ~11/8 = 549.5775{{c}}
-{{Optimal ET sequence|legend=0| 27eg, 38df, 46 }}
-Badness (Sintel): 0.917
 [[Category:Temperament families]]
-[[Category:Pages with mostly numerical content]]
 [[Category:Sensamagic family| ]] <!-- main article -->
-[[Category:Sensamagic| ]] <!-- key article -->
 [[Category:Rank 3]]