Octagar family: Difference between revisions

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The '''octagar family''' of [[temperament]]~~s are~~ [[~~rank-3~~ temperament]]s [[tempering out]] the octagar comma, [[4000/3969]]. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7-odd-limit minimax tuning has fifths and 7's 2.245 cents sharp, with just major thirds.

The '''octagar family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the octagar comma, [[4000/3969]].

[[~~Linear~~ temperament]]s that temper out the octagar comma ~~can be found in~~ [[~~octagar~~ temperaments]].

For [[rank-2 temperament]]s that temper out the octagar comma, see [[Octagar temperaments]].

== Octagar ==

This temperament is ~~also known~~ as ''octagari''.

Octagar is generated by a [[3/2|perfect fifth]] and a minor sixth of [[~]][[63/40]], two of which make ~[[5/2]], and three make an interval short of two octaves by a [[64/63|septimal comma]].

While many 7-limit rank-3 temperaments exhibit a tendency towards tuning flat, a slight sharp tendency is often preferred instead. Octagar provides this; for instance the [[7-odd-limit]] [[minimax tuning]] has [[3/1|3]]'s and [[7/1|7]]'s 2.245 cents sharp, with just [[5/1|5]]'s.

This temperament is catalogued as ''octagari'' in [[Graham Breed]]'s [https://x31eq.com/temper/ Temperament Finder].

[[Subgroup]]: 2.3.5.7

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: Mapping generators: ~2, ~3, ~63/40

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: Angle (63/50, 21/20) = 97.743 degrees

[[Optimal tuning]] ~~(POTE)~~: ~3/2 = 703.~~6224~~, ~21/20 = 89.~~3227~~

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.5353{{c}}, ~3/2 = 703.3499{{c}}, ~63/40 = 792.6380{{c}}

: [[error map]]: {{val| -0.465 +0.930 -1.502 +1.459 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.3764{{c}}, ~63/40 = 792.6299{{c}}

: error map: {{val| 0.000 +1.421 -1.054 +2.311 }}

[[Minimax tuning]]:

* [[7-odd-limit]]

: {{monzo list| 1 0 0 0 | 5/6 1/3 1/2 -1/3 | 0 0 1 0 | 5/6 -2/3 1/2 2/3 }}

: [[~~Eigenmonzo~~ basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5.7/3

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3

* [[9-odd-limit]]

: {{monzo list| 1 0 0 0 | 5/8 1/2 3/8 -1/4 | 0 0 1 0 | 5/4 -1 3/4 1/2 }}

: [[~~Eigenmonzo~~ basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5.9/7

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7

{{Optimal ET sequence|legend=1| 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d }}

[[Badness]]: 0.~~216 × 10<sup>-3</sup>~~

[[Badness]] (Sintel): 0.951

[[Projection pair]]s: 5 - 3969/800, 7 - 27783/4000 to 2.3.7/5

[[Projection pair]]s: <code>5 3969/800 7 27783/4000</code> to 2.3.7/5

{{Databox|[[Hobbit|Hobbit bases]]|

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: Angle (11/7, 22/21) = 97.747 degrees

[[Optimal tuning]] ~~(POTE)~~: ~3/2 = 703.~~8837~~, ~21/20 = 87.~~8919~~

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.6137{{c}}, ~3/2 = 703.6571{{c}}, ~11/7 = 791.5207{{c}}

: [[error map]]: {{val| -0.386 +1.316 -3.659 -2.351 +6.678 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.6783{{c}}, ~11/7 = 791.5168{{c}}

: error map: {{val| 0.000 +1.723 -3.280 -1.632 +7.393 }}

[[Badness]]: 0.~~539 × 10<sup>-3</sup>~~

[[Badness]] (Sintel): 0.647

[[Projection pair]]s: 5 - 242/49, 7 - 21296/3087, 11 - 234256/21609 to 2.3.11/7

[[Projection pair]]s: <code>5 242/49 7 21296/3087 11 234256/21609</code> to 2.3.11/7

[[Associated temperament]]: [[octacot]]

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

Optimal ~~tuning (POTE)~~: ~3/2 = ~~701~~.~~8881~~, ~21/20 = 87.~~4143~~

Optimal tunings:

* WE: ~2 = 1200.5397{{c}}, ~3/2 = 702.2038{{c}}, ~11/7 = 789.6574{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0977{{c}}, ~11/7 = 789.5686{{c}}

~~Optimal ET sequence~~: ~~{{Optimal ET sequence| 12f, 14cf, 15, 26, 29, 41 }}~~

Badness (Sintel): 0.832

== Octasand ==

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[[Optimal tuning]] ~~(POTE)~~: ~3/2 = 703.~~5501~~, ~21/20 = 89.~~3956~~

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.5255{{c}}, ~3/2 = 703.2719{{c}}, ~63/40 = 792.6321{{c}}

: [[error map]]: {{val| -0.475 +0.842 -1.524 +1.578 +0.108 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.2662{{c}}, ~63/40 = 792.6212{{c}}

: error map: {{val| 0.000 +1.311 -1.071 +2.505 +1.061 }}

{{Optimal ET sequence|legend=1| ~~12e~~, ~~14c, 27e~~, 41, 53, 80, 94, 121 }}

{{Optimal ET sequence|legend=1| 27e, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d }}

[[Badness]] (Sintel): 1.20

=== 13-limit ===

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Mapping: {{mapping| 1 0 1 4 -5 0 | 0 1 0 -2 7 4 | 0 0 2 3 -4 -4 }}

Optimal ~~tuning (POTE)~~: ~3/2 = 703.~~5688~~, ~21/20 = 89.~~4032~~

Optimal tunings:

* WE: ~2 = 1199.5112{{c}}, ~3/2 = 703.2823{{c}}, ~63/40 = 792.6491{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.3684{{c}}, ~63/40 = 792.7875{{c}}

{{Optimal ET sequence|legend=0| 27e, 41, 53, 80, 94, 121, 174d, 215, 295d, 336def }}

~~Optimal ET sequence~~: ~~{{Optimal ET sequence| 12e, 14c, 27e, 41, 53, 80, 94, 121 }}~~

Badness (Sintel): 1.03

[[Category:Temperament families]]

[[Category:Octagar family| ]]

~~[[Category:Octagar| ]] ~~

[[Category:Rank 3]]

@@ Line 1: / Line 1: @@
-The '''octagar family''' of [[temperament]]s are [[rank-3 temperament]]s [[tempering out]] the octagar comma, [[4000/3969]]. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7-odd-limit minimax tuning has fifths and 7's 2.245 cents sharp, with just major thirds.
+{{Technical data page}}
+The '''octagar family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the octagar comma, [[4000/3969]].
-[[Linear temperament]]s that temper out the octagar comma can be found in [[octagar temperaments]].
+For [[rank-2 temperament]]s that temper out the octagar comma, see [[Octagar temperaments]].
 == Octagar ==
-This temperament is also known as ''octagari''.
+Octagar is generated by a [[3/2|perfect fifth]] and a minor sixth of [[~]][[63/40]], two of which make ~[[5/2]], and three make an interval short of two octaves by a [[64/63|septimal comma]].
+While many 7-limit rank-3 temperaments exhibit a tendency towards tuning flat, a slight sharp tendency is often preferred instead. Octagar provides this; for instance the [[7-odd-limit]] [[minimax tuning]] has [[3/1|3]]'s and [[7/1|7]]'s 2.245 cents sharp, with just [[5/1|5]]'s.
+This temperament is catalogued as ''octagari'' in [[Graham Breed]]'s [https://x31eq.com/temper/ Temperament Finder].
 [[Subgroup]]: 2.3.5.7
@@ Line 11: / Line 16: @@
 {{Mapping|legend=1| 1 0 1 4 | 0 1 0 -2 | 0 0 2 3 }}
 : Mapping generators: ~2, ~3, ~63/40
@@ Line 20: / Line 24: @@
 : Angle (63/50, 21/20) = 97.743 degrees
-[[Optimal tuning]] (POTE): ~3/2 = 703.6224, ~21/20 = 89.3227
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.5353{{c}}, ~3/2 = 703.3499{{c}}, ~63/40 = 792.6380{{c}}
+: [[error map]]: {{val| -0.465 +0.930 -1.502 +1.459 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.3764{{c}}, ~63/40 = 792.6299{{c}}
+: error map: {{val| 0.000 +1.421 -1.054 +2.311 }}
 [[Minimax tuning]]:
 * [[7-odd-limit]]
 : {{monzo list| 1 0 0 0 | 5/6 1/3 1/2 -1/3 | 0 0 1 0 | 5/6 -2/3 1/2 2/3 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.7/3
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3
 * [[9-odd-limit]]
 : {{monzo list| 1 0 0 0 | 5/8 1/2 3/8 -1/4 | 0 0 1 0 | 5/4 -1 3/4 1/2 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.9/7
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7
 {{Optimal ET sequence|legend=1| 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d }}
-[[Badness]]: 0.216 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.951
-[[Projection pair]]s: 5 - 3969/800, 7 - 27783/4000 to 2.3.7/5
+[[Projection pair]]s: <code>5 3969/800 7 27783/4000</code> to 2.3.7/5
 {{Databox|[[Hobbit|Hobbit bases]]|
@@ Line 56: / Line 64: @@
 : Angle (11/7, 22/21) = 97.747 degrees
-[[Optimal tuning]] (POTE): ~3/2 = 703.8837, ~21/20 = 87.8919
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.6137{{c}}, ~3/2 = 703.6571{{c}}, ~11/7 = 791.5207{{c}}
+: [[error map]]: {{val| -0.386 +1.316 -3.659 -2.351 +6.678 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.6783{{c}}, ~11/7 = 791.5168{{c}}
+: error map: {{val| 0.000 +1.723 -3.280 -1.632 +7.393 }}
 {{Optimal ET sequence|legend=1| 12, 15, 26, 27e, 41, 109e }}
-[[Badness]]: 0.539 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.647
-[[Projection pair]]s: 5 - 242/49, 7 - 21296/3087, 11 - 234256/21609 to 2.3.11/7
+[[Projection pair]]s: <code>5 242/49 7 21296/3087 11 234256/21609</code> to 2.3.11/7
 [[Associated temperament]]: [[octacot]]
@@ Line 75: / Line 87: @@
 Mapping: {{mapping| 1 0 1 4 4 2 | 0 1 0 -2 -2 -1 | 0 0 2 3 4 5 }}
-Optimal tuning (POTE): ~3/2 = 701.8881, ~21/20 = 87.4143
+Optimal tunings:
+* WE: ~2 = 1200.5397{{c}}, ~3/2 = 702.2038{{c}}, ~11/7 = 789.6574{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0977{{c}}, ~11/7 = 789.5686{{c}}
+{{Optimal ET sequence|legend=0| 12f, 14cf, 15, 26, 29, 41 }}
-Optimal ET sequence: {{Optimal ET sequence| 12f, 14cf, 15, 26, 29, 41 }}
+Badness (Sintel): 0.832
 == Octasand ==
@@ Line 86: / Line 102: @@
 {{Mapping|legend=1| 1 0 1 4 -5 | 0 1 0 -2 7 | 0 0 2 3 -4 }}
-[[Optimal tuning]] (POTE): ~3/2 = 703.5501, ~21/20 = 89.3956
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.5255{{c}}, ~3/2 = 703.2719{{c}}, ~63/40 = 792.6321{{c}}
+: [[error map]]: {{val| -0.475 +0.842 -1.524 +1.578 +0.108 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.2662{{c}}, ~63/40 = 792.6212{{c}}
+: error map: {{val| 0.000 +1.311 -1.071 +2.505 +1.061 }}
-{{Optimal ET sequence|legend=1| 12e, 14c, 27e, 41, 53, 80, 94, 121 }}
+{{Optimal ET sequence|legend=1| 27e, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d }}
+[[Badness]] (Sintel): 1.20
 === 13-limit ===
@@ Line 97: / Line 119: @@
 Mapping: {{mapping| 1 0 1 4 -5 0 | 0 1 0 -2 7 4 | 0 0 2 3 -4 -4 }}
-Optimal tuning (POTE): ~3/2 = 703.5688, ~21/20 = 89.4032
+Optimal tunings:
+* WE: ~2 = 1199.5112{{c}}, ~3/2 = 703.2823{{c}}, ~63/40 = 792.6491{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.3684{{c}}, ~63/40 = 792.7875{{c}}
+{{Optimal ET sequence|legend=0| 27e, 41, 53, 80, 94, 121, 174d, 215, 295d, 336def }}
-Optimal ET sequence: {{Optimal ET sequence| 12e, 14c, 27e, 41, 53, 80, 94, 121 }}
+Badness (Sintel): 1.03
 [[Category:Temperament families]]
 [[Category:Octagar family| ]] <!-- main article -->
-[[Category:Octagar| ]] <!-- key article -->
 [[Category:Rank 3]]