Octagar family: Difference between revisions

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The '''octagar family''' of ~~temperaments are~~ [[~~planar~~ temperament]]s tempering out [[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 7s 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]].

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

== Octagar ==

Subgroup: 2.3.5.7

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

[[Comma list]]: [[4000/3969]]

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

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

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

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

Line 15:

Line 23:

: 63/50 length = 0.8966, 21/20 length = 1.0605

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

[[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| 1 0 0 0 ~~}}, {{monzo~~| 5/6 1/3 1/2 -1/3 ~~}}, {{monzo~~| 0 0 1 0 ~~}}, {{monzo~~| 5/6 -2/3 1/2 2/3 }}]

: {{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~~]]~~s (unchanged intervals)~~: 2, 7/~~6, 5/4~~

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

* [[9-odd-limit]]

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

: {{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~~]]~~s (unchanged intervals)~~: 2, 5~~/4,~~ 9/7

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

{{~~Val list~~|legend=1| 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d }}

{{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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== Nakika ==

Subgroup: 2.3.5.7.11

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 100/99, 245/242

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

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

Line 50:

Line 64:

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

[[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]]

Line 65:

Line 85:

Comma list: 100/99, 105/104, 245/242

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

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

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}}

Badness (Sintel): 0.832

== Octasand ==

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 540/539, 2200/2187

[[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| 27e, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d }}

[[Badness]] (Sintel): 1.20

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 540/539

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

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 }}

~~Vals~~: ~~{{Val list| 12f, 14cf, 15, 26, 29, 41 }}~~

Badness (Sintel): 1.03

~~[[Category:Regular temperament theory]]~~

[[Category:Temperament families]]

[[Category:Octagar family| ]]

[[Category:Rank 3]]

@@ Line 1: / Line 1: @@
-The '''octagar family''' of temperaments are [[planar temperament]]s tempering out [[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 7s 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]].
+For [[rank-2 temperament]]s that temper out the octagar comma, see [[Octagar temperaments]].
 == Octagar ==
-Subgroup: 2.3.5.7
+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
 [[Comma list]]: [[4000/3969]]
-[[Mapping]]: [{{val| 1 0 1 4 }}, {{val| 0 1 0 -2 }}, {{val| 0 0 2 3 }}]
+{{Mapping|legend=1| 1 0 1 4 | 0 1 0 -2 | 0 0 2 3 }}
+: Mapping generators: ~2, ~3, ~63/40
-Mapping generators: ~2, ~3, ~63/40
 [[Mapping to lattice]]: [{{val| 0 -1 -2 -1 }}, {{val| 0 -1 0 2 }}]
@@ Line 15: / Line 23: @@
 : 63/50 length = 0.8966, 21/20 length = 1.0605
 : Angle (63/50, 21/20) = 97.743 degrees
+[[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| 1 0 0 0 }}, {{monzo| 5/6 1/3 1/2 -1/3 }}, {{monzo| 0 0 1 0 }}, {{monzo| 5/6 -2/3 1/2 2/3 }}]
+: {{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]]s (unchanged intervals): 2, 7/6, 5/4
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3
 * [[9-odd-limit]]
-: [{{monzo| 1 0 0 0 }}, {{monzo| 5/8 1/2 3/8 -1/4 }}, {{monzo| 0 0 1 0 }}, {{monzo| 5/4 -1 3/4 1/2 }}]
+: {{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]]s (unchanged intervals): 2, 5/4, 9/7
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7
-{{Val list|legend=1| 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d }}
+{{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 38: / Line 52: @@
 == Nakika ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 100/99, 245/242
-[[Mapping]]: [{{val| 1 0 1 4 4 }}, {{val| 0 1 0 -2 -2 }}, {{val| 0 0 2 3 4 }}]
+{{Mapping|legend=1| 1 0 1 4 4 | 0 1 0 -2 -2 | 0 0 2 3 4 }}
 Mapping to lattice: [{{val| 0 1 2 1 2 }}, {{val| 0 -1 0 2 2 }}]
@@ Line 50: / Line 64: @@
 : Angle (11/7, 22/21) = 97.747 degrees
-{{Val list|legend=1| 12, 15, 26, 27e, 41, 109e }}
+[[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 65: / Line 85: @@
 Comma list: 100/99, 105/104, 245/242
-Mapping: [{{val| 1 0 1 4 4 2 }}, {{val| 0 1 0 -2 -2 -1 }}, {{val| 0 0 2 3 4 5 }}]
+Mapping: {{mapping| 1 0 1 4 4 2 | 0 1 0 -2 -2 -1 | 0 0 2 3 4 5 }}
+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 }}
+Badness (Sintel): 0.832
+== Octasand ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 540/539, 2200/2187
+{{Mapping|legend=1| 1 0 1 4 -5 | 0 1 0 -2 7 | 0 0 2 3 -4 }}
+[[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| 27e, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d }}
+[[Badness]] (Sintel): 1.20
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 325/324, 352/351, 540/539
+Mapping: {{mapping| 1 0 1 4 -5 0 | 0 1 0 -2 7 4 | 0 0 2 3 -4 -4 }}
+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 }}
-Vals: {{Val list| 12f, 14cf, 15, 26, 29, 41 }}
+Badness (Sintel): 1.03
-[[Category:Regular temperament theory]]
 [[Category:Temperament families]]
 [[Category:Octagar family| ]] <!-- main article -->
 [[Category:Rank 3]]