Octagar family: Difference between revisions

(5 intermediate revisions by the same user not shown)

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.

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 25:

Line 29:

* [[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]]:

Line 49:

Line 52:

== Nakika ==

Nakika tempers out [[100/99]] and [[245/242]], and is notably [[associated temperament|associated]] with [[octacot]].

[[Subgroup]]: 2.3.5.7.11

Line 66:

Line 71:

* [[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 }}

~~~~

Line 73:

Line 77:

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

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

Scales: [[nakika12]]

Line 83:

Line 85:

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

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

{{Mapping|legend=0| 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}}

~~~~

Line 106:

Line 107:

* [[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 }}

Line 117:

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 }}

{{Mapping|legend=0| 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 }}

Line 128:

Line 127:

Badness (Sintel): 1.03

[[Category:Octagar family| ]]

[[Category:Temperament families]]

[[Category:~~Octagar family| ]] <!~~-~~- main article -->~~

[[Category:Catalogs of rank-3 temperaments]]

~~[[Category:Rank~~ 3]]

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-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
@@ Line 25: / Line 29: @@
 * [[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 }}
-<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.6224{{c}}, ~21/20 = 89.3227{{c}} -->
 [[Minimax tuning]]:
@@ Line 49: / Line 52: @@
 == Nakika ==
+Nakika tempers out [[100/99]] and [[245/242]], and is notably [[associated temperament|associated]] with [[octacot]].
 [[Subgroup]]: 2.3.5.7.11
@@ Line 66: / Line 71: @@
 * [[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 }}
-<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.8837{{c}}, ~21/20 = 87.8919{{c}} -->
 {{Optimal ET sequence|legend=1| 12, 15, 26, 27e, 41, 109e }}
@@ Line 73: / Line 77: @@
 [[Projection pair]]s: <code>5 242/49 7 21296/3087 11 234256/21609</code> to 2.3.11/7
-[[Associated temperament]]: [[octacot]]
 Scales: [[nakika12]]
@@ Line 83: / Line 85: @@
 Comma list: 100/99, 105/104, 245/242
-Mapping: {{mapping| 1 0 1 4 4 2 | 0 1 0 -2 -2 -1 | 0 0 2 3 4 5 }}
+{{Mapping|legend=0| 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}}
-<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8881{{c}}, ~21/20 = 87.4143{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 14cf, 15, 26, 29, 41 }}
@@ Line 106: / Line 107: @@
 * [[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 }}
-<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.5501{{c}}, ~21/20 = 89.3956{{c}} -->
 {{Optimal ET sequence|legend=1| 27e, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d }}
@@ Line 117: / Line 117: @@
 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 }}
+{{Mapping|legend=0| 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}}
-<!-- * POTE: ~2 = 1200.0000{{c}}, ~3/2 = 703.5688{{c}}, ~21/20 = 89.4032{{c}} -->
 {{Optimal ET sequence|legend=0| 27e, 41, 53, 80, 94, 121, 174d, 215, 295d, 336def }}
@@ Line 128: / Line 127: @@
 Badness (Sintel): 1.03
+[[Category:Octagar family| ]] <!-- main article -->
 [[Category:Temperament families]]
-[[Category:Octagar family| ]] <!-- main article -->
+[[Category:Catalogs of rank-3 temperaments]]
-[[Category:Rank 3]]