Octagar family: Difference between revisions

(12 intermediate revisions by 7 users not shown)

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.

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.

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

== Octagar ==

Subgroup: 2.3.5.7

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

[[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 20:

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

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

[[Optimal tuning]] (POTE): ~3/2 = 703.6224, ~21/20 = 89.3227

[[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, 41, 53, 94, 121, ~~162~~, ~~189~~, ~~215~~, ~~230~~ }}

{{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>

Line 38:

Line 45:

== 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 57:

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

[[Optimal tuning]] (POTE): ~3/2 = 703.8837, ~21/20 = 87.8919

[[Badness]]: 0.539 × 10<sup>-3</sup>

Line 65:

Line 74:

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 tuning (POTE): ~3/2 = 701.8881, ~21/20 = 87.4143

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

== Octasand ==

[[Subgroup]]: 2.3.5.7.11

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

[[Optimal tuning]] (POTE): ~3/2 = 703.5501, ~21/20 = 89.3956

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

=== 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 tuning (POTE): ~3/2 = 703.5688, ~21/20 = 89.4032

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

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

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

[[Category:Temperament families]]

[[Category:~~Temperament family~~]]

[[Category:Pages with mostly numerical content]]

[[Category:Octagar family| ]]

[[Category:Octagar| ]]

[[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 [[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.
+[[Linear temperament]]s that temper out the octagar comma can be found in [[octagar temperaments]].
 == Octagar ==
-Subgroup: 2.3.5.7
+This temperament is also known as ''octagari''.
+[[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 20: @@
 : 63/50 length = 0.8966, 21/20 length = 1.0605
 : Angle (63/50, 21/20) = 97.743 degrees
+[[Optimal tuning]] (POTE): ~3/2 = 703.6224, ~21/20 = 89.3227
 [[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, 41, 53, 94, 121, 162, 189, 215, 230 }}
+{{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>
@@ Line 38: / Line 45: @@
 == 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 57: @@
 : Angle (11/7, 22/21) = 97.747 degrees
-{{Val list|legend=1| 12, 15, 26, 29, 41 }}
+[[Optimal tuning]] (POTE): ~3/2 = 703.8837, ~21/20 = 87.8919
+{{Optimal ET sequence|legend=1| 12, 15, 26, 27e, 41, 109e }}
 [[Badness]]: 0.539 × 10<sup>-3</sup>
@@ Line 65: / Line 74: @@
 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 tuning (POTE): ~3/2 = 701.8881, ~21/20 = 87.4143
+Optimal ET sequence: {{Optimal ET sequence| 12f, 14cf, 15, 26, 29, 41 }}
+== 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]] (POTE): ~3/2 = 703.5501, ~21/20 = 89.3956
+{{Optimal ET sequence|legend=1| 12e, 14c, 27e, 41, 53, 80, 94, 121 }}
+=== 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 tuning (POTE): ~3/2 = 703.5688, ~21/20 = 89.4032
-Vals: {{Val list| 12f, 15, 26, 29, 41 }}
+Optimal ET sequence: {{Optimal ET sequence| 12e, 14c, 27e, 41, 53, 80, 94, 121 }}
-[[Category:Regular temperament theory]]
+[[Category:Temperament families]]
-[[Category:Temperament family]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Octagar family| ]] <!-- main article -->
+[[Category:Octagar| ]] <!-- key article -->
 [[Category:Rank 3]]