Olympia: Difference between revisions

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The '''olympia''' ([[monzo]]: {{monzo| 17 -5 0 -2 -1 }}, [[ratio]]: 131072/130977), otherwise known as the '''olympic comma''', is an [[unnoticeable comma|unnoticeable]] [[11-limit]] (specifically [[2.3.7.11 subgroup|2.3.7.11-]][[subgroup]]) [[comma]] measuring about 1.26 [[cent]]s.

The '''olympia''' ([[monzo]]: {{monzo| 17 -5 0 -2 -1 }}, [[ratio]]: 131072/130977), otherwise known as the '''olympic comma''', is an [[unnoticeable comma|unnoticeable]] [[11-limit]] (specifically [[2.3.7.11 subgroup|2.3.7.11-]][[subgroup]]) [[comma]] measuring about 1.26 [[cent]]s. It is the difference between the undecimal quartertone ([[33/32]]) and a stack of two septimal commas (([[64/63]])2), which gives rise to its [[S-expression]] as S642⋅S65. Equivalently, it is the difference between [[22/21]] and a stack of three septimal commas.

It is the difference between the undecimal quartertone ([[33/32]]) and a stack of {{nowrap| two septimal commas (([[64/63]])2) }}, which according to its S-expression comma family categorisation as a lopsided comma (S642⋅S65) is trivial information. Interestingly/nontrivially, tempering it out causes {{nowrap| 1/1 - 64/63 - 33/32 - 22/21 - [[1089/1024]] }} to become equidistant, thereby splitting the [[1089/1024|parapotome]] into four equal parts, as {{nowrap| ([[22/21]])/([[33/32]]) {{=}} [[64/63]] }} and {{nowrap| 1089/1024 * S642 * S65 {{=}} 22/21 * 64/63 }}. Even more interesting is the factorization into two [[13-limit]] [[superparticular ratio]]s: ([[2080/2079]])⋅([[4096/4095]]). These ratios and the olympia itself are the default intervals represented by one, two, and three [[mina]]s in the Olympian level of [[Sagittal notation]], from which it gets its name.

It factors into two [[13-limit]] [[superparticular ratio]]s: ([[2080/2079]])⋅([[4096/4095]]). These ratios and the olympia itself are the default intervals represented by one, two, and three [[mina]]s in the Olympian level of [[Sagittal notation]], from which it gets its name.

== Temperaments ==

[[Tempering out]] this comma in the full 11-limit results in the rank-4 olympic temperament (→ [[Rank-4 temperament #Olympic (131072/130977)]]), or in the 2.3.7.11 subgroup, the rank-3 olympian temperament. Olympic has a very natural 13-limit extension {2080/2079, 4096/4095}. As the comma's order of 11 is one, any 7-limit temperament can be extended to the 11-limit by tempering out this comma, but it works best for temperaments with low complexity and high accuracy in the septimal comma.

[[Tempering out]] this comma in the full 11-limit results in the rank-4 olympic temperament (→ [[Rank-4 temperament #Olympic (131072/130977)]]), or in the 2.3.7.11 subgroup, the rank-3 olympian temperament. In either case, it sets 1–64/63–33/32–22/21–[[1089/1024|(33/32)2]] to become equidistant.

Olympic has a very natural 13-limit extension {2080/2079, 4096/4095}. As the comma's order of 11 is one, any 7-limit temperament can be extended to the 11-limit by tempering out this comma, but it works best for temperaments with low complexity and high accuracy in the septimal comma.

== Etymology ==

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-The '''olympia''' ([[monzo]]: {{monzo| 17 -5 0 -2 -1 }}, [[ratio]]: 131072/130977), otherwise known as the '''olympic comma''', is an [[unnoticeable comma|unnoticeable]] [[11-limit]] (specifically [[2.3.7.11 subgroup|2.3.7.11-]][[subgroup]]) [[comma]] measuring about 1.26 [[cent]]s.
+The '''olympia''' ([[monzo]]: {{monzo| 17 -5 0 -2 -1 }}, [[ratio]]: 131072/130977), otherwise known as the '''olympic comma''', is an [[unnoticeable comma|unnoticeable]] [[11-limit]] (specifically [[2.3.7.11 subgroup|2.3.7.11-]][[subgroup]]) [[comma]] measuring about 1.26 [[cent]]s. It is the difference between the undecimal quartertone ([[33/32]]) and a stack of two septimal commas (([[64/63]])<sup>2</sup>), which gives rise to its [[S-expression]] as S64<sup>2</sup>⋅S65. Equivalently, it is the difference between [[22/21]] and a stack of three septimal commas.
-It is the difference between the undecimal quartertone ([[33/32]]) and a stack of {{nowrap| two septimal commas (([[64/63]])<sup>2</sup>) }}, which according to its S-expression comma family categorisation as a lopsided comma (S64<sup>2</sup>⋅S65) is trivial information. Interestingly/nontrivially, tempering it out causes {{nowrap| 1/1 - 64/63 - 33/32 - 22/21 - [[1089/1024]] }} to become equidistant, thereby splitting the [[1089/1024|parapotome]] into four equal parts, as {{nowrap| ([[22/21]])/([[33/32]]) {{=}} [[64/63]] }} and {{nowrap| 1089/1024 * S64<sup>2</sup> * S65 {{=}} 22/21 * 64/63 }}. Even more interesting is the factorization into two [[13-limit]] [[superparticular ratio]]s: ([[2080/2079]])⋅([[4096/4095]]). These ratios and the olympia itself are the default intervals represented by one, two, and three [[mina]]s in the Olympian level of [[Sagittal notation]], from which it gets its name.
+It factors into two [[13-limit]] [[superparticular ratio]]s: ([[2080/2079]])⋅([[4096/4095]]). These ratios and the olympia itself are the default intervals represented by one, two, and three [[mina]]s in the Olympian level of [[Sagittal notation]], from which it gets its name.
 == Temperaments ==
-[[Tempering out]] this comma in the full 11-limit results in the rank-4 olympic temperament (→ [[Rank-4 temperament #Olympic (131072/130977)]]), or in the 2.3.7.11 subgroup, the rank-3 olympian temperament. Olympic has a very natural 13-limit extension {2080/2079, 4096/4095}. As the comma's order of 11 is one, any 7-limit temperament can be extended to the 11-limit by tempering out this comma, but it works best for temperaments with low complexity and high accuracy in the septimal comma.
+[[Tempering out]] this comma in the full 11-limit results in the rank-4 olympic temperament (→ [[Rank-4 temperament #Olympic (131072/130977)]]), or in the 2.3.7.11 subgroup, the rank-3 olympian temperament. In either case, it sets 1–64/63–33/32–22/21–[[1089/1024|(33/32)<sup>2</sup>]] to become equidistant.
+Olympic has a very natural 13-limit extension {2080/2079, 4096/4095}. As the comma's order of 11 is one, any 7-limit temperament can be extended to the 11-limit by tempering out this comma, but it works best for temperaments with low complexity and high accuracy in the septimal comma.
 == Etymology ==