Olympia: Difference between revisions
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S-expressions are encodings of interval relations. They don't make the interval relations trivial! Also don't discuss temps outside the Temperaments section! 1089/1024 is (33/32)^2 so it's always split in two, and it's split in four since 33/32 is split in two |
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The '''olympia''' ([[monzo]]: {{monzo| 17 -5 0 -2 -1 }}, [[ratio]]: 131072/130977), otherwise known as the '''olympic comma''', is an [[11-limit]] ( | 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 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 == | == Temperaments == | ||
Tempering out this comma in the full 11-limit results in the rank-4 | [[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. | ||
As | 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 == | == Etymology == | ||
The olympia was named by [[Flora Canou]] in 2021, referring to the Olympian level of Sagittal notation. | The olympia was named by [[Flora Canou]] in 2021, referring to the Olympian level of Sagittal notation. | ||
[[Category:Olympic]] | [[Category:Olympic]] | ||
[[Category:Commas named after musical traditions]] | |||