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. It is the difference between the undecimal quartertone ([[33/32]]) and a stack of two septimal commas (([[64/63]])<sup>2</sup>). 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. | 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 {{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. | |||
== Temperaments == | == Temperaments == | ||
Revision as of 18:24, 12 May 2026
| Interval information |
olympic comma
reduced subharmonic
The olympia (monzo: [17 -5 0 -2 -1⟩, ratio: 131072/130977), otherwise known as the olympic comma, is an unnoticeable 11-limit (specifically 2.3.7.11-subgroup) comma measuring about 1.26 cents.
It is the difference between the undecimal quartertone (33/32) and a stack of 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 1/1 - 64/63 - 33/32 - 22/21 - 1089/1024 to become equidistant, thereby splitting the parapotome into four equal parts, as (22/21)/(33/32) = 64/63 and 1089/1024 * S642 * S65 = 22/21 * 64/63. Even more interesting is the factorization into two 13-limit superparticular ratios: (2080/2079)⋅(4096/4095). These ratios and the olympia itself are the default intervals represented by one, two, and three minas 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.
Etymology
The olympia was named by Flora Canou in 2021, referring to the Olympian level of Sagittal notation.