Undecimal octatonic comma: Difference between revisions
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The '''undecimal octatonic comma''' ({{monzo|legend=1| 15 8 0 0 -8 }}, [[ratio]]: 214 990 848 / 214 358 881) is a [[small comma]] in the [[11-limit]] (specifically the [[2.3.11 subgroup]]), measuring about 5.10 [[cent]]s. It is the amount by which a stack of eight [[12/11]] intervals exceeds the octave. | The '''undecimal octatonic comma''' ({{monzo|legend=1| 15 8 0 0 -8 }}, [[ratio]]: 214 990 848 / 214 358 881) is a [[small comma]] in the [[11-limit]] (specifically the [[2.3.11 subgroup]]), measuring about 5.10 [[cent]]s. It is the amount by which a stack of eight [[12/11]] intervals exceeds the octave. | ||
In terms of other commas, it is the difference between the [[Pythagorean comma]] and a stack of two [[Alpharabian comma]]s. In the full 11-limit, it factors into simpler commas as ([[46656/46585]])⋅([[161280/161051]]), the latter being the difference between 46656/46585 and [[9801/9800]]; thus it can be expressed as (46656/46585)<sup>2</sup>/(9801/9800) as well. | |||
== Temperaments == | == Temperaments == | ||
Latest revision as of 11:06, 4 July 2026
| Interval information |
The undecimal octatonic comma (monzo: [15 8 0 0 -8⟩, ratio: 214 990 848 / 214 358 881) is a small comma in the 11-limit (specifically the 2.3.11 subgroup), measuring about 5.10 cents. It is the amount by which a stack of eight 12/11 intervals exceeds the octave.
In terms of other commas, it is the difference between the Pythagorean comma and a stack of two Alpharabian commas. In the full 11-limit, it factors into simpler commas as (46656/46585)⋅(161280/161051), the latter being the difference between 46656/46585 and 9801/9800; thus it can be expressed as (46656/46585)2/(9801/9800) as well.
Temperaments
Tempering out this comma in the 2.3.11 subgroup results in the octatonic temperament. For technical data see 8th-octave temperaments #Octatonic. Since it equates eight of one interval with an octave, it is tempered out if and only if the number of notes in the octave is a multiple of 8.