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| == Temperament == | | == Temperament == |
| [[Tempering out]] this comma leads to the 5-limit version of the [[ennealimmal]] temperament, which remarkably splits the octave into nine equal parts. Since the [[7-limit]] temperament (definable by tempering out both [[2401/2400]] and [[4375/4374]], the two smallest superparticular ratios in the 7-limit) is far more natural to think of than the 5-limit, the 5-limit temperament is only provided below for bookkeeping purposes. | | [[Tempering out]] this comma leads to the 5-limit version of the [[ennealimmal]] temperament, which remarkably splits the octave into nine equal parts. |
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| === Ennealimmal ===
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| : ''For the 7-limit temperament, see [[Septiennealimmal clan #Ennealimmal]].''
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| [[Subgroup]]: 2.3.5
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| [[Comma list]]: {{monzo| 1 -27 18 }}
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| {{Mapping|legend=1| 9 1 1 | 0 2 3 }}
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| : mapping generators: ~27/25, ~5/3
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| [[Optimal tuning]]s:
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| * [[WE]]: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3190{{c}}
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| : [[error map]]: {{val| -0.000 +0.016 -0.023 }}
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| * [[CWE]]: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3191{{c}}
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| : error map: {{val| 0.000 +0.017 -0.023 }}
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| {{Optimal ET sequence|legend=1| 27, 45, 72, 99, 171, 441, 612, 1665, 2277, 2889, 3501, 6390, 16281, 22671, 29061c }}
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| [[Badness]] (Sintel): 0.403
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| == Etymology == | | == Etymology == |
| Ratio
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7629394531250/7625597484987
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| Factorization
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2 × 3-27 × 518
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| Monzo
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[1 -27 18⟩
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| Size in cents
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0.8618262¢
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| Name
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ennealimma
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| Color name
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sy18-3, satritribiyo negative 3rd
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| FJS name
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[math]\displaystyle{ \text{6d}{-3}^{5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5} }[/math]
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| Special properties
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reduced
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| Tenney norm (log2 nd)
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85.5887
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| Weil norm (log2 max(n, d))
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85.5894
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| Wilson norm (sopfr(nd))
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173
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| Comma size
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unnoticeable
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| Open this interval in xen-calc
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The ennealimma, meaning nine limmas, with monzo [1 -27 18⟩, is an unnoticeable 5-limit comma measuring about 0.862 cents. It is the amount by which a stack of nine large limmas falls short of the octave.
Temperament
Tempering out this comma leads to the 5-limit version of the ennealimmal temperament, which remarkably splits the octave into nine equal parts.
Etymology
The name consists of Greek ennea- ("nine") + limma, coined by Paul Erlich and Gene Ward Smith in 2001[1].
See also
References