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The '''parimo''' ([[ratio]]: 1771561/1771470, {{monzo|legend=1| -1 -11 -1 0 6 }}) is an [[Unnoticeable comma|unnoticeable]] [[11-limit]] [[comma]] with a value of approximately 0.09{{cent}}. It is the amount by which an [[octave-reduced]] stack of six [[11/9]] neutral thirds exceeds [[5/3]], as well as the amount by which a stack of three [[243/242|rastmas]] falls short of a [[81/80|syntonic comma]].  
The '''parimo''' is an [[11-limit]] [[unnoticeable comma]] with a ratio of '''1771561/1771470''' = {{Monzo|-1 -11 -1 0 6}} and a value of approximately 0.09 [[cent|cents]]. It is the amount by which an [[octave-reduced]] stack of six [[11/9]] neutral thirds exceeds [[5/3]], as well as the amount by which a stack of three [[243/242|rastmas]] falls short of a [[81/80|syntonic comma]].  


== Temperaments ==
== Temperaments ==
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=== Parimic ===
=== Parimic ===
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 1771561/1771470
[[Comma list]]: 1771561/1771470


[[Mapping]]: [{{val| 1 0 5 0 1 }}, {{val| 0 1 1 0 2 }}, {{val| 0 0 -6 0 -1 }}, {{val| 0 0 0 1 0 }}]
{{Mapping|legend=1| 1 0 5 0 1 | 0 1 1 0 2 | 0 0 -6 0 -1 | 0 0 0 1 0 }}


Mapping generators: ~2, ~3, ~18/11, ~7
: mapping generators: ~2, ~3, ~18/11, ~7


[[POTE generator]]s: ~3/2 = 701.9581, ~18/11 = 852.6073
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.9581, ~18/11 = 852.6073


{{Optimal ET sequence|legend=1| 14c, 17c, 24, 31, 90e, 107cde, 114e, 121, 145, 152, 311, 342, 836, 1178, 1354, 1506, 1817, 1848, 2684, 3665, 4007, 4501, 4843, 6349, 6691, 21921, 28612, 35303, 48343, 63573, 70264e, 76955e, 98876e, 105567e }}
{{Optimal ET sequence|legend=1| 14c, 17c, 24, 31, 90e, 107cde, 114e, 121, 145, 152, 311, 342, 836, 1178, 1354, 1506, 1817, 1848, 2684, 3665, 4007, 4501, 4843, 6349, 6691, 21921, 28612, 35303, 48343, 63573, 70264e, 76955e, 98876e, 105567e }}
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=== Tritomere ===
=== Tritomere ===
Subgroup: 2.3.5.11
[[Subgroup]]: 2.3.5.11


[[Comma list]]: 1771561/1771470
[[Comma list]]: 1771561/1771470


[[Sval]] [[mapping]]: [{{val| 1 0 5 1 }}, {{val| 0 1 1 2 }}, {{val| 0 0 -6 -1 }}]
{{Mapping|legend=2| 1 0 5 1 | 0 1 1 2 | 0 0 -6 -1 }}


Sval mapping generators: ~2, ~3, ~18/11
: sval mapping generators: ~2, ~3, ~18/11


[[POTE generator]]s: ~3/2 = 701.9581, ~18/11 = 852.6073
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.9581, ~18/11 = 852.6073


{{Optimal ET sequence|legend=1| 7, 17c, 24, 31, 69e, 90e, 97, 107ce, 114e, 121, 145, 152, 311, 335, 342, 494, 677, 1019, 1171, 3855, 4349, 5026, 5520, 6691, 7862, 36474, 37645, 44336, 45507, 53369, 61231 }}
{{Optimal ET sequence|legend=1| 7, 17c, 24, 31, 69e, 90e, 97, 107ce, 114e, 121, 145, 152, 311, 335, 342, 494, 677, 1019, 1171, 3855, 4349, 5026, 5520, 6691, 7862, 36474, 37645, 44336, 45507, 53369, 61231 }}

Revision as of 08:25, 20 July 2023

Interval information
Ratio 1771561/1771470
Factorization 2-1 × 3-11 × 5-1 × 116
Monzo [-1 -11 -1 0 6⟩
Size in cents 0.0889308¢
Name parimo
Color name satribilo-agu comma
FJS name [math]\displaystyle{ \text{ddd1}^{11,11,11,11,11,11}_{5} }[/math]
Special properties reduced
Tenney norm (log2 nd) 41.5131
Weil norm (log2 max(n, d)) 41.5132
Wilson norm (sopfr(nd)) 106
Comma size unnoticeable
Open this interval in xen-calc

The parimo (ratio: 1771561/1771470, monzo: [-1 -11 -1 0 6⟩) is an unnoticeable 11-limit comma with a value of approximately 0.09 ¢. It is the amount by which an octave-reduced stack of six 11/9 neutral thirds exceeds 5/3, as well as the amount by which a stack of three rastmas falls short of a syntonic comma.

Temperaments

Tempering it out leads to a form of parimic temperament, which can be said to be the undecimal counterpart of the metric temperament in that the syntonic comma is split into three equal parts in both systems. However, in parimic temperament, the parts represent different intervals – one part represents the rastma, and two represent the biyatisma. The corresponding 2.3.5.11 subgroup temperament is tritomere.

Parimic

Subgroup: 2.3.5.7.11

Comma list: 1771561/1771470

Mapping: [⟨1 0 5 0 1], ⟨0 1 1 0 2], ⟨0 0 -6 0 -1], ⟨0 0 0 1 0]]

mapping generators: ~2, ~3, ~18/11, ~7

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.9581, ~18/11 = 852.6073

Optimal ET sequence: 14c, 17c, 24, 31, 90e, 107cde, 114e, 121, 145, 152, 311, 342, 836, 1178, 1354, 1506, 1817, 1848, 2684, 3665, 4007, 4501, 4843, 6349, 6691, 21921, 28612, 35303, 48343, 63573, 70264e, 76955e, 98876e, 105567e

Badness: 0.0279 × 10-6

Tritomere

Subgroup: 2.3.5.11

Comma list: 1771561/1771470

Subgroup-val mapping: [⟨1 0 5 1], ⟨0 1 1 2], ⟨0 0 -6 -1]]

sval mapping generators: ~2, ~3, ~18/11

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.9581, ~18/11 = 852.6073

Optimal ET sequence: 7, 17c, 24, 31, 69e, 90e, 97, 107ce, 114e, 121, 145, 152, 311, 335, 342, 494, 677, 1019, 1171, 3855, 4349, 5026, 5520, 6691, 7862, 36474, 37645, 44336, 45507, 53369, 61231

Badness: 0.00712 × 10-3

See also

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