2058/2057: Difference between revisions

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'''2058/2057''', the '''xenisma''', is an [[unnoticeable comma|unnoticeable]] [[17-limit]] (also 2.3.7.11.17 [[subgroup]]) [[comma]] measuring about 0.84 [[cent]]s. It is the amount by which a stack of two [[14/11]]'s exceed [[34/21]].  
'''2058/2057''', the '''xenisma''', is a [[17-limit]] (also 2.3.7.11.17 [[subgroup]]) [[comma]] measuring about 0.84 [[cent]]s. It is the amount by which a stack of two [[14/11]]'s exceed [[34/21]].  


== Commatic relations ==
== Commatic relations ==
Line 25: Line 24:
== Temperaments ==
== Temperaments ==
Tempering out this comma in the 17-limit results in the rank-6 '''xenismic temperament''', or in the 2.3.7.11.17 subgroup, the rank-4 '''xenic temperament'''. In either case it enables [[xenismic chords]].  
Tempering out this comma in the 17-limit results in the rank-6 '''xenismic temperament''', or in the 2.3.7.11.17 subgroup, the rank-4 '''xenic temperament'''. In either case it enables [[xenismic chords]].  
=== Xenic ===
[[Subgroup]]: 2.3.7.11.17
[[Comma list]]: 2058/2057
{{Mapping|legend=2| 1 0 0 0 1 | 0 1 0 0 1 | 0 0 1 0 3 | 0 0 0 1 -2 }}
: sval mapping generators: ~2, ~3, ~7, ~11
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.9397, ~7/4 = 968.6818, ~11/8 = 551.4639
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.9446, ~7/4 = 968.6908, ~11/8 = 551.4738
{{Optimal ET sequence|legend=1| 46, 58, 63, 72, 89, 118, 135, 207, 342, 400, 535, 742, 1395, 1930, 2672 }}
=== Xenismic ===
[[Subgroup]]: 2.3.5.7.11.13.17
[[Comma list]]: 2058/2057
[[Mapping]]:<br>
{| class="right-all"
|-
| [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || 1 || ],
|-
| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || 1 || ],
|-
| ⟨ || 0 || 0 || 1 || 0 || 0 || 0 || 0 || ],
|-
| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || 3 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || -2 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
|}
: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.9397, ~5/4 = 386.3137, ~7/4 = 968.6818, ~11/8 = 551.4639, ~13/8 = 840.5277
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.9445, ~5/4 = 386.3208, ~7/4 = 968.6908, ~11/8 = 551.4738, ~13/8 = 840.5389
{{Optimal ET sequence|legend=1| 43, 46, 58, 72, 103, 121, 149, 161, 190g, 224, 270, 311, 354, 400, 460, 581, 742, 814, 935, 1084, 1323, 1395, 2137, 3072e, 3342eg, 4156eg }}*
<nowiki>*</nowiki> [[optimal patent val]]: [[3756edo|3756]]


== Etymology ==
== Etymology ==
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[[Category:Xenismic]]
[[Category:Xenismic]]
[[Category:Commas with unknown etymology]]

Latest revision as of 01:58, 16 November 2024

Interval information
Ratio 2058/2057
Factorization 2 × 3 × 73 × 11-2 × 17-1
Monzo [1 1 0 3 -2 0 -1⟩
Size in cents 0.841426¢
Name xenisma
Color name 17u1uuz32, sululutrizo 2nd
FJS name [math]\displaystyle{ \text{M2}^{7,7,7}_{11,11,17} }[/math]
Special properties superparticular,
reduced
Tenney height (log2 nd) 22.0134
Weil height (log2 max(n, d)) 22.0141
Wilson height (sopfr(nd)) 65
Comma size unnoticeable
Open this interval in xen-calc

2058/2057, the xenisma, is an unnoticeable 17-limit (also 2.3.7.11.17 subgroup) comma measuring about 0.84 cents. It is the amount by which a stack of two 14/11's exceed 34/21.

Commatic relations

This comma is the difference between the following superparticular pairs:

Not to mention some nonsuperparticular but useful relations:

It factors into the following superparticular pairs:

Temperaments

Tempering out this comma in the 17-limit results in the rank-6 xenismic temperament, or in the 2.3.7.11.17 subgroup, the rank-4 xenic temperament. In either case it enables xenismic chords.

Xenic

Subgroup: 2.3.7.11.17

Comma list: 2058/2057

Sval mapping: [⟨1 0 0 0 1], ⟨0 1 0 0 1], ⟨0 0 1 0 3], ⟨0 0 0 1 -2]]

sval mapping generators: ~2, ~3, ~7, ~11

Optimal tunings:

  • CTE: ~2 = 1\1, ~3/2 = 701.9397, ~7/4 = 968.6818, ~11/8 = 551.4639
  • CWE: ~2 = 1\1, ~3/2 = 701.9446, ~7/4 = 968.6908, ~11/8 = 551.4738

Optimal ET sequence: 46, 58, 63, 72, 89, 118, 135, 207, 342, 400, 535, 742, 1395, 1930, 2672

Xenismic

Subgroup: 2.3.5.7.11.13.17

Comma list: 2058/2057

Mapping:

[⟨ 1 0 0 0 0 0 1 ],
⟨ 0 1 0 0 0 0 1 ],
⟨ 0 0 1 0 0 0 0 ],
⟨ 0 0 0 1 0 0 3 ],
⟨ 0 0 0 0 1 0 -2 ],
⟨ 0 0 0 0 0 1 0 ]]
mapping generators: ~2, ~3, ~5, ~7, ~11, ~13

Optimal tunings:

  • CTE: ~2 = 1\1, ~3/2 = 701.9397, ~5/4 = 386.3137, ~7/4 = 968.6818, ~11/8 = 551.4639, ~13/8 = 840.5277
  • CWE: ~2 = 1\1, ~3/2 = 701.9445, ~5/4 = 386.3208, ~7/4 = 968.6908, ~11/8 = 551.4738, ~13/8 = 840.5389

Optimal ET sequence: 43, 46, 58, 72, 103, 121, 149, 161, 190g, 224, 270, 311, 354, 400, 460, 581, 742, 814, 935, 1084, 1323, 1395, 2137, 3072e, 3342eg, 4156eg*

* optimal patent val: 3756

Etymology

The xenisma was named by Margo Schulter in 2000[1].

Notes

  1. ↑ Yahoo! Tuning Group | Re: Kalisma/xenisma (new names?) -- JI tuning and Scala file
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