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. | '''2058/2057''', the '''xenisma''', is an [[unnoticeable comma|unnoticeable]] [[17-limit]] (also 2.3.7.11.17-[[subgroup]]) [[comma]] measuring about 0.841 [[cent]]s. It is the amount by which a stack of two [[14/11]]'s exceed [[34/21]], or equivalently the amount by which a stack consisting of [[49/44]] and 14/11 exceeds [[17/12]]. | ||
== Commatic relations == | == Commatic relations == | ||
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== Temperaments == | == Temperaments == | ||
Tempering out this comma in the 17-limit results in the rank-6 '''xenismic | [[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 === | === Xenic === | ||
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{{Mapping|legend=2| 1 0 0 0 1 | 0 1 0 0 1 | 0 0 1 0 3 | 0 0 0 1 -2 }} | {{Mapping|legend=2| 1 0 0 0 1 | 0 1 0 0 1 | 0 0 1 0 3 | 0 0 0 1 -2 }} | ||
: mapping generators: ~2, ~3, ~7, ~11 | |||
: | |||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = | * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9397{{c}}, ~7/4 = 968.6818{{c}}, ~11/8 = 551.4639{{c}} | ||
* [[CWE]]: ~2 = | * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9446{{c}}, ~7/4 = 968.6908{{c}}, ~11/8 = 551.4738{{c}} | ||
{{Optimal ET sequence|legend=1| 46, 58, 63, 72, 89, 118, 135, 207, 342, 400, 535, 742, 1395, 1930, 2672 }} | {{Optimal ET sequence|legend=1| 46, 58, 63, 72, 89, 118, 135, 207, 342, 400, 535, 742, 1395, 1930, 2672 }} | ||
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = | * [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9397{{c}}, ~5/4 = 386.3137{{c}}, ~7/4 = 968.6818{{c}}, ~11/8 = 551.4639{{c}}, ~13/8 = 840.5277{{c}} | ||
* [[CWE]]: ~2 = | * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9445{{c}}, ~5/4 = 386.3208{{c}}, ~7/4 = 968.6908{{c}}, ~11/8 = 551.4738{{c}}, ~13/8 = 840.5389{{c}} | ||
{{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 }}* | {{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 }}* | ||
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== Etymology == | == Etymology == | ||
This comma was named by [[Margo Schulter]] in 2000<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_12989.html Yahoo! Tuning Group | ''Re: Kalisma/xenisma (new names?) -- JI tuning and Scala file'']</ref>. Its name comes from the Ancient Greek word [[wiktionary: ξένος #Ancient Greek|''xénos'']] ("strange"), which may suggest the strange and wonderful qualities in a [[neogothic]] setting of supraminor and submajor thirds. | |||
== | == References == | ||
[[Category:Xenismic]] | [[Category:Xenismic]] | ||
[[Category:Commas | [[Category:Commas named by translating something into another language]] | ||
Revision as of 12:36, 28 March 2026
| Interval information |
reduced
2058/2057, the xenisma, is an unnoticeable 17-limit (also 2.3.7.11.17-subgroup) comma measuring about 0.841 cents. It is the amount by which a stack of two 14/11's exceed 34/21, or equivalently the amount by which a stack consisting of 49/44 and 14/11 exceeds 17/12.
Commatic relations
This comma is the difference between the following superparticular pairs:
- 364/363 and 442/441
- 441/440 and 561/560
- 936/935 and 1716/1715
- 1225/1224 and 3025/3024
- 1701/1700 and 9801/9800
Not to mention some nonsuperparticular but useful relations:
It factors into the following superparticular pairs:
- 2401/2400 and 14400/14399
- 2080/2079 and 194481/194480
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
Subgroup-val mapping: [⟨1 0 0 0 1], ⟨0 1 0 0 1], ⟨0 0 1 0 3], ⟨0 0 0 1 -2]]
- mapping generators: ~2, ~3, ~7, ~11
- CTE: ~2 = 1200.0000 ¢, ~3/2 = 701.9397 ¢, ~7/4 = 968.6818 ¢, ~11/8 = 551.4639 ¢
- CWE: ~2 = 1200.0000 ¢, ~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
| [⟨ | 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
- CTE: ~2 = 1200.0000 ¢, ~3/2 = 701.9397 ¢, ~5/4 = 386.3137 ¢, ~7/4 = 968.6818 ¢, ~11/8 = 551.4639 ¢, ~13/8 = 840.5277 ¢
- CWE: ~2 = 1200.0000 ¢, ~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*
Etymology
This comma was named by Margo Schulter in 2000[1]. Its name comes from the Ancient Greek word xénos ("strange"), which may suggest the strange and wonderful qualities in a neogothic setting of supraminor and submajor thirds.