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== Temperaments == | == Temperaments == | ||
Tempering out this comma in the full 17-limit gives the rank-6 '''dakotismic temperament''', or in the 2.3.5.7.11.17 subgroup, the rank-5 '''dakotic temperament''', enabling [[dakotismic chords]]. You may find a list of good equal temperaments that support | Tempering out this comma in the full 17-limit gives the rank-6 '''dakotismic temperament''', or in the 2.3.5.7.11.17 subgroup, the rank-5 '''dakotic temperament''', enabling [[dakotismic chords]]. You may find a list of good equal temperaments that support these temperaments below. | ||
=== Dakotismic === | |||
[[Subgroup]]: 2.3.5.7.11.13.17 | [[Subgroup]]: 2.3.5.7.11.13.17 | ||
[[Mapping]]: | [[Mapping]]: <br> | ||
<br>[{{val| 1 0 0 0 0 0 1 }} | [{{val| 1 0 0 0 0 0 1 }}<br> | ||
<br>{{val| 0 1 0 0 0 0 3 }} | {{val| 0 1 0 0 0 0 3 }}<br> | ||
<br>{{val| 0 0 1 0 0 0 -1 }} | {{val| 0 0 1 0 0 0 -1 }}<br> | ||
<br>{{val| 0 0 0 1 0 0 -1 }} | {{val| 0 0 0 1 0 0 -1 }}<br> | ||
<br>{{val| 0 0 0 0 1 0 1 }} | {{val| 0 0 0 0 1 0 1 }}<br> | ||
<br>{{val| 0 0 0 0 0 1 0 }}] | {{val| 0 0 0 0 0 1 0 }}] | ||
Mapping generators: ~2, ~3, ~5, ~7, ~11, ~13 | Mapping generators: ~2, ~3, ~5, ~7, ~11, ~13 | ||
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.2950, ~5/4 = 386.0705, ~7/4 = 968.4704, ~11/8 = 551.8578, ~13/8 = 840.5277 | |||
{{Val list|legend=1| 12f, 14cf, 15g, 19eg, 22, 26, 27eg, 34d, 38df, 41, 46, 58, 72, 121, 140, 171, 190g, 212g, 217, 289, 311, 668, 694g, 740g, 1051dg }}* | {{Val list|legend=1| 12f, 14cf, 15g, 19eg, 22, 26, 27eg, 34d, 38df, 41, 46, 58, 72, 121, 140, 171, 190g, 212g, 217, 289, 311, 668, 694g, 740g, 1051dg }}* | ||
<nowiki>*</nowiki> [[optimal patent val]]: [[861edo|861]] | <nowiki>*</nowiki> [[optimal patent val]]: [[861edo|861]] | ||
=== Dakotic === | |||
[[Subgroup]]: 2.3.5.7.11.17 | |||
[[Sval]] [[mapping]]: <br> | |||
[{{val| 1 0 0 0 0 1 }}<br> | |||
{{val| 0 1 0 0 0 3 }}<br> | |||
{{val| 0 0 1 0 0 -1 }}<br> | |||
{{val| 0 0 0 1 0 -1 }}<br> | |||
{{val| 0 0 0 0 1 1 }}] | |||
Sval mapping generators: ~2, ~3, ~5, ~7, ~11 | |||
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.2950, ~5/4 = 386.0705, ~7/4 = 968.4704, ~11/8 = 551.8578 | |||
{{Val list|legend=1| 12, 14c, 19eg, 22, 27eg, 31g, 34d, 38d, 41, 46, 58, 68, 72, 118, 171, 190g, 193, 212g, 217, 239, 311, 1051dg }} | |||
== Etymology == | == Etymology == | ||
Revision as of 11:24, 23 February 2023
| Interval information |
reduced
595/594, the dakotisma, is a small 17-limit comma measuring about 2.91 cents. Named after Scott Dakota, it is one of the simplest commas tempered out in 311et, a highly notable general-purpose equal temperament.
Commatic relations
This comma identifies itself as the difference between the following superparticular pairs:
- 34/33 and 36/35
- 85/84 and 99/98
- 100/99 and 120/119
- 136/135 and 176/175
- 221/220 and 351/350
- 289/288 and 561/560
- 325/324 and 715/714
- 364/363 and 936/935
- 385/384 and 1089/1088
- 441/440 and 1701/1700
- 442/441 and 1716/1715
- 540/539 and 5832/5831
- 561/560 and 9801/9800
not to mention some nonsuperparticular but useful relations:
It factors into the following superparticular pairs:
not to mention some nonsuperparticular but useful relations:
Temperaments
Tempering out this comma in the full 17-limit gives the rank-6 dakotismic temperament, or in the 2.3.5.7.11.17 subgroup, the rank-5 dakotic temperament, enabling dakotismic chords. You may find a list of good equal temperaments that support these temperaments below.
Dakotismic
Subgroup: 2.3.5.7.11.13.17
Mapping:
[⟨1 0 0 0 0 0 1]
⟨0 1 0 0 0 0 3]
⟨0 0 1 0 0 0 -1]
⟨0 0 0 1 0 0 -1]
⟨0 0 0 0 1 0 1]
⟨0 0 0 0 0 1 0]]
Mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2950, ~5/4 = 386.0705, ~7/4 = 968.4704, ~11/8 = 551.8578, ~13/8 = 840.5277
Dakotic
Subgroup: 2.3.5.7.11.17
Sval mapping:
[⟨1 0 0 0 0 1]
⟨0 1 0 0 0 3]
⟨0 0 1 0 0 -1]
⟨0 0 0 1 0 -1]
⟨0 0 0 0 1 1]]
Sval mapping generators: ~2, ~3, ~5, ~7, ~11
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2950, ~5/4 = 386.0705, ~7/4 = 968.4704, ~11/8 = 551.8578
Etymology
The dakotisma was named by Praveen Venkataramana in 2022 in honor of Scott Dakota's pioneering study of this comma and of its role in 311et.