6656/6655: Difference between revisions
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'''6656/6655''', the '''jacobin comma''', [https://en.xen.wiki/index.php?title=List_of_superparticular_intervals&diff=next&oldid=18864 apparently named] by [[Gene Ward Smith]] in 2014, is a [[13-limit]] (also 2.5.11.13 [[subgroup]]) [[superparticular]] interval of about 0.26{{cent}}. It is the difference between a stack of three [[11/8]] superfourths and one [[13/10]] [[naiadic]] plus an [[octave]]. In terms of commas, it is the difference between [[364/363]] and [[385/384]], between [[2080/2079]] and [[3025/3024]] as well as between [[4096/4095]] and [[10648/10647]]. In the 17-limit, it factors neatly into [[12376/12375]] | '''6656/6655''', the '''jacobin comma''', [https://en.xen.wiki/index.php?title=List_of_superparticular_intervals&diff=next&oldid=18864 apparently named] by [[Gene Ward Smith]] in 2014, is a [[13-limit]] (also 2.5.11.13 [[subgroup]]) [[superparticular]] interval of about 0.26{{cent}}. It is the difference between a stack of three [[11/8]] superfourths and one [[13/10]] [[naiadic]] plus an [[octave]]. In terms of commas, it is the difference between [[364/363]] and [[385/384]], between [[2080/2079]] and [[3025/3024]] as well as between [[4096/4095]] and [[10648/10647]]. In the 17-limit, it factors neatly into ([[12376/12375]])([[14400/14399]]). | ||
== Temperaments == | == Temperaments == | ||
By tempering it out, the '''jacobin temperament''' is defined. | By tempering it out, the '''jacobin temperament''' is defined. Interestingly, [[1789edo]] is an edo that [[support]]s the jacobin temperament. You may find a list of good JI-approximating edos that support this temperament below. Although 1789edo has a unique position due to its number of steps being a hallmark year of the French Revolution, it is more rational to use the other edos for this temperament. | ||
The 17-limit factorization shows us a natural path of extension, also given below. | |||
=== Jacobin === | |||
[[Subgroup]]: 2.3.5.7.11.13 | [[Subgroup]]: 2.3.5.7.11.13 | ||
[[Comma list]]: 6656/6655 | |||
[[Mapping]]: <br> | [[Mapping]]: <br> | ||
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: mapping generators: ~2, ~3, ~5, ~7, ~11 | : mapping generators: ~2, ~3, ~5, ~7, ~11 | ||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 15, 22, 26, 31f, 37, 39df, 41, 46, 63, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1012, 1084, 1236, 1506, 2814, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816ee }} | ||
==== Septendecimal jacobin ==== | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 6656/6655, 12376/12375 | |||
Mapping: <br> | |||
[{{val| 1 0 0 0 0 -9 6 }}, <br> | |||
{{val| 0 1 0 0 0 0 2 }}, <br> | |||
{{val| 0 0 1 0 0 1 2 }}, <br> | |||
{{val| 0 0 0 1 0 0 -1 }}, <br> | |||
{{val| 0 0 0 0 1 3 -2 }}] | |||
Optimal ET sequence: {{Optimal ET sequence| 15g, 22, 37g, 39dfg, 41g, 50, 63g, 72, 111, 152f, 159, 183, 239f, 248, 270, 311, 422, 494, 581, 742, 764, 814, 1075, 1236, 1395, 1506, 2000, 2581, 2814, 2901, 3323, 3395, 8296e, 11691e, 16322ee, 17086cdeeg, 21223cdeefg }} | |||
== See also == | == See also == | ||
Revision as of 10:22, 19 July 2023
| Interval information |
reduced
6656/6655, the jacobin comma, apparently named by Gene Ward Smith in 2014, is a 13-limit (also 2.5.11.13 subgroup) superparticular interval of about 0.26 ¢. It is the difference between a stack of three 11/8 superfourths and one 13/10 naiadic plus an octave. In terms of commas, it is the difference between 364/363 and 385/384, between 2080/2079 and 3025/3024 as well as between 4096/4095 and 10648/10647. In the 17-limit, it factors neatly into (12376/12375)(14400/14399).
Temperaments
By tempering it out, the jacobin temperament is defined. Interestingly, 1789edo is an edo that supports the jacobin temperament. You may find a list of good JI-approximating edos that support this temperament below. Although 1789edo has a unique position due to its number of steps being a hallmark year of the French Revolution, it is more rational to use the other edos for this temperament.
The 17-limit factorization shows us a natural path of extension, also given below.
Jacobin
Subgroup: 2.3.5.7.11.13
Comma list: 6656/6655
Mapping:
[⟨1 0 0 0 0 -9],
⟨0 1 0 0 0 0],
⟨0 0 1 0 0 1],
⟨0 0 0 1 0 0],
⟨0 0 0 0 1 3]]
- mapping generators: ~2, ~3, ~5, ~7, ~11
Optimal ET sequence: 15, 22, 26, 31f, 37, 39df, 41, 46, 63, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1012, 1084, 1236, 1506, 2814, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816ee
Septendecimal jacobin
Subgroup: 2.3.5.7.11.13.17
Comma list: 6656/6655, 12376/12375
Mapping:
[⟨1 0 0 0 0 -9 6],
⟨0 1 0 0 0 0 2],
⟨0 0 1 0 0 1 2],
⟨0 0 0 1 0 0 -1],
⟨0 0 0 0 1 3 -2]]
Optimal ET sequence: 15g, 22, 37g, 39dfg, 41g, 50, 63g, 72, 111, 152f, 159, 183, 239f, 248, 270, 311, 422, 494, 581, 742, 764, 814, 1075, 1236, 1395, 1506, 2000, 2581, 2814, 2901, 3323, 3395, 8296e, 11691e, 16322ee, 17086cdeeg, 21223cdeefg