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| == Temperaments == | | == Temperaments == |
| 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. Miscellaneous temperaments tempering out this comma are collected in [[The Jacobins]]. | | By tempering it out, the '''jacobin temperament''' is defined. Interestingly, [[1789edo]] is an edo that [[support]]s the jacobin temperament - and is consistent to distance 2 on the subgroup 2.5.11.13, the subgroup of the comma. Although 1789edo has a unique position due to its number of steps being a hallmark year of the {{W|French Revolution}}, it is more rational to use the other edos for this temperament. Miscellaneous temperaments tempering out this comma are collected in [[The Jacobins]]. |
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| The 17-limit factorization shows us a natural path of extension, also given below. | | The 17-limit factorization shows us a natural path of extension by adding 12376/12375. Equally, it can be conceptualized as a rank-2 temperament in 2.13/10.11 subgroup, in which case it is called ''jacobin-naiadic'', after [[13/10]] being called a naiadic by interval class. |
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| === Jacobin ===
| | See [[The Jacobins#Jacobin]] for technical data. |
| [[Subgroup]]: 2.3.5.7.11.13 | |
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| [[Comma list]]: 6656/6655
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| [[Mapping]]: <br>
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| {| class="right-all"
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| |-
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| | [⟨ || 1 || 0 || 0 || 0 || 0 || -9 || ],
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| |-
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| | ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || ],
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| |-
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| | ⟨ || 0 || 0 || 1 || 0 || 0 || 1 || ],
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| |-
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| | ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || ],
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| |-
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| | ⟨ || 0 || 0 || 0 || 0 || 1 || 3 || ]]
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| |}
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| : mapping generators: ~2, ~3, ~5, ~7, ~11
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| [[Optimal tuning]]s:
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| * [[WE]]: ~2 = 1199.9887{{c}}, ~3/2 = 701.9663{{c}}, ~5/4 = 386.3430{{c}}, ~7/4 = 968.8484{{c}}, ~11/8 = 551.3967{{c}}
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| * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9661{{c}}, ~5/4 = 386.3383{{c}}, ~7/4 = 968.8455{{c}}, ~11/8 = 551.3980{{c}}
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| {{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 }}
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| [[Badness]] (Sintel): 0.416
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| ==== Septendecimal jacobin ====
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| Subgroup: 2.3.5.7.11.13.17
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| Comma list: 6656/6655, 12376/12375
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| Mapping: <br>
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| {| class="right-all"
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| |-
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| | [⟨ || 1 || 0 || 0 || 0 || 0 || -9 || 6 || ],
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| |-
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| | ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || 2 || ],
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| |-
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| | ⟨ || 0 || 0 || 1 || 0 || 0 || 1 || 2 || ],
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| |-
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| | ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || -1 || ],
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| |-
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| | ⟨ || 0 || 0 || 0 || 0 || 1 || 3 || -2 || ]]
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| |}
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| Optimal tunings:
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| * WE: ~2 = 1199.9890{{c}}, ~3/2 = 701.9675{{c}}, ~5/4 = 386.3466{{c}}, ~7/4 = 968.8454{{c}}, ~11/8 = 551.3945{{c}}
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| * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9670{{c}}, ~5/4 = 386.3410{{c}}, ~7/4 = 968.8434{{c}}, ~11/8 = 551.3964{{c}}
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| {{Optimal ET sequence|legend=0| 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 }}
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| Badness (Sintel): 0.239
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| == Etymology == | | == Etymology == |