6656/6655: Difference between revisions

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| ⟨ || 0 || 0 || 0 || 0 || 1 || 3 || ]]
| ⟨ || 0 || 0 || 0 || 0 || 1 || 3 || ]]
|}
|}
: mapping generators: ~2, ~3, ~5, ~7, ~11


: mapping generators: ~2, ~3, ~5, ~7, ~11
[[Optimal tuning]]s:  
* [[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}}
* [[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}}


{{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 }}
{{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 }}
[[Badness]] (Sintel): 0.416


==== Septendecimal jacobin ====
==== Septendecimal jacobin ====
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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 }}
Optimal tunings:
* 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}}
* 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}}
 
{{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 }}
 
Badness (Sintel): 0.239


== Etymology ==
== Etymology ==

Latest revision as of 06:52, 25 February 2026

Interval information
Ratio 6656/6655
Factorization 29 × 5-1 × 11-3 × 13
Monzo [9 0 -1 0 -3 1⟩
Size in cents 0.2601208¢
Name jacobin comma
Color name Thotrilu-agu comma
FJS name [math]\displaystyle{ \text{m2}^{13}_{5,11,11,11} }[/math]
Special properties superparticular,
reduced
Tenney norm (log2 nd) 25.4007
Weil norm (log2 max(n, d)) 25.4009
Wilson norm (sopfr(nd)) 69
Comma size unnoticeable
Open this interval in xen-calc

6656/6655, the jacobin comma, is an unnoticeable 13-limit (also 2.5.11.13 subgroup) superparticular comma of about 0.26 ¢. It is the difference between a stack of three 11/8 superfourths and one 13/10 naiadic plus an octave (13/5).

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. Miscellaneous temperaments tempering out this comma are collected in The Jacobins.

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 tunings:

  • WE: ~2 = 1199.9887 ¢, ~3/2 = 701.9663 ¢, ~5/4 = 386.3430 ¢, ~7/4 = 968.8484 ¢, ~11/8 = 551.3967 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9661 ¢, ~5/4 = 386.3383 ¢, ~7/4 = 968.8455 ¢, ~11/8 = 551.3980 ¢

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

Badness (Sintel): 0.416

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 tunings:

  • WE: ~2 = 1199.9890 ¢, ~3/2 = 701.9675 ¢, ~5/4 = 386.3466 ¢, ~7/4 = 968.8454 ¢, ~11/8 = 551.3945 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9670 ¢, ~5/4 = 386.3410 ¢, ~7/4 = 968.8434 ¢, ~11/8 = 551.3964 ¢

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

Badness (Sintel): 0.239

Etymology

This comma was apparently named by Gene Ward Smith in 2014 for reasons unknown[1].

See also

References

  1. ↑ Xenharmonic Wiki | List of superparticular intervals (Revision as of 23:46, 13 June 2014 by Wikispaces>genewardsmith)
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