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'''4459/4455''', the '''tristanisma''', is an [[unnoticeable comma|unnoticeable]] [[13-limit]] no-twos [[comma]] measuring roughly a [[cent]] and a half. It is the difference between the [[odd-particular]]s [[245/243]] and [[275/273]], and between [[91/90]] and [[99/98]]. It factors into the [[superparticular ratio]] pairs ([[2080/2079]])⋅([[2401/2400]]) or ([[1001/1000]])⋅([[9801/9800]]).
'''4459/4455''', the '''tristanisma''', is [[13-limit]] no-twos [[unnoticeable comma]] measuring roughly a [[cent]] and a half. It is the difference between the [[Odd-particular|odd-particulars]] [[245/243]] and [[275/273]], and between [[91/90]] and [[99/98]]. It factors into the superparticular ratio pairs ([[2080/2079]])([[2401/2400]]) or ([[1001/1000]])([[9801/9800]]).
 
== Etymology ==
This comma was named by [[User:Tristanbay|Tristan Bay]] in early May 2024 after himself.


== Temperaments ==
== Temperaments ==
=== Tristanismic ===
[[Tempering out]] this comma in the full 13-limit leads to the rank-5 '''tristanismic''' temperament, or in the 3.5.7.11.13 subgroup, the rank-4 '''tristanic''' temperament.  
Tempering out 4459/4455 in the full 13-limit leads to this rank-5 temperament.


[[Subgroup]]: 2.3.5.7.11.13
=== Tristanic ===
[[Subgroup]]: 3.5.7.11.13


[[Comma list]]: 4459/4455
[[Comma list]]: 4459/4455


[[Mapping]]:<br>
{{Mapping|legend=2| 1 0 0 0 4 | 0 1 0 0 1 | 0 0 1 0 -3 | 0 0 0 1 1 }}
{{val| 1 0 0 0 0 0 }}<br>
: mapping generators: ~3, ~5, ~7, ~11
{{val| 0 1 0 0 0 4 }}<br>
{{val| 0 0 1 0 0 1 }}<br>
{{val| 0 0 0 1 0 -3 }}<br>
{{val| 0 0 0 0 1 1 }}


[[Optimal tuning]] ([[CWE]]): 2, ~3 = 1902.065, ~5 = 2786.373, ~7 = 3368.568, ~11 = 4151.449
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1902.0648{{c}}, ~5/3 = 884.3078{{c}}, ~7/3 = 1466.5027{{c}}, ~11/9 = 347.3191{{c}}
* [[CWE]]: ~3 = 1901.9550{{c}}, ~5/3 = 884.3336{{c}}, ~7/3 = 1466.4567{{c}}, ~11/9 = 347.4207{{c}}


{{Optimal ET sequence|legend=1| 8d, 9, 12, 14cf, 17c, 19e, 22, 31f, 41, 58, 72, 118, 121, 130, 190, 198, 270, 581, 711, 981, 1171, 1441, 1752, 2022, 2463d, 2733d }}
[[Optimal ET sequence]]: [[13edt|b13]], [[17edt|b17]], [[22edt|b22]], [[26edt|b26]], [[36edt|b36]], [[39edt|b39]], [[92edt|b92]], [[114edt|b114]], [[131edt|b131]], [[170edt|b170]], [[245edt|b245]], [[284edt|b284]], [[415edt|b415]], [[428edt|b428]], [[559edt|b559]], [[598edt|b598]], [[712edt|b712]], [[843edt|b843]], [[882edt|b882]], [[1725edt|b1725]], [[4332edt|b4332ddf]]


[[Badness]] (Sintel): 0.963
[[Badness]] (Sintel): 0.0885


=== Tristanic ===
=== Tristanismic ===
This is the main temperament that tempers out 4459/4455 in the no-twos 13-limit.
[[Subgroup]]: 2.3.5.7.11.13
 
[[Subgroup]]: 3.5.7.11.13


[[Comma list]]: 4459/4455
[[Comma list]]: 4459/4455


[[Mapping]]:<br>
[[Mapping]]:<br>
{{val|1 0 0 0 4 }}<br>
{| class="right-all"
{{val|0 1 0 0 1 }}<br>
|-
{{val|0 0 1 0 -3 }}<br>
| [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || ],
{{val|0 0 0 1 1 }}
|-
| ⟨ || 0 || 1 || 0 || 0 || 0 || 4 || ],
|-
| ⟨ || 0 || 0 || 1 || 0 || 0 || 1 || ],
|-
| ⟨ || 0 || 0 || 0 || 1 || 0 || -3 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || 1 || 1 || ]]
|}
: mapping generators: ~2, ~3, ~5, ~7, ~11
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0648{{c}}, ~5/4 = 386.3726{{c}}, ~7/4 = 968.5675{{c}}, ~11/8 = 551.4487{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0648{{c}}, ~5/4 = 386.3726{{c}}, ~7/4 = 968.5675{{c}}, ~11/8 = 551.4487{{c}}


[[Optimal tuning]] ([[CWE]]): 3, ~5 = 2786.289, ~7 = 3368.412, ~11 = 4151.331
{{Optimal ET sequence|legend=1| 17c, 19e, 22, 31f, 41, 58, 72, 118, 121, 130, 190, 198, 270, 581, 711, 981, 1171, 1441, 1752, 2022, 2463d, 2733d }}


[[Optimal ET sequence]]: [[9edt|b9]], [[13edt|b13]], [[17edt|b17]], [[22edt|b22]], [[26edt|b26]], [[36edt|b36]], [[39edt|b39]], [[92edt|b92]], [[114edt|b114]], [[131edt|b131]], [[170edt|b170]], [[245edt|b245]], [[284edt|b284]], [[415edt|b415]], [[428edt|b428]], [[559edt|b559]], [[598edt|b598]], [[712edt|b712]], [[843edt|b843]], [[882edt|b882]], [[1725edt|b1725]], [[4332edt|b4332ddf]]
[[Badness]] (Sintel): 0.963


[[Badness]] (Sintel): 0.089
== Etymology ==
This comma was named by [[Tristan Bay]] in early May 2024 after himself.


[[Category:Tristanismic]]
[[Category:Tristanismic]]
[[Category:Commas named after composers]]
[[Category:Commas named after music theorists]]
[[Category:Commas someone named after themselves]]
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