4459/4455: Difference between revisions

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'''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]]).
'''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]]).


== Etymology ==
== Temperaments ==
This comma was named by [[User:Tristanbay|Tristan Bay]] in early May 2024 after himself.
[[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 '''tristanic''' temperament.  


== Temperaments ==
=== Tristanismic ===
=== Tristanismic ===
Tempering out 4459/4455 in the full 13-limit leads to this rank-5 temperament.
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


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[[Mapping]]:<br>
[[Mapping]]:<br>
{{val| 1 0 0 0 0 0 }}<br>
{| class="right-all"
{{val| 0 1 0 0 0 4 }}<br>
|-
{{val| 0 0 1 0 0 1 }}<br>
| [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || ],
{{val| 0 0 0 1 0 -3 }}<br>
|-
{{val| 0 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 || ]]
|}


[[Optimal tuning]] ([[CWE]]): 2, ~3 = 1902.065, ~5 = 2786.373, ~7 = 3368.568, ~11 = 4151.449
[[Optimal tuning]] ([[CWE]]): ~2, ~3 = 1902.065{{c}}, ~5 = 2786.373{{c}}, ~7 = 3368.568{{c}}, ~11 = 4151.449{{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|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 }}
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=== Tristanic ===
=== Tristanic ===
This is the main temperament that tempers out 4459/4455 in the no-twos 13-limit.
[[Subgroup]]: 3.5.7.11.13
[[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 4 }}<br>
{{val|0 1 0 0 1 }}<br>
{{val|0 0 1 0 -3 }}<br>
{{val|0 0 0 1 1 }}


[[Optimal tuning]] ([[CWE]]): 3, ~5 = 2786.289, ~7 = 3368.412, ~11 = 4151.331
[[Optimal tuning]] ([[CWE]]): ~3, ~5 = 2786.289{{c}}, ~7 = 3368.412{{c}}, ~11 = 4151.331{{c}}


[[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]]
[[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): 0.089
[[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]]

Revision as of 10:59, 22 March 2026

Interval information
Ratio 4459/4455
Factorization 3-4 × 5-1 × 73 × 11-1 × 13
Monzo [0 -4 -1 3 -1 1⟩
Size in cents 1.553721¢
Name tristanisma
Color name tholutrizo-agu 3rd, 3o1uz3g3,
tholutrizo-agu comma
FJS name [math]\displaystyle{ \text{dd3}^{7,7,7,13}_{5,11} }[/math]
Special properties reduced
Tenney norm (log2 nd) 24.2437
Weil norm (log2 max(n, d)) 24.245
Wilson norm (sopfr(nd)) 62
Comma size unnoticeable
Open this interval in xen-calc

4459/4455, the tristanisma, is an unnoticeable 13-limit no-twos comma measuring roughly a cent and a half. It is the difference between the 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).

Temperaments

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 tristanic temperament.

Tristanismic

Subgroup: 2.3.5.7.11.13

Comma list: 4459/4455

Mapping:

[⟨ 1 0 0 0 0 0 ],
⟨ 0 1 0 0 0 4 ],
⟨ 0 0 1 0 0 1 ],
⟨ 0 0 0 1 0 -3 ],
⟨ 0 0 0 0 1 1 ]]

Optimal tuning (CWE): ~2, ~3 = 1902.065 ¢, ~5 = 2786.373 ¢, ~7 = 3368.568 ¢, ~11 = 4151.449 ¢

Optimal ET sequence: 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

Badness (Sintel): 0.963

Tristanic

Subgroup: 3.5.7.11.13

Comma list: 4459/4455

Subgroup-val mapping: [⟨1 0 0 0 4], ⟨0 1 0 0 1], ⟨0 0 1 0 -3], ⟨0 0 0 1 1]]

Optimal tuning (CWE): ~3, ~5 = 2786.289 ¢, ~7 = 3368.412 ¢, ~11 = 4151.331 ¢

Optimal ET sequence: b9, b13, b17, b22, b26, b36, b39, b92, b114, b131, b170, b245, b284, b415, b428, b559, b598, b712, b843, b882, b1725, b4332ddf

Badness (Sintel): 0.089

Etymology

This comma was named by Tristan Bay in early May 2024 after himself.

Retrieved from "https://en.xen.wiki/index.php?title=4459/4455&oldid=226648"