4459/4455: Difference between revisions

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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 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]]).


== Temperaments ==
== 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.  
[[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.  
 
=== Tristanic ===
[[Subgroup]]: 3.5.7.11.13
 
[[Comma list]]: 4459/4455
 
{{Mapping|legend=2| 1 0 0 0 4 | 0 1 0 0 1 | 0 0 1 0 -3 | 0 0 0 1 1 }}
: mapping generators: ~3, ~5, ~7, ~11
 
[[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]]: [[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): 0.0885


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


[[Optimal tuning]] ([[CWE]]): ~2, ~3 = 1902.065{{c}}, ~5 = 2786.373{{c}}, ~7 = 3368.568{{c}}, ~11 = 4151.449{{c}}
[[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 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| 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
[[Badness]] (Sintel): 0.963
=== Tristanic ===
[[Subgroup]]: 3.5.7.11.13
[[Comma list]]: 4459/4455
{{Mapping|legend=2| 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{{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]]
[[Badness]] (Sintel): 0.089


== Etymology ==
== Etymology ==

Latest revision as of 11:07, 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 rank-4 tristanic temperament.

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]]

mapping generators: ~3, ~5, ~7, ~11

Optimal tunings:

  • WE: ~3 = 1902.0648 ¢, ~5/3 = 884.3078 ¢, ~7/3 = 1466.5027 ¢, ~11/9 = 347.3191 ¢
  • CWE: ~3 = 1901.9550 ¢, ~5/3 = 884.3336 ¢, ~7/3 = 1466.4567 ¢, ~11/9 = 347.4207 ¢

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

Badness (Sintel): 0.0885

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 ]]
mapping generators: ~2, ~3, ~5, ~7, ~11

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~3/2 = 702.0648 ¢, ~5/4 = 386.3726 ¢, ~7/4 = 968.5675 ¢, ~11/8 = 551.4487 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0648 ¢, ~5/4 = 386.3726 ¢, ~7/4 = 968.5675 ¢, ~11/8 = 551.4487 ¢

Optimal ET sequence: 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

Etymology

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

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