4459/4455

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.