2025/2023

Interval information

Ratio	2025/2023
Factorization	34 × 52 × 7-1 × 17-2
Monzo	[0 4 2 -1 0 0 -2⟩
Size in cents	1.710706¢
Name	fidesma
Color name	17uuryy-3, susuruyoyo negative 3rd
FJS name	[math]\displaystyle{ \text{ddd}{-3}^{5,5}_{7,17,17} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	21.966
Weil norm (log2 max(n, d))	21.9674
Wilson norm (sopfr(nd))	63
Comma size	unnoticeable
S-expression	S15/S17
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2025/2023, the fidesma, is an unnoticeable 17-limit (specifically 3.5.7.17-subgroup) comma with a size of about 1.71 cents. It is the difference between two 17/15 wide wholetones and a 9/7 supermajor third.

It is the superpyth counterpart of 1445/1444, tempered out in any scale where the fifth is sharp enough that two of them approximates 17/15 and four 9/7, most notably 22edo, which is very close to quarter-comma superpyth. However, the fidesma can be conceptualized in other ways, such as being the amount by which two instances of 18/17 exceed 28/25.

It factors into (1701/1700)⋅(2500/2499), and in the 23-limit, (2024/2023)⋅(2025/2024), hinting at an extension of fidesmic to the 2.3.5.7.11.17.23 subgroup.

Temperaments

Tempering out this comma in the 17-limit results in the rank-6 fidesmic temperament, or in the the 3.5.7.17 subgroup, the rank-3 fidic temperament.

Fidic

Subgroup: 3.5.7.17

Subgroup-val mapping: [⟨1 0 0 2], ⟨0 1 0 1], ⟨0 0 2 -1]]

mapping generators: ~3, ~5, ~45/17

Optimal tunings:

• WE: ~3 = 1901.8290 ¢, ~5/3 = 884.3495 ¢, ~45/17 = 1684.4623 ¢
• CWE: ~3 = 1901.9550 ¢, ~5/3 = 884.3084 ¢, ~45/17 = 1684.5363 ¢

Optimal ET sequence: b17, b26, b62, b88, b114, b245, b271, b385, b630, b656, b1015, b1400g, b2415g, b2686gg, b3071dgg

Badness (Sintel): 0.0269

Fidesmic

Subgroup: 2.3.5.7.11.13

Mapping:

[⟨	1	0	0	0	0	0	0	],
⟨	0	1	0	0	0	0	2	],
⟨	0	0	1	0	0	0	1	],
⟨	0	0	0	2	0	0	-1	],
⟨	0	0	0	0	0	1	0	]]

mapping generators: ~2, ~3, ~5, ~45/17, ~11, ~13

Optimal tunings:

• WE: ~2 = 1200.0000 ¢, ~3/2 = 701.8290 ¢, ~5/4 = 386.1785 ¢, ~45/34 = 484.4623 ¢, ~11/8 = 551.3179 ¢, ~13/8 = 840.5276 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8290 ¢, ~5/4 = 386.1785 ¢, ~45/34 = 484.4623 ¢, ~11/8 = 551.3179 ¢, ~13/8 = 840.5276 ¢

Optimal ET sequence: 17cg, 22, 27eg, 34d, 38df, 45efg, 49fg, 50, 60e, 72, 111, 121, 149, 159, 171, 183, 243e, 270, 354, 414, 441, 525, 597, 624, 684, 867, 966g, 1038, 1308g, 1402, 1833g, 2026g, 2551g *

* optimal patent val: 1905

Badness (Sintel): 2.53

Etymology

This comma was named by Aura in 2023. The name comes from both Latin "fidēs" (meaning "chord") and Latin fīdēs (meaning "you will rely on"), which is fitting because those who like more accurate forms of superpyth-like temperaments and scales frequently end up relying on the tempering of this comma for a number of essentially tempered chords.