5120/5103

Interval information

Ratio	5120/5103
Factorization	210 × 3-6 × 5 × 7-1
Monzo	[10 -6 1 -1⟩
Size in cents	5.757802¢
Names	argent comma, 5/7-kleisma
Color name	sry1, saruyo 1sn, sryM, saruyoma
FJS name	[math]\displaystyle{ \text{P1}^{5}_{7} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	24.6391
Weil norm (log2 max(n, d))	24.6439
Wilson norm (sopfr(nd))	50
Comma size	small
S-expression	S8/S9
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5120/5103, the argent comma, 5/7-kleisma, hemifamity comma, or aberschisma, is a small 7-limit comma measuring about 5+3⁄4 ¢. It is the difference between 7/5 (small septimal tritone) and 1024/729 (Pythagorean diminished fifth), or between 10/7 (large septimal tritone) and 729/512 (Pythagorean augmented fourth). It is also the difference between 15/14 and 2187/2048, between 21/20 and 256/243, between 80/63 and 81/64, etc., hence, between the septimal comma of 64/63 and the syntonic comma of 81/80, as well as between 32805/32768, the schisma, and 225/224, the marvel comma.

Temperaments

Tempering it out leads to the rank-3 hemifamity temperament, which splits the septimal quartertone 36/35 into two equal parts, each representing 81/80~64/63. Rank-2 temperaments that temper it out include hemififths, amity, and garibaldi to name three. Edos that temper it out include 12, 29, 34, 41, 46, 53, 58, 87, 94, 99, 111, 140, 145, 152, 239, 292, 391, etc. See Hemifamity family for the rank-3 family where it is tempered out. See Hemifamity temperaments for a collection of rank-2 temperaments where it is tempered out.

Sagittal notation

In the Sagittal system, the downward version of this comma (possibly tempered) is represented by the sagittal  and is called the 7/5 kleisma, or 7/5k for short, because the simplest interval it notates is 7/5, as for example in B-F⁠ ⁠. The upward version is called 5/7k or 7/5k up and is represented by .

Etymology

The name hemifamity was given by Gene Ward Smith in 2005 as a contraction of hemififths and amity.

The name aberschisma, was coined by Tristan Bay in 2024 in reference to groundfault's aberrismic theory, since 5120/5103 is the difference between two of the most common aberrismas in just intonation (81/80 and 64/63).

In 2025, FloraC et al. gave this comma the name argent comma due to the fifth size from the argent temperament being essentially optimal for tuning the full 7-limit temperament (and the 2.3.7/5 subgroup temperament) that tempers only it out.