512/507

Interval information

Ratio	512/507
Factorization	29 × 3-1 × 13-2
Monzo	[9 -1 0 0 0 -2⟩
Size in cents	16.98968¢
Name	tridecimal neutral thirds comma
Color name	3uu1, thuthu 1sn, Thuthu comma
FJS name	[math]\displaystyle{ \text{A1}_{13,13} }[/math]
Special properties	reduced, reduced subharmonic
Tenney norm (log2 nd)	17.9858
Weil norm (log2 max(n, d))	18
Wilson norm (sopfr(nd))	47
Comma size	small
Open this interval in xen-calc

512/507, the tridecimal neutral thirds comma, is a 13-limit (also 2.3.13-subgroup) comma with a value of roughly 17.0 cents. It is significant in just intonation as the amount by which the tridecimal neutral third, 16/13, differs from its fifth complement, 39/32, as well as the amount by which a stack of two 1053/1024 quartertones fall short of the apotome.

Temperaments

Tempering out this comma equates both tridecimal neutral thirds to the true neutral third sqrt(3/2), and is proposed to be called harmoneutral temperament by Vector in 2025, as it tempers the subharmonic neutral third together with its fifth complement. It can thus be seen as the 2.3.13 analogue of 2.3.11 neutral, which tempers out 243/242 and equates 11/9 with its fifth complement, but instead equating 16/13 with 39/32, and preferring sharper fifths over flatter ones.

However, it has significantly more damage than neutral, similarly to how archy has more damage than meantone. For example, the neutral third in 2.3.13 CWE tuning is 7.7 ¢ flat of 16/13, but also 4.4 ¢ sharp of 11/9. This makes it ambiguous, but since 11/9 is simpler and closer, the neutral third is likely to be heard as 11/9 rather than 16/13, which shows the higher damage of harmoneutral. Using a tuning sharper than "optimal", such as 17edo or 78edo, makes the tridecimal nature of the intervals more apparent, but only to an extent. Compare that to in neutral, where the neutral third in 2.3.11 CWE tuning is 3.1 ¢ sharp of 11/9 and 8.9 ¢ flat of 16/13, so it is much more clearly 11/9. Harmoneutral also has much greater errors, with a (2.3.13) 13-odd-limit minimax error of 6.80 ¢ compared to neutral's (2.3.11) 11-odd-limit minimax error of 2.86 ¢.

Another possible option is to temper out both 512/507 and 243/242, and therefore 144/143, leading to namo. This temperament is generated by a neutral third like harmoneutral and neutral, except all of the neutral intervals are now ambiguously 11-limit or 13-limit.

Tempering it out along with 4096/3993 and 361/352 in 2.3.11.13.19 leads to a temperament proposed to be called deltahexacot after its ploidacot.

See also

• Small comma