128/125

Interval information

Ratio	128/125
Factorization	27 × 5-3
Monzo	[7 0 -3⟩
Size in cents	41.05886 ¢
Names	diesis, augmented comma, enharmonic diesis, enharmonic comma
Color name	g32, trigu 2nd, Trigu comma
FJS name	[math]\displaystyle{ \text{d2}_{5,5,5} }[/math]
Special properties	reduced, reduced subharmonic
Tenney norm (log2 nd)	13.9658
Weil norm (log2 max(n, d))	14
Wilson norm (sopfr(nd))	29
Comma size	medium
S-expression	S4/S5
Open this interval in xen-calc

The 41.059-cent interval of 128/125 is called the diesis or augmented comma; it represents the gap between a stack of three 5/4 just major thirds and the octave, or in other words 2/(5/4)³.

Approximations

This interval is fairly accurately represented by a single step in 28-, 31- or 34edo, and by two steps of 53-, 59- or 65edo. In any tuning with pure octaves and just major thirds, such as quarter-comma meantone, it will be exact. Furthermore, in meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called enharmonic diesis or enharmonic comma for this reason.

Temperaments

As a comma

Tempering out this comma leads to augmented temperament. See Augmented family for the family where it is tempered out.

As an interval

If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce 7-limit and 11-limit harmony into 5-limit scales. The simplest way to accomplish this is tempering out 126/125 and 4125/4096 to equate it with 64/63 and 33/32.

Trivia

This interval represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000.

See also

• Diesis (disambiguation page)