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
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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.

Approximation

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.

It 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

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

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)