128/125

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Interval information
Ratio 128/125
Factorization 27 × 5-3
Monzo [7 0 -3
Size in cents 41.058858¢
Names diesis,
augmented comma,
enharmonic diesis,
enharmonic comma
Color name g32, trigu 2nd,
Trigu comma
FJS name [math]\text{d2}_{5,5,5}[/math]
Special properties reduced,
reduced subharmonic
Tenney height (log2 nd) 13.9658
Weil height (log2 max(n, d)) 14
Wilson height (sopfr (nd)) 29
Harmonic entropy
(Shannon, [math]\sqrt{nd}[/math])
~4.51672 bits
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.

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