128/125
Ratio | 128/125 |
Factorization | 2^{7} × 5^{-3} |
Monzo | [7 0 -3⟩ |
Size in cents | 41.058858¢ |
Names | diesis, augmented comma, enharmonic diesis, enharmonic comma |
Color name | g^{3}2, trigu 2nd, Trigu comma |
FJS name | [math]\text{d2}_{5,5,5}[/math] |
Special properties | reduced, reduced subharmonic |
Tenney height (log_{2} nd) | 13.9658 |
Weil height (log_{2} 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
- Diesis (disambiguation page)