256/243
Ratio | 256/243 |
Factorization | 2^{8} × 3^{-5} |
Monzo | [8 -5⟩ |
Size in cents | 90.224996¢ |
Names | Pythagorean limma, Pythagorean diatonic semitone, blackwood comma |
Color name | sw2, sawa 2nd |
FJS name | [math]\text{m2}[/math] |
Special properties | reduced, reduced subharmonic |
Tenney height (log_{2} nd) | 15.9248 |
Weil height (log_{2} max(n, d)) | 16 |
Wilson height (sopfr (nd)) | 31 |
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math]) |
~4.30829 bits |
Comma size | medium |
S-expression | S7 * S8^{2} |
[sound info] | |
open this interval in xen-calc |
The interval 256/243, the Pythagorean limma, or Pythagorean diatonic semitone factors as 2^{8}/3^{5}, is about 90.2 cents in size, and is the diatonic semitone in Pythagorean tuning. It can be generated by stacking five 4/3 just perfect fourths and octave-reducing the resulting interval.
Approximation
This interval is well approximated by any tuning generated with accurate octaves and fifths. For example, 4\53 is a very good approximation.
Temperaments
When this ratio is taken as a comma to be tempered in the 5-limit, it produces the blackwood temperament, and it may be called the blackwood comma. While the blackwood temperament is named after Easley Blackwood Jr, it also coincidentally fits the naming scheme of the comptonesque temperament collection, as Blackwood is the name of a settlement in the old versions of Pandora, a project by Vector. Edos tempering it out include 5edo, 10edo, 15edo, 20edo, 25edo and 30edo. See limmic temperaments for a number of other temperaments where it is tempered out.
Notation
In musical notations that employ the diatonic chain-of-fifths, such as the ups and downs notation, the limma is represented by the distances between B and C, as well as between E and F.
The scale is structured with the following step pattern:
- A to B: whole tone
- B to C: limma
- C to D: whole tone
- D to E: whole tone
- E to F: limma
- F to G: whole tone
- G to A: whole tone
This pattern highlights the placement of the limma intervals between the note pairs above, distinguishing them from the whole tone that occur between the other note pairs.
See also
- 243/128 – its octave complement
- 729/512 – its fifth complement
- 16/15 – the classic (5-limit) diatonic semitone
- Gallery of just intervals
- Medium comma
- Pythagorean tuning