256/243: Difference between revisions

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== Notation ==
== Notation ==
In musical notations that employ the [[chain-of-fifths notation|chain-of-fifths]] along with seven note names, 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.
In musical notations that employ the [[5L 2s|diatonic]] [[chain-of-fifths notation|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:
The scale is structured with the following step pattern:
    A to B: [[9/8|whole tone]]
* A to B: [[9/8|whole tone]]
    B to C: [[256/243|limma]]
* B to C: [[256/243|limma]]
    C to D: [[9/8|whole tone]]
* C to D: [[9/8|whole tone]]
    D to E: [[9/8|whole tone]]
* D to E: [[9/8|whole tone]]
    E to F: [[256/243|limma]]
* E to F: [[256/243|limma]]
    F to G: [[9/8|whole tone]]
* F to G: [[9/8|whole tone]]
    G to A: [[9/8|whole tone]]
* G to A: [[9/8|whole tone]]
This pattern highlights the placement of the limma intervals between B and C, and E and F, distinguishing them from the [[9/8|whole tone]] that occur between the other note pairs.
This pattern highlights the placement of the limma intervals between the note pairs above, distinguishing them from the [[9/8|whole tone]] that occur between the other note pairs.


== See also ==
== See also ==

Revision as of 14:27, 11 August 2024

Interval information
Ratio 256/243
Factorization 28 × 3-5
Monzo [8 -5⟩
Size in cents 90.225¢
Names Pythagorean limma,
Pythagorean diatonic semitone,
blackwood comma
Color name sw2, sawa 2nd
FJS name [math]\displaystyle{ \text{m2} }[/math]
Special properties reduced,
reduced subharmonic
Tenney norm (log2 nd) 15.9248
Weil norm (log2 max(n, d)) 16
Wilson norm (sopfr(nd)) 31
Comma size medium
S-expression S7⋅S82

[sound info]
Open this interval in xen-calc
English Wikipedia has an article on:

The interval 256/243, the Pythagorean limma, or Pythagorean diatonic semitone factors as 28/35, 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. 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:

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

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