2187/2048

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Interval information
Ratio 2187/2048
Factorization 2-11 × 37
Monzo [-11 7
Size in cents 113.68501¢
Names apotome,
Pythagorean chroma,
Pythagorean chromatic semitone,
whitewood comma
Color name Lw1, lawa unison
FJS name [math]\text{A1}[/math]
Special properties reduced,
reduced harmonic
Tenney height (log2 nd) 22.0947
Weil height (log2 max(n, d)) 22.1895
Wilson height (sopfr (nd)) 43
Harmonic entropy
(Shannon, [math]\sqrt{n\cdot d}[/math])
~4.62809 bits
Comma size large

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

2187/2048, the apotome, also known as the Pythagorean chromatic semitone or the Pythagorean chroma, is the chromatic semitone in the Pythagorean tuning. It is the 3-limit interval between seven perfect just fifths (3/2) and four octaves (2/1): 37/211 = 2187/2048, and measures about 113.7¢. Unlike the situation in meantone tunings, it is larger, not smaller, than the corresponding diatonic semitone, which is the Pythagorean minor second of 256/243.

The apotome is associated with sharps (#) and flats (b) in the chain-of-fifths notation. For example, in Pythagorean tuning, C and C# in the same octave are exactly an apotome apart. In tempered tuning systems, the mapping of the apotome dictates the size of sharps and flats. For instance, if the apotome is tempered out, then sharps and flats have no effect on pitch in these systems.

Approximation

This interval is well approximated by any tuning generated with accurate octaves and fifths. For example, 5\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 whitewood temperament, and it may be called the whitewood comma. See apotome family for extensions thereof.

See also