2187/2048

Interval information

Ratio	2187/2048
Factorization	2-11 × 37
Monzo	[-11 7⟩
Size in cents	113.685¢
Names	apotome, Pythagorean chroma, Pythagorean chromatic semitone, whitewood comma
Color name	Lw1, lawa unison
FJS name	[math]\displaystyle{ \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
Comma size	large
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Semitone#Pythagorean tuning

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.

Notation and sharpness

In musical notations that use the cycle of fifths and fourths along with seven note names—such as the ups and downs notation—the apotome is represented by the interval between a note and its sharp counterpart. In this context, the apotome is referred to as the sharpness.

See also

• 4096/2187 – its octave complement
• Gallery of just intervals
• Large comma
• 25/24 – classic chromatic semitone