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]\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{nd}[/math])	~4.27225 bits
Comma size	large
[sound info]
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English Wikipedia has an article on:

Semitone#Pythagorean tuning

2187/2048, the apotome (pronounced /əˈpɒtəmi/, like "a-POT-o'-me"), 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.

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

The apotome is the interval by which a sharp (#) or flat (b) modifies a note in the diatonic 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.

The number of steps an apotome is mapped to in an EDO is referred to as its sharpness.

Etymology

According to the OED, the earliest English use of limma and apotome (alt. spelling "apotomy") with its musical as opposed to mathematical meaning, is in 1694 in A Treatise of the Natural Grounds and Principles of Harmony by Church of England clergyman and natural philosopher William Holder. A relevant quote is "Difference between ... Tone Maj. and Limma. Apotome 2187 to 2048". The words are formed from the Greek, with "apo" meaning "away", "tome" meaning "cut" and limma meaning "remnant". So we begin with a major whole tone; the part cut away is the apotome (chromatic semitone) and the remnant is the limma (diatonic semitone).

See also

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