2187/2048: Difference between revisions

| Sound = jid_2187_2048_pluck_adu_dr220.mp3

{{Wikipedia|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''' or the '''3-limit augmented unison''', 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, [[53edo|5\53]] is a very good approximation.

 

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.

 

The apotome is the interval by which a sharp (#) or flat (b) modifies a note in the [[5L 2s|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.

 

 

According to the OED, the earliest English use of [[limma]] and apotome (alt. spelling "apotomy") with its musical as opposed to mathematical<ref>https://www.scientificlib.com/en/Mathematics/LX/Apotome.html</ref> meaning, is in 1694 in ''A Treatise of the Natural Grounds and Principles of Harmony''<ref>https://ota.bodleian.ox.ac.uk/repository/xmlui/bitstream/handle/20.500.12024/A44132/A44132.html</ref> 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).