2187/2048: Difference between revisions

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'''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]].

'''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]].

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 ==

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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~~ ==

== Notation ==

~~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~~]].

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.

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<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).

== See also ==

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* [[Large comma]]

* [[25/24]] – classic chromatic semitone

== References ==

[[Category:Second]]

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 {{Wikipedia|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]]): 3<sup>7</sup>/2<sup>11</sup> = 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]].
+'''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]]): 3<sup>7</sup>/2<sup>11</sup> = 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 ==
@@ Line 17: / Line 15: @@
 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 ==
+== Notation ==
-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]].
+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.
+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<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).
 == See also ==
@@ Line 25: / Line 28: @@
 * [[Large comma]]
 * [[25/24]] – classic chromatic semitone
+== References ==
 [[Category:Second]]