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]]): 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 ==

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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 employ~~ the [[chain-of-fifths notation~~|chain-of-fifths~~]] ~~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 associated with sharps (#) and flats (b) 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 the [[sharpness]].

== See also ==

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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]]): 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 employ the [[chain-of-fifths notation|chain-of-fifths]] 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 associated with sharps (#) and flats (b) 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 the [[sharpness]].
 == See also ==