2187/2048: Difference between revisions

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'''2187/2048''', the '''Pythagorean 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]].

'''2187/2048''', the '''Pythagorean 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]]. In other words, it is the [[3-limit]] augmented unison, that is, the interval between seven just perfect fifths ([[3/2]]) and four octaves ([[2/1]]): 37/211 = 2187/2048. It measures about 113.7{{c}}. 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 ==

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== 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.

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 [[Whitewood family]] for extensions thereof.

== Notation ==

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]].

The number of steps an apotome is mapped to in an edo is referred to as its [[sharpness]].

== Etymology ==

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 {{Wikipedia|Semitone #Pythagorean tuning}}
-'''2187/2048''', the '''Pythagorean 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]].
+'''2187/2048''', the '''Pythagorean 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]]. In other words, it is the [[3-limit]] augmented unison, that is, the interval between seven just perfect fifths ([[3/2]]) and four octaves ([[2/1]]): 3<sup>7</sup>/2<sup>11</sup> = 2187/2048. It measures about 113.7{{c}}. 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 ==
@@ Line 13: / Line 13: @@
 == 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.
+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 [[Whitewood family]] for extensions thereof.
 == Notation ==
 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]].
+The number of steps an apotome is mapped to in an edo is referred to as its [[sharpness]].
 == Etymology ==