2187/2048: Difference between revisions

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{{Infobox Interval

| Name = apotome, Pythagorean chroma, Pythagorean chromatic semitone, whitewood comma

| Name = Pythagorean apotome, Pythagorean chroma, Pythagorean chromatic semitone, whitewood comma

| Color name = Lw1, lawa unison

| Sound = jid_2187_2048_pluck_adu_dr220.mp3

| Comma = yes

}}

'''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]].

'''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 ==

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).

While ''apotome'' alone refers to this interval, the standard term is ''Pythagorean apotome'', which forms a pair with ''Pythagorean limma''.

== See also ==

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Name = apotome, Pythagorean chroma, Pythagorean chromatic semitone, whitewood comma
+| Name = Pythagorean apotome, Pythagorean chroma, Pythagorean chromatic semitone, whitewood comma
 | Color name = Lw1, lawa unison
 | Sound = jid_2187_2048_pluck_adu_dr220.mp3
 | Comma = yes
 }}
-{{Wikipedia|Semitone#Pythagorean tuning}}
+{{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]]): 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 ==
 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).
+While ''apotome'' alone refers to this interval, the standard term is ''Pythagorean apotome'', which forms a pair with ''Pythagorean limma''.
 == See also ==