1216/1215: Difference between revisions

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'''1216/1215''', the '''password''' or '''Eratosthenes' comma''', is a [[19-limit]] (also 2.3.5.19 subgroup) ~~unnoticeable~~ comma. It is the amount by which [[19/15]] exceeds the [[81/64|Pythagorean major third (81/64)]], or [[20/19]] falls short of the [[256/243|Pythagorean minor second (256/243)]]. It is also the difference between [[19/18]], the undevicesimal semitone and [[135/128]], the major chroma, and in addition, between the [[513/512|undevicesimal comma]] and the [[schisma]].

'''1216/1215''', the '''password''' or '''Eratosthenes' comma''', is an [[unnoticeable comma|unnoticeable]] [[19-limit]] (also 2.3.5.19-[[subgroup]]) [[comma]]. It is the amount by which [[19/15]] exceeds the [[81/64|Pythagorean major third (81/64)]], or [[20/19]] falls short of the [[256/243|Pythagorean minor second (256/243)]]. It is also the difference between [[19/18]], the undevicesimal semitone and [[135/128]], the major chroma, and in addition, between the [[513/512|undevicesimal comma]] and the [[schisma]].

== Commatic relations ==

This comma is the difference between the following superparticular pairs:

* [[76/75]] and [[81/80]]

* [[76/75]] and [[81/80]] *

* [[136/135]] and [[153/152]]

* [[190/189]] and [[225/224]]

* [[256/255]] and [[324/323]]

* [[352/351]] and [[495/494]]

* [[361/360]] and [[513/512]]

* [[361/360]] and [[513/512]] *

* [[456/455]] and [[729/728]]

* [[676/675]] and [[1521/1520]]

* [[1156/1155]] and [[23409/23408]]

<nowiki>*</nowiki> both of these commas are also within the 2.3.5.19 subgroup.

It factors into the following superparticular pairs:

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== Temperaments ==

By tempering out this comma is defined the '''eratosthenes temperament''', which enables the [[eratosthenes chords]]. ~~EDOs~~ supporting this temperament includes {{Optimal ET sequence| 12, 29, 41, 53, 65, 77, 87, 94, 99, 106, 111, 118, 130, 140, 152, 159, 183, 193, 205, 217, 270, 282, 311, 323, 364, 400, 422, 581, and 692}}.

By tempering out this comma is defined the '''eratosthenes temperament''', which enables the [[eratosthenes chords]]. Edos supporting this temperament includes {{Optimal ET sequence| 12, 29, 41, 53, 65, 77, 87, 94, 99, 106, 111, 118, 130, 140, 152, 159, 183, 193, 205, 217, 270, 282, 311, 323, 364, 400, 422, 581, and 692 }}.

== Etymology ==

This comma was named after {{w|Eratosthenes}}, the Ancient Greek polymath. According to an analysis by [[Joseph Monzo]], Eratosthenes made use of this comma by employing "standard" [[3-limit]] pitches in his diatonic [[genus]], but substituting 19-limit ratios which are very close by in pitch in his chromatic and enharmonic genera.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_26618.html Yahoo! Tuning Group | ''ratios in ancient Greek tetrachords (was: Byzantine and Serb Church Music)'']</ref>

== See also ==

* [[Unnoticeable comma]]

* [[List of superparticular intervals]]

== Notes ==

[[Category:Eratosthenes]]

[[Category:Commas named after polymaths]]

[[Category:Commas with unknown etymology]]{{todo|add etymology|comment=add etymology for “password”}}

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 }}
-'''1216/1215''', the '''password''' or '''Eratosthenes' comma''', is a [[19-limit]] (also 2.3.5.19 subgroup) unnoticeable comma. It is the amount by which [[19/15]] exceeds the [[81/64|Pythagorean major third (81/64)]], or [[20/19]] falls short of the [[256/243|Pythagorean minor second (256/243)]]. It is also the difference between [[19/18]], the undevicesimal semitone and [[135/128]], the major chroma, and in addition, between the [[513/512|undevicesimal comma]] and the [[schisma]].
+'''1216/1215''', the '''password''' or '''Eratosthenes' comma''', is an [[unnoticeable comma|unnoticeable]] [[19-limit]] (also 2.3.5.19-[[subgroup]]) [[comma]]. It is the amount by which [[19/15]] exceeds the [[81/64|Pythagorean major third (81/64)]], or [[20/19]] falls short of the [[256/243|Pythagorean minor second (256/243)]]. It is also the difference between [[19/18]], the undevicesimal semitone and [[135/128]], the major chroma, and in addition, between the [[513/512|undevicesimal comma]] and the [[schisma]].
 == Commatic relations ==
 This comma is the difference between the following superparticular pairs:
-* [[76/75]] and [[81/80]]
+* [[76/75]] and [[81/80]] *
 * [[136/135]] and [[153/152]]
 * [[190/189]] and [[225/224]]
 * [[256/255]] and [[324/323]]
 * [[352/351]] and [[495/494]]
-* [[361/360]] and [[513/512]]
+* [[361/360]] and [[513/512]] *
 * [[456/455]] and [[729/728]]
 * [[676/675]] and [[1521/1520]]
 * [[1156/1155]] and [[23409/23408]]
+<nowiki>*</nowiki> both of these commas are also within the 2.3.5.19 subgroup.
 It factors into the following superparticular pairs:
@@ Line 27: / Line 28: @@
 == Temperaments ==
-By tempering out this comma is defined the '''eratosthenes temperament''', which enables the [[eratosthenes chords]]. EDOs supporting this temperament includes {{Optimal ET sequence| 12, 29, 41, 53, 65, 77, 87, 94, 99, 106, 111, 118, 130, 140, 152, 159, 183, 193, 205, 217, 270, 282, 311, 323, 364, 400, 422, 581, and 692}}.
+By tempering out this comma is defined the '''eratosthenes temperament''', which enables the [[eratosthenes chords]]. Edos supporting this temperament includes {{Optimal ET sequence| 12, 29, 41, 53, 65, 77, 87, 94, 99, 106, 111, 118, 130, 140, 152, 159, 183, 193, 205, 217, 270, 282, 311, 323, 364, 400, 422, 581, and 692 }}.
+== Etymology ==
+This comma was named after {{w|Eratosthenes}}, the Ancient Greek polymath. According to an analysis by [[Joseph Monzo]], Eratosthenes made use of this comma by employing "standard" [[3-limit]] pitches in his diatonic [[genus]], but substituting 19-limit ratios which are very close by in pitch in his chromatic and enharmonic genera.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_26618.html Yahoo! Tuning Group | ''ratios in ancient Greek tetrachords (was: Byzantine and Serb Church Music)'']</ref>
 == See also ==
-* [[Unnoticeable comma]]
 * [[List of superparticular intervals]]
+== Notes ==
 [[Category:Eratosthenes]]
+[[Category:Commas named after polymaths]]
+[[Category:Commas with unknown etymology]]{{todo|add etymology|comment=add etymology for “password”}}