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 [[256/255|septendecimal kleisma]] and the [[324/323|nusu comma]], between the [[361/360|go comma]] and~~ the [[513/512|undevicesimal comma]]~~, and between the undevicesimal comma~~ and the [[schisma]].

'''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]].

By tempering it out 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}}.

== Commatic relations ==

This comma is the difference between the following superparticular pairs:

* [[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]]

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

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

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

It factors into the following superparticular pairs:

* [[2431/2430]] and [[2432/2431]]

* [[2080/2079]] and [[2926/2925]]

* [[1729/1728]] and [[4096/4095]]

* [[1540/1539]] and [[5776/5775]]

* [[1225/1224]] and [[165376/165375]]

== 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}}.

== See also ==

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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 [[256/255|septendecimal kleisma]] and the [[324/323|nusu comma]], between the [[361/360|go comma]] and the [[513/512|undevicesimal comma]], and between the undevicesimal comma and the [[schisma]].
+'''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]].
-By tempering it out 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}}.
+== Commatic relations ==
+This comma is the difference between the following superparticular pairs:
+* [[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]]
+* [[456/455]] and [[729/728]]
+* [[676/675]] and [[1521/1520]]
+* [[1156/1155]] and [[23409/23408]]
+It factors into the following superparticular pairs:
+* [[2431/2430]] and [[2432/2431]]
+* [[2080/2079]] and [[2926/2925]]
+* [[1729/1728]] and [[4096/4095]]
+* [[1540/1539]] and [[5776/5775]]
+* [[1225/1224]] and [[165376/165375]]
+== 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}}.
 == See also ==