1216/1215: Difference between revisions

(6 intermediate revisions by 4 users not shown)

Line 5:

}}

'''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 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]].

By tempering it out is defined the '''eratosthenes temperament''', which enables the ~~'''~~eratosthenes chords~~'''~~. ~~EDOs~~ supporting this temperament includes {{~~EDOs~~| 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]]

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

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 }}.

== 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”}}

@@ Line 5: / Line 5: @@
 }}
-'''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 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]].
-By tempering it out is defined the '''eratosthenes temperament''', which enables the '''eratosthenes chords'''. EDOs supporting this temperament includes {{EDOs| 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]]
+<nowiki>*</nowiki> both of these commas are also within the 2.3.5.19 subgroup.
+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 }}.
+== 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”}}