13/12: Difference between revisions

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{{Infobox Interval

~~| Icon =~~

| Name = (lesser) tridecimal neutral second

~~| Ratio = 13/12~~

~~| Monzo = -2 -1 0 0 0 1~~

~~| Cents = 138.57266~~

| Name = tridecimal neutral second

| Color name = 3o2, tho 2nd

~~| FJS name = m2<sup>13</sup>~~

| Sound = jid_13_12_pluck_adu_dr220.mp3

}}

In [[13-limit]] [[~~Just Intonation~~]], '''13/12''' is a '''neutral second''' of about 138.6¢. It is a [[superparticular]] interval, as it is found in the harmonic series between the 13th and the 12th ~~overtone~~ (between [[13/8]] and [[3/2]] in the octave). It is flat of the [[11-limit]] lesser neutral second of [[12/11]] by [[144/143]] (about 12.1¢), and sharp of the 13-limit large semitone of [[14/13]] by [[169/168]] (about 10.3¢).

In [[13-limit]] [[just intonation]], '''13/12''' is the '''(lesser) tridecimal neutral second''' of about 138.6¢. It is a [[superparticular]] interval, as it is found in the harmonic series between the 13th and the 12th harmonics (between [[13/8]] and [[3/2]] in the octave). It is flat of the [[11-limit]] lesser neutral second of [[12/11]] by [[144/143]] (about 12.1¢), and sharp of the 13-limit large semitone of [[14/13]] by [[169/168]] (about 10.3¢).

~~The neutral second in~~ [[~~17edo~~]] ~~is about 141.2¢, about 2~~.~~6¢ sharp of~~ 13/12. ~~Thus~~, ~~if 10\17 (ten degrees of 17edo) is taken to approximate~~ 3/2 ~~and 12\17 taken~~ to ~~approximate 13~~/8, ~~you can generate a~~ 13~~-limit harmonic triad that approximates an~~ 8:12~~:13 chord with a good~~ 13/12.

== Approximation ==

It is approximated to within about 0.11 [[cents]] by the 3-step interval of [[26edo]].

{{Interval edo approximation|{{PAGENAME}}}}

== Temperaments ==

13/12 can be used to generate [[bleu]] temperament in the 2.3.7.11.13 subgroup, mapping [[3/2]] to +5 generators, [[7/4]] to +7 generators, [[11/8]] to +4 generators, and [[13/8]] to +6 generators.

A slightly sharp 13/12 generates [[glacier]] temperament, which equates 5 13/12's to 3/2 like bleu. This temperament has an extension to the 2.3.7.11.13.23.29 subgroup which is more complex but much more accurate than bleu.

== See also ==

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* [[List of superparticular intervals]]

[[Category:~~13-limit]]~~

[[Category:Second]]

~~[[Category:Interval]]~~

~~[[Category:Just interval]]~~

~~[[Category:Ratio~~]]

[[Category:Neutral second]]

~~[[Category:Second]]~~

~~[[Category:Superparticular]]~~

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Icon =
+| Name = (lesser) tridecimal neutral second
-| Ratio = 13/12
-| Monzo = -2 -1 0 0 0 1
-| Cents = 138.57266
-| Name = tridecimal neutral second
 | Color name = 3o2, tho 2nd
-| FJS name = m2<sup>13</sup>
 | Sound = jid_13_12_pluck_adu_dr220.mp3
 }}
-In [[13-limit]] [[Just Intonation]], '''13/12''' is a '''neutral second''' of about 138.6¢. It is a [[superparticular]] interval, as it is found in the harmonic series between the 13th and the 12th overtone (between [[13/8]] and [[3/2]] in the octave). It is flat of the [[11-limit]] lesser neutral second of [[12/11]] by [[144/143]] (about 12.1¢), and sharp of the 13-limit large semitone of [[14/13]] by [[169/168]] (about 10.3¢).
+In [[13-limit]] [[just intonation]], '''13/12''' is the '''(lesser) tridecimal neutral second''' of about 138.6¢. It is a [[superparticular]] interval, as it is found in the harmonic series between the 13th and the 12th harmonics (between [[13/8]] and [[3/2]] in the octave). It is flat of the [[11-limit]] lesser neutral second of [[12/11]] by [[144/143]] (about 12.1¢), and sharp of the 13-limit large semitone of [[14/13]] by [[169/168]] (about 10.3¢).
-The neutral second in [[17edo]] is about 141.2¢, about 2.6¢ sharp of 13/12. Thus, if 10\17 (ten degrees of 17edo) is taken to approximate 3/2 and 12\17 taken to approximate 13/8, you can generate a 13-limit harmonic triad that approximates an 8:12:13 chord with a good 13/12.
+== Approximation ==
+It is approximated to within about 0.11 [[cents]] by the 3-step interval of [[26edo]].
+{{Interval edo approximation|{{PAGENAME}}}}
+== Temperaments ==
+/12 can be used to generate [[bleu]] temperament in the 2.3.7.11.13 subgroup, mapping [[3/2]] to +5 generators, [[7/4]] to +7 generators, [[11/8]] to +4 generators, and [[13/8]] to +6 generators.
+A slightly sharp 13/12 generates [[glacier]] temperament, which equates 5 13/12's to 3/2 like bleu. This temperament has an extension to the 2.3.7.11.13.23.29 subgroup which is more complex but much more accurate than bleu.
 == See also ==
@@ Line 20: / Line 22: @@
 * [[List of superparticular intervals]]
-[[Category:13-limit]]
+[[Category:Second]]
-[[Category:Interval]]
-[[Category:Just interval]]
-[[Category:Ratio]]
 [[Category:Neutral second]]
-[[Category:Second]]
-[[Category:Superparticular]]