14edo: Difference between revisions

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The character of 14edo does not well serve those seeking low-[[limit]] JI approaches, with the exception of [[Subgroup|5:7:9:11:17:19]] (which is quite well approximated, relative to other JI approximations of the low-numbered edos). However, the [[ratio]]s 7/5, 7/6, 9/7, 10/7, 10/9, 11/7, 11/9, and 11/10 are all recognizably approximated, and if you accept that 14edo offers approximations of these intervals, you end up with a low-complexity, high-damage [[11-limit]] temperament where the [[comma]]s listed later in this page are [[tempered out]]. This leads to some of the bizarre equivalences described in the second "Approximate ratios" column in the table.

14et has quite a bit of [[xenharmonic]] appeal, in a similar way to [[17edo|17et]], on account of having three types of 3rd and three types of 6th, rather than the usual two of [[12et]]. Since 14et also has a recognizable 4th and 5th, this makes it good for those wishing to explore alternative triadic harmonies without adding significantly more notes. It possesses a [[triad]]-rich 9-note [[mos scale]] of [[5L 4s]], wherein 7 of 9 notes are [[tonic]] to a subminor, supermajor, and/or neutral triad.

14et has quite a bit of [[xenharmonic]] appeal, in a similar way to [[17edo|17et]], on account of having three types of 3rd and three types of 6th, rather than the usual two of [[12et]]. Since 14et also has a recognizable 4th and 5th, this makes it good for those wishing to explore alternative triadic harmonies without adding significantly more notes. It possesses a [[triad]]-rich 9-note [[mos scale]] of [[5L 4s]], wherein 7 of 9 notes are [[tonic]] to a subminor, supermajor, and/or neutral triad.

=== Prime harmonics ===

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! Approximate [[Harmonic]]s

! Approximate Ratios 1 <ref group="note">{{sg|limit=2.7/5.9/5.11/5.17/5.19/5 [[subgroup]]}}</ref>

! Approximate Ratios 2 <ref group="note">Based on treating 14edo as an 11-limit temperament of {{val| 14 22 32 39 48}} (14c)</ref>

! Approximate Ratios 2 <ref group="note">Based on treating 14edo as an 11-limit temperament of {{val| 14 22 32 39 48}} (14c).</ref>

! Approximate Ratios 3 <ref>~~nearest~~ 15-odd-limit intervals by [[direct approximation]]</ref>

! Approximate Ratios 3 <ref group="note">Nearest 15-odd-limit intervals by [[direct approximation]].</ref>

! colspan="3" | [[Ups and Downs Notation]]

! Interval Type

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|}

~~<references />~~

=== Sagittal notation ===

===Sagittal notation===

This notation uses the same sagittal sequence as [[9edo#Sagittal notation|9-EDO]], is a subset of the notations for EDOs [[28edo#Sagittal notation|28]] and [[42edo#Second-best fifth notation|42b]], and is a superset of the notation for [[7edo#Sagittal notation|7-EDO]].

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== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

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| Island comma

|}

~~<references/>~~

== Scales ==

* 5 4 5 - [[MOS]] of [[2L 1s]]

{{Main|List of MOS scales in {{PAGENAME}}}}

* 4 1 4 1 4 - MOS of [[3L 2s]]

* 3 3 2 3 3 - MOS of [[4L 1s]]

* 3 1 3 3 1 3 - MOS of [[4L 2s]]

* 3 2 2 2 2 3 - [[MODMOS]] of [[2L 4s]]

* 3 1 3 1 3 3 - MODMOS of [[4L 2s]]

* 3 3 1 1 3 3 - MODMOS of 4L 2s; [[Antipental blues]] scale

* 2 2 2 2 1 4 1 - Fennec scale

* 2 2 1 2 2 2 1 2 - MOS of [[6L 2s]]

* 2 1 2 2 2 2 1 2 - MODMOS of [[6L 2s]]

* 2 1 2 1 2 1 2 1 2 - MOS of [[5L 4s]]

* 1 2 1 2 1 1 2 1 2 1 - MOS of [[4L 6s]]

* 1 2 1 1 1 2 1 1 1 2 1 - MOS of [[3L 8s]]

* 1 1 2 1 1 1 1 1 2 1 1 1 MOS of [[2L 10s]]

* 1 1 1 1 1 3 1 1 1 1 1 1 MOS of ~~[[1L 11s]]~~

* 1 1 1 1 1 1 2 1 1 1 1 1 1 MOS ~~of [[1L 12s]]~~

Here are the modes that create MOS scales in 14edo shown on horagrams from Scala, skipping multiples of 14:

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== See also ==

* [[Lumatone mapping for 14edo]]

== Notes ==

== Further reading ==

@@ Line 11: / Line 11: @@
 The character of 14edo does not well serve those seeking low-[[limit]] JI approaches, with the exception of [[Subgroup|5:7:9:11:17:19]] (which is quite well approximated, relative to other JI approximations of the low-numbered edos). However, the [[ratio]]s 7/5, 7/6, 9/7, 10/7, 10/9, 11/7, 11/9, and 11/10 are all recognizably approximated, and if you accept that 14edo offers approximations of these intervals, you end up with a low-complexity, high-damage [[11-limit]] temperament where the [[comma]]s listed later in this page are [[tempered out]]. This leads to some of the bizarre equivalences described in the second "Approximate ratios" column in the table.
-et has quite a bit of [[xenharmonic]] appeal, in a similar way to [[17edo|17et]], on account of having three types of 3rd and three types of 6th, rather than the usual two of [[12et]]. Since 14et also has a recognizable 4th and 5th, this makes it good for those wishing to explore alternative triadic harmonies without adding significantly more notes. It possesses a [[triad]]-rich 9-note [[mos scale]] of [[5L 4s]], wherein 7 of 9 notes are [[tonic]] to a subminor, supermajor, and/or neutral triad.
+et has quite a bit of [[xenharmonic]] appeal, in a similar way to [[17edo|17et]], on account of having three types of 3rd and three types of 6th, rather than the usual two of [[12et]]. Since 14et also has a recognizable 4th and 5th, this makes it good for those wishing to explore alternative triadic harmonies without adding significantly more notes. It possesses a [[triad]]-rich 9-note [[mos scale]] of [[5L&nbsp;4s]], wherein 7 of 9 notes are [[tonic]] to a subminor, supermajor, and/or neutral triad.
 === Prime harmonics ===
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 ! Approximate<br>[[Harmonic]]s
 ! Approximate<br>Ratios 1 <ref group="note">{{sg|limit=2.7/5.9/5.11/5.17/5.19/5 [[subgroup]]}}</ref>
-! Approximate<br>Ratios 2 <ref group="note">Based on treating 14edo as an 11-limit temperament of {{val| 14 22 32 39 48}} (14c)</ref>
+! Approximate<br>Ratios 2 <ref group="note">Based on treating 14edo as an 11-limit temperament of {{val| 14 22 32 39 48}} (14c).</ref>
-! Approximate<br>Ratios 3 <ref>nearest 15-odd-limit intervals by [[direct approximation]]</ref>
+! Approximate<br>Ratios 3 <ref group="note">Nearest 15-odd-limit intervals by [[direct approximation]].</ref>
 ! colspan="3" | [[Ups and Downs Notation]]
 ! Interval Type
@@ Line 217: / Line 217: @@
 |}
-<references />
+=== Sagittal notation ===
-===Sagittal notation===
 This notation uses the same sagittal sequence as [[9edo#Sagittal notation|9-EDO]], is a subset of the notations for EDOs [[28edo#Sagittal notation|28]] and [[42edo#Second-best fifth notation|42b]], and is a superset of the notation for [[7edo#Sagittal notation|7-EDO]].
@@ Line 298: / Line 296: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
@@ Line 431: / Line 430: @@
 | Island comma
 |}
-<references/>
 == Scales ==
-* 5 4 5 - [[MOS]] of [[2L 1s]]
+{{Main|List of MOS scales in {{PAGENAME}}}}
-* 4 1 4 1 4 - MOS of [[3L 2s]]
-* 3 3 2 3 3 - MOS of [[4L 1s]]
-* 3 1 3 3 1 3 - MOS of [[4L 2s]]
-* 3 2 2 2 2 3 - [[MODMOS]] of [[2L 4s]]
-* 3 1 3 1 3 3 - MODMOS of [[4L 2s]]
-* 3 3 1 1 3 3 - MODMOS of 4L 2s; [[Antipental blues]] scale
-* 2 2 2 2 1 4 1 - Fennec scale
-* 2 2 1 2 2 2 1 2 - MOS of [[6L 2s]]
-* 2 1 2 2 2 2 1 2 - MODMOS of [[6L 2s]]
-* 2 1 2 1 2 1 2 1 2 - MOS of [[5L 4s]]
-* 1 2 1 2 1 1 2 1 2 1 - MOS of [[4L 6s]]
-* 1 2 1 1 1 2 1 1 1 2 1 - MOS of [[3L 8s]]
-* 1 1 2 1 1 1 1 1 2 1 1 1 MOS of [[2L 10s]]
-* 1 1 1 1 1 3 1 1 1 1 1 1 MOS of [[1L 11s]]
-* 1 1 1 1 1 1 2 1 1 1 1 1 1 MOS of [[1L 12s]]
 Here are the modes that create MOS scales in 14edo shown on horagrams from Scala, skipping multiples of 14:
@@ Line 490: / Line 473: @@
 == See also ==
 * [[Lumatone mapping for 14edo]]
+== Notes ==
+<references group="note" />
 == Further reading ==