14edo: Difference between revisions

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== Theory ==

~~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~~.

14edo is the double of [[7edo]], and thus contains its flat 686{{C}} fifth, while adding new intervals halfway between each 7edo step. The intervals of 14edo not found in 7edo are [[Backslash notation|1\14]] = 86{{C}}, 3\14 = 257{{C}}, 5\14 = 429{{C}}, 7\14 = 600{{C}}, and their [[octave complement]]s. The 1\14 interval is a small semitone, and its inversion a major seventh, which is suitable for a {{w|leading tone}}. The 3\14 interval can be considered a small [[subminor third]] (or inframinor third), thus bringing a new, distinct flavor from the [[neutral third]] of 7edo, which is 4 steps of 14edo. The 5\14 interval is the [[fifth complement]] of 3\14, and can be considered a [[supermajor third]], so that stacking 3\14 and 5\14 gives the triad 0–3–8 steps (0–257–686{{C}}). Finally, the 7\14 interval is the familiar [[tritone]] found in [[12edo]], as well as every even-numbered [[edo]].

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.

In terms of [[just intonation]], 14edo contains the approximation of [[3/2]] from 7edo. 14edo does not do well in the [[5-limit]], with [[5/4]] being close to halfway between its steps, so that 14edo does not approximate the [[4:5:6]] major triad or the [[10:12:15|1/(6:5:4)]] minor triad accurately. The closest approximation of [[7/4]] is very flat at 11\14 (943{{C}}), so that two of them stack to [[3/1]], meaning that [[49/48]] is [[tempering out|tempered out]], so that 14edo [[support]]s the [[semaphore]] temperament. However, since the 3rd harmonic is flat, the [[7/6]] and [[9/7]] intervals are approximated much more accurately, so that the 0–3–8 steps triad is a usable approximation of [[6:7:9]], and the 0–5–8 steps (0–429–686{{C}}) triad approximates [[14:18:21|1/(9:7:6)]]. The semaphore temperament notably generates the [[mos scale]] with pattern [[5L 4s]] (named ''semiquartal''), which contains many [[~]]6:7:9 and ~1/(9:7:6) triads. In the [[11-limit]], the [[11/8]] interval is tuned very flat and equated with [[4/3]]. However, [[11/9]] is tuned rather accurately, being represented with the 4\14 interval (343{{C}}), so that the [[Neutral (interval quality)|neutral]] triad formed by dividing the perfect fifth in two can be interpreted as a stack of two [[11/9]]'s, thus tempering out [[243/242]]. The neutral third can also be stacked on the supermajor third to get a ~[[7:9:11]] chord.

While prime [[5/1|5]] is poorly approximated, the [[7/5]] and [[11/10]] intervals are approximated fairly well. If we accept these approximations in addition to the ones described earlier, then we end up with a low-complexity, high-damage full [[11-limit]] temperament where many rather large [[comma]]s are tempered out. This leads to some of the bizarre equivalences described in the second "Approximate ratios" column in the table. This mapping uses the (barely) second-best mapping of prime 5, so it is notated with [[wart notation]] as "14c", where c is the 3rd letter of the alphabet, and 5 is the 3rd prime number. While not very accurate as a temperament, this mapping can be used to classify 11-limit intervals, which conveniently tempers out the [[square superparticular]]s of odds 5, 7, 9, and 11, and is the unique mapping to do so.

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. The 5L 4s mos scale is rich in triads, wherein 7 of 9 notes are [[tonic]] to a subminor, supermajor, and/or neutral triad.

14edo also contains an [[omnidiatonic]] scale that can replace the standard diatonic scale, allowing for recognizable triadic harmony using the chords [[6:7:9]] and [[14:18:21]], as well as a neutral chord which can be seen as [[2:sqrt(6):3]].

=== Prime harmonics ===

~~=== Octave stretch ===~~

~~14edo benefits from [[octave stretch]] as harmonics 3, 7, and 11 are all tuned flat. [[22edt]] and [[36ed6]] are among the possible choices.~~

=== Subsets and supersets ===

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

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

! colspan="3" | [[Ups and downs notation]]

! Interval Type

! Audio

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| [[File:piano_1_1edo.mp3]]

|}

=== 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]].

~~<imagemap>~~

~~File:14-EDO_Sagittal.svg~~

~~desc none~~

~~rect 80 0 300 50 [[Sagittal_notation]]~~

~~rect 423 0 583 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]~~

~~rect 20 80 423 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation~~ | ~~limma-fraction notation]]~~

~~default [[File:14-EDO_Sagittal.svg]]~~

~~</imagemap>~~

=== Ivor Darreg's notation ===

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0-3-8-11 = C vE G vB = Cv7 = C down-seven

For a more complete list, see [[Ups and ~~Downs Notation~~ #Chords and Chord Progressions]].

For a more complete list, see [[Ups and downs notation #Chords and Chord Progressions]].

== Approximation to JI ==

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==== Selected 13-limit intervals ====

[[File:14ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Interval mappings ===

=== ~~Zeta peak index~~ ===

{~~| class="wikitable center~~-~~all"~~

{{Q-odd-limit intervals|13.95|apx=val|header=none|tag=none|title=15-odd-limit intervals by 14c val mapping}}

|-

~~! colspan="3"~~ | ~~Tuning~~

~~! colspan="3" | Strength~~

~~! colspan="2" | Closest edo~~

~~! colspan="2" | Integer limit~~

|-

~~! ZPI~~

~~! Steps per octave~~

~~! Step size (cents)~~

~~! Height~~

~~! Integral~~

~~! Gap~~

~~! Edo~~

~~! Octave (cents)~~

~~! Consistent~~

~~! Distinct~~

|-

~~| [[42zpi]]~~

| 13.~~9002525327005~~

| ~~86.3293668353859~~

| ~~4.592177~~

| ~~0.984037~~

~~| 14.097244~~

~~| 14edo~~

~~| 1208.61113569540~~

~~| 7~~

~~| 5~~

|}

== Regular temperament properties ==

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=== Uniform maps ===

=== Rank-2 temperaments ===

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| Sagugu

| Diaschisma

|-

|7

|[[21/20]]

|[-2 1 -1 1⟩

|84.47

|Zogu

|Chroma

|-

| 7

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| Island comma

|}

== Octave stretch or compression ==

14edo benefits from [[octave stretch]] as harmonics 3, 7, and 11 are all tuned flat. [[22edt]], [[ed6|36ed6]] and [[42zpi]] are among the possible choices.

== Scales ==

=== MOS scales ===

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

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[[File:Screen Shot 2020-04-23 at 11.47.30 PM.png|none|thumb|870x870px|5\14 MOS using 1L 1s, 2L 1s, 3L 2s, 3L 5s, 3L 8s]]

=== Beep[9] ===

==== Beep[9] ====

14edo is also the largest edo whose patent val [[support]]s [[beep]] temperament, tempering out the chromatic semitone (21:20), and falling toward the "brittle" (fifths wider than in 9edo) end of that spectrum. beep is one of the simplest 7-limit temperaments, although rather inaccurate (the 7:5 is mapped onto 6\14, over 70 cents flat). Its otonal/major and utonal/minor tetrads are inversions of one another, which allows a greater variety of chord progressions (since different inversions of the same chord may have very different expressive qualities). Despite being so heavily tempered, the tetrads are still recognizable and aren't unpleasant-sounding as long as one uses the right timbres ("bell-like" or opaque-sounding ones probably work best). beep forms enneatonic modes which are melodically strong and are very similar to diatonic modes, only with two mediants and submediants instead of one. Beep[9] has similarities to mavila, slendro, and pelog scales as well.

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* 13\14: Major 9th9: A high, incisive leading tone.

* 14\14: The 10th9 or "enneatonic decave" (i. e. the octave, 2:1).

=== Others ===

* 2 2 2 2 2 2 2 - [[Equiheptatonic]] (exactly [[7edo]])

* 2 2 2 2 1 4 1 - Fennec{{idiosyncratic}} (original/default tuning)

* 1 4 1 2 2 2 2 - Inverse fennec{{idiosyncratic}} (original/default tuning)

* 3 1 4 1 4 1 - Pseudo-[[augmented]]

* 1 4 1 2 1 4 1 - Pseudo-double harmonic minor

== Diagrams ==

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[[File:14edo_mus2.jpg|thumb]]

== Instruments ==

[[Lumatone mapping for 14edo|Lumatone mappings for 14edo]] are available.

== Music ==

== See also ==

* [[~~Lumatone mapping for~~ 14edo]]

* [[MisterShafXen’s take on 14edo harmony]]

== Further reading ==

[[File:Libro_Tetradecafónico.PNG|alt=Libro_Tetradecafónico.PNG|Libro_Tetradecafónico.PNG|thumb|''Tetradecaphonic Scales for Guitar'' cover art.]]

* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Tetradecaphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for Fourteen Equal Divisions of the Octave]''. 2009.

~~== Notes ==~~

~~<references group="note" />~~

[[Category:14edo| ]]

@@ Line 9: / Line 9: @@
 == Theory ==
-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.
+edo is the double of [[7edo]], and thus contains its flat 686{{C}} fifth, while adding new intervals halfway between each 7edo step. The intervals of 14edo not found in 7edo are [[Backslash notation|1\14]] = 86{{C}}, 3\14 = 257{{C}}, 5\14 = 429{{C}}, 7\14 = 600{{C}}, and their [[octave complement]]s. The 1\14 interval is a small semitone, and its inversion a major seventh, which is suitable for a {{w|leading tone}}. The 3\14 interval can be considered a small [[subminor third]] (or inframinor third), thus bringing a new, distinct flavor from the [[neutral third]] of 7edo, which is 4 steps of 14edo. The 5\14 interval is the [[fifth complement]] of 3\14, and can be considered a [[supermajor third]], so that stacking 3\14 and 5\14 gives the triad 0–3–8 steps (0–257–686{{C}}). Finally, the 7\14 interval is the familiar [[tritone]] found in [[12edo]], as well as every even-numbered [[edo]].
-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.
+In terms of [[just intonation]], 14edo contains the approximation of [[3/2]] from 7edo. 14edo does not do well in the [[5-limit]], with [[5/4]] being close to halfway between its steps, so that 14edo does not approximate the [[4:5:6]] major triad or the [[10:12:15|1/(6:5:4)]] minor triad accurately. The closest approximation of [[7/4]] is very flat at 11\14 (943{{C}}), so that two of them stack to [[3/1]], meaning that [[49/48]] is [[tempering out|tempered out]], so that 14edo [[support]]s the [[semaphore]] temperament. However, since the 3rd harmonic is flat, the [[7/6]] and [[9/7]] intervals are approximated much more accurately, so that the 0–3–8 steps triad is a usable approximation of [[6:7:9]], and the 0–5–8 steps (0–429–686{{C}}) triad approximates [[14:18:21|1/(9:7:6)]]. The semaphore temperament notably generates the [[mos scale]] with pattern [[5L 4s]] (named ''semiquartal''), which contains many [[~]]6:7:9 and ~1/(9:7:6) triads. In the [[11-limit]], the [[11/8]] interval is tuned very flat and equated with [[4/3]]. However, [[11/9]] is tuned rather accurately, being represented with the 4\14 interval (343{{C}}), so that the [[Neutral (interval quality)|neutral]] triad formed by dividing the perfect fifth in two can be interpreted as a stack of two [[11/9]]'s, thus tempering out [[243/242]]. The neutral third can also be stacked on the supermajor third to get a ~[[7:9:11]] chord.
+While prime [[5/1|5]] is poorly approximated, the [[7/5]] and [[11/10]] intervals are approximated fairly well. If we accept these approximations in addition to the ones described earlier, then we end up with a low-complexity, high-damage full [[11-limit]] temperament where many rather large [[comma]]s are tempered out. This leads to some of the bizarre equivalences described in the second "Approximate ratios" column in the table. This mapping uses the (barely) second-best mapping of prime 5, so it is notated with [[wart notation]] as "14c", where c is the 3rd letter of the alphabet, and 5 is the 3rd prime number. While not very accurate as a temperament, this mapping can be used to classify 11-limit intervals, which conveniently tempers out the [[square superparticular]]s of odds 5, 7, 9, and 11, and is the unique mapping to do so.
+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. The 5L 4s mos scale is rich in triads, wherein 7 of 9 notes are [[tonic]] to a subminor, supermajor, and/or neutral triad.
+edo also contains an [[omnidiatonic]] scale that can replace the standard diatonic scale, allowing for recognizable triadic harmony using the chords [[6:7:9]] and [[14:18:21]], as well as a neutral chord which can be seen as [[2:sqrt(6):3]].
 === Prime harmonics ===
 {{Harmonics in equal|14}}
-=== Octave stretch ===
-edo benefits from [[octave stretch]] as harmonics 3, 7, and 11 are all tuned flat. [[22edt]] and [[36ed6]] are among the possible choices.
 === Subsets and supersets ===
@@ Line 32: / Line 35: @@
 ! 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 group="note">Nearest 15-odd-limit intervals by [[direct approximation]].</ref>
-! colspan="3" | [[Ups and Downs Notation]]
+! colspan="3" | [[Ups and downs notation]]
 ! Interval Type
 ! Audio
@@ Line 216: / Line 219: @@
 | [[File:piano_1_1edo.mp3]]
 |}
+<references group="note" />
 === 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]].
-<imagemap>
+{{Sagittal chart|}}
-File:14-EDO_Sagittal.svg
-desc none
-rect 80 0 300 50 [[Sagittal_notation]]
-rect 423 0 583 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
-rect 20 80 423 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]
-default [[File:14-EDO_Sagittal.svg]]
-</imagemap>
 === Ivor Darreg's notation ===
@@ Line 256: / Line 253: @@
 -3-8-11 = C vE G vB = Cv7 = C down-seven
-For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]].
+For a more complete list, see [[Ups and downs notation #Chords and Chord Progressions]].
 == Approximation to JI ==
@@ Line 262: / Line 259: @@
 ==== Selected 13-limit intervals ====
 [[File:14ed2-001.svg|alt=alt : Your browser has no SVG support.]]
+=== Interval mappings ===
-=== Zeta peak index ===
+{{Q-odd-limit intervals|14}}
-{| class="wikitable center-all"
+{{Q-odd-limit intervals|13.95|apx=val|header=none|tag=none|title=15-odd-limit intervals by 14c val mapping}}
-|-
-! colspan="3" | Tuning
-! colspan="3" | Strength
-! colspan="2" | Closest edo
-! colspan="2" | Integer limit
-|-
-! ZPI
-! Steps per octave
-! Step size (cents)
-! Height
-! Integral
-! Gap
-! Edo
-! Octave (cents)
-! Consistent
-! Distinct
-|-
-| [[42zpi]]
-| 13.9002525327005
-| 86.3293668353859
-| 4.592177
-| 0.984037
-| 14.097244
-| 14edo
-| 1208.61113569540
-| 7
-| 5
-|}
 == Regular temperament properties ==
@@ Line 322: / Line 291: @@
 === Uniform maps ===
-{{Uniform map|13|13.5|14.5}}
+{{Uniform map|edo=14}}
 === Rank-2 temperaments ===
@@ Line 359: / Line 328: @@
 | Sagugu
 | Diaschisma
+|-
+|7
+|[[21/20]]
+|[-2 1 -1 1⟩
+|84.47
+|Zogu
+|Chroma
 |-
 | 7
@@ Line 430: / Line 406: @@
 | Island comma
 |}
+<references group="note" />
+== Octave stretch or compression ==
+edo benefits from [[octave stretch]] as harmonics 3, 7, and 11 are all tuned flat. [[22edt]], [[ed6|36ed6]] and [[42zpi]] are among the possible choices.
 == Scales ==
+=== MOS scales ===
 {{Main|List of MOS scales in {{PAGENAME}}}}
@@ Line 438: / Line 419: @@
 [[File:Screen Shot 2020-04-23 at 11.47.30 PM.png|none|thumb|870x870px|5\14 MOS using 1L 1s, 2L 1s, 3L 2s, 3L 5s, 3L 8s]]
-=== Beep[9] ===
+==== Beep[9] ====
 edo is also the largest edo whose patent val [[support]]s [[beep]] temperament, tempering out the chromatic semitone (21:20), and falling toward the "brittle" (fifths wider than in 9edo) end of that spectrum. beep is one of the simplest 7-limit temperaments, although rather inaccurate (the 7:5 is mapped onto 6\14, over 70 cents flat). Its otonal/major and utonal/minor tetrads are inversions of one another, which allows a greater variety of chord progressions (since different inversions of the same chord may have very different expressive qualities). Despite being so heavily tempered, the tetrads are still recognizable and aren't unpleasant-sounding as long as one uses the right timbres ("bell-like" or opaque-sounding ones probably work best). beep forms enneatonic modes which are melodically strong and are very similar to diatonic modes, only with two mediants and submediants instead of one. Beep[9] has similarities to mavila, slendro, and pelog scales as well.
@@ Line 457: / Line 438: @@
 * 13\14: Major 9th<sub>9</sub>: A high, incisive leading tone.
 * 14\14: The 10th<sub>9</sub> or "enneatonic decave" (i. e. the octave, 2:1).
+=== Others ===
+* 2 2 2 2 2 2 2 - [[Equiheptatonic]] (exactly [[7edo]])
+* 2 2 2 2 1 4 1 - Fennec{{idiosyncratic}} (original/default tuning)
+* 1 4 1 2 2 2 2 - Inverse fennec{{idiosyncratic}} (original/default tuning)
+* 3 1 4 1 4 1 - Pseudo-[[augmented]]
+* 1 4 1 2 1 4 1 - Pseudo-double harmonic minor
 == Diagrams ==
@@ Line 466: / Line 454: @@
 [[File:14edo_mus2.jpg|thumb]]
+== Instruments ==
+[[Lumatone mapping for 14edo|Lumatone mappings for 14edo]] are available.
 == Music ==
-{{Main|Music in 14edo}}
+{{Main|14edo/Music}}
 {{Catrel|14edo tracks}}
 == See also ==
-* [[Lumatone mapping for 14edo]]
+* [[MisterShafXen’s take on 14edo harmony]]
 == Further reading ==
 [[File:Libro_Tetradecafónico.PNG|alt=Libro_Tetradecafónico.PNG|Libro_Tetradecafónico.PNG|thumb|''Tetradecaphonic Scales for Guitar'' cover art.]]
 * [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Tetradecaphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for Fourteen Equal Divisions of the Octave]''. 2009.
-== Notes ==
-<references group="note" />
 [[Category:14edo| ]] <!-- main article -->