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| ja = 14平均律
| ja = 14平均律
}}
}}
{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2 * 7
{{ED intro}}
| Subgroup = 2.7/5.9/5.11/5.19/5
| Step size = 85.714¢
| Fifth type = [[7edo]] 8\14 685.714¢
| Common uses = chromatic [[7edo]]<br/>semaphore
| Important MOSes = [[semaphore]] 5*2-1*4 (3\14, 1\1)
}}
== Intervals ==


{| class="wikitable center-all right-1 right-2 left-4 left-5"
== Theory ==
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]].
 
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 ===
{{Harmonics in equal|14}}
 
=== Subsets and supersets ===
Since 14 factors into primes as 2 × 7, 14edo contains [[2edo]] and [[7edo]] as subsets.
 
== Notation ==
=== Ups and downs notation ===
{| class="wikitable center-all right-3"
|-
|-
! Degree
! Steps
! Cents
! Cents
! Nearest<br>[[Harmonic]]
! Approximate<br>[[Harmonic]]s
! Approximate<br>Ratios 1 <ref>based on treating 14edo as a 2.7/5.9/5.11/5.17/5.19/5 [[subgroup]]; other approaches are possible.</ref>
! 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>based on treating 14edo as an 11-limit temperament</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>
! colspan="3" | [[Ups and Downs Notation]]
! Approximate<br>Ratios 3 <ref group="note">Nearest 15-odd-limit intervals by [[direct approximation]].</ref>
! colspan="3" | [[Ups and downs notation]]
! Interval Type
! Interval Type
! Audio
|-
|-
| 0
| 0
| 0.000
| 0.000
| 1
| 1
| colspan="2" | 1/1
| 1/1
| 1/1
| 1/1
| unison
| unison
| 1
| 1
| D
| D
| Unison
| Unison
| [[File:piano_0_1edo.mp3]]
|-
|-
| 1
| 1
Line 38: Line 55:
| 67
| 67
| 20/19, 19/18, 18/17
| 20/19, 19/18, 18/17
| 22/21, 28/27, 21/20
| 28/27, 22/21, 21/20
|
| up-unison,<br>down-2nd
| up-unison,<br>down-2nd
| ^1, v2
| ^1, v2
| ^D, vE
| ^D, vE
| Narrow Minor 2nd
| Narrow Minor 2nd
| [[File:piano_1_14edo.mp3]]
|-
|-
| 2
| 2
Line 48: Line 67:
| 71
| 71
| 11/10, 10/9, 19/17
| 11/10, 10/9, 19/17
| 9/8, 10/9, 11/10, 12/11
| 12/11, 11/10, 10/9, 9/8
| 11/10, 10/9
| 2nd
| 2nd
| 2
| 2
| E
| E
| Neutral 2nd, or<br>Narrow Major 2nd
| Neutral 2nd
| [[File:piano_1_7edo.mp3]]
|-
|-
| ·3
| 3
| 257.143
| 257.143
| 37
| 37
| 22/19, 20/17
| 22/19, 20/17
| 7/6, 8/7
| 8/7, 7/6
| 15/13, 7/6
| up-2nd,<br>down-3rd
| up-2nd,<br>down-3rd
| ^2, v3
| ^2, v3
| ^E, vF
| ^E, vF
| Subminor 3rd
| Subminor 3rd
| [[File:piano_3_14edo.mp3]]
|-
|-
| 4
| 4
| 342.857
| 342.857
| 39
| 39
| 11/9, 17/14
| 17/14, 11/9
| 11/9, 5/4, 6/5
| 6/5, 11/9, 5/4
| 11/9
| 3rd
| 3rd
| 3
| 3
| F
| F
| Neutral 3rd
| Neutral 3rd
| [[File:piano_2_7edo.mp3]]
|-
|-
| ·5
| 5
| 428.571
| 428.571
| 41
| 41
| 9/7, 14/11, 22/17
| 22/17, 14/11, 9/7
| 9/7, 14/11
| 14/11, 9/7
| 14/11, 9/7
| up-3rd,<br>down-4th
| up-3rd,<br>down-4th
| ^3, v4
| ^3, v4
| ^F, vG
| ^F, vG
| Supermajor 3rd
| Supermajor 3rd
| [[File:piano_5_14edo.mp3]]
|-
|-
| 6
| 6
| 51<span style="line-height: 1.5;">4.286</span>
| 514.286
| 43
| 43
| 19/14
| 19/14
| 4/3, 11/8
| 4/3, 15/11, 11/8
| 4/3
| 4th
| 4th
| 4
| 4
| G
| G
| Wide 4th
| Wide 4th
| [[File:piano_3_7edo.mp3]]
|-
|-
| ·7
| 7
| 600.000
| 600.000
| 91
| 91
| 7/5, 10/7
| 7/5, 10/7
| 7/5, 10/7
| 7/5, 10/7
| 7/5, 10/7
Line 103: Line 133:
| ^G, vA
| ^G, vA
| Tritone
| Tritone
| [[File:piano_1_2edo.mp3]]
|-
|-
| 8
| 8
Line 108: Line 139:
| 95
| 95
| 28/19
| 28/19
| 3/2, 16/11
| 16/11, 22/15, 3/2
| 3/2
| 5th
| 5th
| 5
| 5
| A
| A
| Narrow 5th
| Narrow 5th
| [[File:piano_4_7edo.mp3]]
|-
|-
| ·9
| 9
| 771.429
| 771.429
| 25
| 25
| 14/9, 11/7, 17/11
| 14/9, 11/7, 17/11
| 14/9, 11/7
| 14/9, 11/7
| 14/9, 11/7
| up-5th,<br>down-6th
| up-5th,<br>down-6th
Line 123: Line 157:
| ^A, vB
| ^A, vB
| Subminor 6th
| Subminor 6th
| [[File:piano_9_14edo.mp3]]
|-
|-
| 10
| 10
| 857.143
| 857.143
| 105
| 105
| 18/11, 28/17
| 8/5, 18/11, 5/3
| 18/11
| 18/11
| 18/11, 8/5, 5/3
| 6th
| 6th
| 6
| 6
| B
| B
| Neutral 6th
| Neutral 6th
| [[File:piano_5_7edo.mp3]]
|-
|-
| ·11
| 11
| 942.857
| 942.857
| 55
| 55
| 19/11, 17/10
| 17/10, 19/11
| 12/7, 7/4
| 12/7, 7/4
| 12/7, 26/15
| up-6th,<br>down-7th
| up-6th,<br>down-7th
| ^6, v7
| ^6, v7
| ^B, vC
| ^B, vC
| Supermajor 6th
| Supermajor 6th
| [[File:piano_11_14edo.mp3]]
|-
|-
| 12
| 12
| 1028.571
| 1028.571
| 29
| 29
| 20/11, 9/5, 34/19
| 19/34, 9/5, 20/11
| 16/9, 9/5, 20/11, 11/6
| 16/9, 9/5, 20/11, 11/6
| 9/5, 20/11
| 7th
| 7th
| 7
| 7
| C
| C
| Neutral 7th, or<br>Wide Minor 7th
| Neutral 7th
| [[File:piano_6_7edo.mp3]]
|-
|-
| 13
| 13
| 1114.286
| 1114.286
| 61
| 61
| 19/10, 36/19, 17/9
| 17/9, 36/19, 19/10
| 21/11, 27/14, 40/21
| 40/21, 21/11, 27/14
|
| up-7th,<br />down-8ve
| up-7th,<br />down-8ve
| ^7, v8
| ^7, v8
| ^C, vD
| ^C, vD
| Wide Major 7th
| Wide Major 7th
| [[File:piano_13_14edo.mp3]]
|-
|-
| ··14
| 14
| 1200.000
| 1200.000
| 2
| 2
| colspan="2" | 2/1
| 2/1
| 2/1
| 2/1
| 8ve
| 8ve
| 8
| 8
| D
| D
| Octave
| Octave
| [[File:piano_1_1edo.mp3]]
|}
|}
<references group="note" />


<references />
=== 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]].


{{Sagittal chart|}}
=== Ivor Darreg's notation ===
[[Ivor Darreg]] wrote in [http://www.tonalsoft.com/sonic-arts/darreg/dar15.htm this article]:
[[Ivor Darreg]] wrote in [http://www.tonalsoft.com/sonic-arts/darreg/dar15.htm this article]:


''The 14-tone scale presents a new situation: while one might use ordinary sharps and flats in addition to conventional naturals for the notes of the 7-tone-equal temperament, it would be misleading and confusing to do so, because there is a 7-tone circle of fifths (admittedly quite distorted) already notatable and nameable as F C G D A E B in the usual manner. But there is no 14-tone circle of fifths. There is simply a second set of 7 fifths in a circle which does not intersect the with the first set. Thus is we think of B-flat and B, or B-natural and F-sharp, the 14-tone-system interval would NOT be a fifth of that system and would not sound like one, since B F would be the very same kind of distorted fifth that C G or A E happens to be in 7 or 14. Our suggestion is to call the new notes of 14, the second set of 7, F* C* G* D* A* E* B*, and use asterisks or arrows or whatever you please on the staff. Or just number the tones as for 13.''
''The 14-tone scale presents a new situation: while one might use ordinary sharps and flats in addition to conventional naturals for the notes of the 7-tone-equal temperament, it would be misleading and confusing to do so, because there is a 7-tone circle of fifths (admittedly quite distorted) already notatable and nameable as F C G D A E B in the usual manner. But there is no 14-tone circle of fifths. There is simply a second set of 7 fifths in a circle which does not intersect the with the first set. Thus is we think of B-flat and B, or B-natural and F-sharp, the 14-tone-system interval would NOT be a fifth of that system and would not sound like one, since B F would be the very same kind of distorted fifth that C G or A E happens to be in 7 or 14. Our suggestion is to call the new notes of 14, the second set of 7, F* C* G* D* A* E* B*, and use asterisks or arrows or whatever you please on the staff. Or just number the tones as for 13.''


The following chart (made by [[Tútim_Dennsuul_Wafiil|TDW]]) shows this recommendation as "standard notation" as well as a proposed alternative.
The following chart (made by [[Tútim Dennsuul Wafiil|TDW]]) shows this recommendation as "standard notation" as well as a proposed alternative.


{| class="wikitable"
{| class="wikitable"
Line 189: Line 240:
|}
|}


== Chord Names ==
== Chord names ==
 
Ups and downs can be used to name 14edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).
Ups and downs can be used to name 14edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).


0-4-8 = C E G = C = C or C perfect<br />
0-4-8 = C E G = C = C or C perfect<br>
0-3-8 = C vE G = Cv = C down<br />
0-3-8 = C vE G = Cv = C down<br>
0-5-8 = C ^E G = C^ = C up<br />
0-5-8 = C ^E G = C^ = C up<br>
0-4-7 = C E vG = C(v5) = C down-five<br />
0-4-7 = C E vG = C(v5) = C down-five<br>
0-5-9 = C ^E ^G = C^(^5) = C up up-five<br />
0-5-9 = C ^E ^G = C^(^5) = C up up-five<br>
0-4-8-12 = C E G B = C7 = C seven<br />
0-4-8-12 = C E G B = C7 = C seven<br>
0-4-8-11 = C E G vB = C,v7 = C add down-seven<br />
0-4-8-11 = C E G vB = C,v7 = C add down-seven<br>
0-3-8-12 = C vE G B = Cv,7 = C down add seven<br />
0-3-8-12 = C vE G B = Cv,7 = C down add seven<br>
0-3-8-11 = C vE G vB = Cv7 = C down-seven
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|Ups and Downs Notation - Chords and Chord Progressions]].
For a more complete list, see [[Ups and downs notation #Chords and Chord Progressions]].


== Just approximation ==
== Approximation to JI ==
=== Selected just intervals by error ===
=== Selected just intervals by error ===
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
! prime 23
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| -16.2
| +42.3
| -26.0
| -37.0
| +16.6
| -19.2
| -40.3
| -28.3
|-
! [[Relative error|relative]] (%)
| 0.0
| -18.9
| +49.3
| -30.3
| -43.2
| +19.4
| -22.4
| -47.1
| -33.0
|}
==== Selected 13-limit intervals ====
==== Selected 13-limit intervals ====
[[File:14ed2-001.svg|alt=alt : Your browser has no SVG support.]]
[[File:14ed2-001.svg|alt=alt : Your browser has no SVG support.]]
=== Interval mappings ===
{{Q-odd-limit intervals|14}}
{{Q-odd-limit intervals|13.95|apx=val|header=none|tag=none|title=15-odd-limit intervals by 14c val mapping}}


=== Temperament measures ===
== Regular temperament properties ==
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 14et.
{| class="wikitable center-4 center-5 center-6"
 
|-
Note: the 14c [[val]] is used for full 7- and 11-limit for lower overall error.
! rowspan="2" | [[Subgroup]]
{| class="wikitable center-all"
! rowspan="2" | [[Comma list]]
! colspan="2" |
! rowspan="2" | [[Mapping]]
! 3-limit
! rowspan="2" | Optimal<br>8ve stretch (¢)
! 7-limit no-5
! colspan="2" | Tuning error
! 11-limit no-5
|-
! 5-limit
! [[TE error|Absolute]] (¢)
! 7-limit
! [[TE simple badness|Relative]] (%)
! 11-limit
|-
|-
! colspan="2" |Octave stretch (¢)
| 2.3.7
| +5.12
| 49/48, 2187/2048
| {{mapping| 14 22 39 }}
| +6.52
| +6.52
| 4.64
| 5.38
|-
| 2.3.7.11
| 33/32, 49/48, 243/242
| {{mapping| 14 22 39 48 }}
| +7.58
| +7.58
| +9.68
| +9.59
| +9.83
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 5.15
| 4.64
| 4.42
| 4.42
| 7.71
| 5.12
| 6.68
| 6.00
|-
![[TE simple badness|relative]] (%)
|5.98
|5.38
|5.12
|8.93
|7.73
|6.94
|}
|}


== Rank two temperaments ==
=== Uniform maps ===
{{Uniform map|edo=14}}
 
=== Rank-2 temperaments ===
* [[List of 14edo rank two temperaments by badness]]
* [[List of 14edo rank two temperaments by badness]]
Here are the modes that create MOS scales in 14edo shown on horagrams from Scala, skipping multiples of 14:
[[File:Screen Shot 2020-04-23 at 11.47.09 PM.png|none|thumb|877x877px|3\14 MOS using 1L 1s, 1L 2s, 1L 3s, 4L 1s, 5L 4s]]
[[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]]
== Scales ==
5 5 4 - [[MOSScales|MOS]] of [[2L_1s|2L1s]]<br />
5 4 5 - [[MOSScales|MOS]] of [[2L_1s|2L1s]]<br />
4 1 4 4 1 - [[MOSScales|MOS]] of [[3L_2s|3L2s]]<br />
4 1 4 1 4 - [[MOSScales|MOS]] of [[3L_2s|3L2s]]<br />
3 3 3 3 2 - [[MOSScales|MOS]] of [[4L_1s|4L1s]]<br />
3 2 3 3 3 - [[MOSScales|MOS]] of [[4L_1s|4L1s]]<br />
3 2 2 2 2 3 - [[MOSScales|MOS]] of [[2L_4s|2L4s]]<br />
2 2 3 2 2 3 - [[MOSScales|MOS]] of [[2L_4s|2L4s]]<br />
'''3 3 1 3 3 1 -''' [[MOSScales|MOS]] of [[4L_2s|4L2s]]<br />
3 1 3 3 1 3 - [[MOSScales|MOS]] of [[4L_2s|4L2s]]<br />
3 1 3 1 3 3 - [[MOSScales|MOS]] of [[4L_2s|4L2s]]<br />
2 2 1 2 2 2 2 1 - [[MOSScales|MOS]] of [[6L_2s|6L2s]]<br />
2 2 2 1 2 2 2 1 - [[MOSScales|MOS]] of [[6L_2s|6L2s]]<br />
'''2 2 2 2 1 2 2 1 -''' [[MOSScales|MOS]] of [[6L_2s|6L2s]]<br />
2 1 2 2 1 2 2 2 - [[MOSScales|MOS]] of [[6L_2s|6L2s]]<br />
2 1 2 1 2 1 2 1 2 - [[MOSScales|MOS]] of [[5L_4s|5L4s]]<br />
2 1 2 1 2 1 2 2 1 - [[MOSScales|MOS]] of [[5L_4s|5L4s]]<br />
2 1 2 1 2 2 1 2 1 - [[MOSScales|MOS]] of [[5L_4s|5L4s]]<br />
2 1 1 2 1 2 1 1 2 1 - [[MOSScales|MOS]] of [[4L_6s|4L6s]]<br />
2 1 1 1 2 1 1 2 1 1 1 - [[MOSScales|MOS]] of [[3L_8s|3L8s]]<br />
'''1 1 2 1 1 1 2 1 1 1 2''' - [[MOSScales|MOS]] of [[3L_8s|3L8s]]<br />
== Harmony ==
The character of 14-EDO does not well serve those seeking low-limit JI approaches, with the exception of 5:7:9:11:17:19 (which is quite well approximated, relative to other JI approximations of the low-numbered EDOs). However, the ratios 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 commas listed at the bottom of this page are tempered out. This leads to some of the bizarre equivalences described in the second "Approximate Ratios" column in the table above.
14-EDO has quite a bit of xenharmonic appeal, in a similar way to 17-EDO, on account of having three types of 3rd and three types of 6th, rather than the usual two of 12-TET. Since 14-EDO 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 5L4s, wherein 7 of 9 notes are tonic to a subminor, supermajor, and/or neutral triad.
=== Titanium[9] ===
14edo is also the largest edo whose patent val supports [[titanium]] temperament, tempering out the chromatic semitone (21:20), and falling toward the "brittle" (fifths wider than in 9edo) end of that spectrum. Titanium 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). Titanium forms enneatonic modes which are melodically strong and are very similar to diatonic modes, only with two mediants and submediants instead of one. Titanium[9] has similarities to mavila, slendro, and pelog scales as well.
Using titanium[9], we could name the intervals of 14edo as follows. The 3, 5, 6, 8, 9, and 11-step intervals are all consonant, while 1, 2, 4, 7, 10, 12, and 13 steps are dissonant. There is no distinction between "perfect" (modulatory) and "imperfect" (major/minor) consonances here; there are enough chords here that root motion may occur by ''any'' consonant interval, and thus ''all'' six consonances are "perfect" intervals, rather than just two of them as in the diatonic system. As in the diatonic scale, the perfect intervals come in pairs separated by a major second, and with a characteristic dissonance between them; in titanium[9] there are three such pairs rather than just one.
* 1\14: Minor 2nd<sub>9</sub>: functions similarly to the diatonic minor second, but is more incisive.
* 2\14: Major 2nd<sub>9</sub>: functions similarly to the diatonic major second, but is narrower and has a rather different quality.
* 3\14: Perfect 3rd<sub>9</sub>: the generator, standing in for 8:7, 7:6, ''and'' 6:5, but closest to 7:6.
* 4\14: Augmented 3rd<sub>9</sub>, diminished 4th<sub>9</sub>: A dissonance, falling in between two perfect consonances and hence analogous to the tritone.
* 5\14: Perfect 4th<sub>9</sub>: technically represents 5:4 but is quite a bit wider.
* 6\14: Perfect 5th<sub>9</sub>: represents 4:3 and 7:5, much closer to the former.
* 7\14: Augmented 5th<sub>9</sub>, diminished 6th<sub>9</sub>: The so-called "tritone" (but no longer made up of three whole tones). Like 4\14 and 10\14, this is a characteristic dissonance separating a pair of perfect consonances.
* 8\14: Perfect 6th<sub>9</sub>: represents 10:7 and 3:2, much closer to the latter.
* 9\14: Perfect 7th<sub>9</sub>: technically represents 5:8 but noticeably narrower.
* 10\14: Augmented 7th<sub>9</sub>, diminished 8th<sub>9</sub>: The third and final characteristic dissonance, analogous to the tritone.
* 11\14: Perfect 8th<sub>9</sub>: Represents 5:3, 12:7 and 7:4.
* 12\14: Minor 9th<sub>9</sub>: Analogous to the diatonic minor seventh, but sharper than usual.
* 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).


== Commas ==
=== Commas ===
14 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes the [[val]] {{val|14 22 33 39 48 52}}.)
14et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 14 22 33 39 48 52 }}.  


{| class="wikitable center-all left-3 right-4"
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
|-
! [[Harmonic limit|Prime<br>Limit]]
! [[Harmonic limit|Prime<br>limit]]
! Comma
! [[Ratio]]<ref group="note">{{rd}}</ref>
! Monzo
! [[Monzo]]
! Cents
! [[Cent]]s
! [[Color notation|Color Names]]
! [[Color name]]
! Name 1
! Name
! Name 2
|-
|-
| 3
| 3
| 2187/2048
| [[2187/2048]]
| |<nowiki> | -11 7 </nowiki>&gt;
| {{monzo| -11 7 }}
| 113.69
| 113.69
| Lawa
| Lawa
| Apotome
| Whitewood comma, apotome
|  
|-
| 5
| [[27/25]]
| {{monzo| 0 -3 2 }}
| 133.24
| Gugu
| Bug comma, large limma
|-
|-
| 5
| 5
| 2048/2025
| [[2048/2025]]
| |<nowiki> | 11 -4 -2 </nowiki>&gt;
| {{monzo| 11 -4 -2 }}
| 19.55
| 19.55
| Sagugu
| Sagugu
| Diaschisma
| Diaschisma
|  
|-
|7
|[[21/20]]
|[-2 1 -1 1⟩
|84.47
|Zogu
|Chroma
|-
|-
| 7
| 7
| 36/35
| [[36/35]]
| |<nowiki> | 2 2 -1 -1 </nowiki>&gt;
| {{monzo| 2 2 -1 -1 }}
| 48.77
| 48.77
| Rugu
| Rugu
| Septimal quarter tone
| Mint comma, septimal quartertone
|
|-
|-
| "
| 7
| 49/48
| [[49/48]]
| |<nowiki> | -4 -1 0 2 </nowiki>&gt;
| {{monzo| -4 -1 0 2 }}
| 35.70
| 35.70
| Zozo
| Zozo
| Slendro diesis
| Sempahoresma, slendro diesis
|
|-
|-
| "
| 7
| 1728/1715
| [[1728/1715]]
| |<nowiki> | 6 3 -1 -3 </nowiki>&gt;
| {{monzo| 6 3 -1 -3 }}
| 13.07
| 13.07
| Triru-agu
| Triru-agu
| Orwellisma
| Orwellisma
| Orwell comma
|-
|-
| "
| 7
| 10976/10935
| [[10976/10935]]
| |<nowiki> | 5 -7 -1 3 </nowiki>&gt;
| {{monzo| 5 -7 -1 3 }}
| 6.48
| 6.48
| Satrizo-agu
| Satrizo-agu
| Hemimage
| Hemimage comma
|
|-
|-
| "
| 7
|  
| <abbr title="140737488355328/140710042265625">(30 digits)</abbr>
| |<nowiki> | 47 -7 -7 -7 </nowiki>&gt;
| {{monzo| 47 -7 -7 -7 }}
| 0.34
| 0.34
| Trisa-seprugu
| Trisa-seprugu
| Akjaysma
| [[Akjaysma]]
| 5\7 octave comma
|-
|-
| 11
| 11
| 99/98
| [[99/98]]
| |<nowiki> | -1 2 0 -2 1 </nowiki>&gt;
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| 17.58
| Loruru
| Loruru
| Mothwellsma
| Mothwellsma
|
|-
|-
| "
| 11
| 243/242
| [[243/242]]
| |<nowiki> | -1 5 0 0 -2 </nowiki>&gt;
| {{monzo| -1 5 0 0 -2 }}
| 7.14
| 7.14
| Lulu
| Lulu
| Rastma
| Rastma
|
|-
|-
| "
| 11
| 385/384
| [[385/384]]
| |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| 4.50
| Lozoyo
| Lozoyo
| Keenanisma
| Keenanisma
|
|-
|-
| 13
| 13
| 91/90
| [[91/90]]
| |<nowiki> | -1 -2 -1 1 0 1 </nowiki>&gt;
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| 19.13
| Thozogu
| Thozogu
| Superleap
| Superleap comma, biome comma
|
|-
|-
| "
| 13
| 676/675
| [[676/675]]
| |<nowiki> | 2 -3 -2 0 0 2 </nowiki>&gt;
| {{monzo| 2 -3 -2 0 0 2 }}
| 2.56
| 2.56
| Bithogu
| Bithogu
| Parizeksma
| Island comma
|
|}
|}
<references group="note" />


== Images ==
== 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.


[[File:14edo_wheel.png|alt=14edo wheel.png|343x343px|14edo wheel.png]]
== Scales ==
=== MOS scales ===
{{Main|List of MOS scales in {{PAGENAME}}}}


== Books ==
Here are the modes that create MOS scales in 14edo shown on horagrams from Scala, skipping multiples of 14:
[[File:Libro_Tetradecafónico.PNG|alt=Libro_Tetradecafónico.PNG|Libro_Tetradecafónico.PNG]]
[[File:Screen Shot 2020-04-23 at 11.47.09 PM.png|none|thumb|877x877px|3\14 MOS using 1L 1s, 1L 2s, 1L 3s, 4L 1s, 5L 4s]]
[[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]]


''Sword, Ron. "Tetradecaphonic Scales for Guitar" IAAA Press. First Ed: June 2009.''
==== 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.


== Compositions ==
Using beep[9], we could name the intervals of 14edo as follows. The 3, 5, 6, 8, 9, and 11-step intervals are all consonant, while 1, 2, 4, 7, 10, 12, and 13 steps are dissonant. There is no distinction between "perfect" (modulatory) and "imperfect" (major/minor) consonances here; there are enough chords here that root motion may occur by ''any'' consonant interval, and thus ''all'' six consonances are "perfect" intervals, rather than just two of them as in the diatonic system. As in the diatonic scale, the perfect intervals come in pairs separated by a major second, and with a characteristic dissonance between them; in beep[9] there are three such pairs rather than just one.
[http://split-notes.com/004/ NANA WODORI] by knowsur


[https://soundcloud.com/overtoneshock/our-pixel-perfect-telephone-discussion-14-edo Our Pixel Perfect Dial Tone of Voice] by Stephen Weigel
* 1\14: Minor 2nd<sub>9</sub>: functions similarly to the diatonic minor second, but is more incisive.
* 2\14: Major 2nd<sub>9</sub>: functions similarly to the diatonic major second, but is narrower and has a rather different quality.
* 3\14: Perfect 3rd<sub>9</sub>: the generator, standing in for 8:7, 7:6, ''and'' 6:5, but closest to 7:6.
* 4\14: Augmented 3rd<sub>9</sub>, diminished 4th<sub>9</sub>: A dissonance, falling in between two perfect consonances and hence analogous to the tritone.
* 5\14: Perfect 4th<sub>9</sub>: technically represents 5:4 but is quite a bit wider.
* 6\14: Perfect 5th<sub>9</sub>: represents 4:3 and 7:5, much closer to the former.  
* 7\14: Augmented 5th<sub>9</sub>, diminished 6th<sub>9</sub>: The so-called "tritone" (but no longer made up of three whole tones). Like 4\14 and 10\14, this is a characteristic dissonance separating a pair of perfect consonances.
* 8\14: Perfect 6th<sub>9</sub>: represents 10:7 and 3:2, much closer to the latter.
* 9\14: Perfect 7th<sub>9</sub>: technically represents 8:5 but noticeably narrower.
* 10\14: Augmented 7th<sub>9</sub>, diminished 8th<sub>9</sub>: The third and final characteristic dissonance, analogous to the tritone.
* 11\14: Perfect 8th<sub>9</sub>: Represents 5:3, 12:7 and 7:4.
* 12\14: Minor 9th<sub>9</sub>: Analogous to the diatonic minor seventh, but sharper than usual.
* 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).


[http://micro.soonlabel.com/0-praxis/audio/August/august_03_thereminnards.mp3 Thereminnards] by Ralph Lewis
=== 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


Pendula (for amplified trombone) by Philip Schuessler
== Diagrams ==
[[File:14edo_wheel.png|alt=14edo wheel.png|343x343px|14edo wheel.png]]


[http://www.freewebs.com/ralphjarzombek/ Music] by Ralph Jarzombek
== Software support ==


[http://home.snafu.de/djwolf/IvorDarregInEagleRock.pdf Ivor Darreg in Eagle Rock] by [[Daniel_Wolf|Daniel Wolf]]
[[File:SA14 for Mus2.zip]]
 
<span style="">[http://micro.soonlabel.com/14-et/daily20110610-sax-Riding_The_L.mp3 Riding the L]</span> by [http://chrisvaisvil.com/?p=943 Chris Vaisvil]
 
<span style="">[http://clones.soonlabel.com/public/micro/jon-lyle-smith/Thorium%20Road.mp3 Thorium Road]</span> by [[Jon_Lyle_Smith|Jon Lyle Smith]]


[http://archive.org/download/tranSentient/tranSentient.mp3 tranSentient] by [[Jon_Lyle_Smith|Jon Lyle Smith]]
[[File:14edo_mus2.jpg|thumb]]


[http://archive.org/download/TheSpectrumOfDesire/the_spectrum_of_desire.mp3 the spectrum of desire] by [[Jon_Lyle_Smith|Jon Lyle Smith]]
== Instruments ==
[[Lumatone mapping for 14edo|Lumatone mappings for 14edo]] are available.


[https://sites.google.com/site/teamouse/home This Way to the Egress] <span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/egress-gpo.mp3 play]</span> by [http://www.io.com/%7Ehmiller/music/index.html Herman Miller]
== Music ==
{{Main|14edo/Music}}
{{Catrel|14edo tracks}}


[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=145852&songID=3680443 Hyperimprovisation 'Tasty'] <span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Barton/Hyperimprovisation%20Tasty.mp3 play]</span> by [[Jacob_Barton|Jacob Barton]]
== See also ==
* [[MisterShafXen’s take on 14edo harmony]]


<span style="">[http://www.h-pi.com/mp3/14ETPrelude.mp3 14ETPrelude]</span> by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]
== 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.


[http://youtu.be/mHyaW1fVWsg Medicine Wheel] by Mark Allan Barnes
[http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/Fourteen_EDO_CBobro_r8b.mp3 Fourteen EDO] by [[Cameron_Bobro|Cameron Bobro]]
== Software Support ==
[[File:SA14 for Mus2.zip]]
[[File:14edo_mus2.jpg|frame|left]]
[[Category:Edo]]
[[Category:14edo| ]] <!-- main article -->
[[Category:14edo| ]] <!-- main article -->
[[Category:Listen]]
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Modes]]
[[Category:Modes]]
[[Category:Teentuning]]
[[Category:Teentuning]]
[[Category:Theory]]
[[Category:Todo:intro]]
[[Category:Todo:unify precision]]
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