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''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.


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== 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).


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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 ==
== JI approximation ==
=== Selected just intervals by error ===
=== Selected just intervals by error ===
==== 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.]]


=== Temperament measures ===
== Regular temperament properties ==
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 14et.  
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 14et.  


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== Rank two temperaments ==
=== Rank-2 temperaments ===
* [[List of 14edo rank two temperaments by badness]]
* [[List of 14edo rank two temperaments by badness]]
Important MOSes include [[semaphore]] 5L4s 221212121 (3\14, 1\1)
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 - [[MODMOS]] 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 - [[MODMOS]] of [[4L_2s|4L2s]]<br />
2 2 1 2 2 2 2 1 - [[MODMOS]] 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 -''' [[MODMOS]] of [[6L_2s|6L2s]]<br />
2 1 2 2 1 2 2 2 - [[MODMOS]] 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]]
2 1 1 1 1 1 2 1 1 1 1 1 Pathological [[MOS]] of [[2L 10s]]
3 1 1 1 1 1 1 1 1 1 1 1 Pathological [[MOS]] of [[1L 11s]]
2 1 1 1 1 1 1 1 1 1 1 1 1 Pathological [[MOS]] of [[1L 12s]]<br />
=== 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.
=== Commas ===
 
14et [[tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 14 22 33 39 48 52 }}.  
* 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).
 
== Commas ==
14 EDO [[tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 14 22 33 39 48 52 }}.  


{| class="commatable wikitable center-all left-3 right-4 left-6"
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<references/>
<references/>
== 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 - [[MODMOS]] 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 - [[MODMOS]] of [[4L_2s|4L2s]]<br />
2 2 1 2 2 2 2 1 - [[MODMOS]] 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 -''' [[MODMOS]] of [[6L_2s|6L2s]]<br />
2 1 2 2 1 2 2 2 - [[MODMOS]] 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]]
2 1 1 1 1 1 2 1 1 1 1 1 Pathological [[MOS]] of [[2L 10s]]
3 1 1 1 1 1 1 1 1 1 1 1 Pathological [[MOS]] of [[1L 11s]]
2 1 1 1 1 1 1 1 1 1 1 1 1 Pathological [[MOS]] of [[1L 12s]]<br />
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]]
=== 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 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).


== Images ==
== Images ==
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* [https://youtu.be/kZJvvESHoRw "Detached and Distant"] by Sevish (from his 2012 album "Sean But Not Heard")
* [https://youtu.be/kZJvvESHoRw "Detached and Distant"] by Sevish (from his 2012 album "Sean But Not Heard")


== Software Support ==
== Software support ==


[[File:SA14 for Mus2.zip]]
[[File:SA14 for Mus2.zip]]
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[[Category:Theory]]
[[Category:Theory]]


[[Category:Todo:intro]]
[[Category:Todo:unify precision]]
[[Category:Todo:unify precision]]
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