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Intervals | Chord Names | Rank two temperaments | Harmony | Notation | Images | Commas | Modes | Books | Compositions

Intervals

Degree
Cents
Approximate
Ratios 1 *
Approximate
Ratios 2 **
Nearest
Harmonic
up/down
notation
Interval Type
0
0
1/1
1/1
1
unison
1
D
Unison
1
85.714
20/19, 19/18, 18/17
22/21, 28/27, 21/20
67
up-unison,
down-2nd
^1, v2
D^, Ev
Narrow Minor 2nd
2
171.429
11/10, 10/9,
19/17
9/8, 10/9,
11/10, 12/11
71
2nd
2
E
Neutral 2nd, or
Narrow Major 2nd

257.143
22/19, 20/17
7/6, 8/7
37
up-2nd,
down-3rd
^2, v3
E^, Fv
Subminor 3rd
4
342.857
11/9, 17/14
11/9, 5/4, 6/5
39
3rd
3
F
Neutral 3rd

428.571
9/7, 14/11,
22/17
9/7, 14/11
41
up-3rd,
down-4th
^3, v4
F^, Gv
Supermajor 3rd
6
514.286
19/14
4/3, 11/8
43
4th
4
G
Wide 4th

600
7/5, 10/7
7/5, 10/7
91
up-4th,
down-5th
^4, v5
G^, Av
Tritone
8
685.714
28/19
3/2, 16/11
95
5th
5
A
Narrow 5th

771.429
14/9, 11/7,
17/11

25
up-5th,
down-6th
^5, v6
A^, Bv
Subminor 6th
10
857.143
18/11
18/11, 8/5, 5/3
105
6th
6
B
Neutral 6th
11·
942.857
19/11, 17/10
12/7, 7/4
55
up-6th,
down-7th
^6, v7
B^, Cv
Supermajor 6th
12
1028.571
20/11, 9/5,
34/19
16/9, 9/5,
20/11, 11/6
29
7th
7
C
Neutral 7th, or
Wide Minor 7th
13
1114.286
19/10, 36/19,
17/9
21/11, 27/14, 40/21
61
up-7th,
down-8ve
^7, v8
C^, Dv
Wide Major 7th
14··
1200
2/1
2/1
2
8ve
8
D
Octave
  • based on treating 14-EDO as a 2.7/5.9/5.11/5.17/5.19/5 subgroup; other approaches are possible.
** based on treating 14edo as an 11-limit temperament

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.

0-4-8 = C E G = C = C or C perfect
0-3-8 = C Ev G = C(v3) = C down-three
0-5-8 = C E^ G = C(^3) = C up-three
0-4-7 = C E Gv = C(v5) = C down-five
0-5-9 = C E^ G^ = C(^3,^5) = C up-three up-five

0-4-8-12 = C E G B = C7 = C seven
0-4-8-11 = C E G Bv = C(v7) = C down-seven
0-3-8-12 = C Ev G B = C7(v3) = C seven down-three
0-3-8-11 = C Ev G Bv = C.v7 = C dot down seven

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.


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Rank two temperaments

List of 14edo rank two temperaments by badness

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 2nd9: functions similarly to the diatonic minor second, but is more incisive.
2\14: Major 2nd9: functions similarly to the diatonic major second, but is narrower and has a rather different quality.
3\14: Perfect 3rd9: the generator, standing in for 8:7, 7:6, and 6:5, but closest to 7:6.
4\14: Augmented 3rd9/diminished 4th9: A dissonance, falling in between two perfect consonances and hence analogous to the tritone.
5\14: Perfect 4th9: technically represents 5:4 but is quite a bit wider.
6\14: Perfect 5th9: represents 4:3 and 7:5, much closer to the former.
7\14: Augmented 5th9/diminished 6th9: 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 6th9: represents 10:7 and 3:2, much closer to the latter.
9\14: Perfect 7th9: technically represents 5:8 but noticeably narrower.
10\14: Augmented 7th9/diminished 8th9: The third and final characteristic dissonance, analogous to the tritone.
11\14: Perfect 8th9: Represents 5:3, 12:7 and 7:4.
12\14: Minor 9th9: Analogous to the diatonic minor seventh, but sharper than usual.
13\14: Major 9th9: A high, incisive leading tone.
14\14: The 10th9 or "enneatonic decave", (i. e., the octave, 2:1).

Notation

Ivor Darreg wrote in 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 following chart (made by TDW) shows this recommendation as "standard notation" as well as a proposed alternative.

Ciclo_Tetradecafonía.png
Intervallic Cycle of 14 steps Equal per Octave


Images


14edo wheel.png

Commas

14 EDO tempers out the following commas. (Note: This assumes the val < 14 22 33 39 48 52 |.)
Comma
Monzo
Value (Cents)
Name 1
Name 2
2187/2048
| -11 7 >
113.69
Apotome

2048/2025
| 11 -4 -2 >
19.55
Diaschisma

36/35
| 2 2 -1 -1 >
48.77
Septimal Quarter Tone

49/48
| -4 -1 0 2 >
35.70
Slendro Diesis

1728/1715
| 6 3 -1 -3 >
13.07
Orwellisma
Orwell Comma
10976/10935
| 5 -7 -1 3 >
6.48
Hemimage


| 47 -7 -7 -7 >
0.34
Akjaysma
5\7 Octave Comma
99/98
| -1 2 0 -2 1 >
17.58
Mothwellsma

243/242
| -1 5 0 0 -2 >
7.14
Rastma

385/384
| -7 -1 1 1 1 >
4.50
Keenanisma

91/90
| -1 -2 -1 1 0 1 >
19.13
Superleap

676/675
| 2 -3 -2 0 0 2 >
2.56
Parizeksma


Modes

5 5 4 - MOS of 2L1s
5 4 5 - MOS of 2L1s

4 1 4 4 1 - MOS of 3L2s
4 1 4 1 4 - MOS of 3L2s
3 3 3 3 2 - MOS of 4L1s
3 2 3 3 3 - MOS of 4L1s
3 2 2 2 2 3 - MOS of 2L4s
2 2 3 2 2 3 - MOS of 2L4s
3 3 1 3 3 1 - MOS of 4L2s
3 1 3 3 1 3 - MOS of 4L2s
3 1 3 1 3 3 - MOS of 4L2s
2 2 1 2 2 2 2 1 - MOS of 6L2s
2 2 2 1 2 2 2 1 - MOS of 6L2s
2 2 2 2 1 2 2 1 - MOS of 6L2s
2 1 2 2 1 2 2 2 - MOS of 6L2s
2 1 2 1 2 1 2 1 2 - MOS of 5L4s
2 1 2 1 2 1 2 2 1 - MOS of 5L4s
2 1 2 1 2 2 1 2 1 - MOS of 5L4s
2 1 1 2 1 2 1 1 2 1 - MOS of 4L6s
2 1 1 1 2 1 1 2 1 1 1 - MOS of 3L8s
1 1 2 1 1 1 2 1 1 1 2 - MOS of 3L8s


Books

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

Compositions

NANA WODORI by knowsur
Thereminnards by Ralph Lewis
Pendula (for amplified trombone) by Philip Schuessler
Music by Ralph Jarzombek
Ivor Darreg in Eagle Rock by Daniel Wolf
Riding the L by Chris Vaisvil
Thorium Road by Jon Lyle Smith
tranSentient by Jon Lyle Smith
the spectrum of desire by Jon Lyle Smith
This Way to the Egress play by Herman Miller
Hyperimprovisation 'Tasty' play by Jacob Barton
14ETPrelude by Aaron Andrew Hunt
Medicine Wheel by Mark Allan Barnes
Fourteen EDO by Cameron Bobro