14edo

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← 13edo14edo15edo →
Prime factorization 2 × 7
Step size 85.7143¢ 
Fifth 8\14 (685.714¢) (→4\7)
Semitones (A1:m2) 0:2 (0¢ : 171.4¢)
Consistency limit 3
Distinct consistency limit 3

14 equal divisions of the octave (abbreviated 14edo or 14ed2), also called 14-tone equal temperament (14tet) or 14 equal temperament (14et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 14 equal parts of about 85.7 ¢ each. Each step represents a frequency ratio of 21/14, or the 14th root of 2.

Theory

The character of 14edo 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 below.

14et has quite a bit of xenharmonic appeal, in a similar way to 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

Approximation of prime harmonics in 14edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -16.2 +42.3 -26.0 -37.0 +16.6 -19.2 -40.4 -28.3 -1.0 -30.7
Relative (%) +0.0 -18.9 +49.3 -30.3 -43.2 +19.4 -22.4 -47.1 -33.0 -1.2 -35.9
Steps
(reduced)
14
(0)
22
(8)
33
(5)
39
(11)
48
(6)
52
(10)
57
(1)
59
(3)
63
(7)
68
(12)
69
(13)

Intervals

Steps Cents Approximate
Harmonics
Approximate
Ratios 1 [1]
Approximate
Ratios 2 [2]
Approximate
Ratios 3 [3]
Ups and Downs Notation Interval Type Audio
0 0.000 1 1/1 1/1 1/1 unison 1 D Unison
1 85.714 67 20/19, 19/18, 18/17 28/27, 22/21, 21/20 up-unison,
down-2nd
^1, v2 ^D, vE Narrow Minor 2nd
2 171.429 71 11/10, 10/9, 19/17 12/11, 11/10, 10/9, 9/8 11/10, 10/9 2nd 2 E Neutral 2nd
3 257.143 37 22/19, 20/17 8/7, 7/6 15/13, 7/6 up-2nd,
down-3rd
^2, v3 ^E, vF Subminor 3rd
4 342.857 39 17/14, 11/9 6/5, 11/9, 5/4 11/9 3rd 3 F Neutral 3rd
5 428.571 41 22/17, 14/11, 9/7 14/11, 9/7 14/11, 9/7 up-3rd,
down-4th
^3, v4 ^F, vG Supermajor 3rd
6 514.286 43 19/14 4/3, 15/11, 11/8 4/3 4th 4 G Wide 4th
7 600.000 91 7/5, 10/7 7/5, 10/7 7/5, 10/7 up-4th,
down-5th
^4, v5 ^G, vA Tritone
8 685.714 95 28/19 16/11, 22/15, 3/2 3/2 5th 5 A Narrow 5th
9 771.429 25 14/9, 11/7, 17/11 14/9, 11/7 14/9, 11/7 up-5th,
down-6th
^5, v6 ^A, vB Subminor 6th
10 857.143 105 18/11, 28/17 8/5, 18/11, 5/3 18/11 6th 6 B Neutral 6th
11 942.857 55 17/10, 19/11 12/7, 7/4 12/7, 26/15 up-6th,
down-7th
^6, v7 ^B, vC Supermajor 6th
12 1028.571 29 19/34, 9/5, 20/11 16/9, 9/5, 20/11, 11/6 9/5, 20/11 7th 7 C Neutral 7th
13 1114.286 61 17/9, 36/19, 19/10 40/21, 21/11, 27/14 up-7th,
down-8ve
^7, v8 ^C, vD Wide Major 7th
14 1200.000 2 2/1 2/1 2/1 8ve 8 D Octave
  1. based on treating 14edo as a 2.7/5.9/5.11/5.17/5.19/5 subgroup; other approaches are possible.
  2. based on treating 14edo as an 11-limit temperament of 14 22 32 39 48] (14c)
  3. nearest 15-odd-limit intervals by direct approximation

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

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).

0-4-8 = C E G = C = C or C perfect
0-3-8 = C vE G = Cv = C down
0-5-8 = C ^E G = C^ = C up
0-4-7 = C E vG = C(v5) = C down-five
0-5-9 = C ^E ^G = C^(^5) = C up up-five
0-4-8-12 = C E G B = C7 = C seven
0-4-8-11 = C E G vB = C,v7 = C add down-seven
0-3-8-12 = C vE G B = Cv,7 = C down add 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.

Approximation to JI

Selected just intervals by error

Selected 13-limit intervals

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Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢)
Tuning error
Absolute (¢) Relative (%)
2.3.7 49/48, 2187/2048 [14 22 39]] +6.52 4.64 5.38
2.3.7.11 33/32, 49/48, 243/242 [14 22 39 48]] +7.58 4.42 5.12

Uniform maps

13-limit uniform maps between 13.5 and 14.5
Min. size Max. size Wart notation Map
13.5000 13.5650 14bbcccddeefff 14 21 31 38 47 50]
13.5650 13.5663 14cccddeefff 14 22 31 38 47 50]
13.5663 13.6470 14cddeefff 14 22 32 38 47 50]
13.6470 13.7140 14cddeef 14 22 32 38 47 51]
13.7140 13.7306 14ceef 14 22 32 39 47 51]
13.7306 13.9173 14cf 14 22 32 39 48 51]
13.9173 13.9970 14c 14 22 32 39 48 52]
13.9970 14.0196 14 14 22 33 39 48 52]
14.0196 14.0702 14e 14 22 33 39 49 52]
14.0702 14.1875 14de 14 22 33 40 49 52]
14.1875 14.1959 14deff 14 22 33 40 49 53]
14.1959 14.3087 14bdeff 14 23 33 40 49 53]
14.3087 14.4264 14bdeeeff 14 23 33 40 50 53]
14.4264 14.4277 14bdddeeeff 14 23 33 41 50 53]
14.4277 14.4577 14bccdddeeeff 14 23 34 41 50 53]
14.4577 14.5000 14bccdddeeeffff 14 23 34 41 50 54]

Rank-2 temperaments

Commas

14edo tempers out the following commas. This assumes the val 14 22 33 39 48 52].

Prime
Limit
Ratio[1] Monzo Cents Color name Name
3 2187/2048 [-11 7 113.69 Lawa Apotome
5 27/25 [0 -3 2 133.24 Gugu Large limma
5 2048/2025 [11 -4 -2 19.55 Sagugu Diaschisma
7 36/35 [2 2 -1 -1 48.77 Rugu Septimal quartertone
7 49/48 [-4 -1 0 2 35.70 Zozo Slendro diesis
7 1728/1715 [6 3 -1 -3 13.07 Triru-agu Orwellisma
7 10976/10935 [5 -7 -1 3 6.48 Satrizo-agu Hemimage
7 (30 digits) [47 -7 -7 -7 0.34 Trisa-seprugu Akjaysma
11 99/98 [-1 2 0 -2 1 17.58 Loruru Mothwellsma
11 243/242 [-1 5 0 0 -2 7.14 Lulu Rastma
11 385/384 [-7 -1 1 1 1 4.50 Lozoyo Keenanisma
13 91/90 [-1 -2 -1 1 0 1 19.13 Thozogu Superleap
13 676/675 [2 -3 -2 0 0 2 2.56 Bithogu Island comma
  1. Ratios longer than 10 digits are presented by placeholders with informative hints

Scales

  • 5 4 5 - MOS of 2L 1s
  • 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:

3\14 MOS using 1L 1s, 1L 2s, 1L 3s, 4L 1s, 5L 4s
5\14 MOS using 1L 1s, 2L 1s, 3L 2s, 3L 5s, 3L 8s

Beep[9]

14edo is also the largest edo whose patent val supports 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.

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.

  • 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 8:5 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).

Diagrams

14edo wheel.png

Software support

File:SA14 for Mus2.zip

14edo mus2.jpg

Music

See also: Category:14edo tracks

See also

Further reading

Libro_Tetradecafónico.PNG
Tetradecaphonic Scales for Guitar cover art.