14edo

← 13edo

14edo

15edo →

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 2^1/14, 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	Approximate Ratios 2	Approximate Ratios 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

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.

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

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

• List of 14edo rank two temperaments by badness

Commas

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

Prime limit	Ratio	Monzo	Cents	Color name	Name
3	2187/2048	[-11 7⟩	113.69	Lawa	Whitewood comma, apotome
5	27/25	[0 -3 2⟩	133.24	Gugu	Bug comma, large limma
5	2048/2025	[11 -4 -2⟩	19.55	Sagugu	Diaschisma
7	36/35	[2 2 -1 -1⟩	48.77	Rugu	Mint comma, septimal quartertone
7	49/48	[-4 -1 0 2⟩	35.70	Zozo	Sempahoresma, 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 comma
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 comma, biome comma
13	676/675	[2 -3 -2 0 0 2⟩	2.56	Bithogu	Island comma

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

Software support

File:SA14 for Mus2.zip

Music

Main article: Music in 14edo

See also: Category:14edo tracks

See also

• Lumatone mapping for 14edo

Further reading

Tetradecaphonic Scales for Guitar cover art.

• Sword, Ron. Tetradecaphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for Fourteen Equal Divisions of the Octave. 2009.