14edo

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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

14edo is the double of 7edo, and thus contains its flat 686 ¢ fifth, while adding new intervals halfway between each 7edo step. The intervals of 14edo not found in 7edo are 1\14 = 86 ¢, 3\14 = 257 ¢, 5\14 = 429 ¢, 7\14 = 600 ¢, and their octave complements. The 1\14 interval is a small semitone, and its inversion a major seventh, which is suitable for a 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 ¢). 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 1/(6:5:4) minor triad accurately. The closest approximation of 7/4 is very flat at 11\14 (943 ¢), so that two of them stack to 3/1, meaning that 49/48 is tempered out, so that 14edo supports 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 ¢) triad approximates 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 ¢), so that the 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 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 commas 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 superparticulars 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 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

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)

Subsets and supersets

Since 14 factors into primes as 2 × 7, 14edo contains 2edo and 7edo as subsets.

Notation

Ups and downs notation

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

Sagittal notation

This notation uses the same sagittal sequence as 9-EDO, is a subset of the notations for EDOs 28 and 42b, and is a superset of the notation for 7-EDO.

Ivor Darreg's 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.

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

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 14edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 14edo (direct approximation, even if inconsistent)

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
11/9, 18/11	4.551	5.3
11/10, 20/11	6.424	7.5
9/7, 14/9	6.513	7.6
15/13, 26/15	9.402	11.0
7/6, 12/7	9.728	11.3
9/5, 10/9	10.975	12.8
11/7, 14/11	11.063	12.9
3/2, 4/3	16.241	18.9
13/8, 16/13	16.615	19.4
7/5, 10/7	17.488	20.4
11/6, 12/11	20.792	24.3
15/11, 22/15	22.665	26.4
13/10, 20/13	25.643	29.9
7/4, 8/7	25.969	30.3
15/8, 16/15	26.017	30.4
5/3, 6/5	27.216	31.8
13/11, 22/13	32.067	37.4
9/8, 16/9	32.481	37.9
13/12, 24/13	32.856	38.3
15/14, 28/15	33.729	39.3
13/9, 18/13	36.618	42.7
11/8, 16/11	37.032	43.2
5/4, 8/5	42.258	49.3
13/7, 14/13	42.584	49.7

15-odd-limit intervals in 14edo (patent val mapping)

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
11/9, 18/11	4.551	5.3
9/7, 14/9	6.513	7.6
15/13, 26/15	9.402	11.0
7/6, 12/7	9.728	11.3
11/7, 14/11	11.063	12.9
3/2, 4/3	16.241	18.9
13/8, 16/13	16.615	19.4
11/6, 12/11	20.792	24.3
13/10, 20/13	25.643	29.9
7/4, 8/7	25.969	30.3
15/8, 16/15	26.017	30.4
9/8, 16/9	32.481	37.9
13/12, 24/13	32.856	38.3
11/8, 16/11	37.032	43.2
5/4, 8/5	42.258	49.3
13/7, 14/13	42.584	49.7
13/9, 18/13	49.097	57.3
15/14, 28/15	51.986	60.7
13/11, 22/13	53.647	62.6
5/3, 6/5	58.498	68.2
15/11, 22/15	63.049	73.6
7/5, 10/7	68.226	79.6
9/5, 10/9	74.739	87.2
11/10, 20/11	79.290	92.5

15-odd-limit intervals by 14c val mapping

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
11/9, 18/11	4.551	5.3
11/10, 20/11	6.424	7.5
9/7, 14/9	6.513	7.6
7/6, 12/7	9.728	11.3
9/5, 10/9	10.975	12.8
11/7, 14/11	11.063	12.9
3/2, 4/3	16.241	18.9
13/8, 16/13	16.615	19.4
7/5, 10/7	17.488	20.4
11/6, 12/11	20.792	24.3
15/11, 22/15	22.665	26.4
7/4, 8/7	25.969	30.3
5/3, 6/5	27.216	31.8
9/8, 16/9	32.481	37.9
13/12, 24/13	32.856	38.3
15/14, 28/15	33.729	39.3
11/8, 16/11	37.032	43.2
13/7, 14/13	42.584	49.7
5/4, 8/5	43.457	50.7
13/9, 18/13	49.097	57.3
13/11, 22/13	53.647	62.6
15/8, 16/15	59.697	69.6
13/10, 20/13	60.072	70.1
15/13, 26/15	76.312	89.0

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 13.8 and 14.2

Min. size	Max. size	Wart notation	Map
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]

Rank-2 temperaments

• List of 14edo rank two temperaments by badness

Commas

14et 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	21/20	[-2 1 -1 1⟩	84.47	Zogu	Chroma
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

Octave stretch or compression

14edo benefits from octave stretch as harmonics 3, 7, and 11 are all tuned flat. 22edt, 36ed6 and 42zpi are among the possible choices.

Scales

MOS scales

Main article: List of MOS scales in 14edo

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).

Others

• 2 2 2 2 2 2 2 - Equiheptatonic (exactly 7edo)
• 2 2 2 2 1 4 1 - Fennec^{[idiosyncratic term]} (original/default tuning)
• 1 4 1 2 2 2 2 - Inverse fennec^{[idiosyncratic term]} (original/default tuning)
• 3 1 4 1 4 1 - Pseudo-augmented
• 1 4 1 2 1 4 1 - Pseudo-double harmonic minor

Diagrams

Software support

File:SA14 for Mus2.zip

Instruments

Lumatone mappings for 14edo are available.

Music

Main article: 14edo/Music

See also: Category:14edo tracks

See also

• MisterShafXen’s take on 14edo harmony

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.