Intervals[edit]

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
3·	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
5·	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
7·	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
9·	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[edit]

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.

File:14ed2-001.svg

14ed2-001.svg

Rank two temperaments[edit]

List of 14edo rank two temperaments by badness

Harmony[edit]

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][edit]

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[edit]

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[edit]

14edo wheel.png

Commas[edit]

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