13edo

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13edo

14edo →

Prime factorization	13 (prime)
Step size	92.3077¢
Fifth	8\13 (738.462¢)
Semitones (A1:m2)	4:-1 (369.2¢ : -92.31¢)
Dual sharp fifth	8\13 (738.462¢)
Dual flat fifth	7\13 (646.154¢)
Dual major 2nd	2\13 (184.615¢)
Consistency limit	3
Distinct consistency limit	3

13 equal divisions of the octave (abbreviated 13edo or 13ed2), also called 13-tone equal temperament (13tet) or 13 equal temperament (13et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 13 equal parts of about 92.3 ¢ each. Each step represents a frequency ratio of 2^1/13, or the 13th root of 2.

Theory

In 13edo, the steps less than 600¢ are narrower than their nearest 12edo approximation, while those greater than 600¢ are wider. This allows for some neat ear-bending tricks, whereby melodic gestures reminiscent of 12edo can quickly arrive at an unfamiliar place.

As a temperament of 21-odd-limit just intonation, 13edo has excellent approximations to the 11th and 21st harmonics, and reasonable approximations to the 5th, 9th, 13th, 17th, and 19th harmonics. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 subgroup temperament, with the 2.5.9.11.13.17.19.21 subgroup being a particularly good example. It has a substantial repertoire of complex consonances for its small size.

Odd harmonics

Approximation of odd harmonics in 13edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+36.5	-17.1	-45.7	-19.3	+2.5	-9.8	+19.4	-12.6	-20.6	-9.2	+17.9
	Relative (%)	+39.5	-18.5	-49.6	-20.9	+2.7	-10.6	+21.0	-13.7	-22.3	-10.0	+19.4
Steps (reduced)		21 (8)	30 (4)	36 (10)	41 (2)	45 (6)	48 (9)	51 (12)	53 (1)	55 (3)	57 (5)	59 (7)

Subsets and supersets

13edo is the sixth prime edo, following 11edo and coming before 17edo.

One step of 13edo is very close to 135/128 by direct approximation (it is a semiconvergent). The 5-limit aluminium temperament (supported by 65edo, 494edo and 1547edo) realizes this proximity through a regular temperament perspective, combining the sound of 13edo with the simplicity of 5-limit JI.

Intervals

13edo chromatic ascending and descending scale on C (MIDI)

#	Cents	Approximated 21-odd-limit Ratios	Erv Wilson	Archaeotonic (Heptatonic 2nd-generated)	Oneirotonic (Octatonic 5th-generated)	26edo names (subset notation)	Fox-Raven (J = 360Hz)	Pseudo-Diatonic Category	Audio
0	0.00	1/1	H	C	C	C	J	Unison
1	92.31	17/16, 18/17, 19/18, 20/19, 21/20, 22/21	β	C#/Db	C#/Db	Cx/Dbb	J#/Kb	Minor second
2	184.62	9/8, 10/9, 11/10, 19/17, 21/19	A	D	D	D	K	Major second
3	276.92	7/6, 13/11, 20/17, 19/16, 22/19	δ	D#/Eb	D#/Eb	Dx/Ebb	L	Minor third
4	369.23	5/4, 11/9, 16/13, 26/21	C	E	E	E	L#/Mb	Major third
5	461.54	13/10, 17/13, 21/16, 22/17	B	E#/Fb	F	Ex/Fb	M	Minor fourth
6	553.85	11/8, 18/13, 26/19	ε	F	F#/Gb	F#	M#/Nb	Major fourth/Minor tritone
7	646.15	16/11, 13/9, 19/13	D	F#/Gb	G	Gb	N	Minor fifth/Major tritone
8	738.46	17/11, 20/13, 26/17, 32/21	γ	G	G#/Hb	G#	O	Major fifth
9	830.77	8/5, 13/8, 18/11, 21/13	F	G#/Ab	H	Ab	O#/Pb	Minor sixth
10	923.08	17/10, 12/7, 22/13, 19/11	E	A	A	A#	P	Major sixth
11	1015.38	9/5, 16/9, 20/11, 34/19, 38/21	α	A#/Bb	A#/Bb	Bb	Q	Minor seventh
12	1107.69	17/9, 19/10, 21/11, 32/17, 36/19, 40/21	G	B/Cb	B	B#/Cbb	Q#/Jb	Major seventh
13	1200.00	2/1	H	C/B#	C	C	J	Octave

Notations

There are seven categories of notation. Only the first two categories are backwards-compatible. They both allow conventional notation to be used, including the staff, note names, relative notation, chord names, etc. And they both allow a piece in conventional notation to be translated to 13edo. They both use the conventional genchain of fifths:

...Db - Ab - Eb - Bb - F - C - G - D -A - E - B - F#- C# - G# - D#...

...d8 - d5 - m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 - A4 - A1...

Except the first version of the second notation swaps sharp and flat, major and minor, and augmented and diminished.

Heptatonic 5th-generated (wide 5th)

13edo can also be notated with ups and downs. If one uses the best fifth, 8\13, the minor 2nd becomes a descending interval! Thus a major 2nd is wider than a minor 3rd, a major 3rd is wider than a perfect 4th, etc. And B is above C, E is above F, A is above Bb, etc. However one can use ups and downs to avoid minor 2nds. Thus A C B D becomes A vB ^C D.

Enharmonic intervals: v⁴A1, ^m2

#	Cents	Up/down notation using the wide 5th of 8\13
0	0	perfect unison	P1	D
1	92	up unison, mid 2nd	^1, ~2	^D, ^^Eb, vvE
2	185	downmajor 2nd, (minor 3rd)	vM2, (m3)	vE, (F)
3	277	major 2nd, upminor 3rd	M2, ^m3	E, ^F
4	369	mid 3rd	~3	^^F, vvF#
5	462	perfect 4th	P4	G
6	554	up 4th, dud 5th	^4, vv5	^G, vvA
7	646	dup 4th, down 5th	^^4, v5	^^G, vA
8	738	perfect 5th	P5	A
9	831	mid 6th	~6	^^Bb, vvB
10	923	downmajor 6th, minor 7th	vM6	vB, C
11	1015	(major 6th), upminor 7th	(M6), ^M7	(B), ^C
12	1108	mid 7th, down 8ve	~7, v8	^^C, vvC#, vD
13	1200	perfect 8ve	P8	D

Step Offset	0	1	2	3	4	5	6	7	8	9
Sharp Symbol
Flat Symbol

Half-sharps and half-flats can also be used, making the ascending scale: D E  vE ^F F  G ^G vA A B  vB ^C C  D

Heptatonic 5th-generated (narrow fifth)

The notational 5th is the 2nd-best approximation of 3/2, 7\13. This is 56¢ flat of 3/2, and the best approximation is 36¢ sharp, noticeably better. But using the 2nd-best 5th avoids the minor 2nd being descending.

There are two ways to do this. The first way preserves the melodic meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the harmonic meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 13edo "on the fly".

The first approach has enharmonic intervals of a trud-augmented 1sn and a downminor 2nd. The second approach has a trup-augmented 1sn and a downmajor 2nd.

#	Cents	Up/down notation using the narrow 5th of 7\13, with major wider than minor			Up/down notation using the narrow 5th of 7\13, with major narrower than minor
0	0	perfect unison	P1	D	perfect unison	P1	D
1	92	up unison, minor 2nd	^1, m2	^D, E	up unison, major 2nd	^1, M2	^D, E
2	185	upminor 2nd, minor 3rd	^m2, m3	^E, Fb	upmajor 2nd, major 3rd	^M2, M3	^E, F#
3	277	downmajor 2nd, upminor 3rd	vM2, ^m3	vE#, ^Fb	downminor 2nd, upmajor 3rd	vm2, ^M3	vEb, ^F#
4	369	major 2nd, downmajor 3rd	M2, vM3	E#, vF	minor 2nd, downminor 3rd	m2, vm3	Eb, vF
5	462	major 3rd, down 4th	M3, v4	F, vG	minor 3rd, down 4th	m3, v4	F, vG
6	554	perfect 4th, down 5th	P4, v5	G, vA	perfect 4th, down 5th	P4, v5	G, vA
7	646	up 4th, perfect 5th	^4, P5	^G, A	up 4th, perfect 5th	^4, P5	^G, A
8	738	up 5th, minor 6th	^5, m6	^A, B	up 5th, major 6th	^5, M6	^A, B
9	831	upminor 6th, minor 7th	^m6, m7	^B, Cb	upmajor 6th, major 7th	^M6, M7	^B, C#
10	923	downmajor 6th, upminor 7th	vM6, ^m7	vB#, ^Cb	downminor 6th, upmajor 7th	vm6, ^M7	vBb, ^C#
11	1015	major 6th, downmajor 7th	M6, vM7	B#, vC	minor 6th, downminor 7th	m6, vm7	Bb, vC
12	1108	major 7th, down 8ve	M7, v8	C, vD	minor 7th, down 8ve	m7, v8	C, vD
13	1200	perfect 8ve	P8	D	perfect 8ve	P8	D

Pentatonic 5th-generated (3L2s)

The degrees are named unison, subthird, fourthoid, fifthoid, subseventh and octoid.

Keyboard: D * F * * G * * A * C * * D (generator = wide 3/2 = 8\13 = perfect 5thoid)

Enharmonic interval: dds3

notes/intervals in melodic order (s = sub-, d = -oid)

0	1	2	3	4	5	6	7	8	9	10	11	12	13
D	D# Fb	F	F#	Gb	G	G#	Ab	A	A# Cb	C	C#	Db	D
P1	A1 ds3	ms3	Ms3	As3 d4d	P4d	A4d	d5d	P5d	A5d ds7	ms7	Ms7	As7 d8d	P8d

notes/intervals in genchain order (s = sub-, d = -oid)

...	-8	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6	7	8	...
...	Fb	Cb	Gb	Db	Ab	F	C	G	D	A	F#	C#	G#	D#	A#	Fx	Cx	...
...	ds3	ds7	d4d	d8d	d5d	ms3	ms7	P4d	P1	P5d	Ms3	Ms7	A4d	A1	A5d	As3	As7	...

Octatonic 5th-generated (5L3s or oneirotonic)

Keyboard: A * B C * D * E F * G H * A (generator = wide 3/2 = 8\13 = perfect 6th)

Enharmonic interval: d2

notes/intervals in melodic order

0	1	2	3	4	5	6	7	8	9	10	11	12	13
A	A# Bb	B	C	C# Db	D	D# Eb	E	F	F# Gb	G	H	H# Ab	A
P1	A1 m2	M2	m3	M3	P4	m5	M5	P6	m7	M7	m8	M8 d9	P9

notes/intervals in genchain order

...	-8	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6	7	8	...
...	D#	A#	F#	C#	H#	E	B	G	D	A	F	C	H	Eb	Bb	Gb	Db	...
...	A1	A6	M3	M8	M5	M2	M7	P4	P1	P6	m3	m8	m5	m2	m7	d4	d8	...

Heptatonic 2nd-generated (6L1s or archaeotonic)

Keyboard: D * E * F * G A * B * C * D (generator = 2\13 = perfect 2nd)

Enharmonic interval: dd2

notes/intervals in melodic order

0	1	2	3	4	5	6	7	8	9	10	11	12	13
D	D# Eb	E	E# Fb	F	F# Gb	G	A	A# Bb	B	B# Cb	C	C# Db	D
P1	A1 d2	P2	m3	M3	m4	M4	m5	M5	m6	M6	P7	A7 d8	P8

notes/intervals in genchain order

...	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6	7	...
...	Db	Eb	Fb	Gb	A	B	C	D	E	F	G	A#	B#	C#	D#	...
...	d8	d2	m3	m4	m5	m6	P7	P1	P2	M3	M4	M5	M6	A7	A1	...

Heptatonic 3rd-generated (3L4s or mosh)

This notation requires ups and downs because 7 perfect thirds octave-reduces to 2 edosteps, not 1.

Keyboard: D E * * F G * * A B * * C D (generator = 4\13 = perfect 3rd)

Enharmonic intervals: vvA1, vm2

notes/intervals in melodic order

0	1	2	3	4	5	6	7	8	9	10	11	12	13
D	E	^E Fb	E# vF	F	G	^G Ab	G# vA	A	B	^B Cb	B# vC	C	D
P1	m2	~2 d3	M2 v3	P3	m4	~4 m5	M4 ~5	M5	P6	^6 m7	A6 ~7	M7	P8

notes/intervals in genchain order

...	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6	7	...
...	Db	Fb	Ab	Cb	E	G	B	D	F	A	C	E#	G#	B#	D#	...
...	d8	d3	m5	m7	m2	m4	P6	P1	P3	M5	M7	M2	M4	A6	A1	...

26edo subset

This notation uses every other note name of 26edo. There are no perfect 4ths or 5ths, only augmented and diminished ones. There are no minor 2nds or augmented 1sns.

There are two versions of the absolute notation. One has only three natural notes (C, D and E) and the other one has only four (F, G, A and B).

Keyboard: D * E * * * * * * * * C * D or * * * F * G * A * B * * * * (generator = 4\26 = 2\13 = major 2nd)

Enharmonic interval: ddd2

notes/intervals in melodic order

	0	1	2	3	4	5	6	7	8	9	10	11	12	13
version #1	D	Dx Ebb	E	Fb	F#	Gb	G#	Ab	A#	Bb	B#	C	Cx Dbb	D
version #2	D#	Eb	E#	F	Fx Gbb	G	Gx Abb	A	Ax Bbb	B	Cb	C#	Db	D#
	P1	d2	M2	d3	M3	d4	A4	d5	A5	m6	A6	m7	A7	P8

notes/intervals in genchain order

	...	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6	7	...
version #1	...	Dbb	Ebb	Fb	Gb	Ab	Bb	C	D	E	F#	G#	A#	B#	Cx	Dx	...
version #2	...	Gbb	Abb	Bbb	Cb	Db	Eb	F	G	A	B	C#	D#	E#	Fx	Gx	...
	...	dd8	d2	d3	d4	d5	m6	m7	P1	M2	M3	A4	A5	A6	A7	AA1	...

Approximation to JI

Selected 13-odd-limit intervals

Tuning by ear

13edo can be approximated by a circle of 64/49 subminor fourths (which can be tuned by tuning two 7/4 subminor sevenths). A stack of 13 of these subfourths closes with an error of +10.526432¢, or +11% of 13-edo's step size.

Approximation to irrational intervals

Acoustic phi

13edo has a very close approximation of acoustic phi (9\13), with only -2.3 cents of error. 23edo and 36edo are even closer, but unlike all closer edos, 13edo has no other interval that represents any ratio from the Fibonacci sequence (3/2, 5/3, 8/5, 13/8, 21/13, etc.) except of course for 1/1 and 2/1. In a way, one could say that 13edo is the only edo that tempers the ratios of the Fibonacci sequence into a single interval.

See also: 9edϕ

Logarithmic phi

As a coïncidence, 13edo also has a very close appoximation of logarithmic phi (21\13), with only -3.2 cents of error.

Not until 144 do we find a better edo in terms of relative error on these two intervals.

However, it should be noted that when we are hearing logarithmic phi, we are in fact hearing 2^ϕ ≃ 3.070. While this interval can still be used in a way or another as a useful tone in a piece of music, it doesn't correspond to anything. When it comes to acoustic phi, we are truly hearing the mathematical constant ϕ ≃ 1.6180.

That being said, logarithmic phi has interesting applications as Metallic MOS, and in particular the fractal-like possibilities of self-similar subdivision of musical scales.

Direct approximation

Interval	Error (abs, ¢)
2ϕ / ϕ	0.858
ϕ	2.321
2ϕ	3.179

Local zeta peak

Main article: 13edo and optimal octave stretching

At the local zeta peak of 13edo, there is an improvement in both acoustic phi and logarithmic phi.

Scales

Main article: 13edo scales

Important mosses (values in parentheses are (period, generator)):

• oneirotonic 5L 3s 22122121 (5\13, 1\1)
• archaeotonic 6L 1s 2222221 (2\13, 1\1)
• lovecraft 4L 5s 212121211 (3\13, 1\1)
• Sephiroth 3L 4s 3131311 (4\13, 1\1)

Due to the prime character of the number 13, 13edo can form several xenharmonic moment of symmetry scales. The diagram below shows five "families" of MOS scales: those generated by making a chain of 2\13 (two degrees of 13edo), 3\13, 4\13, 5\13, & 6\13, respectively.

13edo horograms.pdf

~diagram by Andrew Heathwaite, based on horagrams pioneered by Erv Wilson

Another neat facet of 13edo is the fact that any 12edo scale can be "turned into" a 13edo scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13edo can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected.

Pathological Modes

2 1 1 1 1 2 1 1 1 1 1 2L 9s MOS

3 1 1 1 1 1 1 1 1 1 1 1L 10s MOS

2 1 1 1 1 1 1 1 1 1 1 1 1L 11s MOS

Harmony in 13edo

Contrary to popular belief, consonant harmony is possible in 13edo, but it requires a radically different approach than that used in 12edo (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12edo within 13edo is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13edo. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12edo, since the strongest dissonances in 13edo are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13edo is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the 2*13 subgroup 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26edo.

By this, we can assume that the major ninth of 13edo can be thought of as analogous to the perfect fifth in 12edo and other meantone edos. This means that the major second or major ninth is the most consonant interval next to 2/1 in 13edo followed by 11/8, 5/4 and so on. The 4:5:9 chord can therefore be thought of as a possible basic harmonic triad in 13edo.

The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite close to 2\13. Use this as a generator, and at 7 notes (6L 1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad). These triads and tetrads are likely the most consonant base sonorities available in 13edo and act in a similar way to major/minor triads. However, other sonorities such as Orwell chords are available as well.

Other approaches explored by specific composers and theorists are outlined further down, in the context of more complete tonal systems.

Play the 4:5:9 chord:

Play the 4:5:9:11 chord:

Play the 4:5:9:13 chord:

Play the 4:5:9:21 chord:

Notational and compositional approaches

13edo has drawn the attention of numerous composers and theorists, some of whom have devoted some effort to provide a notation and an outline of a compositional approach to it. Some of these are described below.

The Cryptic Ruse Methods

13edo offers two main candidates for diatonic-like scales: the 6L 1s heptatonic MOS generated by 2\13, and the 5L 3s octatonic MOS. Both of these scales are Rothenberg proper, and bear a slightly-twisted resemblance to the 12edo diatonic scale. Specifically, the 6L 1s scale resembles the 12edo diatonic with one of its semitones replaced with a whole-tone, while the 5L 3s scale resembles the 12edo diatonic with an extra semitone inserted between two adjacent whole-tones.

To facilitate discussion of these scales, Cryptic Ruse has ascribed them names based on H.P. Lovecraft's "Dream Cycle" mythos.

Modes and harmony in the archaeotonic scale

The 2\13-based heptatonic has been named archaeotonic after the "Old Ones" that rule the Dreamlands. Modes of the archaeotonic are named after the individual Old Ones.

A 7-nominal notation is proposed, using the letters A-G. The "C natural" scale is proposed to be degrees 0-2-4-6-8-10-12-(13), with the note "C" tuned to a reference pitch of concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13-edo.

Archaeotonic.png

Treating 13edo as a temperament as proposed above leads to a chord of degrees 0-2-4-6-9 representing the JI harmony 8:9:10:11:13; two such pentads exist in this scale, on E and F. Smaller harmonic units exist as follows: 8:9:10:11 on C, D, E, and F; 8:9:10:13 on E, F and G; 8:9:10 on C, D, E, F, and G; 8:9:11 on C, D, E, and F; 8:9:13 on E, F, G, and A. Finally, on B we have the relatively-discordant 16:17:21:26 (0-1-5-9, or the notes B-C-E-G), which can be octave-inverted into a more concordant 8:13:17:21.

There may be other concordant harmonies possible in this scale that do not represent segments of the harmonic series; further exploration is pending.

Modes and harmony in the oneirotonic scale

The 5\13-based octatonic has been named oneirotonic after the Dreamlands themselves. Modes of the oneirotonic are named after cities in the Dreamlands.

Here an 8-nominal notation is proposed, using letters A-H. The "C natural" scale is proposed to be degrees 0-2-4-5-7-9-10-12-(13), with the note "C" tuned to concert middle C. The modes are laid out in the following table, excerpted from an unfinished paper on 13edo.

There is a great number of potential consonant harmonies in this scale. A dedicated article on harmony and tonality in the oneirotonic scale is forthcoming.

Mapping to standard keyboards

The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 8 (13) 5 10 2 7 12 4 9 1/1. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.

1

6

11

3

8

(13)

5

10

2

7

12

4

9

1

Place in Chain of 738.5 cent intervals

X

*

*

*

*

*

*

*

X

Marked are the octatonic scales (X=Sarnathian)

*

*

*

*

X

*

*

*

*

X

*

*

*

*

*

*

*

*

*

*

*

*

X

*

*

*

*

X

*

*

*

*

D

Eb

E

F

Gb

G

Ab

A

Bb

B

C

Db

D

Keeps the pentatonic scale on the white keys

A

Bb

B

C

Db

D

Eb

E

F

Gb

G

Ab

A

E

F

Gb

G

Ab

A

Bb

B

C

Db

D

Eb

E

B

C

Db

D

Eb

E

F

Gb

G

Ab

A

Bb

B

C

Db

D

Eb

E

F

Gb

G

Ab

A

Bb

B

C

Puts the missing key between a semitone

G

Ab

A

Bb

B

C

Db

D

Eb

E

F

Gb

G

(if that were to be valuable in any way)

The archaeotonic tonality is much simpler to deal with: you just leave out a tone and remember which one. Although, for diatonic use it may be more convenient to put the missing tone between E/F or B/C to keep it on the white keys, with the remaining small step where it looks like it should be.

Regular temperament properties

Uniform maps

13-limit uniform maps between 12.5 and 13.5

Min. size	Max. size	Wart notation	Map
12.5000	12.5661	13bccddeeeffff	⟨13 20 29 35 43 46]
12.5661	12.5743	13bccddeeeff	⟨13 20 29 35 43 47]
12.5743	12.6454	13bccddeff	⟨13 20 29 35 44 47]
12.6454	12.7050	13bcceff	⟨13 20 29 36 44 47]
12.7050	12.8363	13beff	⟨13 20 30 36 44 47]
12.8363	12.8634	13be	⟨13 20 30 36 44 48]
12.8634	12.9341	13b	⟨13 20 30 36 45 48]
12.9341	13.0016	13	⟨13 21 30 36 45 48]
13.0016	13.1066	13d	⟨13 21 30 37 45 48]
13.1066	13.1356	13df	⟨13 21 30 37 45 49]
13.1356	13.1524	13cdf	⟨13 21 31 37 45 49]
13.1524	13.3578	13cdeef	⟨13 21 31 37 46 49]
13.3578	13.3768	13cdddeef	⟨13 21 31 38 46 49]
13.3768	13.4415	13cdddeefff	⟨13 21 31 38 46 50]
13.4415	13.5000	13cdddeeeefff	⟨13 21 31 38 47 50]

Commas

13edo tempers out the following commas. (Note: This assumes the val ⟨13 21 30 36 45 48].)

Prime limit	Ratio	Monzo	Cents	Color name	Name(s)
5	(14 digits)	[-21 3 7⟩	10.06	Lasepyo	Semicomma, Fokker comma
7	1029/1000	[-3 1 -3 3⟩	49.49	Trizogu	Keega
7	525/512	[-9 1 2 1⟩	43.41	Lazoyoyo	Avicennma, Avicenna's Enharmonic Diesis
7	64/63	[6 -2 0 -1⟩	27.26	Ru	Septimal Comma, Archytas' Comma, Leipziger Komma
7	64827/64000	[-9 3 -3 4⟩	22.23	Laquadzo-atrigu	Squalentine
7	3125/3087	[0 -2 5 -3⟩	21.18	Triru-aquinyo	Gariboh
7	3136/3125	[6 0 -5 2⟩	6.08	Zozoquingu	Hemimean comma
11	56/55	[3 0 -1 1 -1⟩	31.19	Luzogu	Undecimal diesis
11	121/120	[-3 -1 -1 0 2⟩	14.37	Lologu	Biyatisma
11	441/440	[-3 2 -1 2 -1⟩	3.93	Luzozogu	Werckisma
13	40/39	[3 -1 1 0 0 -1⟩	43.83	Thuyo	Tridecimal minor diesis
13	105/104	[-3 1 1 1 0 -1⟩	16.57	Thuzoyo	Animist
13	169/168	[-3 -1 0 -1 0 2⟩	10.27	Thothoru	Buzurgisma

Animism

The animist comma, 105/104, appears whenever 3*5*7 = 2^3*13... 13edo does not approximate 3 and 7 individually (26edo does), but 13edo has 21/16 (21 = 3*7) and is also an animist temperament. In 13edo, the 5th harmonic is tuned so flatly that 5/4 = 16/13, leading to some interesting identities. So two scales stand out through this construction:

0 4 5 8 9 13 pentatonic

and

0 1 3 4 5 8 9 10 12 13 nonatonic

Introductory materials

By Inthar

• 13edo example composition (File:13edo 1MC score.pdf)

Oneirotonic Modal Studies

• : Tonal Study in Dylathian
• : Tonal Study in Ultharian
• : Tonal Study in Hlanithian
• : Tonal Study in Illarnekian
• : Tonal Study in Mnarian
• : Tonal Study in Sarnathian
• : Tonal Study in Celephaïsian
• : Tonal Study in Kadathian

Music

Main article: 13edo/Music

See also: Category:13edo tracks

See also

• William Lynch's 13 EDO octaton approach
• 13EDO Scales and Chords for Guitar
• Lumatone mapping for 13edo
• Well-Tempered 13-Tone Clavier (collab project to create 13edo keyboard pieces in a variety of keys and modes)

Further reading

• Sword, Ron. Triskaidecaphonic Scales for Guitar: Practical Theory and Scales on the Thirteen Equal Divisions of the Octave. 2009.