13edo

13edo, or 13 equal divisions of the octave, is the equal tuning featuring steps of (1200/13) ~= 92.308 cents, 13 of which stack to the octave 2/1. It does not approximate many small prime harmonics well at all, so DR-based interpretations may be preferred among 13edo users.

13edo's greatest melodic strength is its proximity to 12edo, whose most important effect is providing an oneirotonic (5L3s, LLsLLsLs) MOS which is a compressed diatonic. For Jaimbee and Inthar's functional harmony and method, see the oneirotonic page.

Tuning theory

Intervals

This page or section deals with proposed concepts. The terminology and concepts used in it are developed by one person or a small group and may lack widespread adoption.

Note: The logic of ground's notation is to preserve the diatonic order of nominals for the stacked oneirotonic subfourth generators, with one additional note: BEADGCFX

Edostep	Cents	Interval region name	ADIN name (Oneirotonic extension)	Oneirotonic TAMNAMS name	Oneirotonic KISS notation	Ground's notation (on A = 440 Hz)	26edo subset notation (on A = 440 Hz)
0	0	Unison	Unison	Perfect 0-(oneiro)step (P0oneis)	1	A	A
1	92.3	Minor 2nd	Minor second	Minor 1-(oneiro)step (m1oneis)	1# / 2b	A# / Cb	Ax / Bbb
2	184.6	Major 2nd	Major second	Major 1-(oneiro)step (M1oneis)	2	C	B
3	276.9	(Sub)minor 3rd	Minor third	Minor 2-(oneiro)step (m2oneis)	3	B	Bx / Cb
4	369.2	(Sub)major 3rd	Major third	Major 2-(oneiro)step (M2oneis) Diminished 3-(oneiro)step (d3oneis)	3# / 4b	B# / Db	C#
5	461.5	Subfourth	Fourth	Perfect 3-(oneiro)step (P3oneis)	4	D	Db
6	553.8	Ultrafourth / Infratritone	Minor tritone	Minor 4-(oneiro)step (m4oneis)	5b	Fb	D#
7	647.2	Ultratritone / Infrafifth	Major tritone	Major 4-(oneiro)step (M4oneis)	5	F	Eb
8	738.5	Superfifth	Fifth	Perfect 5-(oneiro)step (P5oneis)	6	E	E# / Fbb
9	830.8	(Super)minor 6th	Minor sixth	Augmented 5-(oneiro)step (A5oneis) Minor 6-(oneiro)step (m6oneis)	6# / 7b	E# / Gb	F
10	923.1	(Super)major 6th	Major sixth	Major 6-(oneiro)step (M6oneis)	7	G	Fx / Gbb
11	1015.4	Minor 7th	Minor seventh	Minor 7-(oneiro)step (m7oneis)	8b	Xb	G
12	1107.7	Major 7th	Major seventh	Major 7-(oneiro)step (M7oneis)	8	X	Gx / Abb
13	1200	Octave	Octave	Perfect 8-(oneiro)step (P8oneis)	1	A	A

Prime harmonic approximations

Approximation of prime harmonics in 13edo

Harmonic		2	3	5	7	11	13	17	19	23
Error	Absolute (¢)	0.0	+36.5	-17.1	-45.7	+2.5	-9.8	-12.6	-20.6	+17.9
	Relative (%)	0.0	+39.5	-18.5	-49.6	+2.7	-10.6	-13.7	-22.3	+19.4
Steps (reduced)		13 (0)	21 (8)	30 (4)	36 (10)	45 (6)	48 (9)	53 (1)	55 (3)	59 (7)

Edostep interpretations

13edo's edostep functions in the 2.9.5.21.11.13.17.19 subgroup as:

• 17/16
• 18/17
• 19/18
• 20/19
• 21/20 (the interval between 10/9 and 7/6)
• 22/21
• 26/25 (the interval between 5/4 and 13/10)
• 55/52 (the interval between 11/8 and 13/10, and between 5/4 and 13/11)
• 128/121 (the interval between 11/8 and 16/11)

Harmonic series approximations

13edo approximates the following harmonic series chord fairly well (x indicates notes that are harder to approximate):

34:36:38:40:42:x:47:x:52:55:58:61:x:68

This can be derived as follows:

1. the quasi-13edo isoharmonic chord 5:9:13:17:21 => 17:18:x:20:21:x:x:x:26:x:x:x:x:34
2. the simic sixth chord 17:20:26:29 (+1+2+1) => 17:18:x:20:21:x:x:x:26:x:29:x:x:34
3. place 11/8 on harmonic 20 => 34:36:x:40:42:x:x:x:52:55:58:x:x:68
4. use halfway harmonics 19 and 47 => 34:36:38:40:42:x:47:x:52:55:58:x:x:68
5. 61/52 is .6c off from 3\13 => 34:36:38:40:42:x:47:x:52:55:58:61:x:68

Making an over-17 13edo neji thus requires you to choose those three notes:

• The notes resulting in lowest pairwise error in mode 34 are 44, 49, and 64.
• The closest notes in mode 68 are 89, 99, and 129 (which are significantly more complex).
• A less accurate but lower-complexity neji (limited to oneirotonic) is 22:25:26:29:32:34:38:42:44, so one could specifically choose 44, 50, and 64.

Multiples

26edo

Main article: 26edo

39edo

39edo is a Supra (2.3.7.11[17 & 22]) diatonic tuning which has good 11/8 and 9/7 approximations in the mosdiatonic scale, though the 39d val (using the sharp approximation of 7/4) is required. One may favor 39edo over harder Archy tunings for the larger diatonic semitone size.

Approximation of prime harmonics in 39edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	+5.7	+13.7	-15.0	+2.5	-9.8	-12.6	+10.2	-12.9	-14.2	-6.6
	Relative (%)	0.0	+18.6	+44.5	-48.7	+8.2	-31.7	-41.1	+33.1	-41.9	-46.1	-21.4
Steps (reduced)		39 (0)	62 (23)	91 (13)	109 (31)	135 (18)	144 (27)	159 (3)	166 (10)	176 (20)	189 (33)	193 (37)

65edo

65edo is notable as the intersection of Schismic and Wurschmidt. It is a strong 2.3.5.11.19.23.31.47.49 system.

Approximation of prime harmonics in 65edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Error	Absolute (¢)	0.0	-0.4	+1.4	-8.8	+2.5	+8.7	+5.8	-2.1	-0.6	+4.3	-0.4	+7.1	-4.4	+5.4	-0.9
	Relative (%)	0.0	-2.3	+7.5	-47.8	+13.7	+47.1	+31.5	-11.5	-3.2	+23.1	-2.3	+38.6	-24.1	+29.3	-4.8
Steps (reduced)		65 (0)	103 (38)	151 (21)	182 (52)	225 (30)	241 (46)	266 (6)	276 (16)	294 (34)	316 (56)	322 (62)	339 (14)	348 (23)	353 (28)	361 (36)

104edo

See 26edo#104edo.

130edo

See 26edo#130edo.

See also

• The Well-Tempered Triskaidecatonic Clavier (WT13C) project