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, and the JI approximations it does have do not fit very well in a temperament (they fit better in a neji), 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. A functional system for 13edo oneirotonic is provided below.

Tuning theory

Intervals

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	Oneirotonic TAMNAMS name	Fox-Raven notation (on J = 180 Hz)	Ground's notation (on A = 440 Hz)	26edo subset notation (on A = 440 Hz)
0	0	Unison	Perfect 0-(oneiro)step (P0oneis)	J	A	A
1	92.3	Minor 2nd	Minor 1-(oneiro)step (m1oneis)	J# / Kb	A# / Cb	Ax / Bbb
2	184.6	Major 2nd	Major 1-(oneiro)step (M1oneis)	K	C	B
3	276.9	(Sub)minor 3rd	Minor 2-(oneiro)step (m2oneis)	L	B	Bx / Cb
4	369.2	(Sub)major 3rd	Major 2-(oneiro)step (M2oneis) Diminished 3-(oneiro)step (d3oneis)	L# / Mb	B# / Db	C#
5	461.5	Subfourth	Perfect 3-(oneiro)step (P3oneis)	M	D	Db
6	553.8	Ultrafourth / Infratritone	Minor 4-(oneiro)step (m4oneis)	Nb	Fb	D#
7	647.2	Ultratritone / Infrafifth	Major 4-(oneiro)step (M4oneis)	N	F	Eb
8	738.5	Superfifth	Perfect 5-(oneiro)step (P5oneis)	O	E	E# / Fbb
9	830.8	(Super)minor 6th	Augmented 5-(oneiro)step (A5oneis) Minor 6-(oneiro)step (m6oneis)	O# / Pb	E# / Gb	F
10	923.1	(Super)major 6th	Major 6-(oneiro)step (M6oneis)	P	G	Fx / Gbb
11	1015.4	Minor 7th	Minor 7-(oneiro)step (m7oneis)	Qb	Xb	G
12	1107.7	Major 7th	Major 7-(oneiro)step (M7oneis)	Q	X	Gx / Abb
13	1200	Octave	Perfect 8-(oneiro)step (P8oneis)	J	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 2.5.11.13 subgroup as:

• 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 well (x indicates notes that are harder to approximate):

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.

Jaimbee and Inthar's functional system for 13edo

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.

The following system has been developed by Jaimbee and Inthar.

13edo's melodically strongest scale is the oneirotonic MOS (preserving the diatonic property of having at least 2 semitones), so it behooves us to find harmonies that work for it. Since there are certain similarities of oneirotonic to diatonic, we can build off of these similarities to assign functions to oneirotonic degrees.

For a DR-forward framework like this, prefer mellow timbres to bright ones to bring out the DR effect.

Basic chords

The most basic chords in this functional harmony system are:

• Major triad 0-4-7\13: A compressed major triad that sounds desaturated and somewhat bittersweet. Somewhat dubiously +1+1. Oneirotonic provides only two of these triads, so alterations are somewhat frequently used to get a major triad. The major triad has the following important tetrad supersets:
  • 0-2-4-7\13: Reinforces the quasi-DR effect with an extra tone; approximately +1+1+2.
  • 0-4-7-10\13: A compressed dominant tetrad; approximately +1+?+1.
  • 0-4-7-12\13
• Minor +1+2 triad 0-3-8\13: A bright and brooding if somewhat hollow-sounding minor triad. Approximately 17:20:26. The important supersets are:
  • 0-3-8-10: Approximately +1+2+1.
  • 0-3-8-12: Approximately +1+2+2.
  • 0-3-8-11: Something like a minor 7th tetrad.
  • 0-3-8-15
  • 0-3-8-12-15: A concatenation of the minor +1+2 and major +1+1 triads.
• 0-5-9\13: A +1+1 triad and a compressed 2nd inversion major triad. Approximately 13:17:21.
  • 0-5-7-9: Approximately +2+1+1.
  • 0-5-9-12: A compressed major triad on top of a subfourth.
  • 0-5-9-12-15
  • 0-5-7-9-12-15-17
• 0-5-7\13: Compressed sus4. Approximately +2+1.
• 0-4-8\13: "Submajor augmented" triad.
• 0-3-6\13: The most diminished-like triad.