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 accessed by a particular scale like oneirotonic (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

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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 2.5.11.13 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.

Functional patterns

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13edo oneiro enjoys two main (rooted) delta-rational sonorities analogous to major and minor triads: 0-(185)-369-646 ("delta-rational major triad" or just "major"/"Maj") and 0-277-738-923 ("delta-rational minor tetrad" or just "minor"/"min"). One of these chords are on the root in the 6 brightest modes of oneirotonic. In the two darkest modes, I think 0-277-738-1015 or 0-738-1015-277 works well. The chord 0-277-738 will be called "minor triad" or "mintri", and 0-277-646-923 will be called "minor diminished" or "mindim".

A progression on the ascending Celephaïsian scale

A progression on the ascending Melodic Mnarian scale

Adding 923 and 1108 to chords works well, and for jazzy extensions one can add 185, 461, and 646 to the upper octave.

A Mnarian loop with an &8 leading tone at the end

Some motherchords of oneiro modes

Functional chords on each degree

Celephaisian

• 0d: minor
• 1d: minor
• 2d: major
• 3d: minor, mindim
• 4d: minor triad, minor 4ms, minor 6ms, minor 7ms
• 5d: major
• 6d: minor
• 7d: minor triad, minor 4ms, minor 6ms, minor 7ms

Progressions

kd(maj) means the triad 0 369 646 on kd, kd(min) means the tetrad 0 277 738 923 on kd

Common motions:

• 0d(maj or min) → M1d(min)
• 0d(maj or min) → 4d(min) (when ending on 0d this sounds like diatonic V to I)
• 0d(maj or min) → m6d(min)
• 0d(maj) → 5d(maj), 0d(min) → 5d(maj or min) (when ending on 0d this is a "dominant to tonic" motion)
• 0d(maj) → M3d(maj)
• 2d(min) → m1d(maj) → 0d(maj) (pseudo tritone sub)

Functional harmony

Modes can be grouped by their functional properties.

• Dual-fifth: Illarnekian, Celephaïsian, Ultharian
• Dual-fourth: Mnarian, Kadathian, Hlanithian
• Major pJI chord on root: Dylathian, Illarnekian
• Minor pJI chord on root: Celephaïsian, Ultharian, Mnarian, Kadathian
• Lower leading tone: Dylathian, Illarnekian, Celephaïsian
• "Neoclassical functional modes" (loose grouping): Dylathian, Illarnekian, Celephaïsian, Ultharian
• Upper leading tone: Kadathian, Hlanithian, Sarnathian,
• Minor 6-mosstep: Hlanithian, Sarnathian,
• 0 462 831 delta-rational chord on root: Dylathian, Dylydian, Hlanithian,
• "Dorian-like", i.e. no leading tone, 5d is minor, and 6d is major: Ultharian, Mnarian
• 7d is minor: Kadathian, Hlanithian

We'll call degrees that don't have the major delta-rational or the minor delta-rational chord dissonant degrees (keeping in mind that dissonance is a feature a chord has in a musical language rather than a purely psychoacoustic property).

Celephaïsian

Functional chords on each degree:

• 0d: min
• 1d: min
• 2d: Maj
• 3d: min
• 4d: minor triad, minor 4ms, minor 6ms, minor 7ms
• 5d: Maj
• 6d: min
• 7d: minor triad, minor 4ms, minor 6ms, minor 7ms

The main resolving degrees (analogues to dominant in diatonic) are 3d and 5d because of their leading tones.

Motherchord: 0d-m2d-P5d-M6d-(M7d)-M1d-P3d-M5d-(M7d)

Progressions:

• 0d(min) 1d(min) 3d(min or mindim) 0d(min)
• 0d(min) 1d(min) 6d(min) 3d(min or mindim) 0d(min)
• 0d(min) 6d(min) 3d(min or mindim) 0d(min)
• 0d(min) 2d(Maj) 3d(min or mindim) 0d(min)
• 0d(min) 4d(min7) 5d(Maj)/3d(min or mindim) 0d(min)
• 0d(min) 6d(min) 5d(Maj) 0d(min)

Secondary modes:

• 3d Ultharian
• 2d Dylathian
• 5d Illarnekian

Ultharian

• 0d: min
• 1d: minor triad, minor 4ms, minor 6ms, minor 7ms
• 2d: Maj
• 3d: min
• 4d: minor triad, minor 4ms, minor 6ms, minor 7ms
• 5d: min
• 6d: min
• 7d: Maj

Ultharian and Mnarian often behave like Dorian because they lack a leading tone. Resolving degrees: 3d(min), 5d(min), 7d(Maj)

Mnarian

• 0d: min
• 1d: minor triad, minor 4ms, minor 6ms, minor 7ms
• 2d: min
• 3d: min
• 4d: Maj
• 5d: min
• 6d: minor triad, minor 4ms, minor 6ms, minor 7ms
• 7d: Maj

Resolving degrees: 3d(min), 5d(min), 7d(Maj)

Kadathian

• 0d: min
• 1d: Maj
• 2d: min
• 3d: minor triad, minor 4ms, minor 6ms, minor 7ms
• 4d: Maj
• 5d: min
• 6d: minor triad, minor 4ms, minor 6ms, minor 7ms
• 7d: min

Resolving degrees: 2d(min)?, 4d(Maj), 5d(min), 7d(min)

Hlanithian and Sarn

Main tonic chord is 0-3-8-11\13 (min7), works well with m6d(maj) and m7d(min7).