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

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 2.5.11.13 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.

Todo: use ground's oneiro chord names

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:

• Compressed major triad 0-4-7\13 (>maj<): 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 this triad. The compressed major triad has the following important tetrad supersets:
  • 0-2-4-7\13 (>majsus2<): Reinforces the quasi-DR effect with an extra tone; approximately +1+1+2.
  • 0-4-7-10\13 (>dom7<): A compressed dominant tetrad; approximately +1+?+1.
  • 0-4-7-12\13 (>maj<maj7): Approximately +1+1+2.
• The simic triad 0-3-8\13 (sim): A bright and brooding if somewhat hollow-sounding minor triad. Approximately 17:20:26 or +1+2. The important supersets are:
  • 0-3-8-10 (sim6): Approximately +1+2+1.
  • 0-3-8-12 (simmaj7): Approximately +1+2+2.
  • 0-3-8-11 (simmin7): Something like a minor 7th tetrad.
  • 0-3-8-15 (simadd9)
  • 0-3-8-12-15 (simmaj7add9): A concatenation of the minor +1+2 and major +1+1 triads.
• 0-5-9\13 (>IV/I<): 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 (>sus4<): Compressed sus4. Approximately +2+1.
• 0-4-8\13 (>aug<): "Submajor augmented" triad.
• 0-3-6\13 (>dim<): The most diminished-like triad.

Functional patterns

13edo oneiro enjoys two main (rooted) delta-rational sonorities analogous to major and minor triads: 0-(185)-369-646 ("compressed major triad" or just ">maj<") and 0-277-738-923 ("simic sixth" or "sim6"). 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 "simic", and 0-277-646-923 will be called "compressed minor 7th".

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

• I: sim6
• II: sim6
• III: >maj<
• IV: sim6, >min7<
• T: sim, minor 4ms, minor 6ms, minor 7ms
• V: >maj<
• VI: sim6
• VII: sim, minor 4ms, minor 6ms, minor 7ms

Progressions

x(>maj<) means the triad 0 369 646 on degree x, kd(sim6) means the tetrad 0 277 738 923 on kd

Common motions:

• 0d(>maj< or sim6) → M1d(sim6)
• 0d(>maj< or sim6) → 4d(sim6) (when ending on 0d this sounds like diatonic V to I)
• 0d(>maj< or sim6) → m6d(sim6)
• 0d(>maj<) → 5d(>maj<), 0d(sim6) → 5d(>maj< or sim6) (when ending on 0d this is a "dominant to tonic" motion)
• 0d(>maj<) → M3d(>maj<)
• 2d(sim6) → 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
• Compressed major chord on root: Dylathian, Illarnekian
• Simic sixth 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 a >maj< or sim6 chord dissonant degrees (keeping in mind that dissonance is a feature a chord has in a musical language rather than a purely psychoacoustic property).

Dylathian

The below uses I, II, III, IV, T, V, VI, VII degrees/functions and ground's notation for oneirotonic nominals. Interval names are in ADIN.

Starting with Dylathian, we'd find compressed major chords on the I and IV degrees, while simic chords appear on the II, T, VI, and VII degrees. For the III and V degrees, you get a chord of edo steps 0-3-9-11, which is the third type of DR tetrad, which could be viewed as an inversion of the compressed dominant tetrad.. Alternatively, you could also play a DR chord of scale degrees 0-5-9 on the third degree, and in some contexts it may be favorable (see below).

For each of these chords, we can associate functions with them. The simplest of these relationships is between the root compressed major chord and the compressed major chord on the perfect fourth. By adding octaves on certain notes, we can recreate the familiar dominant cadence from diatonic, only now on the IV rather than the V. The most simple of these progressions would look something like this (in ground's notation):

1. B-D-F-G-F
2. E-G-X-C-D
3. B-D-F-G-F

Or, in 13edo steps:

1. 0-2-4-7-17
2. 5-7-9-12-18
3. 0-2-4-7-17

In this cadence, the fifth on E is so narrow that it creates a leading tone relative to the root, and by playing the octaves above E, you can create a minor tritone that wants to resolve inwards to the compressed major chord on the root. The presence of the octaves above the major third helps drive this resolution, but can be omitted. Another neat effect is that given the dominant is now on the IV, then the II simic sixth chord would be exactly halfway to the dominant, making it the mediant. It also has a much nicer simic sixth chord on it compared to the III 1st inversion compressed major chord, making it more akin to how the mediant works in diatonic.

We can also relate other chords to the dominant, mediant and tonic. The relative minor is more or less exactly analogous to diatonic, being a minor third below the tonic (in the case of B Dylathian, it would be A Celephaïsian, the minor VI). The mediant can also function as a secondary dominant for resolutions to the relative minor; the highest note in the II minor chord is one semitone below the minor third of the VI minor chord. By playing the octave above certain notes, resolving between the two modes is pretty simple.

1. B-D-F-G-B
2. `D-`E-`A-`C-C
3. `A-`B-`E-`G-B

In 13edo steps:

1. 0-2-4-7-13
2. `2-`5-`10-`12-12
3. `10-0-5-7-13

The ` denotes playing an octave lower than the root.

The II (D minor) and VI (A minor) would probably sound the smoothest when played in a lower register than the tonic (B major) as notated, but if you want to move upwards from the root it still works. Resolving from the relative minor (A minor) to the tonic (B major) is a pretty weak but still usable resolution. Another neat resolution is moving from the III (inverted major) to the dominant IV.

If you play the 0-5-9 chord on the third degree, the lowest note will be a semitone lower than the lowest in the dominant, and the highest note will be a semitone higher than the highest in the dominant. By either extending the 0-5-9 chord to 0-3-5-9, or simplifying the dominant chord to a 0-4-7 chord, you can drive this resolution very powerfully, and this could either create a chain of strong resolutions going iii (inverted major)-IV-I, or it could help drive resolutions to the Ilarnekian mode above (in this case, E Ilarnekian).

Technically you wouldn't have to extend or simplify any of these chords, but the triad next to all the tetrads feels somewhat empty. All in all, using this technique you could probably simplify all the tetrads down to 0-4-7 and 0-3-8 for major and minor, respectively. These would help since the 0-3-5-9 chord doesn't have much of a DR effect, while the simplified major and minor still have a DR effect, though a bit weaker than the tetrads. The vi inverted major (9-12-18-20) chord also has some neat features, as it functions as an inversion of the dominant IV chord. It also doesn't need any extensions with octaves to work well unlike the dominant chord, so it could be seen as a more tense version of dominant. Since it also drives the resolution up by a minor third, the same tetrad on the III could be used to drive a resolution to a major V, helping to shift the key center from Q to M#. If done twice, this resolution can shift your key center up a minor third from Q -> G# -> F#, which gives the progression a really jazzy feel.

The only chord we haven't covered now would be the minor T (7-10-15-17). This chord has a much weaker relationship to the other chords, so it doesn't have any strong directionality. However, it does share some notes with a few important chords, notably the I chord and the relative minor on the VI. A resolution to either of these will be similarly strong, that is to say, not very strong. In this case it could also be seen as a secondary mediant which is not the relative minor, about halfway between the I chord and the VII chord a octave above it, and either of these resolutions would probably sound fine in most contexts. This gives it a role completely unlike any of the functions in traditional diatonic. It also works pretty well as a setup for the V inverted major, so in a progression it can help add some flair or beef to the resolution.

Ilarnekian

To start with the basics, Ilarnekian is just Dylathian with a flattened 6th. In E Ilarnekian, you'd get: E G X A C B D F E

With Ilarnekian being the second major mode (after Dylathian), we'd get the same I chord, E>maj< One of the most immediate effects we'd see, however, is that the dominant IV is now IV>min7<. It still can function as a dominant, though only with the added octave above the 4th, and slightly weaker than the Dylathian dominant cadence. Another interesting thing is that the IV-I cadence is now simultaneously a minor plagal and a dominant cadence, radically different from anything in diatonic.

The II chord is again simic sixth, with it now driving a secondary resolution to D Ultharian instead of D Celephaïsian. The III inverted major chord is also still inverted compressed major, and also still drives a pretty good resolution up to the simic sixth IV, though admittedly weaker than Dylathian.

The T chord is still simic sixth, and still functions as a secondary mediant.

It gets interesting again when looking at the compressed major V chord.

By playing the V lower than the tonic and playing the octave above the root of the V chord, you get an entirely new approach to a dominant chord. The third of the V is one semitone below the tonic, and the octave above the root is one semitone above the fifth of the tonic chord. This drives a very strong inward resolution that resembles dominant in diatonic slightly more than the dominant IV chord in Dylathian, and a lot more than the compressed minor IV chord in Ilarnekian. It would look something like this:

• E G X C
• `B `D `F G B
• E G X C

In 13edo steps:

• 0-2-4-7
• `8-`10-`12-2-8
• 0-2-4-7

with ` again notating playing an octave lower than the starting chord.

Moving on, the VI chord would be simic sixth driving the resolution to the IV simic sixth, in the same way Dylathian's II simic sixth drives the resolution to the relative minor. It would also function as the Ilarnekian relative minor, in this case D Ultharian. The vii chord would be an inverted major chord, and would drive a resolution to the tonic pretty well. This comes from the fact that the 0-3-9-11 chord would have the minor third become the major second of the tonic, the root move up a semitone to the tonic, and the perfect fifth move down a semitone to become the tritone of the tonic chord. The minor sixth in the chord could be omitted to make the resolution stronger, but the chord would sound much more dissonant.

Celephaïsian

Functional chords on each degree:

• I: sim6
• II: sim6
• III: >maj<
• IV: sim6
• T: sim, minor 4ms, minor 6ms, minor 7ms
• VI: >maj<
• VII: sim6
• VIII: sim, minor 4ms, minor 6ms, minor 7ms

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

Progressions:

• Isim6 IIsim6 IV(sim6 or >min7<) Isim6
• Isim6 IIsim6 VIsim6 IV(sim6 or >min7<) Isim6
• Isim6 VIsim6 IV(sim6 or >min7<) Isim6
• Isim6 III>maj< IV(sim6 or >min7<) Isim6
• Isim6 Tsim7 V>maj< IV(sim6 or >min7<) Isim6
• Isim6 VIsim6 V>maj< Isim6

Secondary modes:

• IV Ultharian
• III Dylathian
• V Illarnekian