13edo: Difference between revisions

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replaced that weirdly specific KISS notation with a more general KISS scheme
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* 2d(min) → m1d(maj) → 0d(maj) (pseudo tritone sub)
* 2d(min) → m1d(maj) → 0d(maj) (pseudo tritone sub)


==== Modal harmony ====
==== Functional harmony ====
Modes can be grouped by their functional properties.
Modes can be grouped by their functional properties.
* Dual-fifth: Illarnekian, Celephaïsian, Ultharian
* Dual-fifth: Illarnekian, Celephaïsian, Ultharian
Line 242: Line 242:
* 7d is minor: Kadathian, Hlanithian
* 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).
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).
===== Dylathian =====
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Starting with Dylathian, we'd find major chords on the first and fourth degree, while minor chords appear on
the second, fifth, seventh and eighth degrees. For the third and sixth degrees, you get a chord of scale degrees
0-3-9-11, which is the third type of pseudo-JI tetrad³, which we could call "augmented". Alternatively, you could
also play a pseudo-JI chord of scale degrees 0-5-9³ on the third degree, and in some contexts it may be favorable
for reasons I'll get to soon.
For each of these chords, we can associate functions with them. The simplest of these relationships is between
the root major chord and the major chord on the perfect fourth.
By adding nonaves (commonly called octaves) on certain notes, we can recreate the familiar dominant cadence
from diatonic, only now on the fourth degree rather than the fifth. The most simple of these progressions would
look something like this:
Q-J-K-M-K
L-M-N-P-L
Q-J-K-M-K
Or, in number notation:
0-2-4-7-17
5-7-9-12-18
0-2-4-7-17
In this cadence, the fifth on L is so narrow that it creates a leading tone relative to the root, and by playing the
nonave above L, you can create a narrow tritone that wants to resolve inwards to the major chord on the root.
The presence of the nonave 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 minor chord would be exactly
halfway to the dominant, making it the mediant. It also has a much nicer minor chord on it compared to the iii
augmented 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 Q Dylathian, it would be O
Celephaïsian, the minor vii). 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 vii minor
chord. By playing the nonave above certain notes, resolving between the two modes is pretty simple.
Q-J-K-M-Q
`J-`L-`O-`P-P
`O-`Q-`L-`M-Q
Or, in number notation:
0-2-4-7-13
`2-`5-`10-`12-12
`10-0-5-7-13
<`> notates playing a nonave lower than the root.
The ii (J minor) and vii (O minor) would probably sound the smoothest when played in a lower register than the
tonic (Q major) as notated, but if you want to move upwards from the root it still works.
Resolving from the relative minor (O minor) to the tonic (Q major) is a pretty weak but still usable resolution.
The minor sixth in the relative minor chord (L) is a semitone above the major third of the tonic major chord (K),
and the minor third of the relative minor chord is the root of the tonic (Q).
Another neat resolution is moving from the iii+ to the dominant IV.
If you play the 0-5-9 chord on the third degree (K-N-Q), 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 (K-M-N-Q), or simplifying the dominant chord to a 0-4-7 chord
(L-N-P), you can drive this resolution very powerfully, and this could either create a chain of strong resolutions
going iii+-IV-I, or it could help drive resolutions to the Ilarnekian mode above (in this case, L Ilarnekian).
Technically you wouldn't have to extend or simplify any of these chords, but the triad next to all the tetrads feels
kinda 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 pseudo-JI
effect, while the simplified major and minor still have a pseudo-JI effect, though a bit weaker than the tetrads³.
The vi+ (augmented vi, N-P-L-M / 9-12-18-20) chord also has some neat features, as it sort of functions as an
inversion of the dominant IV chord. It also doesn't need any extensions with nonaves to work super 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+ (augmented iii) could be used
to drive a resolution to a major sharp 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->M#->K#, which gives the progression a really
jazzy feel.
The only chord we haven't covered now would be the minor v (M-O-J-K, 7-10-15-17). This chord has a much
weaker relationship to the other chords, so it doesn't have any super strong directionality. However, it does share
some notes with a few important chords, notably the I chord and the relative minor on the vii. 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 IX chord a
nonave 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, and it seems obvious this would happen since
there's now eight notes instead of seven. It also works pretty well as a setup for the augmented vi, 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 Q Ilarnekian, you'd get:
Q J K L M Nb O P
With Ilarnekian being the second major mode (after Dylathian), we'd get the same I chord, Q major. One of the
most immediate effects we'd see, however, is that the dominant IV is now a minor iv. It still can function as a
dominant, though only with the added nonave 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 also minor, with it now driving a resolution to O Ultharian instead of O Celephaïsian. The iii+
chord is also still augmented, and also still drives a pretty good resolution up to the minor iv, though admittedly
weaker than dylathian.
The v chord is still minor, and still functions as a secondary mediant.
It gets interesting again when looking at the major VI chord.
By playing the VI lower than the tonic and playing the nonave above the root of the VI chord, you get an
entirely new approach to a dominant chord. The third of the VI is one semitone below the tonic, and the
nonave 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 minor dominant iv chord in Ilarnekian.
It would look something like this:
Q J K M
`Nb `O `P J Nb
Q J K M
Or, in number notation:
0-2-4-7
`8-`10-`12-2-8
0-2-4-7
With <`> again notating playing a nonave lower than the starting chord.
If I'm not mistaken, I think Inthar's approach to this scale uses this as its main dominant chord and Ilarnekian as
it's main major mode, due to this being slightly closer to the diatonic sound. That approach would make the iv a
mediant, and you could probably extrapolate the rest from there. I'll leave the specifics to Inthar, but I thought it
was worth noting.
Moving on, the vii chord would be minor, and would be the minor chord driving the resolution to the minor iv,
in the same way the Dylathian minor ii drives the resolution to the relative minor. It would also function as the
Ilarnekian relative minor, in this case O Ultharian.
The viii+ chord would be augmented, 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 major sixth move down a semitone to become the fifth of the tonic
chord. The minor eighth in the chord could be omitted to make the resolution stronger, but the chord would
sound much more dissonant,-->
===== Celephaïsian =====
===== Celephaïsian =====
Functional chords on each degree:
Functional chords on each degree:

Revision as of 15:25, 18 January 2026

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.

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

This is a problematic page or section. It lacks sufficient justification, content, or organization, and is subject to future overhaul or deletion.

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).

Dylathian
Ilarnekian
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).

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