13edo/Inthar's approach

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This page describes User:Inthar's approach to 13edo, mainly based on oneirotonic (5L 3s).

Standing assumptions

The Fox-Raven absolute pitch system is used in this article to name 5L 3s pitches: The notation used in this article is J Ultharian (LsLLsLsL) = JKLMNOPQJ, with reference pitch N = 261.6255653 Hz. I denote raising and lowering by a chroma (L − s) by # and b.

I'll use the TAMNAMS standard to denote oneirotonic intervals and degrees (independent of their absolute pitch).

The chain of perfect 3-steps becomes: ... Pb Kb Nb Q L O J M P K N Q# L# O# ...

Thus the 13edo gamut is as follows:

J/Q# J#/Kb K/Lb L/K# L#/Mb M M#/Nb N/Ob O/N# O#/Pb P Q Q#/Jb J

Oneirotonic modes

Mode UDP Name
LLsLLsLs 7|0 Dylathian (də-LA(H)TH-iən)
LLsLsLLs 6|1 Illarnekian (ill-ar-NEK-iən)
LsLLsLLs 5|2 Celephaïsian (kel-ə-FAY-zhən)
LsLLsLsL 4|3 Ultharian (ul-THA(I)R-iən)
LsLsLLsL 3|4 Mnarian (mə-NA(I)R-iən)
sLLsLLsL 2|5 Kadathian (kə-DA(H)TH-iən)
sLLsLsLL 1|6 Hlanithian (lə-NITH-iən)
sLsLLsLL 0|7 Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn"



We call the LSLLLSLS pattern (independently of modal rotation) archeodim, because the "LLL" resembles the archeotonic scale in 13edo and the "LSLSLS" resembles the diminished scale. Archeodim is the most important oneirotonic modmos pattern (a modmos is a mos with one or more alterations), because it allows one to evoke certain diatonic modes where three whole steps in a row are important (Dorian, Phrygian, Lydian or Mixo) in an octatonic context. The mos would not always be able to do this because it has at most two consecutive large steps. Archeodim modes exist in all oneirotonic tunings, since they use the same large and small steps as the oneirotonic scale itself.

Mode UDP Name
LLLsLsLs 7|0 #3d Dylydian (Dylathian #3d)
LLsLsLsL 6|1 b7d Illarmixian (Illarnekian b7d)
LsLLLsLs 5|2 #5d Celdorian (Celephaïsian #5d)
sLLLsLsL 4|3 b1d Ulphrygian (Ultharian b1d)
LsLsLLLs 3|4 #7d Mnionian (Mnarian #7d)
sLsLLLsL 0|7 #6d Sardorian (Sarnathian #6d)
LsLsLsLL 3|4 b6d Mnaeolian (Mnarian b6d)
sLsLsLLL 0|7 #5d Sarlocrian (Sarnathian b5d)

The cyclic order of archeodim modes are:

  1. LSLLLSLS is Celdorian
  2. SLLLSLSL is Ulphrygian
  3. LLLSLSLS is Dylydian
  4. LLSLSLSL is Illarmixian
  5. LSLSLSLL is Mnaeolian
  6. SLSLSLLL is Sarlocrian
  7. LSLSLLLS is Mnionian
  8. SLSLLLSL is Sardorian

Notably the cyclic order is almost exactly the "circle" of diatonic modes except the "circle" breaks at Sardorian-Celdorian.

In 18edo, the augmented 3-mosstep (LLL) is a 600¢ tritone, which can be used as a dissonance.

Modes of the harmonic Mnarian scale

We call the scale LsLsLsLAs (with A = L + chroma) the harmonic Mnarian scale. The table below shows the modes of Harmonic Mnarian.

Mode UDP Name
LsLsLsAs 3|4 b6d #7d Mnarian b6d #7d
sLsLsAsL 0|7 b5d #7d Sarnathian b5d #7d
LsLsAsLs 3|4 #5d #7d Mnarian #5d #7d
sLsAsLsL 0|7 #4d #6d Sarnathian #4d #6d
LsAsLsLs 5|3 #3d #5d Celephaïsian #3d #5d
sAsLsLsL 4|3 b1d #3d Ultharian b1d #3d
AsLsLsLs 7|0 #1d #3d Dylathian #1d #3d
sLsLsLsA 0|7 b5d b7d Sarnathian b5d b7d

13edo oneiro chordal theory

Favorite modes: Dylathian (22122121), Illarnekian (22121221), Celephaisian (21221221), Mnionian (21212221), Dylydian (22212121), Harmonic Mnarian (21212131)?

We will primarily discuss functional harmony using delta-rational chords, treating these separately from counterpoint. h/t to tprice and Jaimbee for the delta-rational chords presented here and for other discussions.

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


  • 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

Mnionian (Melodic Mnarian)

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


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)

Modal harmony

Modes can be grouped by their functional properties.

  • Dual-fifth: Illarnekian, Celephaïsian, Ultharian; Illarmixian, Ulphrygian
  • Dual-fourth: Mnarian, Kadathian, Hlanithian; Mnionian, Mnaeolian
  • Major pJI chord on root: Dylathian, Illarnekian; Dylydian, Illarmixian
  • Minor pJI chord on root: Celephaïsian, Ultharian, Mnarian, Kadathian; Mnionian, Sardorian, Ulphrygian
  • Lower leading tone: Dylathian, Illarnekian, Celephaïsian; Dylydian, Mnionian
  • "Neoclassical functional modes" (loose grouping): Dylathian, Illarnekian, Celephaïsian, Ultharian; Mnionian, Mnaeolian(?)
  • Upper leading tone: Kadathian, Hlanithian, Sarnathian, Sardorian, Sarlocrian
  • Minor 6-mosstep: Hlanithian, Sarnathian, Mnaeolian, Sarlocrian
  • 0 462 831 delta-rational chord on root: Dylathian, Dylydian, Hlanithian, Mnaeolian
  • "Dorian-like", i.e. no leading tone, 5d is minor, and m7d 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).


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)


  • 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


  • 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)


  • 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)


  • 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).