User:Inthar/5L 3s

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This page describes User:Inthar's approach to 5L 3s (aka oneirotonic).

Now with new theory (thanks to tprice and Jaimbee)

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

The 18edo gamut is notated as follows:

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

The 21edo gamut:

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

Modes

Oneirotonic modes are named after cities in the Dreamlands.

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"


Alterations

Archeodim

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 &4 Dylydian (Dylathian #4)
LLsLsLsL 6|1 @8 Illarmixian (Illarnekian b8)
LsLLLsLs 5|2 &6 Celdorian (Celephaïsian #6)
sLLLsLsL 4|3 @2 Ulphrygian (Ultharian b2)
LsLsLLLs 3|4 &8 Mnionian (Mnarian #8)
sLsLLLsL 0|7 &7 Sardorian (Sarnathian #7)
LsLsLsLL 3|4 @7 Mnaeolian (Mnarian b7)
sLsLsLLL 0|7 @6 Sarlocrian (Sarnathian b6)

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.

Other modmosses

  • LSLSLSAS Harmonic Mnarian

13edo oneiro chordal theory

Favorite modes: Dylathian (22122121), Illarnekian (22121221), Celephaisian (21221221), Mnionian (21212221)

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

13edo oneiro enjoys two main (rooted) pseudo-JI sonorities analogous to major and minor triads: 0-(185)-369-646 ("pseudo-JI major triad" or just "major"/"Maj") and 0-277-738-923 ("pseudo-JI 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

Functional chords on each degree

Celephaisian

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

Mnionian (Melodic Mnarian)

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

Progressions

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

Common motions:

  • 0md(maj or min) → M1md(min)
  • 0md(maj or min) → 4md(min) (when ending on 0md this sounds like diatonic V to I)
  • 0md(maj or min) → m6md(min)
  • 0md(maj) → 5md(maj), 0md(min) → 5md(maj or min) (when ending on 0md this is a "dominant to tonic" motion)
  • 0md(maj) → M3md(maj)
  • 2md(min) → m1md(maj) → 0md(maj) (pseudo tritone sub)

Modulations

Modal harmony

Modes can be grouped by their functional properties.

  • Major pJI chord on root: Dylathian, Illarnekian; Dylydian, Illarmixian
  • Leading tone: Dylathian, Illarnekian, Celephaïsian; Dylydian, Mnionian
  • "Classical functional modes" (loose grouping): Dylathian, Illarnekian, Celephaïsian, Ultharian; Mnionian, Mnaeolian(?)
  • Minor pJI triad on root: Celephaïsian, Ultharian, Mnarian, Kadathian; Mnionian, Sardorian, Ulphrygian
  • Upper leading tone: Kadathian, Hlanithian, Sarnathian, Sardorian, Sarlocrian
  • Minor 6-mosstep: Hlanithian, Sarnathian, Mnaeolian, Sarlocrian
  • 0 462 831 pseudo-JI chord on root: Dylathian, Dylydian, Hlanithian, Mnaeolian

We'll call degrees that don't have the major pseudo-JI or the minor pseudo-JI 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

  • 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 3md and 5md because of their leading tones.

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)

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

It often behaves like Dorian because it lacks a leading tone. Resolving degrees: 3d(min), 5(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), 5(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), 5(min), 7d(min)

18edo oneiro theory

Favorite modes: Mnionian (31313331), Mnaeolian (31313133), Ulphrygian (13331313), Celdorian (31333131), Sardorian (13133313)

The biggest difference from 13edo oneiro is that the P0ms-M2ms-m4ms chord (0 400 667) is now dissonant.

A progression on ascending 18edo Mnaeolian

21edo oneiro theory

Favorite modes: Dylathian (33233232), Sarn (23233233)

21edo has the soft oneirotonic (5L 3s) mos with generator 8\21; in addition to the naiadics (457.14¢) and extremely sharp fifths (742.85¢) that generate it, it has neutral thirds (instead of major thirds as in 13edo oneirotonic), neogothic minor thirds, and meantone-like diatonic semitones. The 4-steps (4-step intervals) are more tritone-like than fifth-like, unlike in 13edo, although they do have a consonant, even JI-like quality to them. In terms of JI, it mainly approximates 16:23:30 (R-Lmos5-L7ms). Importantly, the sharp fifth is now harmonically much more fifth-like than the flat fifth, unlike in 13edo and harder tunings. The main consonances are now chords using seconds and sevenths (in addition to thirds and sixths). The relatively consonant 114¢ and 1086¢ intervals may help to mask some of the more dissonant sonorities such as the naiadic and the sharp fifth, and work for chill and lo-fi styles.

  • 0-342-629: buzzy with sine-like timbres
  • 0-343-457
  • 0-171-457
  • 0-286-400
  • 0-171-457-743
  • 0-114-571
  • 0-286-571
  • 0-286-686
  • 0-457-571

Seconds, tritones and sevenths are consonances in addition to thirds and sixths. The naiadic is a semiconsonance. The sharp fifth is fifth-like but is still dissonant. The triad root-third-629 can resolve to a more consonant chord such as 0-343-457.

It is suggested to use R-2ms-mos5-6ms carefully in 21edo oneiro - it sounds more like a dissonant, unstable 12edo diminished 7th, unlike in 13edo and harder oneiro tunings where they have a more discernible (though complex) identity.

Nejis

A curated list of nejis.

  • 26:29:32:34:38:42:44:49:52 Tridecimal Dylathian
  • 26:29:30:34:38:42:44:49:52 Tridecimal Celdorian
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