5L 4s/Inthar's approach

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This page describes my approaches to the semiquartal (5L4s) mos or generator framework, firstly a contrapuntal approach and secondly based on a fictional tradition inspired by world musics such as maqam.


Features of hard-of-basic (basic to superhard) semiquartal counterpoint:

  1. The main difficulty is that melodic motions and timings are very unlike what we're used to in heptatonic counterpoint (stepwise motion doesn't end up where we expect it to).
  2. Fortunately, semiquartal has plenty of small steps and relatively dissonant intervals such as the supermajor third, which assists with melodic movement.
  3. 5L 4s counterpoint is also easier than in mosses such as 5L 3s and 4L 3s which do not have perfect fifths, in that consonance of basic triads is stable under inverting or revoicing.

The notation used is conventional meantone circle-of-fifths notation plus ^/v (the small step of parahard semiquartal, 1\19 in 19edo and 1\24 in 24edo), which satisfies B^ = Cv. A semifourth can be written C-D^ or C-Ebv.


These are all the intervals that occur in semiquartal. Cent values are in 19edo.

  • minor 1-step: 63c, dissonant (tends to resolve to unison)
  • major 1-step: 189c, suspended (can resolve down or up)
  • perfect 2-step: 253c, consonant
    • aka semifourth, ultraminor 3rd
  • augmented 2-step: 379c, consonant
    • aka major 3rd
  • minor 3-step: 316c, consonant
    • aka minor 3rd
  • major 3-step: 442c, dissonant (can resolve down to mosthird or up to perfect fourth)
    • aka supermajor 3rd
  • minor 4-step: 505c, suspended
    • aka perfect fourth
  • major 4-step: 633c, dissonant (tends to resolve up to perfect fifth)
  • minor 5-step (semi-)dissonant (can resolve down to perfect fourth)
  • major 5-step: 693c, consonant
    • aka perfect fifth
  • minor 6-step: 758c, dissonant (can resolve down to perfect fifth or up to moseighth)
    • aka semitenth, ultraminor 6th
  • major 6-step: 884c, consonant
    • aka major 6th
  • minor 7-step: 821c, consonant
    • aka minor 6th
  • major 7-step: 947c, consonant
    • aka semitwelfth, harmonic 7th
  • minor 8-step: 1011c, suspended (can resolve down, less tendency to resolve up)
  • major 8-step: 1137c, dissonant (tends to resolve up to octave)
  • perfect 9-step: 1200, octave

Be especially careful with naiadics or semitenths in low registers.

Contrary motion

Unison opening up to 5th

Unison opening up to 4th

How to deal with supermajor thirds

Stepwise motions

Descending stepwise motion spanning supermajor thirds (major 3-steps) are effective small resting points. Ascending stepwise motion is less stable unless you can make the context stabilize the supermajor third above the basepoint (e.g. as the tonic, or part of a supermajor triad sonority).

Machaut cadences
  • G^ C > G D, G^ C D^ > G D E^
  • A^ D > G^ D^, A^ D E^ > G^ D^ F
Supermajor to semifourth

Parallel perfect 7-steps and perfect 2-steps

7-steps work best for parallel motion within the octave. Parallel 2-steps should be voiced as tenths (11-steps) or wider voicings.

Other motions

  • Major to semifourth: D G^ B^ > D^ G C
  • Semiquartal dominant: G B^ D E^ > G C D^/E



  • C Ebv G (Ultraminor triad)
  • C Fv G (Supermajor triad)
  • C G Bbv (Ultraminor 7th chord)
  • C Fbv A (Superaugmented triad)


  • C Ebv Fv G
  • C Ebv F G
  • C Fv G Av
  • C Ebv G Av
  • C Gbv Bb (Locrian 7th): approximates 5:7:9 in 33edo semiquartal


Semiquartal enjoys pentachordality in that every mode of it consists of two conjunct or two disjunct pentachords.

2-step pentachords

  • LLSS Aug-2-Step Semiquartal Pentachord
  • LSLS Major Alternating "
  • SLLS Large-Centered "
  • LSSL Small-Centered "
  • SLSL Minor Alternating "
  • SSLL Dim-2-Step "

Add a fifth (major 5-step; 685.714c in 14edo, 694.737c in 19edo, 678.261c in 23edo) on top for strength. For a Locrian feel add a minor or diminished 5-step (600c in 14edo, 568.421c in 19edo, 626.087c in 23edo but 573.913c might be better) instead.

Neutralized pentachords (in 19edo and 28edo)

Enharmonic pentachords (in 23edo)