Modal union

This page is about a type of scale. For a collection of scales, see Dome.

The modal union of a periodic scale is the scale obtained by superimposing all its modes on the same tonal center. The resulting scale contains every mode of the original scale on the tonic.

The modal union of an n-tone scale has at most 2n−1 distinct tones; it has fewer tones if there are enharmonic equivalents. For example, the basic diatonic scale of 12edo has 7 tones, but its modal union only has 12 tones instead of 13, because the augmented fourth of the lydian mode is enharmonically equivalent to the diminished fifth of the the locrian mode. However, in other tunings of the diatonic scale, such as 17edo (hard diatonic) or 19edo (soft diatonic), the modal union has 13 tones, and the enharmonic notes differ by a diesis.

Examples

Soft diatonic scale (3:2 5L 2s, 19edo)

     Lydian: D  -  -  E  -  -  F# -  -  G# -  A  -  -  B  -  -  C# - (D)
     Ionian: D  -  -  E  -  -  F# -  G  -  -  A  -  -  B  -  -  C# - (D)
 Mixolydian: D  -  -  E  -  -  F# -  G  -  -  A  -  -  B  -  C  -  - (D)
     Dorian: D  -  -  E  -  F  -  -  G  -  -  A  -  -  B  -  C  -  - (D)
    Aeolian: D  -  -  E  -  F  -  -  G  -  -  A  -  Bb -  -  C  -  - (D)
   Phrygian: D  -  Eb -  -  F  -  -  G  -  -  A  -  Bb -  -  C  -  - (D)
    Locrian: D  -  Eb -  -  F  -  Gb -  -  -  A  -  Bb -  -  C  -  - (D)

Modal union: D  -  Eb E  -  F  F# Gb G  G# -  A  -  Bb B  -  C  C# - (D)

Hard archeotonic scale (3:1 6L 1s, 19edo)

    Ryonian: J  -  -  K  -  -  L  -  -  M  -  -  N& -  -  O& -  -  P&(J)
 Karakalian: J  -  -  K  -  -  L  -  -  M  -  -  N& -  -  O& P  -  - (J)
   Lobonian: J  -  -  K  -  -  L  -  -  M  -  -  N& O  -  -  P  -  - (J)
Horthathian: J  -  -  K  -  -  L  -  -  M  N  -  -  O  -  -  P  -  - (J)
  Oukranian: J  -  -  K  -  -  L  M@ -  -  N  -  -  O  -  -  P  -  - (J)
  Tamashian: J  -  -  K  L@ -  -  M@ -  -  N  -  -  O  -  -  P  -  - (J)
Zo-Kalarian: J  K@ -  -  L@ -  -  M@ -  -  N  -  -  O  -  -  P  -  - (J)

Modal union: J  K@ -  K  L@ -  L  M@ -  M  N  -  N& O  -  O& P  -  P&(J)

Basic left-handed diasem scale (3:2:1 5L 2M 2s, 21edo)

        LH Lydian: J  -  -  K  -  -  L& M  -  -  N^ -  O  -  -  P& Q  -  -  R^ - (J)
        LH Ionian: J  -  -  K  L  -  -  M  -  N  -  -  O  -  -  P& Q  -  -  R^ - (J)
          LH Mixo: J  -  -  K  L  -  -  M  -  N  -  -  O  P  -  -  Q  -  R  -  - (J)
 LH Bright Dorian: J  K@ -  -  L  -  Mv -  -  N  -  -  O  P  -  -  Q  -  R  -  - (J)
LH Bright Aeolian: J  K@ -  -  L  -  Mv -  -  N  O@ -  -  P  -  Qv -  -  R  -  - (J)

   LH Dark Dorian: J  -  -  K  -  L^ -  -  M^ -  -  N& O  -  -  P& -  Q^ -  -  R&(J)
  LH Dark Aeolian: J  -  -  K  -  L^ -  -  M^ N  -  -  O  -  P^ -  -  Q^ -  -  R&(J)
      LH Phrygian: J  -  Kv -  -  L^ -  -  M^ N  -  -  O  -  P^ -  -  Q^ R  -  - (J)
       LH Locrian: J  -  Kv -  -  L^ Mv -  -  N  -  Ov -  -  P^ -  -  Q^ R  -  - (J)

      Modal union: J  K@ Kv K  L  L^ Mv M  M^ N  N^ Ov O  P  P^ Qv Q  Q^ R  R^ R&(J)

& = +2\21 | ^ = +1\21 | v = −1\21 | @ = −2\21

Properties

For any MOS scale with a nL 1s step pattern in any non-basic tuning, the modal union is a child MOS scale with either an (n+1)L ns or an nL (n+1)s step pattern. The archeotonic example above shows that the modal union of 6L 1s is 6L 7s for a hard tuning, while it would be 7L 6s for a soft tuning.

For any n-tone MOS scale tuned as a subset of an equal tuning with fewer than 2n−1 tones per period, the modal union is the tuning itself. The case of the 12edo diatonic scale mentioned above is an example of that property. It could also be generalized to non-MOS scales with a different restriction on the number of tones for the tuning.