Diagrams of modalities

Introduction

Diagrams of 12-tone system modalities and some their parts ( Greek scales tetrachords and middle age scales pentachords and hexachords).

We would like to display structure of modality in a simple way. We will take advantage of following expression:
halftone + fifth + major third = octave; i.e. 1 + 7 + 4 = 12
Fifth and major third represent extreme continuity, halftone extreme impulse.

We will write the intervals in a fictive coordinated system: the major thirds horizontally, the halftones vertically, the fifths on diagonals.

For example the Bartok's modality, i.e. modality with number 1371 and distance scheme 222121(2) should have the following diagram.

                 +
              +
        +  +  +
        +
     +

This modality has 12-transpositions, as we can see when we put the diagram on the list of note symbols:

      c   d   e   f   g   gis ais
      cis dis f   fis gis a   h
      d   e   fis g   a   ais c
       ....
      h   cis dis e   fis g   a

For example this transposition:

                  + ais
              + f
    + gis + c + e
    + g
+ d

Diagrams show us inner symmetry of modalities (natural modality, gypsy, Messiaen's and Lendvai's modalities,...). Some modalities are mutualy symmetrical (for example harmonic major and minor modality).

We can distinguish the good-ordered modalities (natural, harmonic, gipsy,..) from an inconsistent ones (whole-tone modality).

The diagrams of modalities should be also useful for displaying structures of more complicated modalities (modalities of systems with order greater then 12) or tuning systems.

Tetrachords and tetratonics

4 - tones structures
122(7) * 53* Dorian tetrachord

      +  +
      +
         +

212(7) * 45* Phrygian tetrachord

      +
      +
         +
      +

221(7) * 43* Lydian tetrachord

      +
         +
      +  +

131(7) * 51* Chromatic Mezotetrachord

      +  +
      +  +

151(5) * 195* Lendvai type modality

            +
         +
         +
      +

Pentachords and pentatonics

5 - tones structures
2122(5) * 181* Dorian pentachord

              +
           +
     +  +
     +

1222(5) * 213* Phrygian pentachord

           +   +
           +
     +  +

2221(5) * 171* Lydian pentachord

           +  +
           +
     +  +

2212(5) * 173* Ionic pentachord

              +
           +  +
        +
     +

2131(5) * 179* Semiditonic pentachord

           +
     +  +
     +  +

1312(5) * 205* Semiditonic pentachord

        +  +
        +  +
     +

2322(3) * 661* Chinese modality

                  +
               +
         +  +
      +

2312(4) * 333* Indian Hindola

          +  +
             +
       +  +

2141(4) * 355* Japan

          +  +
       +  +
       +

2411(4) * 327* Bardos modality

       +  +
       +
    +  +

2321(4) * 331* Korean type

            +
      +  +
      +
   +

1412(4) * 397* Korean type

         +
      +  +
   +  +

Hexachords and hexatonics

6 - tones structures
22122(3) * 693* Guidon hexachord

              +
        +  +
     +  +
   +

22222(2) *1365* Whole-tones modality

   +  +  +
   +  +  +

23211(3) * 663* Blues

        +
     +     +
     +  +
     +

13131(3) * 819* Augmented

               +
            +  +
         +
      +  +

22122(3) * 693* Natural type

              +
        +  +
     +  +
  +

22212(3) * 725* Natural type

12222(3) * 853* Prometheus

       +  +  +
          +
    +  +

23121(3) * 667* Indian Dipaka

       +  +
    +     +
    +  +

22221(3) * 683* Indian Sviraga

       +  +
    +
    +  +  +

Heptatonics

7 - tones structures
221221(2) *1387* Natural

           +  +
         +  +
      +  +
   +

121221(3) * 859* Harmonic minor

            +
      +  +
   +  +  +
   +

121311(3) * 871* Indian Bhairava

      +  +
   +  +  +
   +  +

121212(3) * 877* Whistle

       +  +
    +     +
    +  +
       +

222211(2) *1367* Combined

        +  +  +
        +
  +  +  +

222121(2) *1371* Bartok modality

              +
           +
     +  +  +
     +
  +

212121(3) * 731* Diminished

      +
      +  +
   +     +
   +  +
      +

113211(3) * 925* Greek chromatic

      +
  +   +   +
  +   +
  +

122121(3) * 875* Harmonic major

111213(3) * 623* Debussy

      +
      + +
    + + +
      +

112311(3) * 743* Indian type

Octatonics

8 - tones structures
2122111(2) *1455* Greek Complete

           +  +
        +  +
     +  +
  +  +

1212121(2) *1755*

     +    +
     +  +
  +     +
  +  +

2121121(2) *1467* Major-Minor

           +  +
     +  +  +
     +  +
  +

2212111(2) *1391* Spanish

              +
           +  +
        +  +  +
     +

2211111(3) * 703* Blues type

   +  +
   +  +  +
      +
   +  +

1111212(3) *1005* Blues type

  +  +
     +  +
     +  +
  +  +

1122111(3) * 943* Free constructed

        +  +
     +  +
  +  +  +
     +

1113111(3) * 975* Messiaen type

        +  +
     +  +
     +  +
  +  +

2112211(2) *1495* Messiaen type

           +
        +  +  +
        +
  +  +  +

Nonatonics

9 - tones structures
21112211(1) *1519* Blues type

     +  +
     +  +  +
        +  +
     +  +

11211211(2) *1911* Messiaen type

    +  +  +
    +  +  +
    +  +  +

21111121(2) *1531* Blues type


Music theory