Annex 2

Annex 3

10. Cyclical chord/scale classes

It would seen logical to think that the number of possible inversions of any chord must be the same as to the different number of notes it contains, or that the number of modes of a scale would also be the same as the number of notes of the mode. This is not always true on account of the internal cyclical chord/scale classes; in those cases there are some inversions or modes that coincide. For this reason the number of chords in the N5 level of equivalence (see Figure 73) is not always the same as the number of chord classes in N6 level multiplied by the number of notes. This is only true for the chord/scale classes which never have a cyclical structure, i.e. the chords/scales of 5, 7 and 11 notes.

In this annex (Table 8) there are represented all the cyclical and semi-cyclical chord/scale classes (the left number means its classification in the chord/scale Tables), that is to say, all the chords or scales that have some or all of their inversions or modes with an identical structure —concerning the internal intervals.

The modes à transpositions limitées introduced by O. Messiaen in his book Technique de mon langage musical (1944) are in fact some of these cyclical chord/scale classes. Messiaen named these modes from 1 to 7. Mode 1 is equivalent to 6-notes chord/scale class #51 (scale of tones), mode 2 is related to 8-notes chord/scale #43 (octatonic scale), mode 3 is related to 9-notes chord/scale #19, mode 4 to 8-notes chord/scale #8, mode 5 to 6-notes chord/scale #78, mode 6 to 8-notes chord/scale #30 and mode 7 to 10-notes chord/scale #3.

Messiaen named these scales as modes of limited transposition because if they are successively transposed by semitones there is always a moment where the same scale (the same notes) is found —obviously, before getting to the transported octave—. For example, if the transposition of scales of mode 2 (8-43) is made by minor third the result is always the same notes or mode 1 (6-51) by major 2nds., and so on. All this is also true for all the modes in Table 8.

Tabla 8

Annex 4