Cyclical chord-mode 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 of different 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-mode classes; in those cases there are some inversions that coincide. For this reason the number of chords in the N5 level of equivalence (see Fig. 12) 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 chords-mode classes which never have a cyclical structure, i.e. chords of 5, 7 and 11 notes.
In this appendix (Table 5) there are represented all the cyclical and semi-cyclical chord-mode classes (the left number means its classification in the chord-mode 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 distance of tones–.
In some cases several enharmonies have been undertaken in the notation of the representative in the chord-mode class tables so as to offer a better scale clearness. For the same reasons the chromaticisms in each cycle have not been changed.
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-mode classes. Messiaen named these modes from 1 to 7. Mode 1 is equivalent to 6-notes chord-mode class #51 (scale of tones), mode 2 is related to 8-notes chord-mode #43 (octatonic scale), mode 3 is related to 9-notes chord-mode #19, mode 4 to 8-notes chord-mode #8, mode 5 to 6-notes chord-mode #78, mode 6 to 8-notes chord-mode #30 and mode 7 to 10-notes chord-mode #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 5.