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Myhill’s Property

I am trying to get my head around this idea in the context of the diatonic scale (diatonic collection).   First, I hope I am beginning to understand the definition, in terms of c- and d-distances (just stating that there are two possible intervals for each “generic” interval does not really explain it for me.)  Next, what is its importance? Coming from a jazz perspective, I feel it somehow exemplifies the importance/interaction of the diatonic collection and “melodic minor collection” in jazz and perhaps other music forms.

If anyone reading this has any idea what I am talking about, please comment and help me out!

 

Diatonic and Diabolic Scale Collections

The “diatonic collection” describes the set of scales possible using the interval series most commonly used in the major scale i.e. tone-tone-semitone-tone-tone-tone-semitone- (TTSTTTS).  This concept is also described in terms of “modes” of the diatonic major scale. (Ionian, dorian etc.)  This is illustrated in the figure below (acknowledgements to Timothy Johnson for the scale circle or “clock”.)

The figure shows the diatonic collection, and also what I call the “alternate collection” – I can’t find an acknowledged term for this separate collection. Variously referred to as the ascending (or jazz) melodic minor collection, or derivations of Ravel or Pomeroy scales, I think it is important to emphasize the variety of scales in the collection, so that, for example, the jazz “altered” scale is as vital a feature as is the melodic minor, just as, say, the natural minor scale is inherent to the diatonic collection.

Coming to my particular interest in these scale collections:

First to say, my discussion is limited to seven-note (heptatonic) scales, and to scales containing only tones and semitones.  Of course there are other vital note (pitch-class) series – including harmonic minor, pentatonic, and whole-tone.  But these I believe are slightly different cases, and can be said to be “more than scales”. (Perhaps a future post on this very arguable point, particularly the significance of the whole-tone scale).

This being said, the diatonic and alternate collections are the only heptatonic collections practicable in common western music.   Any other combinations of tones and semitones would be strange, involving consecutive semitones.   This finding is surprising to me, especially as I read about music set theory and the seemingly almost endless possibilities of scales/series (e.g. these collections are called 7-35 and 7-34 in Forte’s classification).  That is why I call the TSTSTTT collection the “alternate collection”.

In future posts I will discuss the interesting differences between the diatonic and alternate collections, both theoretical (“evenness”, Myhill’s property etc.) and perhaps more practical, in terms of tritones, fifths, derived triads etc.