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What are the hexachordal systems discussed in The Study of Counterpoint?

I would like an explanation of them and how they relate to the modal first species exercises written out by Joseph.

The student write out exercises in modes C, D, E, F, G, A. The first species are written in modes does that mean that the diatonic system hadn't yet been introduces and accepted?

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I originally posted in the comments, but since time has passed, I think it more appropriate to be shown below:

Hexachords are chords or pitch-class sets that contain six notes.

A hexachordal system refers to exactly what you outlined: use of the first six modes of the diatonic scale - which had been established long before Fux came and wrote his text. The seventh mode (Locrian) was not used because it does not naturally produce the interval of a perfect-fifth, and therefore renders authentic cadences impossible. The scale is omitted because it is essentially too difficult to work with without considerable intervention.

I should add here that different types of hexachords were employed and that you should refer to page 31 of the text for further clarification.

Hope that helps.

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  • "and therefore renders authentic cadences impossible" The Locrian mode is missing long before authentic cadences exist. Fux states: "Quia quintam Consonantiam non habet, nec adeo tonum constituere potest".
    – lmat - Reinstate Monica
    Commented Mar 16, 2015 at 20:25
  • @Limited Atonement - The Locrian mode was also missing long before composers started thinking about writing with specific harmonic progressions.
    – jjmusicnotes
    Commented Mar 16, 2015 at 23:16

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