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I am studying Fux's Gradus Ad Parnassum, and I've gotten to the end of the first species. Before I move on to the second, I'd like to try and write a bunch of cantus firmi that I can then harmonize, for practice. The book left me with many questions about how this might be done, so I looked at a few other books and websites. Now I'm sifting through that information (not all of which is in agreement) and trying to put together a set of guidelines. I still have questions, however, and I'm hoping you can help me with some of them.


The first deals with outlined intervals. I have read that augmented, diminished, and seventh intervals are not to be outlined, but the Lydian cantus firmus by Fux outlines an ascending minor seventh between notes 5 and 8:




Does this mean the outlined m7 is allowed, but not the M7?


Another interval allegedly forbidden in outline is the d5, but there's one in Salzer & Schachter's C Aeolian cantus (figure 1-21c) between notes 5 and 8:




Is this a mistake? Or is it just that the CF was purposely not written according to "strict" rules?


I've carefully analyzed the cantus firmi of Fux, Jeppesen, Salzer & Schachter, Schoenberg, and Schenker, and these are the only questionable outlined intervals I could find. At this point I am leaning toward formulating my rule for outlined intervals as follows: no augmented intervals, no diminished intervals, (even the d5) and no major seventh intervals, but minor seventh intervals are allowed. Does that seem reasonable? Thanks in advance!

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To make things clear, instead of modern major/minor 7-degree scales, old music used three overlapping 6-degree scales:


1.) normal hexachord: C ut, D re, E mi, F fa, G sol, A la

2.) soft hexachord: F ut, G re, A mi, Bb fa, C sol, D la

3.) hard hexachord: G ut, A re, B mi , C fa, D sol, E la


"Mi contra fa" was considered bad, and usually avoided between different hexachords, both melodically and harmonically: E mi contra Bb fa, B mi contra F fa, B mi contra Bb fa. (Note that C, D, G and A exist in all hexachords.)

E-Bb and B-F are diminished 5ths (or augmented 4ths) while Bb-B is an augmented prime.

So while they were bad intervals, I think 7th was not, especially when using slow (long) notes.

Here is an example of a Ionian cantus firmus outlining a major 7th (lower notes in capital):

C, G, A, G, F, G, c, e, d, c


About your second cantus... In later music (~1600-) diminished intervals and 7-degree scales became more and more common, so later theorists usually allow those intervals. They sound better in downward direction, and can be resolved by a rising semitone, as in Salzer's Aeolian cantus. Anyway, "strict rules" are very arbitrary and differs from theorist to theorist.


Good luck!




PS.: Please use cut time for Fux's examples, otherwise 2nd species won't work (no weak beats).


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