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If augmented fifths and minor sixths are enharmonically equivalent why does johan fux in his book study of counterpoint allow minor sixths but not augmented fifths if they sound the same? How does one distinguish between the two an augmented fifth and minor sixth when composing?

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Fux does allow them in counterpoint. As I pointed out in one of my comments, confusion comes from voice-relationships:

A minor-sixth is of course allowable between two voices because it is an imperfect consonant interval.

A minor-sixth is not allowable within the same voice because it is a leap greater than a perfect-fifth and is therefore inexcusable in strict counterpoint.

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If you're looking at Alfred Mann's translation, he's using standard notation.

This is one of those cases where using standard notation can be really helpful.

In standard notation, something is a fifth if you count 4 lines/spaces up from the starting note.

Fifth

It is a sixth interval is you count 5 lines/spaces up from the starting note.

Sixth

In the natural modes, there are no naturally occurring augmented fifths. However, there are a few natural occuring minor sixths.

In the book, fux says to avoid non-diatonic notes if possible and if you sharpen a note to get an augmented sixth chord, you are adding a non-diatonic note to the music.

In the book, tritones are harmonically forbidden by fux. He either avoid tritones, or he sharpens the fifth(or flattens to lower note), thereby making it a perfect fifth. Most of the time he uses accidentals to change a tritone to a perfect fifth, or at the end of the phrase where it is necessary to get the leading tone.

By the way: For melodies, skips should not be larger than a fifth, OR you can have skips of an octave OR you can have skips of minor sixths ONLY if its going up. I don't recall seeing a skip of greater than an octave but I'd like to know if there is. Tritone skips are also forbidden.

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