In Second Species Composition, how do you traverse an octave between downbeats

Question

From what I understand you can have an octave leap from one downbeat to the next in first species as long as it rebounds with a step in the opposite direction.

However, in programming second species, I can find no rule on what acceptable half notes I can have:

An escape tone (2nd-7th) would cause a leap of a 7th which is note allowed...

A change of register (Up a 9th, down a 2nd) would require a leap of greater than an octave which isn't allowed.

I don't see anything about leaps of a 6th and 3rd or a 5th and a 4th. Are those allowed?

Or are octave leaps between downbeats just not allowed in second species?

Answer 1

Assuming you mean second-species counterpoint in the 16th-century Renaissance style, the rule is generally (and here I quote from Harold Owen's Modal and Tonal Counterpoint, p. 30):

Two or more consecutive leaps in one direction are rare. When they do occur, they outline a consonant harmony, usually a major or minor triad.

Furthermore, note that leaps of sixths in this style are almost exclusively minor sixths. This means that the only possible way of traversing an octave would be by M3/m6, P4/P5, or P5/P4. As such, consecutive octave downbeats created by consecutive leaps would be very rare.

However, Owen says that:

In sacred music of this period a single pitch may be used as many as three times in succession.

Thus it might be possible to leap an octave within the course of a single measure and then repeat that upper pitch on the downbeat of the next measure before proceeding by step in the opposite direction. This would create consecutive octave downbeats, but not as a result of consecutive leaps.