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I've searched for this question online many times, but never found anything close to an answer:

Given a cantus firmus, is it always possible to do counterpoint with it?

The rules don't appear to answer the question with an easy yes, but someone may have written a mathematical/musical paper proving this statement. Does anyone know of a reference?

Later: I am currently working with Peter Westergaard's formulation of counterpoint, so those are the rules I'd be most interested in, with (for the moment) first species, two line counterpoint. If there is a reference to any formulation (Fux, etc.), I'd be interested in that, too.

  • I can imagine a simple tenor derived of e.g. where it is hard to write a satisfactory counterpoint of 2 voices (first species) . S. The „primitive“ tune of the first phrase Zion hört die Wächter singen C-E-G-G-G-G-A-G = triad + changing sixth. But what Bach has written to this tune has become one of the most famous c.p. – Albrecht Hügli Oct 18 at 6:08
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I think this question can only be answered if you know not only the constraints on species counterpoint (as the OP clarified to Westergaard's formulation), but also the constraints on what constitutes a "proper" cantus firmus.

Westergaard (I'm assuming we're talking about his tonal theory book) is working with a set of rules of species counterpoint that are designed to introduce tonal theory. Thus, there are a lot of unwritten assumed constraints on his first-species counterpoint that are basically laying the groundwork for coherent harmonic progressions (as Fux does as well). If you literally read only the rules he officially states for first species, and you assume a cantus firmus that doesn't wildly violate basic tenets for melody or bass line he has elsewhere laid out, I don't see how you could possibly fail to have a possible counterpoint line. In some cases, it may not be very musical and might not incorporate the tonal characteristics he seems to be aiming for, but in terms of strict adherence to his species constraints, I don't see how you could generate a cantus firmus that follows his melodic principles and wouldn't have a counterpoint solution. (Westergaard's constraints are a lot more lax than many authors.)

For an example of what I mean by "unmusical," one could simply imagine a cantus firmus ending with a stereotypical octave leap down, then fourth up for a cadence, but then insisting on creating a counterpoint below that CF. The result would sound absurd, but I don't think it would technically violate Westergaard's explicit constraints on first species. It would certainly violate the spirit and intent behind his method, though.

Introducing further constraints would lessen the chances of producing something unmusical but still "acceptable" to the explicit rules, but if you did so, then the reasonable constraints on a "valid" cantus firmus should probably be tightened up too, resulting in little chance of a CF with no solution.

This latter idea has been thoroughly investigated by computer programs modeling solutions since the 1950s. (A brief summary of that history, with some useful sources at the end, can be found here.) One of the most useful articles that I recall reading on this topic is David Lewin's "An Interesting Global Rule for Species Counterpoint" (In Theory Only 6/8, March 1983). Lewin's "global rule" produces more generally "musical" solutions for a given CF, but it does place heavy constraints on solutions. In general there, one needs to work backwards from the final cadence, which is the most constrained and trickiest part. Again, I think Lewin mostly presupposes a "reasonable" cantus firmus that doesn't violate general late Renaissance melodic principles. But, for example, if you tried to combine Lewin's rules with some of Westergaard's more "tonal" cantus firmi, and then insisted on doing something silly like using Westergaard's bass-line-like cantus firmus as the top melody, then of course you couldn't find a viable solution. But in that case, Westergaard's CF would violate Lewin's melodic principles to begin with, so I'm not sure that's a valid test.

So, in general, I'd say even with severely constrained species rules, as long as the cantus firmus is "reasonable" (and following the same severe constraints in its own generation), counterpoint solutions are probably available. However, to speak of mathematical proof would require a more formal statement of the rules. While the rules for generating counterpoint are usually fairly explicit in many species methods, the rules for creating a "valid" cantus firmus are often not stated as explicitly, making this question impossible to answer for many cases.

  • I've sung some actual renaissance pieces where the tenors did the octave-down-fourth-up figure while the basses sang a descending scale. Surprising, but not actually absurd. I don't know what Amy later theorist would have made of it, however. – phoog Oct 18 at 2:25
  • @phoog - Sure. I know precisely the kind of cadences you're talking about, which were even more common in the 1400s, and often involved voice-crossing. I doubt those moments you mention were only two-voice first-species, though. Were they? Following Westergaard's rules, assuming tenor line G-G-C, you'd need a bass line something like C(12th below)-E-C, which is an absurd way to end a 2-voice cadence (either tonally or in Renaissance style). Obviously there are lots of things you can get away with contrapuntally with more voices and other rhythms. – Athanasius Oct 18 at 2:35
  • Correction on previous comment: "Amy" should be "any." There was no voice crossing of the bass and the tenor, but I'm sure the tenor was crossing other voices, because, as you correctly surmise, this was not two-voice counterpoint. And, come to think of it, the last tenor interval might have been an ascending fifth (so an octave leap from a 10th above the bass to a 3rd above the bass, then jumping up a 5th as the bass descends a 2nd). I wish I could remember what piece it was; I just remember being struck by how bass-like the tenor line was. – phoog Oct 18 at 3:01
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There is some mathematical analysis about the subject. There were a few cantus fermi that could be proved not to have a satisfactory counterpoint. Generally, most melodies do admit of some type of counterpoint.

  • "Proved"? How on Earth can you they formally define "satisfactoriness"? – Kilian Foth Oct 17 at 9:28
  • In the paper, they looked at whether one could avoid all parallels and poor melodic skips. It's not that one cannot write a counterpoint that sounds good; one cannot write a counterpoint that doesn't violate certain (arbitrary) constraints. – ttw Oct 17 at 11:42
  • After I posted my original question, I realized that I'd left this unclear. Currently, I am working with Peter Westergaard's formulation, so I could specify using that to formulate 1st-species 2-line counterpoint. – Christopher Heckman Oct 17 at 22:37
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Given a cantus firmus, is it always possible to do counterpoint with it?

By the time we get to Handel, he says yes, even if his countersubjects are often mostly rests.

We'd have to move this question to mathoverflow to find someone dogged enough to find a counterexample, a c.f. so pathological that counterpoint would be impossible.

You may have to qualify what kind of canti firmi you consider relevant.

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    And mathoverflow would refer me back here. 8-) I've developed an interest in algorithmic music composition recently, and a lot of what I read/work on falls between the cracks. – Christopher Heckman Oct 18 at 6:25
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I can imagine a tenor derived from e.g. an early invocation that doesn‘t fit to a note-to-note c.p. But such practices are historical nonsense: trying to apply the rules of Fux to music that was not „written“ for polyphony.

Because:

First step to train the rules of Fux would be to compose a good c,f.

The aim of studying c.p. must be to overcome the speciesc and break out of the rules of Fux.

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