The defintion of melody I outlined was fairly wide in one sense.

http://www.truthagainsttheworld.com/essays/melody

It allows for a great deal of leeway in the virtually limitless ways in which concrete elements (which notes, rhythms, etc.) can be applied in creating a melody. In some ways, the qualification of “perceivable as a self-contained entity” is dependent upon the ability of any given individual to identify it as such on their own. It is very elastic but not infintely so. The parameters would have to be limited by the epistemological limit of the human mind. Thus we may need more specialized help in defining that aspect.

By saying “healthy variety”, I mean at a minimum it should have some variety. It should not have only one note, or only step wise motion, or only skip motion. It should not have only one rhythmic element; it should have at least two different rhythmic values present. Again, there must be a limit to amount of variety. Too much would make it disintegrate into unintelligability. I don’t know where that limit is, a serious philosopher working with a physiologist might be able to idenfity this.

The central (in my estimation) & thus most important characteristic of a melody would have to be that it implies harmonic motion & as a consequence is goal-directed. The easiest (most direct) way to accomplish this is in the form of the “question/answer phrases”. This is where I suggest the use of half-cadence & full candence. A melody does not necessarily have to contain these explicitly; they can be implied. In many of Beethoven’s pieces, for example, he presents half-cadence after half-cadence moving from key to key. Thus, he prolongs the resolution only finally providing a full candence at the end of the piece. The point is that there should be at minimum at least two chords. Implying the tonic & the dominant chords is the most direct manner of implying motion.

Since can be difficult to discuss without reference to actual music I have compiled several examples.

Consider the first part of Beethoven’s “Ode to Joy” theme as a complete unit:

Beethoven’s “Ode To Joy” Theme

Now let’s remove all variation of rhythm making all the notes of the same rhythmic value:

Beethoven’s “Ode To Joy” Theme With No Rhythmic Variation

Notice that melodically it still contains harmonic goal-directed motion. But making all the half notes quarter notes results in a kind of loopy effect & makes it less effective. It is still a melody but variation of rhythm enhances one’s ability to perceive it as a “unit” that starts, goes somewhere & definitively ends. Without rhythmic variation it is not only monotonous but actually more difficult to “hear” the melody. I have read studies that suggest the evolution of melody is in some ways a result of mimicing speech patterns.  I find that a plausible premise.

Next, let’s remove the goal directed component harmony. In it’s bare-bones presentation there are only two chords referenced in the melody – the tonic (the “one” chord being a chord built on the first note of the key/scale) & the dominant (the “five” chord being built on the fifth). Beethoven actually uses more chords than this in harmonizing the melody, but this is a simplified version for easier analysis. What you will hear in the next example is the only the notes of the melody that are part of the tonic chord; if a note was part of the dominant chord I did not include it. By doing this it is possible to hear what I mean about goal-directed motion being provided by harmonic changes & motion. No harmonic variety equals no goal-directed motion.

Beethoven’s “Ode To Joy” Theme Without Implied Harmony

It possible for your mind to “insert” (imagine) the necessary harmonic background though even when listening to this example. So, to go a step further, this example simply bangs away on the tonic chord under the melody making it harder to imagine harmonic motion:

Beethoven’s “Ode To Joy” Theme Without Any Harmony

Notice this could conceivably be the beginning statement of a melody that then went to another chord. Also, notice this could be considered a “theme” of sorts. It could be a melodic fragment that will be used in various ways throughout a larger composition. On it’s own, even though it is pleasant, it doesn’t really do much or “go anywhere”.

Also consider the idea of the half cadence/full cadence. First the melody with only half cadences:

Beethoven’s “Ode To Joy” Theme With Only Half Cadence

Notice that is sounds as if stuck in a loop, endlessly open with no definite end.

Now with only full cadences:

Beethoven’s “Ode To Joy” Theme With Only Full Cadence

Notice it definitely ends. But without the half cadence first the ending is not fully prepared (or “set up”). It sounds like a melody on it’s own but this is due to the fact that there is some harmonic motion going on in the phrase to begin with.

The important point here is that certain notes of scale/key imply certain chords and as a result harmonic motion. Implying the right note (at the right time) is crucial in giving the melody a sense of direction. Obviously, by “right note” I do not mean there is only one option or choice available at any given time. Sometimes the choices are virtually limitless; other times, though, they are. It depends what the composer wants to accomplish. An extreme example would be at the end of a piece; if you want to end a piece definitively your only melodic option is the root note.

On the subject of tempo I have the melody played very fast:

Beethoven’s “Ode To Joy” Theme Too Fast

Since you know it is supposed to be the “Ode to Joy” you might actually be able to identify it after a few listens. So let’s make it absurd:

Beethoven’s “Ode To Joy” Theme Way Too Fast

I am not aware of any pieces that move faster than eighth notes at 300 beats per minute (bpm) that are discernable. Most of the faster pieces I have ever heard top out at around 250 bpm. The absurd example is 32nd note triplets at 250bpm. This works out to about 50 notes per second. I can play about 12 notes per second in a linear pattern & about 30 notes per second in arpeggiated patterns (sweeping technique!). But if the info is not relatively simple it gets hard to identify. If I am just repeating a single scale or chord it’s easy. If it’s more complex harmonically it’s harder to discern. For example, listen to any of Coltrane’s “sheets of sound” solos, Allan Holdsworth or Shawn Lane guitar solos.

What I have tried to show here is that many different elements are responsible for a melody qua melody; not just one. In my estimation the goal-directed harmonic implication of any given musical line is the most important, but even this is still dependent upon the other elements to give a melody substance.