Definition of a step in the (melodic) minor scale? (According to the definition in Kostka - Tonal Harmony and examples therein)

Question

The short version of my question is:

Assuming the definition of the minor scale in Kostka's Tonal Harmony textbook (Eighth edition, chapter 4), which is that:

there is, in a sense, one minor scale that has two scale steps 6 and 7, that are variable. That is, there are two versions of 6 and 7, and both versions will usually appear in a piece in the minor mode.

how is then a step vs a skip/leap defined? Is, for example, 6 to raised 6 (raised here means in comparison to natural minor) a step, or does step mean the scale degree has to change: e.g., from 6 to 7 (I assume the latter is correct [more details below])? Is "7 to raised 6" considered a step?

Some more information from the respective section (example numbers from the Kostka book are given in brackets):

In an example (Nr. 4-2) the note material is written out for the root E as: E(1),F#(2),G(3),A(4),B(5),C(6),C#(raised 6),D(7),D#(raised 7),E(1). This example is not called a 'scale' by the book, and there is a bracket above 6 and raised 6 (same for 7 and raised 7). One could thus infer that these are the options you have for scale degrees 6 and 7, rather than it being a 10 note scale (which of course would be a strange way to view it).

Now a section follows stating that:

Melodically, the most graceful thing for raised 6 and raised 7 to do is to ascend by step, whereas 6 and 7 [meaning natural minor scale degrees] tend naturally to descend by step

Then some examples are given (Nr. 4-3 and 4-4, from Bach's Well Tempered Clavier), where, e.g., the sequences "7,6,5" or "raised 7, 1 (above previous note)" occur (which is perfectly in line with the above statement). Then it is stated that

If a 6 or 7 is left by leap instead of by step, there will generally be an eventual stepwise goal for that scale degree, and the 6 and 7 will probably be raised or left unaltered according to the direction of that goal.

and then some examples (also from Bach - Well Tempered Clavier) follow that gave me some trouble:

One is the sequence "raised 7,5,raised 6, raised 7, 1 (above previous note)". It is stated that the first note (raised 7) is left by leap (which would more precisely be a skip i think?) and thus the question if the 7 gets raised or not depends on the "eventual stepwise goal". Now this is indicated to be the 1 at the end of the sequence, thus the initial 7 is raised. Enclosed in this sequence there is a "raised 6, raised 7" (going to 1(above previous note)). Why is the raised 6 not the eventual stepwise goal for the first 7 (which would thus be unaltered)? Is it because unaltered 7 to raised 6 is not a step? Does this mean that it's only a step between 6 and 7 if both are raised/unaltered?

While writing all this down things got a lot clearer, but the question on what exactly the conditions for a step are, are not completely clear to me. The book also does not give much explanation for the statements regarding this example. I, from what I tried to figure out, would assume that indeed both 6 and 7 have to be raised/unaltered the same way. This would however forbid the unaltered 7 in a sequence "7,raised 6, raised 7, 1 (above previous note)" since it is not leaving by "step" (since one is unaltered and the other one is raised) and thus would target the 1 and would be raised (because of the 1 at the end of the sequence) even though for "(raised) 7, raised 6" it is then also not a leap/skip which in the example given initiates the conditioning of the raised/unaltered option to the "eventual stepwise goal"? This seems to me, to be a special case not mentioned in the book.

The section in the book makes it clear that the mechanism explained above is not a hard rule (in fact some examples from famous pieces are given where this "rule is broken"). Nonetheless, it points to the given example and makes the statements i described above, so i assume there is some logic to it.

Answer 1

Moving from scale degree 6 to scale degree 7, or from 7 to 6, is always a "step" regardless any alterations to one or both of those notes. "Step" is a shorthand for "the interval of a second", and "skip" is shorthand for "any interval larger than a second".

Examples:

• raised 6 to flat 7 is a "minor second", thus "a step".
• raised 6 to raised 7 is a "major second", also "a step".
• flat 6 to raised 7 is an "augmented second", so, too, "a step".

The melodic minor scale is a model for how minor is sometimes used in compositions: 6 and 7 raised when the melody is ascending; 6 and 7 not raised when the melody is descending. For more on the melodic minor scale, see Why does the melodic minor scale turn into natural minor when descending?.

Answer 2

how is then a step vs a skip/leap defined?"

Not necessarily strictly. It's reasonable to call the interval between the lowered sixth degree and the raised seventh degree a leap, and it is reasonable to call it a step. It depends on the context.

Some of those steps are also half steps, and some are whole steps. Moving from the raised sixth degree to the lowered, or vice versa, is also a half step. You may have to disambiguate by indicating whether you are talking about steps on the diatonic scale or not.

But one thing you will generally never find in common practice period music is that interval in a melodic context, so the question is not really pertinent to the analysis. Any voice that has the lower sixth degree in one chord will generally resolve down by half step in the next, and any voice with the higher seventh degree will generally resolve up by half step. If they go somewhere else, it won't be to move by an augmented second.