Schumann posthumous variation V's choice on descending the semitone or whole tone

Question

Here is the starting part of Schumann posthumous variation V,

I am interested in knowing the "logic" or the "reasoning" why and how Schumann chose the descending tones of the following middle melody (marked in blue and green):

What the middle melody shows in the blue color has:

flat B -> A -> flat A -> flat G -> flat G -> F -> E -> F, (Melody 1)

which descends by

semitone, semitone, whole tone, stay same tone, semitone, semitone, ascend semitone back to F. (1)

What the middle melody shows in the green color has:

F -> flat E -> flat E -> flat D -> flat D -> C -> B -> C, (Melody 2)

which descends by

whole tone, stay same tone, whole tone, stay same tone, semitone, semitone, ascend semitone back to F. (2)

My question is, for the purpose of making a more symmetric and perhaps also mystic sound, why not Schumann chose in the middle melody shows in the green color as:

F -> E -> flat E -> flat D -> flat D -> C -> B -> C, (Melody 3)

which descends by

semitone, semitone, whole tone, stay same tone, semitone, semitone, ascend semitone back to F. (3)

In comparison, the choice of melody (1) and (3) seems more natural, more natural (in my opinion) than the choice of Schumann's choice on the melody (1) and (2). Could we give a good reasoning/explanation behind?

Answer 1

TL;DR

1. Measures 1–4 comprise two, 2-bar segments.
2. • Within each 2-bar segment, the final melodic note of each harmony is repeated as the first note of the following harmony. Thus, the Eb (third note of bar 4) must be preceded by an Eb (second note of bar 4).
   • For every harmony, the melody contains at least one chord tone. Thus, since the first note of bar 4 (F) is not a chord tone, the second note must be (as the chord changes with the third melody note).

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Both the above and below explanations are summarized in the below graphic.

Legend

• Harmonic groupings are indicated by color.
• Chord tones within a harmonic grouping have full-size note heads.
• Repeated notes across harmonic boundaries are indicated by dotted slurs.

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Reframe the question

The question boils down to why, on measure 4's second melodic note, did Schumann write an Eb rather than an Fb?

Built into this question is the tacit assumption that the important element is the interval between the first and second pitches in each measure. (Measure three begins with a half step. Why doesn't measure four?) However, the problem is best answered by reframing the question as why did Schumann not immediately precede measure 4's third melodic note with a note a step above?

Melodic breakdown

Beginning in measure 1, consider each sequence of four melodic pitches, with particular attention to the second and third of each group (in bold).

measure/beats	pitches	beat 2–3 interval
m. 1, b. 1–2	Bb — Ab — Gb — F	Descending whole step
m. 1, b. 3–4	Eb — Db — C — Cb	Descending half step
m. 2, b. 1–2	Cb — Bb — A — Bb	Descending half step
m. 2, b. 3–4	Bb — Ab — G — Ab	Descending half step
m. 3, b. 1–2	Bb — Bbb — Ab — Gb	Descending half step
m. 3, b. 3–4	Gb — F — E — F	Descending half step
m. 4, b. 1–2	F — Eb — Eb - Db	Unison
m. 4, b. 3–4	Db — C — B — C	Descending half step

It's clear that in all but one case, there is a descending step from the second to third pitch. However, this reveals the presence of two intervallic outliers: the measure 4 unison already in question, but also the measure 1 descending whole step. So now the question becomes why are two intervallic relationships different from all the others?

Harmonic breakdown

Next, consider the harmonic structure of the first four measures:

measure/beats	chord	RNA
m. 1, b. 1–4	Db	I
m. 2, b. 1–2	Gb	IV
m. 2, b. 3–4	Db	I
m. 3, b. 1–2	Ab7	V7
m. 3, b. 3–4	Bb-7	ii7/V
m. 4, b. 1	Ab	C[6-4]/V
m. 4, b. 2	Eb7	V7/V
m. 4, b. 3–4	Ab7	V7

Notice the harmonic rhythm: 4 beats, 2 beats, 2 beats || 2 beats, 1 beat, 1 beat, 2 beats. The pipes denote the division of the two, 2-bar segments comprising the four bars being considered.

All harmonies last two beats except the first measure and the first two beats of the fourth measure — corresponding exactly to the two intervallic outliers. Nearly the entire first measure is devoted to a descending scale (Db major/Bb minor) over a single harmony (Db major), and the melodic notes stick to that scale; thus, the descending whole step. However, nearly every time there is a chord change, Schumann ends the first chord and begins the second with a unison melody note.

The "nearly" points up another seeming anomaly: why do the chord transitions all involve unison melody notes except from measure 2 into measure 3?

This is due to the internal structure of the phrase, shown above. Measure 2 is the end of one motivic segment, and Measure 3 is the beginning of a new motivic segment. This is evidenced, in fact, by the ascending step between the two measures, which returns the music in m. 3 to the same pitch(-class) as started the first measure (Bb).

Another reason

For every harmony, its accompanying melodic segment contains at least one chord tone. Since measure 4's beat 1 and beat 2 comprise different harmonies, either the first or second notes of beat 1's melody must be a chord tone. Since the first note is F — a non-chord tone — the second must be. Thus, the second note can't be Fb — a non-chord tone.