What did Schoenberg mean with this?

Question

A couple of years ago I decided to formalize my harmony studies (I'm a mathematician) by reading Schoenberg's book Theory of Harmony, however I left it since it felt a little too difficult for my current knowledge. Since then, I've been teaching myself harmony from different books and a couple of weeks ago decided to give a second try to this book, and even though I can follow the ideas in a fairly fast way now I seem to be loosing something because I still can't understand the last sentence of the next paragraph (I also added the examples quoted in the paragraph just to give it more context):

Context:

So far, Schoenberg has stated several (mostly temporary) rules in order to introduce a scheme which allows to link chords in a way that respects the nature of the harmonic series, however in this paragraph he states that in the minor mode one can not resolve (using his temporary rules) I7 to any triad which contain a raised interval (that is IV# and V#) because "g cannot go to f#". This claim seems a little odd to me since the rules that I can recall and that are related to this are the following:

1. The seventh degree must go down a step, and
2. The tonic must go up a fourth.

In this scenario the reasonable resolution should be I7 - IV, where IV should be major (and thus moving g, the seventh of a, to f#, the major third of d). With that said my question would be: What is the "rule" that does not allows us to use this cadence to resolve I7?

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Disclaimers:

1. So far, resolutions have taken place by means of diatonic triads in the corresponding mode, so if that where a rule I'd understand that resolutions in general wouldn't be possible by means of any triad with a raised interval (since they do not belong to the diatonic triads of the minor mode), however in the same paragraph they state the resolution II7 - V, where V is major, which contradicts the latter reasoning (unless V being major where some sort of exception to the rule, which would actually make a lot of sense, since, well, it is the dominant chord of the minor mode).
2. For the people that reads this question but have not read the book and does not understand the notation, it's because I'm using the notation of the book, which is pretty different from the one in most jazz harmony books I've seen so far.

Answer 1

Why g can't move to f#

Schoenberg follows the rules of the melodic minor scale, earlier in the chapter explaining this as the primary melodic evolution of the use of the minor mode. The rules for resolution of raised and unraised pitches are laid out in his four laws of the pivot tones.

It's worth noting, although Schoenberg doesn't mention it, that the movement between raised six and unraised seven is the melodic motion that distinguishes dorian from minor.

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Detailed explanation

All references are to Arnold Schoenberg, Theory of Harmony University of California Press, 1983) (Google Books, accessed 5 Jan 2022)

So far, Schoenberg has stated several (mostly temporary) rules in order to introduce a scheme which allows to link chords in a way that respects the nature of the harmonic series

This is true for major, but not for minor.

The minor mode is thus purely synthetic, a product of art, and attempts to represent it as something given in nature [i.e., the harmonic series] are pointless. (page 95)

Regarding Disclaimer #1

One can see in Schoenberg's minor-mode examples of diatonic triads (e.g., page 99, example 41) and seventh chords (page 108, example 59), that he does admit as chords both with raised sixth and/or seventh scale degrees.

However, earlier in the chapter, Schoenberg makes clear that the sixth and seventh degrees must be used in pairs — either both raised or both not raised — when proceeding from one of those scale degrees to the other, effectively restricting the melodic progression of chord tones to the melodic minor scale (see pp. 97–98). This is formally laid out in his four laws of the pivot tones (page 98).

First pivot tone, g#: g# must go to a; for g# is only used for the sake of the leading tone progression. Under no circumstances may g or f follow g#, nor may g# go to f# (at least for now).

Second pivot tone, f#: f# must go to g#, for it appears only for the sake of the g#. Under no circumstances may g follow [f#], nor, of course [may] f.

Third pivot tone, g: g must go to f, because it belongs to the descending form of the scale. Neither f# nor g# may follow it.

Fourth pivot tone, f: f must go to e, because it belongs to the descending form of the scale. f# may not follow it [nor may g#, which is explained in the earlier passages on pages 97 and 98].

Thus, at this point in the book moving between f and g# or between f# and g are both forbidden both ascending and descending. A chord in which g is the seventh, it may not resolve to f#.