Modulation

To the dominant: the submediant is the pivot — vi of the home key, ii of the new — and the first ♯ in the figures is the moment the journey becomes audible.

To the dominant from a flat key: the same pivot, and the ♯ that raises the new leading tone lands on a natural letter — B♮, one semitone up from the key’s B♭.

To the subdominant: one ♭ un-crowns the home tonic — the b7 figure turns I into the new key’s dominant seventh, and the old leading-tone voice falls.

To the subdominant from a sharp key: the b7 lowers the key’s own F♯ back to F♮ — the soprano walks G, F♯, F♮, E as the old key is let down gently.

To the relative minor: the subdominant pivots — IV of the home key, VI of the new — and the ♯ manufactures the leading tone the minor mode never owns.

The same descent into the relative minor from G major: the pivot is C, and D♯ — a sharp the signature never promised — makes E minor real.

From minor to the relative major: no new accidental at all — the ♯ simply stops coming, and the unraised seventh becomes the new dominant’s root.

The same costless slide from D minor to F major: the pivot is the minor subdominant, the new dominant needs no accidental, and the close is suddenly bright.

The Rule of the Octave carried to five sharps: the same harmonized scale, the same figures, and a hand that must now find A♯ without hesitating.

The descending Rule four flats deep: the 4/2 still arrives on the falling fourth degree, and the bass still pays the seventh it carries — in any key whatsoever.