When the melody is composed from little pieces called melodic themes M1, M2, M3 etc and each one of them or a small number of them (e.g. M1, M2) , have the same underlying chord C1, then we have a particular simple and interesting relation between the chords C1, C2 , C3 and the melodic themes (M1, M2), (M3, M4) ,...etc. This is not the case when the melodic themes start at one chord and end to the next, that we usually call in the book, as "external melodic Bridges" . We are in the case of "internal melodic Bridges". This relation is based on the pitch translations of the melodic themes and of the chords. Actually this is also a scheme of composition of melodies based on small melodic themes (see post 9), when the chord progression is given or pre-determined.
Other translations of the melodic themes can be during the same underlying chord, and are obviously of an interval of 3rd.
Now even when we are at external melodic bridges e.g. M1 which starts at underlying chord C1 and ends in underlying chord C2, even then this homomorphism is of use! The way to make it work is to take the range of the melodic theme (usually starting and ending note as simplicial submelody) equal as interval to the interval of the roots of the underlying chords C1, C2.
Chord progressions that two successive chords are always either 1) an interval of 4th , that is successive n the wheel of 4ths 2) Relative chords where major turns to minor and vice-versa, thus roots-distance an interval of 3rd 3) Chromatic relation , in other words the roots differ by a semitone
are best chord progressions for parallel translations of melodic themes by intervals of octave, 4th-5th, 3rd and semitone.