Modulation through unprepared cadence 4/6 chord of any new tonality

The simplest modulation: just take the tonic (cadence) 4/6 chord of any new tonality, and reinforce it by adding dominant 7 and the tonic chord. From whatever tonality to whatever tonality you modulate in that way, it will all sound well.

This should be taught as the foundation for modulation. This exercise expands the sense of tonality, and destroys whatever mental auditory limitations the student of harmony may have before the beginning of learning.

Any minor chord can be interpreted as a subdominant in a major or minor tonality

Take any natural minor chord in the initial tonality, or minorize any major chord, and re-interpret it as a subdominant of the new tonality. Follow up with a cadence 6/4 chord, dominant, and the tonic.

You can use this technique to modulate into a major OR minor tonality.

This is based on the sensation that the subdominant gravitates better if it's minor, which sounds very natural even in the major tonality.

A minor 6-chord works really well, but it could be any other permutation.

To make this type of modulation more interesting, you can add a small third below the main tone of the minor chord (turning it into the si-re-fa-la type of 7-chord) or think of it as adding a large second at the top (making it a 5/6 chord). In that case, it becomes a second diminished 7-chord (or 5/6) in a minor tonality, or a second diminished (harmonic) 7-chord (or 5/6) in a major tonality. By "harmonic" I mean the subdominant with the lowered third.

Any minor chord can be interpreted as a second in a major tonality

There are three natural minor chords in a major tonality: second, third, and sixth. Any of these chords can be interpreted as the second. Obviously, the second, interpreted as the second, will keep you in the same tonality. But the third minor, interpreted as the second, will allow you to modulate into the major tonality one tone up from the current, and the sixth minor, interpreted as the second minor, will allow you to modulate into dominant major tonality.

So, with only the natural minors, using this technique, you can modulate to 2 tonalities.

You can also "minorize" the major chords – the tonic, the subdominant, and the dominant – and that will give you three more tonalities to modulate to: low ("napolitan") seventh major, low third major, and the major subdominant.

The total is 5 major tonalities that can be modulated directly from a major.

We can use the same technique to modulate from a minor tonality. There are three natural minor chords in a minor tonality: tonic, subdominant and dominant. The tonic, interpreted as the second in a major, allows us to modulate into the seventh natural major. The subdominant, interpreted as the minor second, allows us to modulate into the parallel major. Finally, the natural minor dominant, interpreted as the minor second, allows us to modulate into major subdominant.

Additionally, we can "minorize" the natural majors: the third, the sixth, and the seventh. This gives us the possible modulations to the low second "napolitan" major via minorized third, the low fifth (= high fourth, via "minorized" sixth), and into the sixth from the minorized natural seventh.

Additionally, we have at least two more possibilities with the "napolitan" major (low second major in both major and minor tonalities). In a major, we can take the low second major, "minorize" it, and then modulate to the seventh natural major. In a minor, we can take the low second, minorized it, and modulate into the high ("rised") seventh.

The easiest in all those cases is to use the sixth-chord.

Modulation through any minor chord or 6-chord intepreted as minor subdominant

Any minor chord or minor 6-chord could be interpreted as minor subdominant, after which it's tonic cadence 4/6 can be taken, and reinforced.

Modulation through any small minor 5/6 or 3/4 chord 1 tone lower

You can use a small minor 5/6 (polu-umenschenny; maly umenschenny 5/6) to modulate to a major or minor a tone lower than it's prima. For example: from re-fa-la-si (the 5/6 of si-re-fa-la) you can go to A major or A minor. This opens many more opportunities for modulation, because you can find many natural minor chords in any tonality and add a great second at the top, or can turn any major chord into a minor, and add a great second at the top – and then modulate. This is to be analyzed and used in composition.