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The submediant chord is a distant pre-dominant chord. It typically moves to another pre-dominant, such as ii or IV, but sometimes goes directly to V. Movement to V is not a common progression.
Voice-leading follows all the preceding methods the discussed. When going to the ii, it is best to try to retain the common tone and move the other two voices to the nearest chord tones in the next chord. When going to the IV (or iv), it is best to try to retain both common tones and move the remaining voice to the nearest chord tones in the next chord.
Chords that commonly precede the VI are I, V and iii (III, to be discussed later). When moving from V to VI in minor, it is important to move the leading-tone up to the tonic to avoid an augmented 2nd. This leads to the VI chord having a doubled third, rather than a doubled root. This can also be done in Major, but is not necessary since there is no lowered 6th scale degree in major.
When V-vi (VI) occurs, it is called a deceptive progression (or deceptive resolution), since we expect V to go to the tonic chord. When it occurs at a cadence, it is referred to as the Deceptive Cadence. It is usually followed by an authentic cadence soon after.