Notes, chords and keys related by perfect fifth are discussed in detail in various sections of TonalityGUIDE, but major and minor thirds also play an important role - particularly in harmonic progressions.
One of the most common modulations is between relative major and minor, and this relationship is explained in the relevant section in terms of the number of notes in common between the two scales.
Looking at the third relationship as a harmonic progression, another relationship becomes clear. As can be seen below, diatonic triads a third apart differ in only one of their notes.
As discussed in the section on diatonic triads, the tonic, subdominant and dominant (I, IV and V ) are particularly important for establishing a sense of key and mode. Because they is only one note different, the two chords a third either side of I, IV and V can be substituted for them in progressions more easily than other chords.
Some theorists and analysts have suggested that I - IV - V - I is the progression that establishes a sense of key in the simplest way. The third relationship between ii and IV would in this way account for the strength of the progression I - ii - V - I, and the interrupted cadence (V - vi) could also be explained in this way.
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