Secondary dominants are those chordal indications concocted by theorist who do not accept a unique identifier for the dominant. Therefore, in cases where a dominant occurs somewhere besides the fifth note of a scale, a ‘theory’ must be devised to account for its missing identifier. The term ‘dominant’ is defined in dictionaries and music dictionaries as a note or chord on the fifth note of a scale. However, there should be a second contrasting definition, that of a chord identity with characteristic intervals of a major 3rd and a minor 7th with an identifier, the letter ‘x’. That second definition is nowhere to be found in any dictionary or music dictionary except the New Music Dictionary Encyclopedia, anmde.com. ‘x’ is used by several writers but without explanation or definition. It conflicts with no other concepts in music, so is a valid, usable symbol for the chord’s identity. A dominant on the super-tonic for example indicated as a ‘secondary dominant’ is written ‘V7/V’, where ‘V7’ is the identity, and ‘V’ is ‘tonicized’ so that there is a kind of ‘V-I’ relationship between the ‘V7’ and the tonicized ‘V’. Using the symbol ‘x’, a dominant on the super tonic may be written, ‘II’ and ‘x’; ‘II’ being the function of the chord and ‘x’ being its identifier. It’s simple and direct.
Secondary leading-tone dominants are dominants with missing roots. Traditional theory apparently does not subscribe to the theory of the missing root, so diminished or half-diminished chords found on other than the leading-tone are described as ‘secondary leading-tone chords’. The theory of the missing root states that a minor 3rd will produce a ‘sympathetic’ interval two octaves and a major 3rd below. ‘F-sharp’ to ‘A’, a minor 3rd will produce tone, ‘D’ making a ‘D’ major triad without its written root. A diminished-seventh chord on ‘F-sharp’; F#–A–C–Eb then may be indicated as a dominant (x) on the super-tonic ‘D’ in ‘C’ major. Both the diminished and half-diminished chords begin with a minor 3rd with the missing (phantom) root two octaves and a major 3rd below. In a minor key, however, the super-tonic (II) may be either diminished as a triad, or as a half-diminished chord, normally. Note well however that any chord of any identity may be used on any chord within any scale. For example the super-tonic in a major key may be dominant (x), minor, half-diminished, etc. And, there are chords that suggest or imply an identity without the required characteristic interval, including the missing root. A chord with a missing root may have its root appear in the next beat or the next measure, or not at all. The theory then, may be called ‘implied roots’, ‘implied dominants’, or ‘implied chords’.
Ralph Carroll Hedges, B.Ed., B.Mus., M.M.