The tritone is one of the more important items in music. It is the fundamental part of the dominant. The tritone interval is made up of three steps, hence ‘tri-tone’; C-D, D-E, and E-F#. Therefore, C-F# is a tritone, also indicated as an ‘augmented 4th’ (#4). The inversion of this tritone is, F#-C, a diminished 5th (b5).
The following example is from the Bach Kleine Praeludium, #2 in ‘C’ major. The first measure start with a ‘C’ major triad. The major 3rd ‘E’ is its characteristic interval. A minor a 7th, ‘B-flat’ appears on the fourth beat. This creates a dominant (x) harmony. It is not ‘secondary’. The third measure shows another dominant (x) over a ‘C’ pedal. The functional progression of the first four measures is I-IV-V-I. The fifth measure shows another dominant (x) on the super-tonic (II). Its characteristic intervals are the major 3rd ‘F-sharp’, and the minor 7th ‘C’. It is again, not ‘secondary’ to anything. It is what it is, and where it is. The super-tonic (II) moves on to ‘V’, which is major.
The tritone is vital for the recognition of dominants without a root (ox). The following is taken from the Chopin Nocturne Op 9 nr 2 in Eb major. The second beat of the first measure shows a dominant minor 9th without a root (oxm9) The ‘o’ indicates zero (no) root, and ‘m9’ is the fourth note of the chord. The tritone is ‘D’ to ‘A-flat’, the major 3rd and minor 7th of the dominant (ox m9), with its root, ‘B-flat’ missing. ‘C-flat’ is the minor 9th, over an ‘Eb’ pedal.
The following is from the Mozart sonata K 283 in G major, ‘Presto’ movement. It shows Mozart’s use of the tritone in the dominant (x) with and without a root. It is total genius, and far ahead of his time.
The dominant (x) on the 3rd beat of measure 245 contains the tritone in the lower stave, with the root ‘B’ on the upper stave. III-VI is normal Circle progression. The dominant (x) on the 3rd beat of measure 47 contains the tritone in the lower stave with the root in the upper stave on the sub-mediant, VI. This progresses to the super-tonic (II) dominant (x) without a root on the second beat of measure 248, that progresses to a dominant (x) on the dominant (V) on the fourth beat, again without a root.
Measure 249 contains a tritone on the second beat that is the dominant (x) on the tonic (I) without a root that progresses to the sub-dominant (IV), a ‘C’ major chord with the 3rd as the lower-most note, that progresses to the dominant (x) on the dominant (V) in measure 250 with the root as the lower-most note and the tritione ‘F-sharp to ‘C’ in the upper stave. The ‘E’ minor chord on the sub-mediant (VI) moves (progresses) to the super-tonic (II), a minor chord with the root in the upper stave. This moves on the to dominant (V) as a major chord that progresses to ‘G’ with no other chord tone other than the 5th on the third beat.
The third beat of measure 253 contains a tritone, ‘A-sharp’ to ‘E’ that represents the augmented-sixth and major 3rd of the dominant (x) on the sub-dominant (IV). This in turn progresses to a cadence on the mediant (III) in measure 254. The dominant (x) on the third beat of measure 255 contains its tritione in the upper stave. This retains the same mediant (III) as in the previous measure. ‘E’ in the lower stave anticipates the ‘E’ dominant chord (x) in measure 256. III-VI is normal Circle progression.
The tritone on the second beat in measure 256 completes the dominant (x) on ‘E’, the sub-mediant. The third beat contains a dominant (x) on the super-tonic (II) with the tritone in the upper stave, and the root in the lower stave. This progresses to the dominant (V) in measure 257 with the complimentary tritone on the second beat completing the dominant (x). The remaining chords are self-explanatory.
The above is but a small snippet within this ‘presto’ movement. There is no greater example of Mozart’s complete command of the harmonic usages of the tritone, and dominants with and without a root. Please note that the chord identities, M, x, and m, are indicated between the staves and chord functions (positions) below the staves, I, II, V, VI, etc.
As one may see, the chord progressions are fairly standard, VI-II-V-I, or may be step-wise, IV-III, etc. The use of lower case Roman numerals must never be used, viz., ‘ii’, ‘iii’, etc. as it leads to confusion and is not consistent with chords that are note minor, viz., ‘vii’. In addition, the theory of the ‘secondary dominant’ is rendered useless since ‘x’ may be used as the identifier of the dominant quality wherever it may be found, therefore the term ‘secondary’ may be eliminated as also useless. In addition, ‘V’ is not the ‘primary’ dominant (x) in a minor key, ‘VII’ is. However, ‘primary’ and ‘secondary’ are irrelevant.
Ralph Carroll Hedges, B.Ed., B.Mus., M.M.
 Any augmented interval inverts to a diminished interval, and v.v. – good to remember! As they are spelled differently, they are ‘enharmonic’, i.e. the same keys on the piano, therefore the same distance; a tritone.
 The theory of the ‘missing root’ (sum and difference tone) may be found in Helmholtz, ‘On the Sensations of Tone’ pg 152 ff, and Ulehla, ‘Contemporary Harmony’ pg 114 ff, and Giuseppi Tartini, “Trattato di musica secondo la vera scienza dell’armonia'” (Padua, 1754), and on the web, (see ‘Tartini’s tone’ in Wikipedia), also see, ‘sum and difference tones’. No current theory text includes this information. The indication of the dominant identity, ‘x’ is not used. In addition, there is no indication for the missing root, therefore it is included here as ‘o’ in front of ‘x’: ‘ox’.
 The use of ‘figured bass’ is useless in this movement. Figured bass was developed during the Baroque period as a ‘short-hand’ for the keyboardist in ‘realizing’ an accompaniment for the Baroque orchestra. Composers did not combine the figures with Roman numerals, as is the erroneous practice of current theorists. In place of figured bass simply indicate what part of the chord is the lower-most note, e.g., ‘the third’, ‘the seventh’, etc.
 See the Chopin Nocturne op 55 nr 1 in F minor. The second chord is a normal dominant (x) on the leading-tone (VII).