Modulation: moving from one key area to another key area; CM to FM, EbM to GM, for two such examples. The process of doing this is the subject of this article. First of all, avoid indications of modulations whenever possible. If there are several measures of a new key, then a modulation may be in order. However, if there are only a few chords that seem to be in a different key, avoid the indication of a modulation. The example below is from the JS Bach ‘Kleine Praeludium’ nr 2 in ‘C’ major. The first measure is on the tonic (I). When a ‘B-flat’ is added, it becomes a dominant (x) on the tonic (I). ‘C’ may move to ‘F’ in the following measure, but it is short lived. An indication of a modulation is not appropriate. Also, an indication of a ‘secondary dominant’ is completely inappropriate. When one adopts ‘x’ as the symbol for the dominant identity, a dominant (x) on the tonic (I) works well. The tonic moves to the sub-dominant in the second measure. There are no conflicts.
The following is from the same Bach Preludes. Nr 3 in ‘C’ minor shows a true modulation after the tenth measure. The Prelude begins in the key of ‘C’ minor, and continues in the key of ‘C’ minor for the first ten bars. However, at the eleventh measure it modulates to the key of the Mediant (III), ‘G’ minor, and stays in the key of ‘G’ minor for the remaining of the Prelude. The symbol, ‘ox’ indicates a dominant (x) without a root (o). ‘F-sharp’ is the major 3rd, and ‘C’ is the minor 7th of the dominant (x) on ‘D’, the missing root. The major 3rd and minor 7th are the characteristic intervals of the dominant (x) with or without a root. ‘D’ is the dominant of the new key of ‘G’ minor. The root, ‘D’ does finally appear at the end of measure 12.
The following is from the Chopin Prelude Op 28 nr 3 in G major. Measure 17 shows a modulation to the key of the Sub-Dominant (VI), ‘C’ major with a dominant (x) on the dominant (V). The key of ‘C’ major continues for the next six measures.
Measure 23 modulates back to the key of the Tonic (I), ‘G’ major, as ‘F-sharp’ is no longer cancelled. ‘C’ is the sub-dominant of ‘G’ major. The passage is taken from the ‘C’ major scale, the sub-dominant of ‘G’ major. Since it contains an augmented 4th, ‘F-sharp’, the scale is Lydian.
As a principle stick with functional indications of the prevailing key, unless there are several measures in a new key, then a modulation might be appropriate. Always try for differing analytic solutions so you don’t get stuck in a ‘one-way’ street.
Ralph Carroll Hedges, B.Ed., B.Mus., M.M.
 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’. Diminished seventh chords such as ‘F#o’ are not functionable under any circumstance, such as a ‘#IVo’ since all intervals are of the same size; step and a half. Therefore, no one note may lay claim to a root.
 There are two definitions of ‘dominant’. They are; 1. The fifth note of any scale. 2. A unique chord quality with the major 3rd and minor 7th as its characteristic intervals, with ‘x’ as its unique symbol.