To modulate a scale as smoothly as possible, we must change only one of its notes at a time. We call this process granular modulation, and we explore it using four common scales: major, melodic minor, harmonic major, and harmonic minor. The table below lists all the possibilities. For modulations that change two or more notes, see Heptatonic Alterations.

We focus on these four scales exclusively because they have unusual properties. The major and melodic minor are the only heptatonic scales that combine semitones and whole tones, and don't contain multiple semitones in a row. Similarly, the harmonic major and harmonic minor are the only heptatonic scales that combine semitones and whole tones with a single minor third, and don't contain multiple semitones in a row. Scales that contain multiple semitones in a row are problematic because the cycle of thirds can't be applied to them without ambiguities. The four scales explored above all implement the cycle of thirds consistently, which makes them suitable for building chords in thirds, as is customary in traditional harmony.

The cycle of fifths is a well-known example of granular modulation. Sharping the fourth of the major scale modulates to the major key up a fifth, and repeating this process cycles through all twelve keys. Many other granular cadences can be constructed by selecting modulations from the table above, using various combinations of the four scales we're considering. An example is given below.

The granular modulations table was derived by analyzing all the possible alterations of the major and melodic minor scales, as shown below. Alterations that cause duplicate notes are marked N/A. In the pitch sets, T=10 and E=11.