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.

Granular Modulations
FromToAlteration
MajorI Harmonic majorb6
I Melodic minorb3
II Melodic minor#1
IV Majorb7
V Major#4
VI Harmonic minor#5
Melodic minorI Harmonic minorb6
I Major#3
V Harmonic major#4
bVII Majorb7
Harmonic MajorI Harmonic minorb3
I Major#6
IV Melodic minorb7
Harmonic MinorI Harmonic major#3
I Melodic minor#6
III Majorb7

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.

A Granular Cadence
ChordScale nameScale tonesAlteration
Fmaj7C MajorC D E F G A B#1
B-7b5D Melodic minorC# D E F G A B#4
Fmaj7#5A Harmonic majorC# D E F G# A B#6
Dmaj7A MajorC# D E F# G# A B#1
G#-7b5B Melodic minorC# D E F# G# A# Bb6
Dmaj7#5B Harmonic minorC# D E F# G A# Bb7
Gmaj7D MajorC# D E F# G A Bb7
Cmaj7G MajorC D E F# G A Bb7

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.

Major Scale (1 2 3 4 5 6 7)
AlterationScale nameBuilt onPitch setForte #
NoneMajorI[0,2,4,5,7,9,E]7-35
b1N/A
#1Melodic minorII[1,2,4,5,7,9,E]7-34
b2  [0,1,4,5,7,9,E]7-30B
#2Neapolitan minorIII[0,3,4,5,7,9,E]7-30A
b3Melodic minorI[0,2,3,5,7,9,E]7-34
#3N/A
b4N/A
#4MajorV[0,2,4,6,7,9,E]7-35
b5  [0,2,4,5,6,9,E]7-29B
#5Harmonic minorVI[0,2,4,5,8,9,E]7-32A
b6Harmonic majorI[0,2,4,5,7,8,E]7-32B
#6  [0,2,4,5,7,T,E]7-29A
b7MajorIV[0,2,4,5,7,9,T]7-35
#7N/A

Melodic Minor Scale (1 2 b3 4 5 6 7)
AlterationScale nameBuilt onPitch setForte #
NoneMelodic minorI[0,2,3,5,7,9,E]7-34
b1N/A
#1Neapolitan majorII[1,2,3,5,7,9,E]7-33
b2Neapolitan majorI[0,1,3,5,7,9,E]7-33
#2N/A
b3N/A
#3MajorI[0,2,4,5,7,9,E]7-35
b4  [0,2,3,4,7,9,E]7-27
#4Harmonic majorV[0,2,3,6,7,9,E]7-32B
b5  [0,2,3,5,6,9,E]7-31
#5  [0,2,3,5,8,9,E]7-31
b6Harmonic minorI[0,2,3,5,7,8,E]7-32A
#6  [0,2,3,5,7,T,E]7-27
b7MajorbVII[0,2,3,5,7,9,T]7-35
#7N/A