Continuing on from my first article in this series, which can be found below, I will now show how the most simple of mathematical operations, and by operations I am referring to the basic functions in maths that make up larger, more complex concepts, can be utilised in order to understand the intricacies in even the most complex music. These are simply addition and subtraction, nothing more or less. These operations are all that are needed to understand the functions of music and indeed how numbers themselves function. But more on that later.

The previous post showed that instead of visualising a circle of fifths when modulating through the keys, the simple operation of (+1) and (-1) can be used. Now, the benefit of this mathematical approach is that not only does it apply to the Major keys and scales, but indeed all of the other 66 scales in existence too. Let me explain.

When a piece of music changes from one scale it is using to another, say from the Major scale to the Harmonic minor for example, this is accomplished by the same simple mathematical operation that I introduced the readers to in my first part of this series, namely, by the simple process of addition.

This is accomplished by the specific alteration of the note on degree 5 of the scale, the dominant, and more precisely, it is by the addition of a semitone to the root of the mode on the 5th degree, the Mixolydian. For example, in a piece in C Major, the change to the relative minor of A minor is accomplished most easily and efficiently by raising the note of G to G#. This is represented simply as:

G Mixolydian (+1) = G# Alt bb7 (mode VII of A Harmonic minor)

That is all that is needed, a single (+1) on the correct degree of the scale, in this case Mixolydian on degree V, and the alteration to the Harmonic minor on the relative minor degree VI is the result.

What about the return from using the Harmonic minor scale to the relative Major or any one of its modes? This is the exact opposite. When the music is using the Harmonic minor scale, the simplest way back to the Major scale is to flatten the seventh degree of the scale. So in A Harmonic minor that is G# flattened to G and the modes that this effects are G# Alt bb7 becomes G Mixolydian. This is shown below with a minus sign representing the subtraction of a semitone from the root of the Alt bb7 mode:

G# Alt bb7 (-1) = G Mixolydian

The Ionian mode of the Major scale is mode I, and if its root is raised by a semitone, the result is a Melodic scale on the second degree of the Major scale. C Ionian becomes C# Altered or C# Superlocrian, depending on your preference of names, which is mode VII of D Melodic which resolves to D minor, and that is shown accompanied by the simple mathematical operation of:

C Ionian (+1) = C# Altered (D Melodic mode VII)

The D Dorian mode in the key of C Major (D E F G A B C) changes to D# Alt bb37 when its root is raised, which is mode VII of E Neapolitan minor and is the quickest and most efficient way of modulating to the key or emphasizing the chord of E minor, which is shown below:

D Dorian (+1) = D# Alt bb37 (E Neapolitan minor mode VII)

The same mathematical operations of (+1) and (-1) apply to all modulations and changes between scales themselves. The place of resolution is simply a recognition of the scale that arises after the mathematical operation and then the knowledge of where it is leading to is straightforward. This becomes far more complex when the resolution is interrupted by another mathematical operation of (+1) and (-1) which then turns the music in another direction, which when continued, can create music in the style of J. S. Bach, Takemitsu, Brouwer etc or can lead into polytonal, atonal and jazz concepts. This is where the next article on this subject will begin. Thanks for reading.

For those interested in knowing more, my book The Modal Method of Music (M3) is available here:

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