Scale formulas such as the following

are a convenient means of viewing differences of melodic structure between scales, arpeggios and modes. The main point to keep in mind is that these formulas are related to the *major* scale and not the scale which you are trying to construct. For example, the above formula is for the natural minor scale. However, applying the formula to a minor scale, in this case A minor, would produce the following notes:

clearly not the desired note sequence. Applying the formula to A *major *

produces the correct sequence of notes (notice all the sharps are lowered, becoming naturals). If we wanted to find the notes of D natural minor we would apply the formula to the D major scale which would produce the following notes

Notice that the B of the D major scale is now lowered to give B flat. Using formulas in this way tells us that the natural minor scale is the same as the major scale except the third, sixth and seventh degrees are lowered.

The formula for the Harmonic minor scale is:

Which, when applied to the A major and D major scales respectively, would produce the following notes

A – B – C – D – E – F – G sharp, and D – E – F – G – A – B flat – C sharp. Because the seventh degree of the harmonic minor scale’s formula has no accidental, means the seventh degree is retained as the raised seventh degree of the harmonic minor scale.

The formula for the ascending melodic minor scale would be:

Notice, when it is applied to the A major scale, both the sixth and seventh degrees of the major scale are retained, A – B – C – D – E – F sharp – G sharp, while the third of the major scale is lowered.

**Arpeggios**

Formulas are also used for arpeggios. For example, to produce the notes for a minor ninth arpeggio, the formula is

which, when applied to an A major scale

produces the correct notes for that arpeggio.

The formula for a half diminished seventh arpeggio would be

while the formula for a diminished seventh arpeggio would be

if this formula is applied to the D major scale we would get the following notes

D – F – A flat – C flat. Remember that the double flat sign means a note is flattened twice. So, a sharpened note becomes a flat:

As mentioned previously, the main reason for using scale formulas is to show differences in the structure of scales, arpeggios and modes. In the previous example we know by looking at the formula that the diminished seventh has the same notes as the half diminished seventh except the seventh in the diminished seventh is a double flat. Of course the same knowledge can be acquired simply by learning the intervallic relationships of the various scales, arpeggios and modes. Scale formulas can, however, be useful for quickly explaining the structure of less well known scales.

**Exotic Scales**

Formulas can be very handy when using scales which are not so common. For example, the Hirajoshi scale’s formula

can be used to construct the scale on different root notes,

A – B – C – E – F

D – E – F – A – B flat

or to compare its structure with the structure of a natural minor scale,

or, harmonic minor scale

In this way we can very quickly see that the Hirajoshi scale has the same structure as the natural minor and harmonic minor scales but lacks the seventh degree.

**Modes**

Formulas for modes are used in the same way as for other scales and arpeggios. For example, the formula for the Mixolydian mode is

which tells us that the Mixolydian mode is the same as a major scale but the seventh degree is lowered. Applying this to a G major scale would give us the notes G – A – B – C – D – E – F.

The Dorian mode’s formula

tells us that the Dorian mode is the same as the major scale except the third and seventh degrees are lowered. Applying this to a C major scale would give us the notes C – D – E flat – F – G – A – B flat.

The formulas of the remaining modes of the major scale are

Phrygian

Lydian

Locrian

Ionian has the same formula as a major scale while Aeolian has the same formula as the natural minor scale. The formulas for modes of the minor scales will be discussed at another time.