Scales

With the major scale there is a standard way we construct it, which is by taking some root note and then following the following pattern:

W W H W W W H

Where W stands as a whole tone (a jump of distance 2) and H stands as a half tone (a jump of distance 1). So we can re-write that as:

2 2 1 2 2 2 1

So starting on C (0*), we get the following notes:

0* (+2) 2* (+2) 4* (+1) 5* (+2) 7* (+2) 9* (+2) 11* (+1) 0*

Then instead of taking jumps with respect to the previous note, we could consider them with respect to the root of the scale and get these intervals:

0 2 4 5 7 9 11 0

Since

• 0 = 0
• 2 = 2
• 4 = 2 + 2
• 5 = 2 + 2 + 1
• 7 = 2 + 2 + 1 + 2
• 9 = 2 + 2 + 1 + 2 + 2
• 11 = 2 + 2 + 1 + 2 + 2 + 2
• 0 = 2 + 2 + 1 + 2 + 2 + 2 + 1

If we were to set our anchor note to the root of the scale, then these intervals are now anchor intervals which means that, this collection of ais defines the major scale.

So if we were to make an F major scale, then we would take 5* (F) and generate the following notes from it

• 5* + 0 = 5* (F)
• 5* + 2 = 7* (G)
• 5* + 4 = 9* (A)
• 5* + 5 = 10* (Bb)
• 5* + 7 = 0* (C)
• 5* + 9 = 2* (D)
• 5* + 11 = 4* (E)

Notice here we're adding a number to a note, which produces another note.

If we consider the minor scale, we have this pattern

W H W W H W W

In terms of numbers, that is

2 1 2 2 1 2 2

And with respect to the root note, the ai collection is:

0 2 3 5 7 8 10 0

Scale Name	AI Collection
Major	0 2 4 5 7 9 11
Minor	0 2 3 5 7 8 10
Harmonic Minor	0 2 3 5 7 8 11
Melodic Minor	0 2 3 5 7 9 11
Whole Tone Scale	0 2 4 6 8 10
Acoustic Scale	0 2 4 6 7 9 10

Additionally there many other scales which can be found here, and we will have a separate discussion about modes later on.

Something that's interesting to note is that a minor and major scale can define the same set of notes as each other. The canonical example is that of 0* major and 9* minor, where the 0* major generates the notes 0* 2* 4* 5* 7* 9* 11*, and the 9* minor defines the note collection of 9* 11* 0* 2* 4* 5* 7* which are both the same.

In general, what that's saying is that given some note x*, that x* major is the same as (x-3)* minor, which can be seen in generality as:

Major: (x + 0)*, (x + 2)*, (x + 4)*, (x + 5)*, (x + 7)*, (x + 9)*, (x + 11)*

Minor: (x - 3 + 0)*, (x - 3 + 2)*, (x - 3 + 3)*, (x - 3 + 5)*, (x - 3 + 7)*, (x - 3 + 8)*, (x - 3 + 10)*

Which simplifies down to

(x + 9)*, (x + 11)*, (x + 0)*, (x + 2)*, (x + 4)*, (x + 5)*, (x + 7)*

When re-arranged yields:

(x + 0)*, (x + 2)*, (x + 4)*, (x + 5)*, (x + 7)*, (x + 9)*, (x + 11)*

Thus it is now clear as to why these two relative interval collections which differ can generate a set of notes which do not differ.