Scales

Major Scales

The first scale that we will discuss is the major scale. The major scale is constructed with this formula: $W$’s represent whole steps and $h$’s represent half steps.

$$ \begin{matrix} W & W & h & W & W & W & h \end{matrix} $$

Mathematical Stability

The previous pattern is the way it is because it is the only way to perfectly combine the three most fundamental and stable intervals in all of music: the Octave, the Perfect Fifth, and the Major Third.

When you strike a piano key, it produces a frequency (a sound wave). When you strike two keys, your ear perceives the relationship between their two frequencies. Consonance (a pleasing, stable sound) happens when the frequencies are in simple, whole-number ratios.

The most stable interval is the Octave ($Do$ to the next higher $Do$).

• If the first $Do$ vibrates at 100Hz, the higher $Do$ vibrates at 200 Hz.
• The 2:1 ratio is so simple that our brains perceive the two notes as fundamentally the same. The Octave is the boundary of the scale.

The second most stable interval is the Perfect Fifth ($Do$ to $Sol$).

• If $Do$ is 100Hz, $Sol$ is 150Hz.
• The 3:2 ratio is what ancient Greek mathematicians like Pythagoras used to build the basis of music theory. This interval is so stable that it is the backbone of the scale.

The third critical, stable ratio is the Major Third ($Do$ to $Mi$).

• This interval (5:4) is the crucial mathematical relationship that gives the chord its bright, “major” quality.

The Major Scale is essentially the collection of seven notes required to connect the C Major chord ($\mathbf{C}$, $\mathbf{E}$, $\mathbf{G}$) to the next C in the smoothest, most consonant way.

If you start on the Tonic (I) and insist on placing the most stable notes (III, IV, V, VIII) in their correct positions, the pattern $W-W-H-W-W-W-H$ is forced to emerge.

Note	Position in Scale	Why the Step is $\mathbf{W}$ or $\mathbf{H}$	Resulting Pattern
C	$I$ (Tonic)	The starting point.	-
D	$II$	$W$ (Whole Step) is required for smooth connection.	$\mathbf{W}$
E	$III$	Must be a Major Third (stable $5:4$ ratio). This placement forces the steps from C to E to be $W-W$.	$W-\mathbf{W}$
F	$IV$	Must be a Perfect Fourth (stable $4:3$ ratio). To maintain this highly consonant interval, the space between E and F must be the smallest possible step: a Half Step.	$W-W-\mathbf{H}$
G	$V$	Must be a Perfect Fifth (stable $3:2$ ratio). This is the strongest support for the Tonic. The remaining distance from F must be $W$.	$W-W-H-\mathbf{W}$
A	$VI$	$W$ (Whole Step)	$W-W-H-W-\mathbf{W}$
B	$VII$	$W$ (Whole Step)	$W-W-H-W-W-\mathbf{W}$
C	$VIII$	Must be the Octave (stable $2:1$ ratio). This forces the seventh note (B) to be only a Half Step away, creating the necessary tension that pulls the ear to the final resolution.	$W-W-H-W-W-W-\mathbf{H}$

$Do$ Major Scale

Let’s build a $Do$ Major Scale.

1. Our starting note will be $Do$
2. From the $Do$ we will take a whole step to $Re$
3. From the $Re$, we will take another whole step to $Mi$
4. Next, we will go up a half step to $Fa$
5. From $Fa$, the whole step will take us to $Sol$
6. Next is another whole step to $La$
7. The last whole step takes up to $Si$
8. Finally, the half step return us to $Do$

So $Do$ major is composed by:

$$ \begin{matrix} & W & & W & & h & & W & & W & & W & & h \\ Do & \Leftrightarrow & Re & \Leftrightarrow & Mi & \Leftrightarrow & Fa & \Leftrightarrow & Sol & \Leftrightarrow & La & \Leftrightarrow & Si & \Leftrightarrow & Do \\ \end{matrix} $$

$Mi_\flat$ Major Scale

1. Our starting note will be $Mi_\flat$
2. From the $Mi_\flat$ we will take a whole step to $Fa$
3. From the $Fa$, we will take another whole step to $Sol$
4. Next, we will go up a half step to $La_\flat$
5. From $La_\flat$, the whole step will take us to $Si_\flat$
6. Next is another whole step to $Do$
7. The last whole step takes up to $Re$
8. Finally, the half step return us to $Mi_\flat$

So $Mi_\flat$ major is composed by:

$$ \begin{matrix} & W & & W & & h & & W & & W & & W & & h \\ Mi_\flat & \Leftrightarrow & Fa & \Leftrightarrow & Sol & \Leftrightarrow & La_\flat & \Leftrightarrow & Si_\flat & \Leftrightarrow & Do & \Leftrightarrow & Re & \Leftrightarrow & Mi_\flat \\ \end{matrix} $$

$Re$ Major Scale

For our final scale, we will build the $Re$ Major Scale

1. Our starting note will be $Re$
2. The first whole step takes us from $Re$ to $Mi$
3. The second whole step takes us to $Fa_\sharp$
4. The half step takes us to $Sol$
5. The whole step takes us to $La$
6. From $La$, the whole step takes us to $Si$
7. From $Si$, the whole step takes us to $Do_\sharp$
8. Finally, the half step takes us back to $Re$

So $Re$ major is composed by:

$$ \begin{matrix} & W & & W & & h & & W & & W & & W & & h \\ Re & \Leftrightarrow & Mi & \Leftrightarrow & Fa_\sharp & \Leftrightarrow & Sol & \Leftrightarrow & La & \Leftrightarrow & Si & \Leftrightarrow & Do_\sharp & \Leftrightarrow & Re \\ \end{matrix} $$