Pitches

Figure: Position of the middle C and the concert pitch on a piano keyboard

Pitch classes and octaves

Based on the octave equivalence phenomenon, music classifies pitches using pitch classes and octaves. For instance, the pitch A4 belongs to the pitch class A and the octave 4. Pitch classes and octaves are most easily visualized when looking at a piano keyboard. A piano keyboard consists of repeating groups of keys. In each group of keys, there are 7 white and 5 black keys, a total of 12 keys. The 7 white keys produce the tones with the pitch classes C, D, E, F, G, A and B. The 5 black keys produce the tones with the pitch classes C#, D#, F#, G# and A#. The symbol # is called the sharp and tells us that, for instance, the pitch class C# (C sharp) is obtained by raising the pitch class C for a half of the distance between the pitch classes C and D. 

The next figure shows the pitch classes in one of the repeating groups of the keys on a piano keyboard.

Figure: The pitch classes in one of the repeating groups of the keys on a piano keyboard

Each repeating group of the keys on a piano keyboard produces the tones that belong to the same octave. Octaves are numbered in the form of 0, 1, 2, 3, 4, 5, 6, 7 and 8. From the next figure, we can easily calculate the number of keys on a piano keyboard. We have seven full octaves (the octaves 1 – 7), three keys from the octave 0 and one key from the octave 8. That is 7 * 12 + 3 + 1 = 88 keys.

Figure: A piano keyboard and octaves

In the next figure, we can see all the piano keys that produce the pitch class C, that is, the keys that produce the pitches C1, C2, C3, C4, C5, C6, C7 and C8.

Figure: A piano keyboard and the pitch class C

Pitches and frequencies

By definition, every pitch class contains pitches with frequencies that can be obtained from each other by multiplying or dividing them with the number 2 (the octave principle). As an example, we can easily calculate the frequencies of the pitch class A. We already said that the concert pitch is A4 and that it has the frequency of 440Hz. Therefore, A5 has double the frequency of A4, which is 440 Hz * 2 = 880 Hz. A6 has double the frequency of A5, which is 880 Hz * 2 = 1760 Hz. A3 has half the frequency of the A4, which is 440 Hz / 2 = 220 Hz. A2 has half the frequency of A3, which is 220 Hz / 2 = 110 Hz etc. 

Equidistant tones are tones whose frequencies are equally distanced. Frequencies of equidistant tones form what is mathematically called a geometric progression i.e. the frequency of the next tone is obtained by multiplying the frequency of the previous one with a constant. 

In the modern tuning system, the so-called equal temperament tuning system, 12 pitch classes equally divide an octave. Thus, the pitch classes form 12 equidistant tones and their frequencies can be obtained by multiplication with 12th root of 2 which is approximately 1.0595. As an example, we can easily calculate the frequency of A#4 from the frequency of A4, by multiplying the frequency of A4 with 12th root of 2, which is 440 Hz * 1.0595, to arrive at  approximately 466.16 Hz.

Pitch class spellings

Each pitch class has more than one spelling. Pitch class spellings are different spellings that represent the same pitch classes. Pitch class can be spelled using the sharp symbol, as we have already done for the pitch classes corresponding to the black keys on a piano. However, pitch classes are often spelled also by using the flat, double sharp and double flat symbols.

The symbol b is called the flat. It tells us that, for instance, the pitch class Db is obtained by lowering the pitch class D for a half of the distance between the pitch classes C and D. Therefore, we can also say that the black keys on a piano produce the tones with the pitch classes Db, Eb, Gb, Ab and Bb. These are the same pitch classes as before (C#, D#, F#, G# and A#) only now they are differently spelled. The symbol x is called the double sharp, and it is equivalent to applying the sharp two times. The symbol bb is called the double flat, and it is equivalent to applying the flat two times.

The next figure shows pitch class spellings. Pitch class spellings that are shown on the same piano key represent the same pitch class.

Figure: Pitch class spellings

Pitch spellings

Similarly, we can define pitch spellings as different spellings that represent same pitches. For instance, G#4 and Ab4 are two different pitch spellings that represent the same pitch.

Half step and whole step

The distance between two pitches can be measured using half steps and whole steps.

The half step is the distance between two adjacent pitches i.e. pitches of tones that are produced by pressing two adjacent piano keys. For instance, the distance between the pitches C4 and C#4 is equal to one half step. Of course, we can say the same for the distance between the pitches C4 and Db4 since Db4 and C#4 are two pitch spellings that represent the same pitch. Most of the adjacent keys on the piano form a pair of a white and black key. However, two white keys can also be adjacent therefore the distance between the pitches of their tones can be equal to one half step. For instance, the distance between the pitches E4 and F4 is equal to one half step.

The whole step is the distance that is equal to two half steps. For instance, the distance between the pitches C4 and D4 is equal to one whole step. It is the sum of the half step between the pitches C4 and C#4 and the half step between the pitches C#4 and D4.

As an example, we can determine the distance between the pitches F4 and A#4. By looking at the piano keyboard, it is obvious that the distance is equal to two whole steps and one half step (or five half steps).