Using Algebra to Find an Octave’s Pitch

Finding an Octave’s Pitch Algebraically
The one consistent interval used in music across all cultures is known as the octave. In music, the octave is known as the interval between one pitch and another that is either half or double its frequency (Hz). For example, if we were to take the pitch of middle C (262 Hz) and find its above octave equivalent we would get C5 (524 Hz). Middle C’s below octave equivalent is C3 (131 Hz). This pattern of doubling and halving the frequency of C will produce the respective octaves.

To learn more about what frequency and Hz are click here.

The concept of each successive octave’s frequency can be found through the exponential algebraic equation: y = m(2^x). The symbol “^” means “to the power of.” This means that the equation will use “2” to the power of “x”. In this case “x” is the exponent and two is known as the base. Although sometimes the symbol “^” is used to describe the relationship between the base and exponent, it is more often seen in superscript form.

Here is a diagram of the exponent in superscript form:

Diagram of the Exnent 2^4 and it's parts as well as its function

In other words, if x = 2 then it will be 2 * 2. If x = 3, it will be 2 * 2 * 2. If x = 4, it will be 2 * 2 * 2 * 2. The use of x as the

In this example, m is the lowest possible frequency that we can produce from the pitch C, which is 4.088. Or simply, m = lowest octave of pitch C. Furthermore, x is the C’s (SDN) scientific designation number (the boldened number: C-2, C-1, C0, C1, C2, etc.) plus 3. Or simply, x = ({C}SDN + 3). y is the frequency of the pitch chosen.

Here is an example:

Say I want to find the frequency of the pitch C7. (frequency = Hz)

Hz of pitch = (Hz of C-2) * (2) ^ [ (SDN of C7) + 3]
y = m*2^x
y = 4.088 * 2 ^ (7+3)
y = 4.088 * 2 ^ (10)
y = 4.088 * 1,024
y = 4186.112 Hz
And one can see the answer corresponds with the octave chart below:

Chart of pitches and their corresponding frequencies

This equation will also work for all twelve pitches in western music! One just needs to use the proper bass pitch (m) and SDN of the new pitch. For example, the musical note A’s lowest pitch is A-2 at 6.875 Hz. A-2 will be m. Then add “2” to the SDN of the A’s frequency you wish to find and you will have your answer. It is important to explain why “2” is added to A’s SDN instead of “3.” The reason is that “3” is ONLY to be added to the SDN of C. All of the other pitches in music will have “2” added to their SDN. C is the exception because it is where we reset the number that designates the octave we are using.

For example, we list notes as “A3, B3, C4, D4 etc.” or “G5, A5, B5, C6, D6, etc.” In the musical context, C determines which octave we are at. That is why it is an exception.

Back to finding a pitch’s frequency.

Here is another example of finding a pitch in a “non-C” musical note:

Say I want to find the frequency of the pitch A4.

Hz of pitch = (Hz of A-2) * (2) ^ [ (SDN of A4) + 2]
y = m*2^x
y = 6.875 * 2 ^ (4+2)
y = 6.875 * 2 ^ (6)
y = 6.875 * 64
y = 440.000 Hz

One other important note to add is that not all musical notes can be found at the Octocontra level (Octocontra means the “-2” in C-2, A-2, G-2, etc.). If this is the case simply divide the Subsubcontra level of the note (the “-1” in C-1, B-1, Ab-1, etc.) by 2 and use this as a substitute for the Hz of whichever note was chosen.

The concept of musical scales is where the octave presents itself cross-culturally. For example, a western major scale will go from C4 to C5 which spans an octave. In Gamelan music, A Javanese musical style where intervals differ from Western music, the Sléndro scale also completes itself at an octave. In Hindustani music, Northern Indian Classical Music, the Raga scales are also divided in between octaves. The octave is the basic interval that sets the foundation for scales and harmonies cross-culturally.

Below are examples of the scales mentioned above:

C Major Scale (Western Music)

A Western Major C Scale

Sléndro scale (Gamelan Music)

A Slendro scale on Gamelan Music, notated on a western musical staff

(the +’s and -’s indicate frequencies outside of the Western tonal system)

Asavari Thaat Scale (Hindustani Music)

The Asavari Thaat Scale of Hindustani music, notated on a western musical staff.

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