Octave in Engineering and Physics Terminology

The word octave is derived from the word octo, meaning eight. However, in engineering, physics, and music, an octave is defined as an interval where the frequency doubles or is halved. For instance, the note A3 to A4 is considered a one-octave increase, where the frequency of A4 is actually twice the frequency of A3.

So, why is a doubling of frequency called an octave, which is related to the number eight? The answer is that it has become a conventional term, though there is a logical reason. In music tradition, the distance from one note to the same note an octave above it, for example from C to C', passes through eight notes, including the starting and ending notes:

C - D - E F - G - A - B C'

Or, from A to A', the following notes are present:

A - B C - D - E F - G - A'

As you can see, the interval from C to C' and A to A' contains eight notes. This is why a doubling of frequency eventually got the label one octave.

Mathematical Calculation of Octaves

Mathematically, the value of an octave is presented in a simple equation. The ratio between two frequencies, say F1​ and F2​, can be expressed as:

Where:

• N is the octave value.

• F1​ and F2​ are the two frequencies being compared.

Note: The log operator can be base 10 or base e (natural logarithm, ln). In this article, log refers to the base-10 logarithm.

Example 1:

• $F_1=440$ Hz, $F_2=880$ Hz

• F2​ / F1​=(880 / 440)=2

• $N=\frac{\log (2)}{\log (2)\; /}=1$

• This represents an increase of 1 octave.

Example 2:

• $F_3=300$ Hz and $F_4=1200$ Hz. How many octaves is F4​ above F3​?

• $F_4/F_3=1200\; /\; 300=4$

• Since $4=\; 2{^}^2$, therefore the value of N is 2.

• Thus, 1200 Hz is 2 octaves above 300 Hz.

Example 3:

• $F_1=400$ Hz, $F_2=50$ Hz. How many octaves is F2​ below F1​?

• $N=\frac{\log (F_1/F_2)}{\log (2)\; /}=\frac{\log (400\; /\; 50)}{\log (2)\; /}=\frac{\log (8)}{\log (2)\; /}$

• $N=\frac{0.903}{0.301}=3$

• The frequency of 50 Hz is 3 octaves below 400 Hz. This can also be seen through gradual increases: 

• 50 Hz to 100 Hz (1 octave),
• 100 Hz to 200 Hz (1 octave), and 
• 200 Hz to 400 Hz (1 octave). 

        The total increase is 3 octaves.

 

Example 4: Application in Graphic Equalizers

• Let's look at a 5-band graphic equalizer. Its center frequencies are typically: 62 Hz, 250 Hz, 1 KHz, 4 KHz, and 16 KHz.

• The octave distance between the center frequencies is:

  • From 62 Hz to 250 Hz: $N=\frac{\log (250/62)}{\log (2)}\approx 2$ octaves.

  • From 250 Hz to 1 KHz: $N=\frac{\log (1000/250)}{\log (2)}=2$ octaves.

  • From 1 KHz to 4 KHz: $N=\frac{\log (4000/1000)}{\log (2)}=2$ octaves.

• Since the distance between each band is 2 octaves, this type of equalizer is also called a 2-octave equalizer. 

• It's important to note that while the octave distance is the same, the absolute frequency distance is not (e.g., 250 - 62 = 188 Hz, while 1000 - 250 = 750 Hz).

Example 5: 1/3 Octave Equalizer

• In professional audio, a 31-band equalizer is often used, also known as a 1/3 octave equalizer. This is because the audio spectrum from approximately 15 Hz to 20 KHz (about 10 octaves) is divided into 30 or 31 bands, resulting in approximately 1/3 of an octave per band.

• A value of 1/3 octave means the ratio of the nearest center frequencies is $F_1/F_2={2^}^N={2^\{}^{1/3\}}\approx 1.26$ times.

• The ISO standard for the center frequencies of a 1/3 octave graphic equalizer are: 

• 20, 25, 31.5, 40, 50, 63, 80, 100, 125, 160, 200, 
• 250, 315, 400, 500, 630, 800, 1000, 1250, 1600, 2000, 
• 2500, 3150, 4000, 5000, 6300, 8000, 10000, 12500, 16000, 20000 Hz.

Hopefully, this slightly mathematical article is easy to understand and helpful.

TABIK !

(This post is parallel to the status on the FaceBookGroup The Art of Electronics with the same Topic)