Frequency, (one-third) octave, and decade

Frequency, (one-third) octave, and decade

You are all familiar with graphic equalizers like the DBX 231S below. You may have wondered why the frequencies for each band are defined as they are shown on this equalizer. This DBX 231S is a dual channel, 31-band 1/3-octave constant Q frequency band equalizer. Do you want to understand where these frequencies come from and what this 1/3-octave implies? Then, read on. 

Similarly, again in the context of equalizing, you may have seen figures like below. Where does the strange division (in decades) of frequencies come from? How to interpret the 6dB roll-off per octave? Want to know? Read on.

Decades

A decade is a unit for measuring frequency ratios on a logarithmic scale. One decade corresponds to a ratio of 10 between two numbers for example 100Hz and 1000Hz. In the next table, I show the frequencies that are a decade apart. For the actual calculation I started at 1000Hz (see R script below) and determined the decades below and above this frequency. The next decade would be 100kHz, which is of no practical use for us and, thus, I stopped at 10,000Hz.  The table also shows the octave bands (from to ) with the bandwidth () being defined as

This ‘relative’ bandwidth is constant for the decade bands. The center frequency () is the geometric mean of the lowest and highest frequency of each band (.  The geometric mean is often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature.

and can directly be calculated from the center frequency: and . In this calculate we use the square root of 10, which is the multiplication factor used to determine the succeeding frequencies.

In the plot below I show the frequencies (black solid lines) and the frequency bands (dotted blue lines) on a linear frequency scale (x-axis). The lower frequencies are very close together since the full range of the scale (0 to 10kHz) is large. This makes immediately clear why it is better to use a logarithmic scale to make these plots.

In the plot below I now show the frequencies (black solid lines) and the frequency bands (dotted blue lines) on a logarithmic frequency scale (x-axis).  A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way.  On this scale the difference between pairs of ticks is not constant as on the linear scale. For the figurer below I performed a transformation. Now we see that the center frequencies are at equal distances from each other and it is also clear that on this scale the lower and upper frequencies of each band are equidistant. This figure is far more easy to read.

In principle it is possible to also plot other frequencies on the logarithmic scale, which I have done below. We see here, for example, that the distance between 2 and 3Hz is the same as the distance between 20 and 30Hz, which is the same as the distance between 20 and 30kHz. We also see that the frequencies become more compressed on the x-axis if they become larger.

Octaves

Next to decades, frequencies are often displayed as octave bands. An octave is a doubling of the frequency. Thus, instead of a multiplication factor of 10, we now have a multiplication factor of 2. The principles are the same as for the decades. If we start at a frequency of 1kHz then we can construct the octave bands shown in the table below. The BW is defined identical as in the case of the decades and the center frequency is again the geometric mean of the lower and higher frequencies.

However, the calculation of and from the center frequency is slightly different: and since our multiplication factor is 2.

In the plot below, I have plotted the octave bands on a linear scale (black lines: center frequencies; dotted blue lines: octave bands).

In the plot below the frequencies are shown on a logarithmic scale but this time the axis is transformed by .

One-third octaves

Finally, we regularly see one-third octave bands. A one-third octave is a logarithmic unit of frequency ratio equal to either one third of an octave (major third). One-third octave” is defined as one tenth of a decade, corresponding to a frequency ratio of , or one tenth of a decade (base 10; ANSI S1.6-2016)

This frequency ratio defines the 10 intervals to bridge a decade. Thus, starting at 100Hz we would get the following 10 subsequent frequencies:

125.8925412
158.4893192
199.5262315
251.1886432
316.227766
398.1071706
501.1872336
630.9573445
794.3282347
1000

In a similar way we can also define the one-third octave band as the frequency ratio of (base2; ISO 18405:2017). It is easy to show that these definitions are approximately the same:

We need to solve for in:

Thus, the calculation of subsequent frequencies is .

The calculation of and from the center frequency is now:

since our multiplication factor is . This is equal to

and .

Applying all of this, we get the third-octave band table below. I have not rounded the (center) frequencies, but in general these frequencies are rounded to avoid dealing with non fractional numbers. These are then referred to as the “preferred values” (ISO3: 1973)

Below the one-third octave bands on a logarithmic scale.

Boundaries of bands

Above we have seen that and can directly be calculated from the center frequency: and . In this calculate we use the square root of 10, which is the multiplication factor used to determine the succeeding frequencies. This is easily checked if we realize that, for decades,

with being our multiplication factor, and realizing that the high boundary of the first frequency band () and the low boundary of the subsequent () frequency band should be equal.  Thus, we obtain:

In a similar way we can derive the multiplication factors for the (one-third) octave bands.

Geometric mean

Above we showed that is the geometric mean of the boundary frequencies, i.e., . This can also easily be checked starting from the same set of equations:

Reorganizing both equations give:

Substituting the second equation in the first  gives

Realizing that results in

R script

I made a small R script to generate the tables and figures shown above. R is a free software environment for statistical computing and graphics. You can download it [here]. If you consider using this script then you also may want to install RStudio [here].

Frequency, Octave, decade plots (R script) 13.53 KB 83 downloads

See also Frequency, (one-third) octave, and decade GitHub ...

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References

• Wikipedia one-third octave
• Wikipedia Octave
• https://courses.physics.illinois.edu/phys406/sp2017/Lab_Handouts/Octave_Bands.pdf
• https://apmr.matelys.com/Standards/OctaveBands.html

Specification of (fractional) octaves (ANSI S1.11.2004) (pdf) 2.41 MB 168 downloads

Specification for octave-band and fractional-octave-band analog and digital filters...

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