Octave Band

An octave band is a frequency band that spans one octave (). In this context an octave can be a factor of 2 or a factor of . An octave of 1200  cents in musical pitch (a logarithmic unit) corresponds to a frequency ratio of

A general system of scale of ''octave bands'' and ''one-third octave'' bands has been developed for frequency analysis in general, most specifically for acoustics. A ''band'' is said to be an octave in width when the upper band frequency is approximately twice the lower band frequency.

Fractional octave bands

A whole frequency range can be divided into sets of frequencies called ''bands'', with each band covering a specific range of frequencies. For example, radio frequencies are divided into multiple levels of band divisions and subdivisions, and rather than octaves, the highest level of radio bands (VLF, LF, MF, HF, VHF, etc.) are divided up by the wavelengths' power of ten (''decads'', or ''decils'') that is the same for all radio waves in the same band, rather than the power of two, as in analysis of acoustical frequencies.

In acoustical analysis, a one-third octave band is defined as a frequency band whose upper band-edge frequency (  or  ) is the lower band frequency (  or  ) times the tenth root of ten, or  : The first of the one-third octave bands ends at a frequency 125.9% higher than the starting frequency for all of them, the ''base frequency'', or approximately 399  musical cents above the start (the same frequency ratio as the musical interval between the notes –. The second one-third octave begins where the first-third ends and itself ends at a frequency or 158.5% higher than the original starting frequency. The third-third, or last band ends at or 199.5% of the base frequency.

Any useful subdivision of acoustic frequencies is possible: Fractional octave bands such as or of an octave (the spacing of musical notes in 12 tone equal temperament) are widely used in acoustical engineering.

Analyzing a source on a frequency by frequency basis is possible, most often using Fourier transform analysis.

Octave bands

Calculation

If $\ f_\mathsf\$ is the center frequency of an octave band, one can compute the octave band boundaries as

:$\ f_c = \sqrt f_\mathsf = \frac\ ,$

where $\ f_\mathsf\$ is the lower frequency boundary and $\ f_\mathsf\$ the upper one.

Naming

:

Note that 1000.000 Hz, in octave 5, is the nominal central or reference frequency, and as such gets no correction.

One-third octave bands

Base 2 calculation

:
%% Calculate Third Octave Bands (base 2) in Matlab
fcentre = 10^3 * (2 .^ ( 18:133))
fd = 2^(1/6);
fupper = fcentre * fd
flower = fcentre / fd

Base 10 calculation

:
%% Calculate Third Octave Bands (base 10) in Matlab
fcentre = 10.^(0.1.* 2:43
fd = 10^0.05;
fupper = fcentre * fd
flower = fcentre / fd

Naming

Due to slight rounding errors between the base two and base ten formulas, the exact starting and ending frequencies for various subdivisions of the octave come out slightly differently.

:

Normally the difference is ignored, as the divisions are arbitrary: They aren't based on any clear or abrupt change in any crucial physical property. However, if the difference becomes important – such as in detailed comparison of contested acoustical test results – either all parties adopt the same set of band boundaries, or better yet, use more accurately written versions of the same formulas that produce identical results. The cause of the discrepancies is deficient calculation, not a distinction in the underlying mathematics of base 2 or base 10: An accurate calculation with an adequate number of digits, would produce the same result regardless of which base logarithm used.

See also

* octave
* octave (electronics)

References

Acoustics

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