octave band

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){{fact|date=March 2024}} 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 ( {{math|f{{sub|2}} }} or {{mvar|f}}{{sub|max}} ) is the lower band frequency ( {{math|f{{sub|1}} }} or {{mvar|f}}{{sub|min}} ) times the tenth root of ten,IEC 61260-1:2014 or {{math|1.2589}} : 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 {{sc|c}}–{{sc|e}}. The second one-third octave begins where the first-third ends and itself ends at a frequency {{nobr| {{math|1.2589 ² {{=}} 1.5849 ×}} ,}} or 158.5% higher than the original starting frequency. The third-third, or last band ends at {{nobr|  {{math|1.2589 ³ {{=}} 1.9953 ×}} ,}} or 199.5% of the base frequency.

Any useful subdivision of acoustic frequencies is possible: Fractional octave bands such as {{sfrac| 1 | 3 }} or {{sfrac| 1 | 12 }} of an octave (the spacing of musical notes in 12 tone equal temperament) are widely used in acoustical engineering.{{cite web |title=Octave-band center frequencies |website=cross-spectrum.com |department=audio articles |url=http://www.cross-spectrum.com/audio/articles/center_frequencies.html |access-date=2017-11-23 |url-status=live |archive-url=https://web.archive.org/web/20170514010527/http://www.cross-spectrum.com/audio/articles/center_frequencies.html |archive-date=2017-05-14 |df=dmy-all}}

Analyzing a source on a frequency by frequency basis is possible, most often using Fourier transform analysis.{{cite web |title=Basics |department=The fast Fourier transform (FFT)|series=Support / know-how |website=nti-audio.com |url=https://www.nti-audio.com/en/support/know-how/fast-fourier-transform-fft |access-date=2024-01-09 |df=dmy-all}}

If $\backslash \; f\_\backslash mathsf\{c\}\backslash$ is the center frequency of an octave band, one can compute the octave band boundaries as

:$\backslash \; f\_c\; =\; \backslash sqrt\{2\}\; f\_\backslash mathsf\{min\}\; =\; \backslash frac\{\backslash \; f\_\backslash mathsf\{max\}\backslash \; \}\{\backslash \; \backslash sqrt\{2\backslash \; \}\backslash \; \}\backslash \; ,$

where $\backslash \; f\_\backslash mathsf\{min\}\backslash$ is the lower frequency boundary and $\backslash \; f\_\backslash mathsf\{max\}\backslash$ the upper one.

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

{{main|One-third octave}}

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%% Calculate Third Octave Bands (base 2) in Matlab

fcentre = 10^3 * (2 .^ ([-18:13]/3))

fd = 2^(1/6);

fupper = fcentre * fd

flower = fcentre / fd

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%% Calculate Third Octave Bands (base 10) in Matlab

fcentre = 10.^(0.1.*[12:43])

fd = 10^0.05;

fupper = fcentre * fd

flower = fcentre / fd

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.

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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.{{clarify|how can the result in any one band be identical when the band limits are not?|date=March 2024}}

• octave
• octave (electronics)

{{reflist|25em}}

Category:Acoustics

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