# Octave band

An octave band is a frequency band that spans one octave (Play). In this context an octave can be a factor of 2 or a factor of 10$^{0.301}$. An octave of 1200 cents in musical pitch (a logarithmic unit) corresponds to a frequency ratio of   ⁠2/ 1 ⁠ ≈ 10$^{0.301}$ .
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 ( f$_2$ or f$_{max}$ ) is the lower band frequency ( f$_1$ or f$_{min}$ ) times the tenth root of ten, or 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 C–E. The second one-third octave begins where the first-third ends and itself ends at a frequency 1.2589 ² = 1.5849 × , or 158.5% higher than the original starting frequency. The third-third, or last band ends at   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 ⁠ 1 / 3 ⁠ or ⁠1/ 12 ⁠ 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 {c}}\$ is the center frequency of an octave band, one can compute the octave band boundaries as
    $\ f_{c}={\sqrt {2}}f_{\mathsf {min}}={\frac {\ f_{\mathsf {max}}\ }{\ {\sqrt {2\ }}\ }}\ ,$
where $\ f_{\mathsf {min}}\$ is the lower frequency boundary and $\ f_{\mathsf {max}}\$ the upper one.

### Naming

    Band number Nominal frequency Calculated frequency A-weight adjustment  −116 Hz15.625 Hz  031.5 Hz31.250 Hz −39.4 dB  163 Hz62.500 Hz −26.2 dB  2125 Hz125.000 Hz −16.1 dB  3250 Hz250.000 Hz −8.6 dB  4500 Hz500.000 Hz −3.2 dB  51 kHz1000.000 Hz     0   dB  62 kHz2000.000 Hz +1.2 dB  74 kHz4000.000 Hz +1.0 dB  88 kHz8000.000 Hz −1.1 dB  916 kHz16000.000 Hz −6.6 dB
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 .^ ([-18:13]/3)) 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.*[12:43]) 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.
    Band numberNominal frequencyBase 2 calculated frequencyBase 10 calculated frequency 116 Hz15.625 Hz15.849 Hz 220 Hz19.686 Hz19.953 Hz 325 Hz24.803 Hz25.119 Hz 431.5 Hz31.250 Hz31.623 Hz 540 Hz39.373 Hz39.811 Hz 650 Hz49.606 Hz50.119 Hz 763 Hz62.500 Hz63.096 Hz 880 Hz78.745 Hz79.433 Hz 9100 Hz99.213 Hz100 Hz 10125 Hz125.000 Hz125.89 Hz 11160 Hz157.490 Hz158.49 Hz 12200 Hz198.425 Hz199.53 Hz 13250 Hz250.000 Hz251.19 Hz 14315 Hz314.980 Hz316.23 Hz 15400 Hz396.850 Hz398.11 Hz 16500 Hz500.000 Hz501.19 Hz 17630 Hz629.961 Hz630.96 Hz 18800 Hz793.701 Hz794.43 Hz 191 kHz1000.000 Hz1000 Hz 201.25 kHz1259.921 Hz1258.9 Hz 211.6 kHz1587.401 Hz1584.9 Hz 222 kHz2000.000 Hz1995.3 Hz 232.5 kHz2519.842 Hz2511.9 Hz 243.150 kHz3174.802 Hz3162.3 Hz 254 kHz4000.000 Hz3981.1 Hz 265 kHz5039.684 Hz5011.9 Hz 276.3 kHz6349.604 Hz6309.6 Hz 288 kHz8000.000 Hz7943.3 Hz 2910 kHz10079.368 Hz10 kHz 3012.5 kHz12699.208 Hz12.589 kHz 3116 kHz16000.000 Hz15.849 kHz 3220 kHz20158.737 Hz19.953 kHz
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.