Octave band

Source: Wikipedia, the free encyclopedia.

An octave band is a frequency band that spans one

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

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,

power of ten (decads, or decils)[citation needed
] 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 ( f2 or fmax ) is the lower band frequency ( f1 or fmin ) times the tenth root of ten,

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.[5]

Analyzing a source on a frequency by frequency basis is possible, most often using Fourier transform analysis.[6]

Octave bands

Calculation

If is the center frequency of an octave band, one can compute the octave band boundaries as

where is the lower frequency boundary and the upper one.

Naming

Band
number 		Nominal
frequency[7] 		Calculated
frequency 		A-weight
adjustment
 −1 16 Hz 15.625 Hz
  0 31.5 Hz 31.250 Hz   −39.4 dB
  1 63 Hz 62.500 Hz   −26.2 dB
  2 125 Hz 125.000 Hz   −16.1 dB
  3 250 Hz 250.000 Hz   −8.6 dB
  4 500 Hz 500.000 Hz   −3.2 dB
  5 1 kHz 1000.000 Hz       0   dB
  6 2 kHz 2000.000 Hz   +1.2 dB
  7 4 kHz 4000.000 Hz   +1.0 dB
  8 8 kHz 8000.000 Hz   −1.1 dB
  9 16 kHz 16000.000 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

Main article: One-third octave

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
number		 Nominal
frequency
Base 2

calculated
frequency
Base 10

calculated
frequency
1 16 Hz 15.625 Hz 15.849 Hz
2 20 Hz 19.686 Hz 19.953 Hz
3 25 Hz 24.803 Hz 25.119 Hz
4 31.5 Hz 31.250 Hz 31.623 Hz
5 40 Hz 39.373 Hz 39.811 Hz
6 50 Hz 49.606 Hz 50.119 Hz
7 63 Hz 62.500 Hz 63.096 Hz
8 80 Hz 78.745 Hz 79.433 Hz
9 100 Hz 99.213 Hz 100 Hz
10 125 Hz 125.000 Hz 125.89 Hz
11 160 Hz 157.490 Hz 158.49 Hz
12 200 Hz 198.425 Hz 199.53 Hz
13 250 Hz 250.000 Hz 251.19 Hz
14 315 Hz 314.980 Hz 316.23 Hz
15 400 Hz 396.850 Hz 398.11 Hz
16 500 Hz 500.000 Hz 501.19 Hz
17 630 Hz 629.961 Hz 630.96 Hz
18 800 Hz 793.701 Hz 794.43 Hz
19 1 kHz 1000.000 Hz 1000 Hz
20 1.25 kHz 1259.921 Hz 1258.9 Hz
21 1.6 kHz 1587.401 Hz 1584.9 Hz
22 2 kHz 2000.000 Hz 1995.3 Hz
23 2.5 kHz 2519.842 Hz 2511.9 Hz
24 3.150 kHz 3174.802 Hz 3162.3 Hz
25 4 kHz 4000.000 Hz 3981.1 Hz
26 5 kHz 5039.684 Hz 5011.9 Hz
27 6.3 kHz 6349.604 Hz 6309.6 Hz
28 8 kHz 8000.000 Hz 7943.3 Hz
29 10 kHz 10079.368 Hz 10 kHz
30 12.5 kHz 12699.208 Hz 12.589 kHz
31 16 kHz 16000.000 Hz 15.849 kHz
32 20 kHz 20158.737 Hz 19.953 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 10: An accurate calculation with an adequate number of digits, would produce the same result regardless of which base logarithm used.[clarification needed
]

See also

References

  1. ISBN 978-0-471-25293-1. Archived from the original
    on 5 December 2017 – via Google books.
  2. ^ IEC 61260-1:2014[full citation needed]
  3. ^ IANSI S1-6-2016[full citation needed]
  4. ^ IEC 61260-1:2014
  5. ^ "Octave-band center frequencies". audio articles. cross-spectrum.com. Archived from the original on 14 May 2017. Retrieved 23 November 2017.
  6. ^ "Basics". The fast Fourier transform (FFT). nti-audio.com. Support / know-how. Retrieved 9 January 2024.
  7. ^ "Specification for Octave, Half-Octave, and Third Octave Band Filter Sets" (PDF). resource.org. p. 13. ANSI S1.11. Retrieved 7 March 2018.


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