In Engineering applications, sound spectrums are usually represented in octave or one-third octave frequency bands rather than in narrow frequency bands. This frequency representation is linked to the perception of sound by a human ear and it allows a compression of the amount of information. An important part of the information is however lost when converting results from narrow frequency bands to octave or one-third octave frequency bands.

Each octave and 1/3 octave bands are identified by a middle frequency $f_{0}$, a lower frequency bound and an upper one defined as:

Octave1/3 octave
Middle frequency$f_{0}$$f_{0} Lower freq. boundf_{0}/2^{1/2}$$f_{0}/(2^{1/2})^{1/3}=f_{0}/2^{1/6}$
Upper freq. bound$f_{0}\times 2^{1/2}$$f_{0}\times(2^{1/2})^{1/3}=f_{0}\times 2^{1/6}$

The reference middle frequency in acoustics is 1 000 Hz. Other middle frequencies $F_0$ are computed by successive multiplications or divisions of this reference value with $2$ or $2^{1/3}$ for octave and third-octave bands respectively. To avoid dealing with non fractional numbers, "preferred values" [ISO3] are substituted to these operation results.

Bounds are computed using the exact value of the operation and not its preferred value.

In a general manner the middle frequencies of an n-th octave bands are obtained by successive multiplication or division of 1 000 Hz with $2^{1/n}$. The lower and upper frequnecy bounds of a given n-th octave bands with middle frequency $f_0$ are given by $f_{0}/(2^{1/2})^{1/n}$ and $f_{0}\times(2^{1/2})^{1/n}$ respectively.

# Example

The two tables below summarized the octave and the 1/3 octave middle frequencies and bounds computed from expressions reported above.

Octave middle freq.      Octave band
15.6 (preferred: 16)11.0 - 22.1
31.3 (preferred: 31.5)22.1 - 44.2
62.5 (preferred: 63)44.2 - 88.4
12588.4 - 176.8
250176.8 - 353.6
500353.6 - 707.1
1 000707.1 - 1 414.2
2 0001 412.2 - 2 828.4
4 0002 828.4 - 5 656.9
8 0005 656.9 - 11 313.7
16 00011 313.7 - 22 627.4

1/3 octave middle freq.     1/3 octave band
15.6 (preferred: 16)13.9 - 17.5
19.7 (preferred: 20)17.5 - 22.1
24.8 (preferred: 25)22.1 - 27.8
31.3 (preferred: 31.5)27.8 - 35.1
39.4 (preferred: 40)35.1 - 44.2
49.6 (preferred: 50)44.2 - 55.7
62.5 (preferred: 63)55.7 - 70.2
78.7 (preferred: 80)70.2 - 88.4
99.2 (preferred: 100)88.4 - 111.4
125111.4 - 140.3
157.5 (preferred: 160)140.3 - 176.8
198.4 (preferred: 200)176.8 - 222.7
250222.7 - 280.6
315280.6 - 353.6
396.9 (preferred: 400)356.6 - 445.4
500445.4 - 561.2
630561.2 - 707.1
793.7 (preferred: 800)707.1 - 890.9
1 000890.9 - 1 122.5
1 259.9 (preferred: 1 250)1 122.5 - 1 414.2
1 587.4 (preferred: 1 600)1 414.2 - 1 781.8
2 0001 781.8 - 2 244.9
2 519.8 (preferred: 2 500)2 244.9 - 2 828.4
3 174.8 (preferred: 3 150)2 828.4 - 3 563.6
4 0003 563.6 - 4 489.8
5 039.7 (preferred: 5 000)4 489.8 - 5 656.9
6 349.6 (preferred: 6 300)5 656.9 - 7 127.2
8 0007 127.2 - 8 979.7
10 079.4 (preferred: 10 000)8 979.7 - 11 313.7
12 699.2 (preferred: 12 500)11 313.7 - 14 254.4
16 00014 254.4 - 17 959.4
20 158.7 (preferred: 20 000)17 959.4 - 22 627.4

The figure below represents the conversion from narrow frequency bands (range: [200 - 4 000] Hz with a frequency step of 4 Hz) to one-third octave frequency bands of the sound absorption coefficient for a 30 mm-thick fibrous material measured at normal incidence in a standing wave tube.
The measured values of the sound absorption coefficient contained in a 1/3 octave band are averaged; the obtained mean value of the sound absorption coefficient is then reported at preferred value of the 1/3 octave middle frequency .

# Script

% function [one_third_freq,band] = one_third_octave(frequencies,measurements)
%
% Narrow bands to one-third octave bands representation.
%
% Example: [one_third_freq_preferred,band] = narrow_to_one_third_octave(frequencies,alpha_diffuse)
%
% The input parameters are:
% o frequencies: frequency values
%   (with a fixed of variable frequency step),
% o measurements: acoustic absorption coefficent values
%   (corresponding to the frequency vector defined above).
%
% The output parameters are:
% o one_third_freq: center frequencies of 1/3 octave bands,
% o bands: values of the acoustic absorption coefficent in 1/3 bands
%
% Copyleft 2007-2011 luc.jaouen@matelys.com
% cf. APMR on the web,
% Standards/OctaveBands.html