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^[1]^{[full citation needed]} or a factor of 10^0.301.^[2]^{[full citation needed]}^[3]^{[full citation needed]} 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)^{[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 (  $f 2$ or $f$ _max ) is the lower band frequency (  $f 1$ or $f$ _min ) times the tenth root of ten,^[4] 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 $1.2589 ² = 1.5849 \times$ , or 158.5% higher than the original starting frequency. The third-third, or last band ends at $1.2589 ³ = 1.9953 \times$ , 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 $\ 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^[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

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 2 or base 10: An accurate calculation with an adequate number of digits, would produce the same result regardless of which base logarithm used.^{[clarification needed]}

References

^ Crocker (1997). [no title cited]. John Wiley & Sons. p. 1325. ISBN 978-0-471-25293-1. Archived from the original on 5 December 2017 – via Google books.
^ IEC 61260-1:2014^{[full citation needed]}
^ IANSI S1-6-2016^{[full citation needed]}
^ IEC 61260-1:2014
^ "Octave-band center frequencies". audio articles. cross-spectrum.com. Archived from the original on 14 May 2017. Retrieved 23 November 2017.
^ "Basics". The fast Fourier transform (FFT). nti-audio.com. Support / know-how. Retrieved 9 January 2024.
^ "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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[1] Crocker (1997). [no title cited]. John Wiley & Sons. p. 1325. 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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Fractional octave bands

Octave bands

Calculation

Naming

One-third octave bands

Base 2 calculation

Base 10 calculation

Naming

See also

References