AC-04Acoustic Fundamentals

Octave & Fractional-Octave Bands

Centre and edge frequencies for 1/1 to 1/24-octave bands per IEC 61260, with filter shapes.

Inputs

Bandwidth

Frequency of interest

Band centre

398.11Hz

nominal 400 Hz

Lower edge f₁

354.81Hz

Upper edge f₂

446.68Hz

Bandwidth

91.87Hz

Q = 4.33

1/3-octave band map31 bands, 20 Hz–20 kHz

Filter shapes around the selected band

Selected bandNeighbours

x	Nominal	f₁	f_m exact	f₂
-17	20 Hz	17.78	19.953	22.39
-16	25 Hz	22.39	25.119	28.18
-15	31.5 Hz	28.18	31.623	35.48
-14	40 Hz	35.48	39.811	44.67
-13	50 Hz	44.67	50.119	56.23
-12	63 Hz	56.23	63.096	70.79
-11	80 Hz	70.79	79.433	89.13
-10	100 Hz	89.13	100	112.2
-9	125 Hz	112.2	125.89	141.3
-8	160 Hz	141.3	158.49	177.8
-7	200 Hz	177.8	199.53	223.9
-6	250 Hz	223.9	251.19	281.8
-5	315 Hz	281.8	316.23	354.8
-4	400 Hz	354.8	398.11	446.7
-3	500 Hz	446.7	501.19	562.3
-2	630 Hz	562.3	630.96	707.9
-1	800 Hz	707.9	794.33	891.3
0	1 kHz	891.3	1,000	1,122
1	1.25 kHz	1,122	1,258.9	1,413
2	1.6 kHz	1,413	1,584.9	1,778
3	2 kHz	1,778	1,995.3	2,239
4	2.5 kHz	2,239	2,511.9	2,818
5	3.15 kHz	2,818	3,162.3	3,548
6	4 kHz	3,548	3,981.1	4,467
7	5 kHz	4,467	5,011.9	5,623
8	6.3 kHz	5,623	6,309.6	7,079
9	8 kHz	7,079	7,943.3	8,913
10	10 kHz	8,913	10,000	11,220
11	12.5 kHz	11,220	12,589	14,130
12	16 kHz	14,130	15,849	17,780
13	20 kHz	17,780	19,953	22,390

§ Theory of operation

Constant-percentage bandwidth

Octave-band analysis splits the spectrum into filters whose bandwidth is a fixed fraction of their centre frequency, matching the roughly logarithmic way we perceive pitch. In a 1/b-octave system every band has the same relative bandwidth and the same number of bands per decade.

Base-10 definition (IEC 61260-1 / ANSI S1.11)

\begin{aligned} G &= 10^{3/10} \approx 1.99526 \qquad \text{(octave ratio)} \\ b\ \text{odd}:\quad f_m &= 1000 \cdot G^{x/b} \\ b\ \text{even}:\quad f_m &= 1000 \cdot G^{(2x+1)/(2b)} \\ f_1 &= f_m \cdot G^{-1/(2b)} \qquad f_2 = f_m \cdot G^{+1/(2b)} \end{aligned}

where x is an integer band number. The exact mid-band frequencies are irrational, so for 1/1 and 1/3 octaves the standard assigns rounded nominal labels (31.5, 63, 125…). Base-10 is used rather than base-2 so that exactly ten 1/3-octave bands fit in a decade and frequencies repeat every decade.

Bandwidth and Q

1/1 octave: bandwidth ≈ 70.7 % of $f_m$ , Q ≈ 1.41
1/3 octave: bandwidth ≈ 23.1 %, Q ≈ 4.32 - close to the critical bands of hearing above 500 Hz
1/12 octave: bandwidth ≈ 5.8 % - one semitone

Filter shape

Real analysers implement each band as a band-pass filter. The response plotted here is a 3rd-order Butterworth band-pass (the classic Class 1 implementation), with −3 dB at the band edges. Adjacent bands cross at −3 dB so that energy is conserved when bands are summed.