The “standard” centre frequencies for 1/3 octave bands are based upon the Preferred Numbers. These date from the 19th century when Col. Charles Renard (1849–1905) was given the job of improving captive balloons used by the military to observe enemy positions. This work resulted in what are now known as Renard numbers. Preferred Numbers were standardised in 1965 in British Standard BS2045:1965 Preferred Numbers and in ISO and ANSI versions in 1973. Preferred numbers are not specific to third octave bands. They have been used in wide range of applications including capacitors & resistors, construction industry and retail packaging.
In BS2045 these preferred numbers are called the R5, R10, R20, R40 and R80 series. The 'R' is used to acknowledge the original work of Renard in the 1870's. The relationship is
|Preferred Series No||R10||R20||R40||R80|
The basis of audio fractional octave bands is a frequency of 1000Hz. There are two ISO and ANSI approved ways in which the exact centre frequencies may be defined. One scheme is the base2 method where the ratio between 2 exact centre frequencies is given by 2^(1/N) with N as 3 for 1/3 octaves and so on. The other method is the base 10 method where the ratio is given by 10^(3/[10N]). This ratio may also be written as 2^(3/[10Nlog2]). For nearly all practical purposes both ratios are the same but tones at band edges can be interesting and may appear to be in different octave bands. The base 2 one is simpler to use (and is often favoured by non-engineering programmers!), but the base 10 one is actually sounder numerically.
One very good reason for using base 10 is that all the exact centre frequencies are the same for each decade. This is not the case for the base 2 frequencies.
As an example (using base 2) the theoretical centre frequency of the 1/3 octave below 1000 is found by dividing by 2^(1/3). This is 793.7005... . Using base 10 the corresponding centre frequency is 794.3282... . In both cases the nearest preferred frequency is 800Hz so that is what the band is called. When working out the edge band frequencies for a 1/3 octave then these are respectively
upper = centre * 21/6
lower = centre / 21/6
where the centre frequency is the exact one not the preferred one. For (1/N)th octave the relationship is simply
upper = centre * 21/2N
lower = centre / 21/2N
If we use the base 2 method and find the centre frequency of the third octave band 10 steps below we get 99.21257... Hz, but with base 10 we get exactly 100.0Hz. If we continue further down to 10Hz and 1Hz then the base 2 centre frequencies are 9.84313...Hz and 0.97656...Hz respectively. The base 10 values are at 10Hz and 1Hz of course. The point to notice is that theses low centre frequencies now differ by approximately (1/24)th of an octave between the two methods.
Generally in audio work we are not too concerned about the very low frequencies. It does explain, however, why the standards use the 1kHz rather than the logical 1Hz as the reference centre frequency. If the 1Hz was used as the reference centre frequency then there would be serious discrepancies between the two schemes at 1kHz, which is very important acoustically. It is also interesting to note that third octave band numbering does use 1Hz as the reference point. We have 1Hz = 100 is third octave band 0, 10Hz = 101 is band 10, 100Hz = 102 is band 20, 1000Hz = 103 is band 30 and so on.
Preferred Values 1Hz to 10Hz, 1/24th Octave
The R80 table above gives the 1/24th octave preferred frequencies. For 1/12th skip one to get 1.0, 1.06, 1.12 etc. For 1/6 skip three to give 1.0, 1.12, etc. For 1/3 then skip seven to get 1.0, 1.25 and so on.
So for 1/3 octave bands we obtain the well known sequence: 1.0, 1.25, 1.6, 2.0, 2.5, 3.15, 4.0, 5.0, 6.3, 8.0
A. Van Dyck. Preferred numbers. Proceedings of the Institute of Radio Engineers, volume 24, pages 159-179 (February 1936)
British Standard BS2045:1965 Preferred Numbers
ISO 3-1973, Preferred Numbers – Series of Preferred Numbers.
ISO 17-1973, Guide to the Use of Preferred Numbers and of Series of Preferred Numbers.
ISO 497-1973, Guide to the Choice of Series of Preferred Numbers and of Series Containing More Rounded Values of Preferred Numbers.
ANSI Z17.1-1973, American National Standard for Preferred Numbers.
Preferred Number on Wikipedia
Preferred Numbers from Sizes.com
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Dr Colin Mercer is Chief Signal Processing Analyst at Prosig and has responsibility for signal processing and its applications. He was formerly at the Institute of Sound and Vibration Research (ISVR), Southampton University where he founded the Data Analysis Centre. He is a Chartered Engineer and a Fellow of the British Computer Society.