Octave-Band Center Frequencies

The following is a posting that appeared in the alt.sci.phyics.acoustics newsgroup. While every acoustician is familiar with the traditional octave band center frequencies (250 Hz, 500 Hz, 1000 Hz etc) and their fractional relatives, it is less clear as to how these values are derived. This post describes how the 1/6 and 1/10 octave-band center frequencies are derived, and this basically summarizes ANSI Standard S1.6-1984.

I want to thank Dr. Mercer for allowing me to publish his post on my site. You can also find the original post at Google Groups.

From: "Colin Mercer" <colin.mercer@prosig.com>
Newsgroups: alt.sci.physics.acoustics
Subject: Re: center frequencies for 1/12, 1/24 octaves
Date: Mon, 27 Mar 2000 10:39:20 +0100
Organization: Prosig Ltd.

The "standard" centre frequencies for 1/3 Octaves are based upon the Preferred Numbers. These date from around 1965. They are not specific to third octaves. The only reference we have is to British Standard BS2045:1965 Preferred Numbers. I expect there are equivalent ISO and ANSI versions. In BS2045 these preferred numbers are called the R5, R10, R20, R40 and R80 series. The relationship is

Preferred Series No R10 R20 R40 R80

1/N Octave 1/3 1/6 1/12 1/24

Steps/decade 10 20 40 80

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 found. The method you refer to 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. The base 2 one is simpler to use but the base 10 one is actually sounder numerically.

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. 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 * 2^1/6
lower = centre / 2^1/6

where the centre frequency is the exact one not the preferred one. The same goes for other bandwidths using the appropriate factors.

Preferred Values 1Hz to 10Hz, 1/24th Octave

1.00 1.602.50 4.006.30

1.03 1.65 2.584.12 6.50

1.061.70 2.654.25 6.70

1.091.75 2.72 4.376.90

1.12 1.802.804.50 7.10

1.151.85 2.90 4.62 7.30

1.18 1.90 3.00 4.75 7.50

1.22 1.95 3.07 4.87 7.75

1.25 2.00 3.15 5.00 8.00

1.28 2.06 3.25 5.15 8.25

1.32 2.12 3.35 5.30 8.50

1.36 2.18 3.45 5.45 8.75

1.40 2.24 3.55 5.60 9.00

1.45 2.30 3.65 5.80 9.25

1.502.36 3.75 6.00 9.50

1.552.43 3.87 6.15 9.75

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.

Regards Colin Mercer
Prosig Ltd.