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.