center frequencies for 1/12, 1/24 octaves
Mike Blommer
and 1/24th octave filters? I know there are theoretical relations (2^(1/12)
or 10^(1/40)) for the ratio of two adjacent 1/12th octave bands, but these
values don't match what I've seen in various signal analysis tools (just
like 1/3rd octave center frequencies don't exactly follow the 2^(1/3) or
10^(1/10) rule).
I know of the ANSI S1.11 for 1/3rd octave filters (where the "preferred
center frequencies" are listed), but I have not found anything similar for
1/12th or 1/24th.
Thanks in advance for any help.
Mike
Colin Mercer
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 1.60 2.50 4.00 6.30
1.03 1.65 2.58 4.12 6.50
1.06 1.70 2.65 4.25 6.70
1.09 1.75 2.72 4.37 6.90
1.12 1.80 2.80 4.50 7.10
1.15 1.85 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
1.50 2.36 3.75 6.00 9.50
1.55 2.43 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
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