Q factor bandwidth per octave filter conversion and converter - quality factor Q to bandwidth in octaves and vice versa convert filter BW

Relation between Q factor and bandwidth BW

Conversion: 'bandwidth per octave' N to quality factor Q

Q factor to 'bandwidth per octave' N (octave bandwidth)
Also called bandwidth in octaves. (Bandwidth BW = f₂ − f₁)

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Conversion formula: 'octave bandwidth' N to quality factor Q:

Q factor N to Q - sengpielaudio

Conversion formula: Quality factor Q to 'octave bandwidth' N:

Q factor Q to N log - sengpielaudio

Also known is this longer formula with 4 Qs; see its development at:
Bandwidth in octaves versus Q in bandpass filters − RaneNote 170

Q factor Q to N log - sengpielaudio

Frequency ratio of an octave:

Ratio Formula - sengpielaudio

Formula to convert quality factor Q to 'octave bandwidth' N,
but with logarithmus naturalis:

Q factor Q to N log - sengpielaudio

And the very short formula to convert quality factor Q
to 'bandwidth per octave' N, but with sinh^-1:

Q factor Q to N - sengpielaudio

The Q Factor - Sengpielaudio

Conversion chart or table of 'octave bandwidth' N to quality factor Q:

BW per octave	Filter Q	BW per octave	Filter Q	BW per octave	Filter Q	BW per octave	Filter Q
1/80	115.4	1	1.41	4	0.267	7	0.089
1/60	86.6	1 1/4	1.12	4 1/4	0.242	7 1/4	0.082
1/50	72.1	1 1/3	1.04	4 1/3	0.234	7 1/3	0.079
1/40	57.7	1 1/2	0.92	4 1/2	0.220	7 1/2	0.075
1/30	43.3	1 2/3	0.82	4 2/3	0.207	7 2/3	0.071
1/25	36.1	1 3/4	0.78	4 3/4	0.200	7 3/4	0.068
1/20	28.9	2	0.67	5	0.182	8	0.063
1/16	23.1	2 1/4	0.58	5 1/4	0.166	8 1/4	0.058
1/12	17.3	2 1/3	0.56	5 1/3	0.161	8 1/3	0.056
1/10	14.4	2 1/2	0.51	5 1/2	0.152	8 1/2	0.053
1/8	11.5	2 2/3	0.47	5 2/3	0.143	8 2/3	0.050
1/6	8.7	2 3/4	0.45	5 3/4	0.139	8 3/4	0.048
1/5	7.2	3	0.40	6	0.127	9	0.044
1/4	5.8	3 1/4	0.36	6 1/4	0.116	9 1/4	0.041
1/3	4.3	3 1/3	0.35	6 1/3	0.113	9 1/3	0.039
1/2	2.9	3 1/2	0.33	6 1/2	0.106	9 1/2	0.037
2/3	2.1	3 2/3	0.30	6 2/3	0.100	9 2/3	0.035
3/4	1.9	3 3/4	0.29	6 3/4	0.097	9 3/4	0.034
						10	0.031

Q factor as a function of the bandwidth in octaves N

Bandwidth in octaves N	Q factor
3.0	0.404
2.0	0.667
1.5	0.920
1.0	1.414
2/3	2.145
1/2	2.871
1/3	4.318
1/6	8.651
1/12	17.310

A high filter quality means narrow-band filtering (notch), with a large Q factor.
This results in steep filter flanks with a small bandwidth.

A low filter quality means broad-band filtering, with a small Q factor.
This results in flat filter flanks with a large bandwidth.

Notice:
A low Q factor = broad band (wide) bandwidth or high Q factor = narrow band (small) bandwidth.

Calculating the 3 dB cut-off frequencies; given center frequency f₀ and the Q factor
Interrelationship of 'octave bandwidth' N and the quality factor Q
Formulas for conversion of bandwidth in octaves to quality factor
Questions on "Parametric filter adjustment"
Conversion table Q to N and N to Q for parametric filters
Excel conversion - quality factor Q to bandwidth in octaves N
Filter Slope (dB/oct) is not Bandwidth

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