Q factor bandwidth in octaves filter calculator formula bandwidth - quality factor Q to bandwidth BW width octave convert filter BW octave mastering slope dB/oct steepness EQ equalizer ??

Bandpass filter (BPF)
● Relation between Q  factor and bandwidth  BW ●

Filter conversion: 'bandwidth in octaves' N to quality factor Q
and Q factor to 'bandwidth in octaves' N (octave width)    Q = f₀/BW

Bandwidth BW = f₂ − f₁= f₀/Q       Equalizer EQ bandpass filter
Q factor = quality factor       Bandwidth BW of a filter band
The multiplicative inverse or the reciprocal of the
quality factor 1/Q is called the dissipation factor d (damping)
People use 'Q' and 'bandwidth' interchangeably, though they're not.
Defining the bandwidth as the −3 dB points cannot be correct for a boost gain of 3 dB or less.

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Given BW in octaves to find Q and given Q to find BW in octaves.

BW = Δf = f₂ − f₁ = f₀/Q f₁ = f₀²/f₂ = f₂ − BW f₂ = f₀²/f₁ = f₂ + BW Q = f₀/BW

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

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

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

Frequency ratio of an octave:

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

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

The Q Factor

Conversion chart or table
'bandwidth in octaves' N to quality factor Q

BW in octaves	Filter Q	BW in octaves	Filter Q	BW in octaves	Filter Q	BW in octaves	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	8.65	2 3/4	0.45	5 3/4	0.139	8 3/4	0.048
1/5	7.20	3	0.40	6	0.127	9	0.044
1/4	5.76	3 1/4	0.36	6 1/4	0.116	9 1/4	0.041
1/3	4.32	3 1/3	0.35	6 1/3	0.113	9 1/3	0.039
1/2	2.87	3 1/2	0.33	6 1/2	0.106	9 1/2	0.037
2/3	2.14	3 2/3	0.30	6 2/3	0.100	9 2/3	0.035
3/4	1.90	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	Filter Q factor
3.0 wide	0.404 low
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 small	17.310 high

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

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.

The larger the Q the more narrow the resonance peak.
The smaller the Q the more broad the resonance peak.

The Q factor or the bandwidth does
not tell the "steepness" in dB/oct.

Slope in dB/oct = steepness of the filter flanks
● Only with high pass and low pass filters − not with bell curves ●

Note: The Q factor (quality factor) or the bandwidth is not convertable to the "slope" as dB/oct.
There are mastering equalizers with false information regarding the filter setting as
"Slope in dB/octave" and not Q factor (width), see:
Filter slope or steepness (dB/oct) is not bandwidth = Slope in dB/oct or steepness of filter slopes is not the 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 or steepness (dB/oct) is not Bandwidth
Adding decibels of one-third octave bands to level of one octave band

With "quality" is not meant how valuable the signal is. It is meant the quality of the filter.
At a filter with flat slopes many frequencies are influenced around the cutoff frequency.
The filter has therefore a larger bandwidth. The so-called quality factor is given with a low number specified. If the filter has steep slopes, its bandwidth is smaller.
Here a few frequencies below and above its cutoff frequency are affected and the quality factor Q is specified as a high number.

Why is the bandwidth and the cutoff frequency found at a level of "−3 dB"?
Full width at half maximum (FWHM).
That is the point where the energy (power) is fallen to the 1/2 value or 0.5 = 50 percent of the initial energy quantity.
There the voltage is fallen to the value of √(1/2) = 1/√2 or 0.71 = 70.1 percent of the initial voltage as field quantity. A 3 dB voltage drop is a decrease of 29,29 % to 70,71 %.

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