Sones and phons loudness level sone phon dBA dB sone dba calculator fan noise sound volume psychoacoustics

• Calculator: Correlation between sones und phons •

The subjective perceived sound volume and the artificial term loudness.
The volume or loudness of a noise is given in sone. A sone is equivalent
to 40 phons, which is 40 dBSPL of a sine wave of 1000 Hz.

Loudness level L_N in phons and Loudness N in sones
Loudness N in sones and Loudness level L_N in phons (volume)

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Linear conversion valid between 40 phons and 120 phons

For loudness level L_N > 40 phons: loudness N in sones = 2^{[(LN in phons − 40)/10]}
For loudness N > 1 sone: loudness level L_N in phons = 10 × log₂(N in sones)

Conversion only valid between 8 phons and 40 phons

For loudness level L_N < 40 phons: loudness N in sones = (L_N in phons / 40)^2.86 − 0.005
For loudness N < 1 sone: loudness level L_N in phons = 40 × (N in sones)^0,35

According to Stanley Smith Stevens' definition, 1 sone is equivalent to 40 phons,
which is defined as the loudness level of a pure 1 kHz tone at L_N = 40 dBSPL,
or dBA, but only (!) for a sine wave of 1 kHz and not for broadband noise.

Relation between loudness N in sones and loudness level L_N in phons

Dynamic range	phons	sones
	120	256
	110	128
fff	100	64
ff	90	32
f	80	16
---	70	8
p	60	4
pp	50	2
ppp	40	1
	32	1/2
	25	1/4
	19	1/8
	14	1/16
	11	1/32
	9	1/64

Sones-Phons-Bar

Loudness N (sone) and loudness level L_N (phon), as shown
here, is easily converted into one another, but with the
psychoacoustic perceived loudness and the objectively
measured sound pressure in pascals or its level in dBSPL
or dBA this is rather not possible. There is no formula.
The typical question: "How to convert 0.5 sone to decibel
(dB)?" cannot be answered. Only a pure tone of 1 kHz
measured in phons is equivalent to dB-SPL.

The volume of a sound is a subjective perception. To "measure" loudness,
the volume of a 1,000 hertz reference tone is adjusted until it is perceived by
listeners to be equally as loud as the sound being "measured". The loudness
level, in phons, of the sound is then equal to the sound-pressure level, in
decibels.
Between the loudness N in sone and the loudness level L_N in phon we have
the following connection (ISO-recommendation ISO/R 131-1959):
Loudness N = 2^{(LN − 40)/10} or loudness level L_N = 40 + 10 × lb N.
"lb" means logarithm base 2. 10 × lb N = 10 × log₂(N)

The sone is a unit of perceived loudness after a proposal of Stanley Smith
Stevens (1906-1973) in 1936. In acoustics, loudness is a subjective measure
of the sound pressure. One sone is equivalent to 40 phons, which is defined
as the loudness of a 1 kHz tone at 40 dBSPL. The number of sones to
a phon was chosen so that a doubling of the number of sones sounds to
the human ear like a doubling of the loudness, which also corresponds to
increasing the sound pressure level by 10 dB, or increasing the sound
pressure by a factor 3.16 (= √10). At frequencies other than 1 kHz, the
measurement in sones must be calibrated according to the frequency
response of human hearing, which is of course a subjective process.
The study of apparent loudness is included in the topic of psycho acoustics.
Volume in acoustics is used as a synonym for loudness. It is a common term
for the amplitude or the level of sound.

To be fully precise, a measurement in sones must be qualified by the optional
suffix G, which means that the loudness value is calculated from frequency
groups, and by one of the two suffixes F (for free field) or D (for diffuse field).

Notice - Comparing dBA and sone: Sone to dBA cannot be
converted linear. Sone is not dBA and dBA is not sone.
There is no correlation between sound pressure level SPL as
broadband measuring and dBA. To know the measuring distance
and the frequency content of the signals could also be important.

With the following table you can try to convert roughly, but be cautious using
these psychoacoustic values. The frequency composition of the signal
amplitude is always unknown. The measuring point is important for the value
of measure.
The following chart is, however, for the dBA values no accurate
knowledge − it's more a guess.

sones	phons	dBA	sones	phons	dBA
0.1	17.9	20.5	1.8	48.5	34.8
0.2	22.8	21.5	1.9	49.3	35.3
0.3	26.2	22.5	2.0	50.0	35.8
0.4	29.0	23.5	2.1	50.7	36.4
0.5	31.4	24.4	2.2	51.4	37.0
0.6	33.5	25.3	2.3	52.0	37.5
0.7	35.3	26.3	2.4	52.6	38.0
0.8	37.0	27.2	2.5	53.2	38.4
0.9	38.6	28.2	2.6	53.8	38.8
1.0	40.0	29.2	2.7	54.3	39.3
1.1	41.4	30.2	2.8	54.9	39.8
1.2	42.6	31.1	2.9	55.4	40.2
1.3	43.8	32.0	3.0	55.9	40.6
1.4	44.9	33.0	3.1	56.3	41.1
1.5	45.9	33.5	3.2	56.8	41.5
1.6	46.9	33.9	3.3	57.2	42.0
1.7	47.7	34.4	3.4	57.7	42.5

A formula with a cautious try to convert sones to decibels:
dBA = 33.22 × log (sones) + 28 with a possible accuracy of ± 2 dBA
or sones = 10^[(dBA − 28) / 33.22]

Conversion sone loudness to phons loudness level - sengpielaudio

The phon is a unit of perceived loudness level, which is a subjective measure
of the strength (not intensity) of a sound. At a frequency of 1 kHz, 1 phon is
defined to be equal to 1 dB of sound pressure level above the nominal
threshold of hearing, the sound pressure level SPL of 20 µPa (micropascals) =
2×10⁻⁵ pascal (Pa). Our ears as sensors cannot convert sound intensities and
powers, they can only use the sound pressure changes between 20 Hz and
20,000 Hz. At other frequencies, the phon departs from the decibel, but is
related to it by a frequency weighting curve (equal-loudness contour) that
reflects the frequency response of human hearing. The standard curve for
human hearing is the A-weighted curve (the equal-loudness contour for a
40 dB stimulus at 1 kHz), but others are in use.

sones	1	2	4	8	16	32	64	128	256
phons	40	50	60	70	80	90	100	110	120

The "unit" phon has been largely replaced by the dBA (A-weighted decibel),
though many old textbooks and instructors continue to use the phon.

Note: "Set the volume of the radio double as loud or half as loud." Who does
not know, how to do this, is a normal person. Psycho-acousticians are telling
us, that it has to be 10 dB level difference. Try to cool your hot coffee to the
point "half as hot" - and think it over. Your own feeling may be much different
to other persons.

An increase from 6 dB to 10 dB is perceived by most listeners as "double"
the volume. These sensations are highly subjective, meaning that different
people will hear this different ways, and "twice as loud" is a much harder thing
to guess than something.
The human perception of loudness is perceived differently from each subject. In other words it is one’s own perception of sound and it is subjective of sound
pressure level SPL.

Sound Level Comparison Chart with Factor

Table of sound level dependence and the change of the respective factor to subjective
volume (loudness), objective sound pressure (voltage), and sound intensity (acoustic power)
How many decibels (dB) level change is double, half, or four times as loud?
How many dB to appear twice as loud (twofold)? Here are all the different factors.
Factor means "how many times" or "how much" ... Doubling of loudness.

Level Change	Volume Loudness	Voltage Sound pressure	Acoustic Power Sound Intensity
+40 dB	16	100	10000
+30 dB	8	31.6	1000
+20 dB	4	10	100
+10 dB	2.0 = double	3.16 = √10	10
+6 dB	1.52 fold	2.0 = double	4.0
+3 dB	1.23 fold	1.414 fold = √2	2.0 = double
- - - - ±0 dB - - - -	- - - - 1.0 fold - - - -	- - - - 1.0 fold - - - -	- - - - 1.0 fold - - - -
−3 dB	0.816 fold	0.707 fold	0.5 = half
−6 dB	0.660 fold	0.5 = half	0.25
−10 dB	0.5 = half	0.316	0.01
−20 dB	0.25	0.100	0.01
−30 dB	0.125	0.0316	0.001
−40 dB	0.0625	0.0100	0.0001
Log. quantity	Psycho quantity	Field quantity	Energy quantity
dB Change	Loudness multipl.	Voltage multiplier	Power multiplier

For a 10 dB increase of the sound level we require ten times more power from the amplifier.
This increase of the sound level means for the sound pressure a lifting of the factor 3.16.
Loudness and volume are highly subjective. That belongs to the domain of psychoacoustics.

Is 10 dB or 6 dB sound level change for a doubling or halving of the loudness (volume) correct?
About the connection between sound level and loudness, there are various theories. Far spread is still the
theory of psycho-acoustic pioneer Stanley Smith Stevens, indicating that the doubling or halving the
sensation of loudness corresponds to a level difference of 10 dB. Recent research by Richard M. Warren,
on the other hand leads to a level difference of 6 dB. *) This means that a double sound pressure
corresponds to a double loudness. The psychologist John G. Neuhoff found out that for the rising level our
hearing is more sensitive than for the declining level. For the same sound level difference the change of
loudness from quiet to loud is stronger than from loud to quiet.
It is suggested that the sone scale of loudness reflects the influence of known experimental biases and
hence does not represent a fundamental relation between stimulus and sensation.
*) Richard M. Warren, "Elimination of Biases in Loudness Judgments for Tones"

How do you convert sound quantities to decibel values?
How many decibels is twice (double, half) or three times as loud?
How does the sound decrease with distance?

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