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# Sound Quality Metrics: Loudness and Sones

Siemens Experimenter

How loud is a sound?

Many people think to quantify how loud a sound is by using a decibel value. Decibel (dB) values accurately represent the amplitude of a sound, but they do not accurately represent the perceived loudness of a sound.

In fact, there is a separate sound quality metric called loudness (with units sones or phons) which gives a much better representation of how humans perceive the level of a sound.

The human hearing domain

The human hearing domain is shown in Figure 1. Tracing the lower limit (the hearing threshold), it is evident that the lower limit of hearing varies with frequency. The threshold has different values at different frequencies.

Figure 1: The human hearing domain is both frequency and level dependent. The dip at about 4000Hz is because the air volume in the ear canal goes into resonance at that frequency.

Notice the dip in Figure 1 between 3000Hz-5000Hz. Humans can hear particularly well at these frequencies. In general, humans can hear sounds at lower decibel levels between 3 kHz and 5 kHz than at any other frequency.

For example, humans are able to hear a 10dB sound at 5 kHz but we cannot hear a 10dB sound at 50Hz.

Figure 2: A tone at 50Hz and 10dB is inaudible. A tone at 5000Hz and 10dB is audible.

The dB value is the same at both frequencies, but the perceived loudness is very different – one is audible, the other is inaudible. Clearly dB alone is insufficient to represent the perceived loudness of a sound.

Loudness as a Sound Metric

The loudness metric is based off of perceived loudness. Thus the metric was developed with a jury of humans (unlike decibels which is simply a math equation).

Each curve on the chart below represents a curve of equal loudness for sinusoidal tones. As frequency changes along a curve, the dB value also must change to result in an equally loud sound.

Figure 3: The curves of equal loudness originally developed by Fletcher-Munson in 1933.

To develop this metric, a jury of humans with normal hearing was gathered. The jury would listen to a tone at 1000Hz and a particular dB level. Then, a second tone would be played at a different frequency. The level of the second tone would be altered until it sounded equally as loud as the 1000Hz tone.

For example, a 1000Hz tone was played at 40dB. Then a 100Hz tone was played and the volume was adjusted until it sounded equally as loud to the jury as the 1000Hz tone. The 100Hz tone would have to be playing at 52dB to sound equally as loud as the 1000Hz tone at 40dB (see Figure 4).

Figure 4: To develop the curves, two tones were played and the levels adjusted until they sounded equally as loud.

This jury experiment was repeated until all frequencies and 13 decibel ranges within the human hearing domain were included. This resulted in several curves of equal loudness.

A curve of equal loudness represents all the frequencies and dB levels that are perceived as equally as loud. Looking below, every frequency and dB level that lies on the pink curve is perceived as equally as loud. This means that a 100Hz tone at 50dB sounds equally as loud as a 20Hz tone at 92dB. They can both be expressed as 1 sone or 40 phons.

Figure 5: The curve of equal loudness represent dB and frequency values that are perceived as equally loud.

At 1000Hz, the curves of equal loudness are 10dB, or 10 phons, apart. This means that at 1000Hz, an increase of 10dB corresponds to a doubling in perceived loudness.

The “rule of thumb” that a 10dB increase corresponds to a doubling in perceived loudness does not hold true at all frequencies. At 30Hz a doubling in perceived loudness only requires a 5dB increase. However, a doubling of sone level does correspond to a doubling of perceived loudness at all frequencies. This can be seen in Figure 6.

Figure 6: The sone scale is frequency independent. A doubling of sones equates to a doubling of perceived loudness at any frequency.

The sone scale is linear, so no matter the frequency or level, 2 sones is twice as loud as 1 sone, and 4 sones is twice as loud as 2 sones.

The Loudness Unit: Phons and Sones

Loudness level can be expressed in sones or phons, which are both units of loudness.

The phon is a unit of loudness that represents equal loudness to a 1000Hz tone. This means that if a tone is 90 phons it is equally loud as a 1000Hz tone at 90dB. If a 125 Hz tone is 47 phons, it is equally as loud as a 1000Hz tone at 47dB.

The sone is the other unit of equal loudness. The conversion between phons and sones is below:

The sone is typically preferred over the phon because it is a linear unit. This means that if the sone value triples, the perceived loudness triples. This holds true across the entire frequency range.

Phons do not scale linearly with perceived loudness. An increase of 10 phons represents a doubling in perceived loudness.

The sone scale is perhaps more intuitive.

• The difference between 60 and 70 phones is not a logical doubling in perceived loudness.
• The difference between 4 sones and 8 sones is a logical doubling in perceived loudness.

Practical Example: Vacuums

To demonstrate the value of sones versus decibels, listen to the video below. It is recommended to listen with high quality headphones rather than laptop speakers.

Sounds from two different vacuums will play. The playback order is as follows:

• Vacuum A: 4 seconds
• Vacuum B: 4 seconds
• Vacuum A: 4 seconds
• Vacuum B: 4 seconds

Listen to the recording and decide for yourself which vacuum sounds louder.

(view in My Videos)

Many listeners remark that vacuum brand A sounds louder than vacuum brand B. Many say this is due to the “loud” tone in vacuum brand A.

Figure 7: Spectrums of vacuum brand A and brand B. Brand A has a “loud” tone at around 5500Hz.

Look at the table of results below.

Figure 8: Comparison of decibel and sone results from vacuum comparison.

The results indicate:

• Vacuum brand A and brand B have virtually the same decibel level.
• Vacuum brand A has a higher sone level than vacuum brand B.

In fact, the sone level calculated considers vacuum B to be perceived as ~30% quieter than vacuum brand A! The sone calculation more closely matches human perception of the sound.

A-Weighting

It would be unfair to talk about loudness and decibels without mentioning the A-weighted decibel scale.

In an attempt to make decibels more closely match human perception of loudness, the A-weighted decibel scale was developed. Essentially, the A-weighting curve adjusts the dB level at different frequencies to make it more closely match the perceived loudness.

Depending on the frequency, the dB level is either attenuated or accentuated. This new dB value, the A-weighted dB value, is supposed to more closely match the perceived loudness.

Figure 9: A-weighting curve - dB of attenuation versus frequency.Imagine measuring a 50 Hz tone. According to the A-weighting scale below, the measured dB value should be subtracted by 30dB to more closely match human perception of the sound (red dot). Alternatively, imagine measuring a 5000 Hz tone. According to the A-weighting scale, the measured value does not need to be adjusted (green dot).

Figure 10: The A-weighting scale attenuates or accentuates dB measurements based on frequency. Red dot is 30 dB of attenuation. Green dot is 0 dB of attenuation.

If the A-weighted curve is compared to the human hearing threshold, you can see that it attempts to follow the shape of the threshold, but due to limitation of creating filters in analog circuits, the curve had to be simplified. Furthermore, the A-weighting curve does not change as a function of sound level, like the equal loudness curves do.

Figure 11: The A-weighting curve is much simpler than the equal loudness curves.The A-weighting decibel value is more representative of perceived loudness than linear decibels. However the loudness metric is still the superior qualifier of perceived loudness due to the more detailed adjustment of the equal loudness curves.

If we added the A-weighted decibel value to the vacuum results chart from above it would appear as follows:

Figure 12: A results comparison using dB, dB(A), and sones.

So, the A-weighted decibels showed more of a difference than the linear decibels, but still not as much of a difference as sones.

Loudness Standards

There are different standards for the calculation of sones that can be used:

• Stevens Mark VI and VII – Professor Stanley Smith Stevens of Harvard published his Mark VI and Mark VII standards for sone calculations in 1961 and 1975 respectively.
• ISO532B – Standard was first published in 1975 and incorporated masking calculations proposed by Professor Eberhard Zwicker.
• DIN 45631 – The German standards organization, Deutches Insitit fur Normung (DIN), also maintains a commonly referenced standard for the calculation of loudness.

These methods outlined in these standards can produce different results for loudness. All methods for calculating loudness are available in Simcenter Testlab.

Calculating Loudness in Simcenter Testlab

There are two main places to calculate loudness in Simcenter Testlab (previously called LMS Test.Lab): in the display and in throughput processing.

Calculating Loudness in a Display

1. In a display with a sound spectrum plotted, right click on the legend and select “Options”.

Figure 13: Select “Options…”.

2. In the “Calculated Content Tab”, select the type of loudness calculation to be performed. Available calculations include ISO532B Free Field, ISO 532B Diffuse Field, Stevens 6, and Stevens 7. Click the “Add to Selection” arrow to calculate the metric. Check on the “Unit Label” box to display the unit in the legend.

Figure 14: Choose the metric to be calculated.3. The sone value for the sound spectrum will be displayed.

Figure 15: The value and unit will be displayed in the legend.

Calculating Loudness in Signature Throughput Processing

To calculate loudness in Signature Throughput Processing, turn on three add-ins: ANSI-IEC Octave filtering, Signature Throughput Processing, and Sound Quality Metrics.

Figure 16: ANSI-IEC Octave filtering requires 23 tokens. Sound Quality Metrics requires 33 tokens. Signature Throughput Processing requires 36 tokens.

In Simcenter Testlab Time Data Processing worksheet, under “Section Settings”, turn on the loudness metrics to be calculated. Note that time varying loudness according to DIN 45631, the most popular loudness formulation, is only available in Throughput Processing.

Figure 17: Throughput processing -> Section Settings -> Psychoacoustic Metrics

More information about processing Time Varying Loudness according to the DIN standard, and calculating a N10 value from it, see the Forum post: How do I calculate N10 Time Varying Loudness in Simcenter Testlab?.

A specific loudness spectrum can also be calculated.  See the Forum post: How to Calculate a Specific Loudness spectrum in Simcenter Testlab?

Conclusion

The loudness metric is a great way to more closely match people’s perception of hearing. Decibels, and even A-weighted decibels, do not adjust enough for the ears dependency on level and frequency to give a representative value of the perceived loudness.

Questions? Email jacklyn.kinsler@siemens.com, post a reply, or contact Siemens PLM GTAC support.