Loudness

Sound loudness is a subjective term describing the strength of the ear's perception of a sound. It is intimately related to sound intensity but can by no means be considered identical to intensity. The sound intensity must be factored by the ear's sensitivity to the particular frequencies contained in the sound. This is the kind of information contained in equal loudness curves for the human ear. It must also be considered that the ear's response to increasing sound intensity is a "power of ten" or logarithmic relationship. This is one of the motivations for using the decibel scale to measure sound intensity. A general "rule of thumb" for loudness is that the power must be increased by about a factor of ten to sound twice as loud. To more realistically assess sound loudness, the ear's sensitivity curves are factored in to produce a phon scale for loudness. The factor of ten rule of thumb can then be used to produce the sone scale of loudness. In practical sound level measurement, filter contours such as the A, B, and C contours are used to make the measuring instrument more nearly approximate the ear.

Since "loudness" is a subjective measurement of perception, one must be careful about how much accuracy you attribute to it. But though ff is much louder than p in dynamic level, it is not 1000x louder, so one must attempt to develop a scale of loudness that comes closer to mapping your ear's perception. The "rule of thumb" for loudness is one way to attempt that.

"Rule of Thumb" for Loudness

A widely used "rule of thumb" for the loudness of a particular sound is that the sound must be increased in intensity by a factor of ten for the sound to be perceived as twice as loud. A common way of stating it is that it takes 10 violins to sound twice as loud as one violin. Another way to state the rule is to say that the loudness doubles for every 10 phon increase in the sound loudness level. Although this rule is widely used, it must be emphasized that it is an approximate general statement based upon a great deal of investigation of average human hearing but it is not to be taken as a hard and fast rule.

Why is it that doubling the sound intensity to the ear does not produce a dramatic increase in loudness? We cannot give answers with complete confidence, but it appears that there are saturation effects. Nerve cells have maximum rates at which they can fire, and it appears that doubling the sound energy to the sensitive inner ear does not double the strength of the nerve signal to the brain. This is just a model, but it seems to correlate with the general observations which suggest that something like ten times the intensity is required to double the signal from the inner ear.

One difficulty with this "rule of thumb" for loudness is that it is applicable only to adding loudness for identical sounds. If a second sound is widely enough separated in frequency to be outside the critical band of the first, then this rule does not apply at all.

While not a precise rule even for the increase of the same sound, the rule has considerable utility along with the just noticeable difference in sound intensity when judging the significance of changes in sound level.

Adding Loudness

When one sound is produced and another sound is added, the increase in loudness perceived depends upon its frequency relative to the first sound. Insight into this process can be obtained from the place theory of pitch perception. If the second sound is widely separated in pitch from the first, then they do not compete for the same nerve endings on the basilar membrane of the inner ear. Adding a second sound of equal loudness yields a total sound about twice as loud. But if the two sounds are close together in frequency, within a critical band, then the saturation effects in the organ of Corti are such that the perceived combined loudness is only slightly greater than either sound alone. This is the condition which leads to the commonly used rule of thumb for loudness addition.

Critical Band

When two sounds of equal loudness when sounded separately are close together in pitch, their combined loudness when sounded together will be only slightly louder than one of them alone. They may be said to be in the same critical band where they are competing for the same nerve endings on the basilar membrane of the inner ear. According the the place theory of pitch perception, sounds of a given frequency will excite the nerve cells of the organ of Corti only at a specific place. The available receptors show saturation effects which lead to the general rule of thumb for loudness by limiting the increase in neural response.

If the two sounds are widely separated in pitch, the perceived loudness of the combined tones will be considerably greater because they do not overlap on the basilar membrane and compete for the same hair cells. The phenomenon of the critical band has been widely investigated.

Backus reports that this critical band is about 90 Hz wide for sounds below 200 Hz and increases to about 900 Hz for frequencies around 5000 Hertz. It is suggested that this corresponds to a roughly constant length on the basilar membrane of length about 1.2 mm and involving some 1300 hair cells. If the tones are far apart in frequency (not within a critical band), the combined sound may be perceived as twice as loud as one alone.

Critical Band Measurement

For low frequencies the critical band is about 90 Hz wide. For higher frequencies, it is between a whole tone and 1/3 octave wide.

Center Freq (Hz) Critical bandwidth (Hz) 100 90 200 90 500 110 1000 150 2000 280 5000 700 10000 1200 Rossing 2nd Ed p74