34 SECTIONAL ADDRESSES 



frequency, where the greatest intensity that can be tolerated (the threshold 

 of pain) is some lo million million (lo^^) times that corresponding to 

 the threshold of hearing. In such circumstances, we turn, as always, to 

 a geometrical rather than an arithmetical scale, and the unit adopted 

 for the purpose is the bel, which is a ratio signifying a lo-fold increase 

 in intensity, power, or energy. Two bels signify a loo-fold increase, 

 three bels a looo-fold increase, and so on. 



Equipped with such seven-league boots, and starting at a zero approxi- 

 mating to the threshold of hearing, we can traverse the entire auditory 

 intensity range for a medium-frequency note, in as few as thirteen 

 geometrically progressive steps. But the steps are too big for practical 

 convenience, and so it is usual to speak of a range of 130 decibels, which 

 provides a serviceable energy scale. Arithmetically, a decibel (db) 

 denotes approximately a 5/4 energy increase (i.e. antilog i/io), two decibels 

 a (5/4)^ increase, three decibels a (5/4)' = a 2-fold increase, ... 10 

 decibels a (5/4)^° = a lo-fold increase, i.e. a bel. More generally, two 

 similar sounds of intensities / and /q and corresponding acoustical pressures 

 p and pQ are said to differ in intensity by n decibels when 



« = 10 log 10 {Ijh) 

 or « = 20 log 10 (plPn) 



If /q or pQ corresponds to some selected zero, then n becomes the number 

 of decibels above that zero level. 



Thus provided with an acoustical intensity scale, we can proceed to 

 set up a loudness scale which is based on the accepted ability of the average 

 individual to compare and match loudness. To this end (just as in 

 photometry we make use of a standard candle) we need a standard sound ; 

 and for the purpose a pure reference tone is chosen which, on the British 

 Standard Scale, has a frequency of 1000 cycles per second. We also 

 require a zero of loudness at or near the threshold of hearing, and this is 

 arbitrarily adopted as corresponding to a pressure of 0-0002 dyne per 

 sq. cm. If now we operate the reference tone by successively increasing 

 decibel steps of energy, the associated changes of loudness are expressed 

 in numerically identical steps on a scale ofphons. That is, if the reference 

 tone is excited by an intensity of n decibels above the zero, the loudness 

 is n phons. The equivalent loudness of any other sound or noise is 

 evaluated by matching it by ear under specified conditions against the 

 suitably adjusted reference tone, the numerical value of the latter in 

 phons then giving the equivalent loudness of the sound to be measured. 

 Thus by this procedure we have set up a subjective scale of equivalent 

 loudness, the unit being the phon. 



It happens that a phon corresponds roughly to the smallest difference 

 of loudness which can be detected by alternate listening, in the case of a 

 sound of medium frequency and moderate loudness. Experience shows, 

 too, that for many loud noises of common occurrence the loudness level 

 in phons is approximately equal to the intensity level in decibels — a 

 convenient relation for many purposes. 



A number of different zeros of loudness have unfortunately been 

 employed in the past, e.g. i millidyne per sq. cm., which results in 



