The purpose of this series of web pages is to introduce you to the concept of logarithmic scales, and in particular to the decibel scales commonly used in acoustics to measure loudness.  There are four web documents (including this one) that cover the necessary topics.  The second document is a mathematical review of logarithms. Even if you are confident in your mathematical skills with logs, it is worth trying some of the worked examples just to verify you know how to use your calculator!  After going through all of the documents you should take the associated online quiz.  It is your responsibility to build your understanding to the appropriate level to attempt the quiz.  I am available via e-mail, after class, and in office hours to answer your questions.  Start reviewing the material early so that you will have sufficient time to get your questions answered.  Good Luck.

Logarithmic versus Linear Scales

The purpose of the following paragraphs is to explain the difference between a linear and a logarithmic scale and to indicate conceptually why we use logarithmic scales as the measurement method in stimuli/sensation situations such as quantifying loudness.

First, we must clearly understand the idea of a linear scale.  Our experience of the everyday world makes us familiar with linear scales.  A simple example is a ruler used to quantify distance.  Each centimeter step on the ruler is equally spaced, that is, the distance between 0 cm and 1 cm is the same as the distance between 11 cm and 12 cm.  This concept may seem obvious and trivial; however, contrast this with the following example of the logarithmic nature of the the frequency separation of the octaves on a piano keyboard.

Like the centimeter spacings on a ruler, the keys on a piano are equally spaced apart; thus, playing two notes an octave apart requires the same spread of the hand at the bass end of the piano as at the treble end.  Now consider the frequency difference between the lowest and highest octaves on a piano.  We know that to raise a note an octave we must double its frequency.  Note that this is a multiplicative process rather than an additive process.  What difference does this make?  Consider the frequencies of the successive octaves of the lowest note on a piano.  On my piano at home the lowest note is A.  This A is four octaves below the A above middle C.  Some quick math should convince you that if A above middle C has a frequency of 440 Hz then the lowest A has a frequency of 27.5 Hz (divide 440 by 2 four times).  The octaves of the notes A thus have frequencies given by the sequence 27.5, 55, 110, 220, 440, 880, 1760, 3520 Hz.  Note that at the bass end the frequency difference between the notes of an octave is 55-27.5=27.5 Hz.  In contrast at the highest octave the frequency difference is 3520-1760=1760 Hz.  Even though we would perceive each interval as an octave the frequency difference between the notes in the two cases is vastly different because of the multiplicative relation between the notes in an octave.  We will see that not only are the octaves related by a multiplicative factor but also the individual semitones that make up a scale, but that's the subject of later lectures.

The key point to understand from the previous example is that the difference between a logarithmic and a linear scale rests on whether the natural steps increase in an additive fashion (linear scale) or in a multiplicative fashion (logarithmic scale).  As a quick self test identify which of the following sequences are linear and which are logarithmic.  The answers are given at the bottom of the page.

1.     1, 2.5, 4, 5.5, 7, 8.5, 10...
2.     3, 9, 27, 81, 243...
3.     10, 100, 1000, 10000 ...
4.     50, 550, 1050, 1550, 2050...

Logarithmic Nature of Loudness

You may well be wondering why we are spending so much time developing an understanding of logarithmic scales.  Well, it so happens that our senses respond in a fashion that is logarithmic (at least to a pretty good approximation).  We are concerned here primarily with the sense of hearing; however, a very similar relation holds for vision.  Sensation is the reaction of our ear/brain to an incoming stimulus (e.g. a sound).  Now it is possible to precisely measure stimuli because (in the case of sound) the size of the stimulus is related to the pressure amplitude of the sound wave.  A microphone can translate the pressure variations into an electrical signal that can be digitized and measured precisely.  In contrast, the sensation produced by a sound is a subjective quantity.  Sensation can only be determined by asking a human test subject questions such as "Which of these sounds is louder?".  These experiments are referred to as being in the realm of psychophysics, i.e. the combination of psychological testing methods with physics.

One of the early workers in the field who explored the psychophysical relation between sensation and stimulus was named Fechner.  He documented the logarithmic nature of the senses in a book The Elements of Psychophysics published in 1860.  The relation between stimulus and sensation--often called Fechner's law--states that "As stimuli are increased by multiplication, sensation increases by addition."  For example, if a series of sounds are played with quantitatively measured stimuli that increase logarithmically in the ratio 1, 2, 4 ,8, 16 our ear/brain combination would measure a loudness increase that went up in equal steps.  In other words the last sound would be perceived as roughly 5 times as loud as the first sound rather than 16 times as loud as would be implied from the quantitative measurement.  I should stress here that the multiplicative factor between steps is not necessarily 2, I just chose the number for this example.  The important point is that our perception of loudness is logarithmic not linear.

You may wonder why we would evolve with a logarithmic response to loudness.  One reason is related to the fact that our ability to hear spans an enormous range of pressure amplitudes--a vary large dynamic range.  A logarithmic response helps to compress this range so that our response to variations in weak sounds is similar to the response to variations in loud sounds.   We will explore this compression in some of the following modules.  First, we will review the mathematics of logarithms in the next section.

 

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