8: Pitch, Loudness and Timbre
The mechanism of human hearing does not operate as a perfect scientific instrument. In this chapter we relate a few subjective measurements of sound (things people report after hearing a sound) to objective, scientific measurements (measurements made made in a laboratory using scientific instruments). The three subjective quantities of pitch, loudness and timbre are related to laboratory measurements of a sound wave's fundamental frequency, amplitude and waveform, respectively.
Pitch, fundamental frequency, v = f λ, loudness, sound intensity (in W/m2), sound intensity level (SIL in dB), decibels (dB), just noticeable difference (loudness and frequency), timbre.
The main component that gives us the perception of the pitch of a musical note is the fundamental frequency, measured in Hertz. In the modern musical scales used today the piano note middle C has a frequency of 261.63 Hz (we will look at how scales are constructed a bit later). This chart has the frequencies of all musical notes, based on a frequency of 440 Hz for the note labeled A4 (A above middle C on the piano). As we will see in the next chapter, musical sounds are usually composed of many frequencies but the fundamental frequency gives us the basic quality we perceive as pitch.
Recall that for both sound and light, frequency times wavelength equals speed (v = f λ) but the speeds of light and sound are very different (v = 3×108 m/s for light v = 344 m/s for sound). We can use the equation to find that a 440 Hz sound wave has a wavelength of 0.78 m. In the case of light, frequency tells us the color of light. Green light for example lies in the frequency range 525 THz to 575 THz (T is tera or 1012). A 525 THz electromagnetic signal has a wavelength of 571 nm (n is nano = 10-9). The previous chart also has the wavelengths of notes on the musical scale in centimeters.
A person whose hearing is not damaged can hear frequencies as low as 20Hz and as high as 20,000Hz. But very few people today can hear this range of frequencies. Exposure to normal sounds in everyday modern life tends to do at least some damage to most peoples hearing at an early age. Hearing is also affected by normal aging.
Notice that many animals, including dogs, can hear frequencies in the ultrasonic range (above 20,000 Hz). Dog whistles used to train dogs have frequencies between about 23,000 Hz and 54,000 Hz so dogs (and many other animals) can hear them but humans cannot.
- Video/audio examples:
- Article by Peter L. Tyack in Physics Today about Human-generated sound and marine mammals.
- Christian Huygens in 1693 noticed that the fountain at Chantilly, France produced an audible pitch. He determined this was the result of the echo of sound from the fountain being reflected off of a set of nearby steps. The steps were about half a meter in depth, causing sound to return from each subsequent step with a time delay (period) of 1m divided by 340 m/s (vt = d). A set of pulses with this period will have a frequency of 340 Hz. This is sometimes called a repetition pitch.
- Sounds reflected from the steps of the Mayan ruins at Chichenitza also produce a pitch but the pitch changes over time, due to the height of the steps. Sound from the bottom of the stairs has a repetition pitch depending on just the depth of the first step. Sound reflecting off the upper steps has a different repetition pitch because of the angle (the sound travels along more of a hypotenuse connecting the edge of one step to the next, rather than the shorter distance from the edge directly to the back of the step). This lower repetition pitch also takes longer to return because of the further distance to the higher steps. The result turns a handclap into a chirp: Sounds; Spectrogram. We will talk about spectrograms later.
Generally the loudness of a sound is related to the amplitude of the sound wave; a wave with bigger variations in pressure generally sounds louder. For any type of wave the energy carried by the wave is proportional to amplitude squared. This means doubling the amplitude increases the power by a factor of four (two squared). But the amount of energy reaching your ear also depends on the frequency since a wave with more oscillations per second (higher frequency) will mean the same amplitude hits your eardrum more often. Sound intensity is defined to be the energy per second (power in Watts) reaching a given area (measured in square meters). Normal conversation has an intensity of about 10-6 W/m2.
The human ear is an amazing instrument that can detect intensities as low as 10-12 W/m2 and can hear intensities as high as 103 W/m2 (although this is loud enough to cause damage to the ear). To make this huge range easier to write down, a second scale of loudness was created called the sound intensity level, measured in decibels. The relationship between sound intensity, I measured in Watts per meter squared and sound intensity level (SIL) measured in decibels (dB), is given by SIL = 10 log (I/Io). Here log is the logarithm and Io = 10-12 W/m2 is a reference sound intensity at about the threshold of human hearing.
Here are a few examples and rules of thumb for converting intensity (W/m2) into intensity levels (in dB):
- A 10 fold increase in intensity equals an addition of 10dB. So going from a car horn to a jackhammer multiplies the intensity by 10 (1 W/m2 to 10 W/m2) but adds 10 dB to the intensity level (120 dB to 130 dB).
- A two fold increase in intensity (twice as loud in W/m2) equals and addition of 3 dB to the SIL. Suppose one trombone produces a sound level of 40dB. How loud are four trombones? Doubling the number of trombones to two adds 3dB, doubling again to four adds 3dB more so the new sound level is 46dB.
- Suppose the sound intensity is 100 W/m2. What is the sound level? I = 10 log (100/10-12) = 10 log (1014) = 10*14 = 140 dB.
- Suppose the sound level is 110 dB. What is the sound intensity? 110 dB = 10 log (I/10-12). Divide both sides by 10 to get 11 = log (I/10-12). Now take inverse log 11 (same as 1011) to get 1011 = I/-12. Multiply both sides by -12 to get 0.1 W/m2 = I.
- A Sound Conversion web site that converts between sound level, sound pressure and sound intensity.
Both sound intensity (W/m2) and sound intensity level (SIL) are numbers that can be measured precisely in the laboratory (objective). The human ear, however, is an imperfect measuring instrument. We hear better at a mid-range of frequencies than we do at very low or very high frequencies. The phon scale is a subjective measurement of loudness. This scale is arrived at by asking real humans to compare the loudness of different notes and an average is taken for many people (subjective). The units of the phon are the same as SIL units; the Decibels (dB).
The diagram below (modified from an MIT OpenCourseWare graph) relates sound intensity level (SIL, measured in dB with laboratory instruments), pressure (measured in W/m2 with laboratory instruments) and phons (human perception). A SIL of 110 dB is considered painful while a SIL of 0 is at the threashold of hearing. If our ears were the same as laboratory instruments the lines would go straight across. The phon scale and the SIL scale do give approximately the same number in dB but only for frequencies around 1000 Hz. In other words our subjective perception of loudness and the laboratory measurement agree but only for sounds with a frequency of 1000 Hz.
Notice there is a dip in all the curves between 1000 Hz and 5000 Hz indicating we are more sensitive to these frequencies and this is true for all loudness readings. For example suppose we perceive a sound at 4000 Hz to be 45 dB (phons) (labeled by a blue X in the diagram). The chart shows that at this loudness and frequency the dB reading in the laboratory is actually around 36 dB (dotted line to the SIL axis). So we perceive a sound of 36 dB (measured in the lab) as being much louder (45 dB) if it occurs at 4000 Hz. This is not surprising once you realize these are important frequencies for human speech; our hearing mechanism is built to hear human voices better than sound with much higher or much lower frequencies. This greater sensitivity around 3500 Hz is due to the tube resonance of the auditory canal (see chapter 12 for tube resonance and chapter 10 for a picture of the auditory canal).
It is also the case that intensity has an effect on perceived frequency; the same laboratory frequency will appear to be a slightly different frequency if the intensity is different. High frequencies are perceived to be a slightly higher pitch than normal if they are very loud. Low frequencies are perceived to be slightly lower than expected if they are very loud. Medium loudness doesn't change the perceived pitch very much.
The above curves are very much like the frequency response curves of microphones and speakers. No microphone has the same sensitivity to all frequencies and no speaker reproduces all frequencies equally well, as we will see in Chapter 18 on electronics. Likewise our hearing does not have the same sensitivity at all frequencies.
8c: Just Noticeable Difference
Two other differences in human hearing as compared to laboratory measurements are Just Noticeable Difference in frequency (JND Hz) and the Just Noticeable Difference in loudness (JND dB). If a group of people are asked to decide if two frequencies are the same or slightly different most people can tell if the frequency is different by 1 Hz for low frequency sounds. So the JND (Hz) for a 500 Hz sound is about 1 Hz; most of us can tell the difference between 500 Hz and 501 Hz. At frequencies above 2000 Hz however, most people start having more difficulty telling two frequencies apart. For example at 4000 Hz the JND (Hz) is about 8 Hz meaning that the two frequencies must be about 8 Hz apart before the notes sound different; we can't distinguish a 4000 Hz pitch from a 4001 Hz or even a 4007 Hz pitch. Probably for this reason no musical instrument produces fundamental frequencies above 5000 Hz; we wouldn't be able to tell if the instrument was in tune.
If a group of people are asked to decide if two tones are or are not the same loudness, it turns out that the majority of them will make different decisions depending on the frequency of the note and the initial loudness. It is easier to tell if two sounds are the same loudness when they are both very loud. For example most people can tell if the SIL level changes by 0.5 dB when the sound is at 80 dB but need a change of 1.5 dB to detect a difference if the sound is at 40 dB to start with. There is also a slight difference in the perception of loudness differences at different frequencies, which is not surprising given the difference in perception at different frequencies (the phon scale, above).
- Video/audio examples:
- An online test for JND in frequency. Take the test. Record your answers (we will compare everyone's response in class). What did you find out about your own Just Noticable Difference in frequency?
8d: Timbre (the first time)
If a trumpet and a clarinet play the same note we can still tell the difference between the two instruments. Likewise, different voices sound different even when singing the same note. Why? We now know that if they are playing or singing the same pitch the fundamental frequency is the same for both so it is not the pitch that enables us to tell the difference. These differences in the quality of the pitch are called timbre and depend on the actual shape of the wave which in turn depends on the other frequencies present and their phases. Pure tones such as from a tuning fork have a pure sine wave shape and a single frequency. However the notes from musical instruments and voices are more complex and normally contain many frequencies, as we will see in the next chapter. We will also come back to other aspects of the human perception of sounds in Chapter 10 on Perception. For now the main point is that the subjective perception of pitch, loudness and timbre are each related to more than one quantity that can be measured in the laboratory. The following diagram shows some of the connections between objective (laboratory) measurements and subjective perception.
Notice that our perception of loudness is mainly determined by the intensity of the sound (energy per second per square meter) but also is influenced by frequency and waveform of the sound. Likewise our perception of pitch is mainly determined by the fundamental frequency but also influenced by intensity and waveform. Finally, timbre is determined by waveform (which is determined by the other frequencies present and their phases) with influences from intensity and the fundamental frequency. As we will see later (in a demo in class) the duration of a sound also affects how we perceive its pitch, loudness and timbre.
Pitch is primarily determined by the fundamental frequency of a note. Perceived loudness is related to the intensity or energy per time per area arriving at the ear. Timbre is the quality of a musical note and is related to the other frequencies present. Laboratory instruments measure the fundamental frequency in Hz and sound intensity in W/m2 of a sound wave as independent properties. As we will see we can also measure the other frequencies present which determines the waveform. Our hearing mechanisms, on the other hand, perceive the subjective qualities of timbre, pitch and loudness of a musical note. The objectively measured quantities are related to the subjective perceptions but the relationship is not precise. For example we perceive loudness differently for different frequencies. Our ears are better at distinguishing differences in frequency (JND Hz) at low frequencies than high. And we distinguish loudness differences better for loud sounds (JND dB). As we will see there are several other interesting features of our hearing system that make the perception of sound different from measurements made in the lab.
- End of chapter exercises: Pitch, Loudness, Timbre.