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| This applet lets you hear and see what different frequencies and amplitudes look/sound like. Try to see if you can predict what a tone will sound like before you change it.
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Frequency, Pitch and Intervals
What is frequency? Essentially, it’s a measurement of how often a given event repeats in time. If you subscribe to a daily paper, then the frequency of paper delivery could be described as once per day, seven times per week. When we talk about the frequency of a sound, we’re referring to how many times a particular pattern of amplitudes repeats during one second.
Not all waveforms, or physical vibrations (in fact almost none of them!), repeat exactly. But many vibratory phenomena, especially those in which we perceive some sort of pitch, repeat more or less regularly. If we assume that in fact they are repeating, we can measure the rate of repetition, and we call that the waveform’s frequency.
As we'll discuss below, we measure frequencies in cycles per second, or hertz (Hz). Press the moving soundwave icon to hear 2 sine tones: one at 400 Hz and one at 800 Hz.
Figure .x Two sinewaves. The frequency of the red wave is twice that of the blue one, but their amplitudes are the same. It would be difficult or impossible to actually hear this as two distinct tones, since the octaves fuse into one sound.
Sinewaves
A sinewave is a good example of a repeating pattern of amplitudes, and in some ways the simplest. That's why they're sometimes referred to as simple harmonic motion. Let's arbitrarily fix the amplitude scale to be from -1 to 1, so the sinewave goes from 0 to 1 to 0 to -1 to 0. If the complete cycle of the sinewave’s curve takes one second to occur, then we say that it has a frequency of one cycle per second (cps), or one Hertz (abbreviated as Hz or kHz for 1,000Hz. By the way, this measure is named after a scientist, not a rental car agency).
The frequency range of sound (or more accurately, human hearing) is usually given as 0Hz to 20kHz, but our ears don’t fuse very low frequency oscillations (0-20 Hz, called the infrasonic range) into a pitch. Low frequencies just sound like beats. These numbers are a bit fuzzy: some people hear pitches as low as 15 Hz, others can hear frequencies significantly higher than 20 kHz. A lot depends on the amplitude, the timbre, and other factors. The older you get (and the more rock n’ roll you listened to!), the more your ears become insensitive to high frequencies (a natural biological phenomenon called presbycusis).
Installed
Pulses change into fused sounds with a discernible pitch at a certain rate, which is somewhere between 15-25 Hz. This applet lets you move a pulse's frequency in and out of that range. Where do you hear it fuse into a pitch?
source
lowest freqency (Hz)
highest frequency (Hz)
piano
27.5
4,186
female speech
140
500
male speech
80
240
compact disc
0
22,050
human hearing
20
20,000
The Period of a Waveform
When we talk about frequency in music we are referring to how often a sonic event happens in its entirety over the course of a specified time segment. For a sound to have a perceived frequency, though, it must be periodic (repeat in time). The period of an event is the length of time it takes to occur, so it's clear that the two concepts (periodicity, frequency) are related, if not pretty much equivalent.
The period of a repeating waveform is the length of time it takes to go through one cycle. The frequency is sort of the inverse, how many times it repeats that cycle per unit time. We can understand the periodicity of sonic events just like we understand that the period of a daily newspaper delivery is one day — but one period per day would be a very low sonic frequency (can you compute what it would be in cycles per second?).
Since a 20Hz tone by defintion, is a cycle that repeats 20 times a second, then in 1/20th of a second, one cycle goes by, so it has a period of 1/20 or .05 of a second. Now the "thing" that repeats is one basic unit of this regularly repeating wave — like a sinewave (at the beginning of this section there's a picture of two of them together). It's not too hard to see that the time it takes for one copy of the basic wave to reoccur (or move through whatever medium it is in) is proportional to the distance from crest to crest (or any two successive corresponding points for that matter).
This distance is called the wavelength of the wave (or of the periodic function). In fact, if you know how fast the wave is moving, then it is easy to figure out the wavelength from the period.
Physically, the period is inversely proportional to the wavelength. Wavelength is a spatial measure that says how far the wave travels in space in one period. We measure it in distance, not time. The speed of sound (s) is about 345 meters/second. To find the wavelength (w) for a sound of a given frequency, first we invert the frequency (1/f) to get it’s period (p) then we use the simple formula:
Press the soundwave above to hear a gong. (The musicians above are from STSI Bandung a music conservatory, in West Java, Indonesia participating in a recording session for a piece called mbuh by the contemporary composer Suhendi. This recording can be heard on the CD Asmat Dream: New Music Indonesia Lyrichord Compact Disc # 7415.)
Figure .x Very low musical sounds can have very long wavelengths: some Central Javanese (from Indonesia) gongs vibrate at around 8 - 10 Hz, and as such, their wavelengths are on the order of 30 - 35 meters. Look at the size of the gong in this photo! It makes some very low sounds.
Using the above formula, we find that the wavelength of a 1Hz tone is 345 meters, which makes sense, since a 1Hz tone has a period of 1 second, and sound travels 345 meters in one second! That's pretty far, until you realize that since these waveforms are usually symmetrical, if you were standing, say, at 172.5 meters from a vibrating object making a 1 Hz tone, and right behind you was a perfectly reflective surface, it's entirely possible that the negative portion of the waveform might cancel out the positive and you'd hear nothing! This is a rather extreme (and completely hypothetical) example, but it is true that wave cancellation is a common physical occurrence, though it depends on a great many parameters (but in fact, it's always happening in some way).
Pitch
Musicians usually talk to each other about the frequency content of their music in terms of pitch, or sets of pitches, called scales (actually, musicians usually talk to each other about money, or when they're getting fed, but let's pretend that they exist on a slightly higher plane). You’ve probably heard someone mention a g-minor chord, a blues scale, or a symphony in C, but has anyone ever told you about the new song they wrote with lots of 440s in it? We hope not!
Humans tend to recognize relative relationships, not absolute physical values. And when we do, those relationships (especially in the aural domain) tend to be logarithmic. That is, we don't perceive the difference (subtraction) of two frequencies, but the ratio (division).
This means that it is much easier for most humans to hear or describe the relationship or ratio between two frequencies than it is to name the exact frequencies they are hearing. And in fact, for most of us, the exact frequencies aren’t even very important — we recognize "Row, Row, Row, Your Boat" regardless of what frequency it is sung at, as long as the relationships between the notes are more or less correct. The common musical term for this is transposition — we hear the tune correctly no matter what key it's sung in.
In other words, if we perceived frequency differences rather than frequency ratios. Note that the contour (the "up and down-ness" of the pitches) remains, but intuitively, we tend not to recognize the operation of addition in frequency, but multiplication. That's what logarithmic perception is all about. Installed
This applet shows what would happen if a simple melody were transposed linearly, as opposed to logarithmically.
Although pitch is directly related to frequency, it’s not the same! As we pointed out earlier, and similar to what we saw in Section 1.2 when we discussed amplitude and loudness, frequency is a physical, or acoustical phenomenon. Pitch is perceptual (or psychoacoustic, cognitive, or psychophysical). It turns out that the way we organize frequencies into pitches is somewhat surprising (but no surprise if you understood what we talked about in the previous section): we require more and more change in frequency to produce the same perceptual change in pitch.
Once again, this is all part of that logarithmic perception "thing" we've been yammering on about, because the way we describe that increase is by logarithms and exponentials. Here's a simple example (we promise not to mention Bubba again!): the difference to our ears between 101 and 100 Hz is much greater than the difference between 1001 and 1000 Hz. We don't hear a change of 1 Hz for each, we hear a change of 1001/1000 (= 1.001) as compared to a much bigger change of 101/100 (= 1.01).
Intervals, Octaves
So we don’t really care about the linear, or arithmetic differences between frequencies, we are almost solely interested in the ratio of two frequencies. We call those ratios intervals, and almost every musical culture in the world has some term for this concept. In western music, the 2:1 ratio is given a special importance, and called an octave.
It seems clear (though not totally unarguable) that most humans tend to organize the frequency spectrum between 20 and 20 kHz roughly into octaves, which means powers of 2. That is, we perceive the same pitch difference between 100 and 200 Hz as we do between 200 and 400 Hz., 400 and 800 Hz, and so on. In each case, the ratio of the two frequencies is 2:1.We sometimes call this base-2 logarithmic perception. Many theorists believe that the octave is somehow fundamental to, or innate and hard-wired, in our perception, but this is difficult to prove. It's certainly common throughout the world, though a great deal of approximation is tolerated, and often preferred!
An octave is a very special frequency relationship that seems to have near universal importance, but nobody is exactly sure why (though a lot of people think they know)! Installed
An octave quiz. Do you prefer your octaves pure, or just a little bit "off"? Most people seem to "prefer" octaves (2/1 frequency ratios) that are just a little bit wide.
In almost all musical cultures, pitches are named not by their actual frequencies, but as general categories of frequencies in relationship to other frequencies, all a power of 2 apart. For example, A is the name given to the pitch on the piano or clarinet with a frequency of 440Hz as well as 55Hz, 110Hz, 220Hz, 880Hz, 1760Hz, and so on . The important thing is the ratio between the frequencies, not the distance — 55Hz to 110Hz is an octave that happens to span 55Hz, yet 50Hz to 100Hz is also an octave, even though it only covers 50Hz. But, if an orchestra tunes to a different A (as most do, nowadays, for example, to middle A = 441Hz or 442Hz to sound higher and brighter), those frequencies will all change to be multiples/divisors of the new absolute A.
We say that the frequencies in the blue graph are rising linearly because to get each new frequency we simply add 110Hz to the last frequency — the change in frequency is always the same. However, to get octaves we must double the frequency, meaning that the difference (subtractively) in frequency between two adjacent octaves is always increasing. There is no permission for this.
IMAGE .x The red graph shows a series of octaves starting at 110 Hz (an A) — each new octave is twice as high as the last one.The blue graph shows linearly increasing frequencies, also starting at 110Hz.
One thing is clear however: to have pitch, we need frequency, and thus periodic waveforms. Each of these three concepts implies the other two. This is very important when we discuss, below, how frequency is not just used for pitch, but also in determining timbre. To get some sense of this, consider that the highest note of a piano is around 4kHz. What about the rest of the range, the almost 18kHz of available sound? It turns out that this larger frequency range is used by the ear to determine a sound’s timbre.
We will discuss timbre in the next section.
Before we move on to timbre though, we should mention that pitch and amplitude are related. When we hear sounds, we tend to compare them, and think of their amplitudes, in terms of loudness. The perceived loudness of a sound depends on a combination of factors, including the sound’s amplitude, and frequency content. For example, given two sounds of very different frequencies, but at exactly the same amplitude, the lower frequency sound will often seem softer! Our ear tends to amplify certain frequencies and attenuate others.
Fletcher-Munson Curves
For example, for a 50 Hz tone to sound as loud as a 1000 Hz tone at 20 dB, it needs to sound at about 55 dB. These curves are surprising, and tell us some important things about how our ears evolved. While these curves are the result of rather gross data generalization, and will of course vary depending on the different sonic environments to which listeners are accusomed, they also seem to be rather surprisingly accurate across cultures. Perhaps they represent something that is hardwired rather than learned. They are widely used by audio manufacturers to make equipment more efficient, and sound more realistic. There is no permission for this graphic.
Figure .x Equal-loudness contours (often refered to as Fletcher-Munson curves) are curves which tell us how much intensity is needed at a certain frequency in order to produce the same perceived loudness as a tone at a different frequency. They're sort of loudness isobars. If you follow one line (which meanders across intensity levels and frequencies), you're following an equal loudness contour.
When looking at the figure for the Fletcher-Munson curves note the way the curves start high in the low frequencies, dip down in the mid-frequencies, and swing back up again. What does this mean? Well, humans need to be very sensitive to the mid-frequency range. That's how, for instance, you can tell immediately if your mom's upset when she calls you on the phone (which cuts off everything above around 7 kHz.).
Most of the sounds we need to recognize for survival purposes occur in the mid-frequency range. Low frequencies are not too important for survival (unless of course you need to hear that herd of caribou approaching from a few miles away). The nuances and tiny inflections in speech and most of sonic reality tend to happen in the 500-2k range, and we have evolved to be extremely sensitive there (though it's hard to say which came first, the evolution of speech of the evolution of our sensitivity to speech sounds. Ouch... our heads are starting to hurt!).
This mid-frequency range sensitivity is probably a universal human trait, and not all that culturally dependent. So, if you're travelling on a trip in Outer Slobovia, you may not be able to understand the person at the table in the cafe next to you, but if you whistle the Fletcher-Munson curves, you'll both have a great time together.
Installed
Fletcher Munson Example