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| Applet x
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. Installed |
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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, 7 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.
Figure .x Two sine waves. 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 here this as two distinct tones, since the octaves would fuse into one sound. Press the moving soundwave icon to hear 2 sine tones: one at 400 Hz and one at 800 Hz.
Sine Waves
Pulses change into frequencies 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? Installed
Applet x
A sine wave 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 arbitarily fix the amplitude scale to be from -1 to 1, so the sine wave 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 Hertz (abbreviated as Hz or kHz for 1,000Hz).
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, maybe what you had for breakfast! 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).
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 periods in music we are referring to how often a sonic event happens in its entirety over the course of a specified time segment. We can the 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 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 of period 1/20 or .05 of a second. Now the "thing" that repeats is one basic unit of this regularly repeating wave -- like a sine wave (there's a picture of two of them above). It's not too hard to see that the time it takes for one copy of the basic wave to move across should be proportional to the distance from crest to crest (or any two successive corresponding points for that matter). This is called the wavelength of the wave (or 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 proportional to what we refer to as the wavelength. Wavelength is a spatial measure that says how long 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 given frequency, first we invert the frequency (1/f) to get it’s period (p) then we use the simple formula:
Look at the size of the gong in this photo! It makes some very low sounds, and it is enitrely possible that the musician playing it doesn't hear the lowest portions of its total sound. Press the soundwave below 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 35 - 35 meters. Interestingly, if you sit too close to them, you can't really hear them, because you're "inside" the peaks and troughs of the waveform.
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 withing 345 meters (or even half that distance) from a vibrating object making a 1 Hz tone, you'd be standing in the middle of the actual waveform!
Pitch
This applet shows what would happen if a simple melody were transposed linearly, as opposed to logarithmically.In other words, if we perceived frequency differences rather than frequency ratios. Note that contour remains, but intuitively, we tend not to recognize the operation of addition in frequency, but multiplication. Installed
Applet x
Musicians usually talk to each other about the frequency content of their music in terms of pitch, or sets of pitches, called scales. 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)? Probably not!
Humans tend to recognize relationships, not absolutes. And when we do, those relationships (especially in the aural domain) tend to be ratiometric (or sometimes referred to as logarithmic, which is a slightly different kind of thing, and we'll talk about that below). That is, we don't perceive the difference (subtraction) of two frequencies, but the ra Color tio (division). 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.
Although pitch is directly related to frequency, it’s not the same! As we pointed out earlier, frequency is a physical, or acoustical phenomenon. Pitch is perceptual (or psychoacoustic, or psychophysical). It turns out that the way we organize frequencies into pitches is somewhat surprising: we require more and more change in frequency to produce the same perceptual change in pitch. This is called logarithmic perception.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 as compared to a change of 101/100 (much bigger!).
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 around the world has some term for this concept. In western music, the 2:1 ratio is given a special importance, and called an octave.
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.An octave is a very special frequency relationship that seems to have near universal importance, but nobody is exactly sure why! Installed
Applet x
It seems clear (though not totally unarguable) that most humans tend to organize the frequency spectrum between 20 and 20 kHz roughly into powers of 2 (octaves). 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. We sometimes call this logarithmic perception (base 2). Many theorists believe that the octave is somehow fundamental to, or 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!
In most western European music, pitches are named not by their actual frequencies, but as general categories of frequencies all a power of 2 apart. For example, ‘A’ is the name given to the pitch 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, to sound higher and brighter), those frequencies will all change to be multiples/divisors of the new absolute 'A' (maybe 441, or 442 Hz).
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. We say that the frequencies in the blue graph are rising linearly because to get each new frequency we simply add 100Hz 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 in frequency between two adjacent octaves is always increasing.
One thing we san say after all this is: 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 in determining timbre. To get some sense of this, consider that the highest note of a piano is around 4 kHz. What about the rest of the range, the almost 14 kHz 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 its 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
Figure .x Equal-loudness contours (often refered to as Fletcher-Munson curves) are curves which tell us how much amplitude is needed at a certain frequency in order to produce the same perceived loudness as a tone at a different frequency. 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 are surprising, but widely used by audio manufacturers to make equipment more efficient, and sound more realistic.
When looking at the Fletcher-Munson curves figure 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 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 for survival purposes, and more importantly, recognizing those sounds, happen in that range. Low frequencies are not too important for survival (unless of course you need to hear that herd of caribou approaching fom a few miles away). But 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. This may be a universal human thing, 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.
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Fletcher Munson example