The Basics: Sound, Vibration, Frequency, Pitch, and Amplitude
Sound is created by the vibration of some physical object. That object may be a solid, liquid, or even a gas. All physical objects have a property called elasticity that comes into play to create vibrations. Elasticity determines how an object that is deformed by some force will tend to restore itself to its original position. A diving board is a good example of an object that is deformed by the diver and which restores itself to its original form after the diver leaves the board. The movement of the board from its rest position to the maximum deformation and then back to the rest position is called a half cycle. The elasticity of the object again comes into play as the inertia of the moving object actually causes it to move past its rest position and to deform itself again in the opposite direction of the initial force. The motion from the original position to the maximum deformation in the opposite direction and then back to the original position is another half cycle. This back and forth vibration occurs at a rate that depends on the physical makeup of the object being vibrated. A complete vibration occurs when an object completes two half cycles. The rate of motion of the object is measured in cycles per second, which is the definition of frequency - the number of cycles per second at which an object is vibrating.
The pitch of a sound is determined solely by the frequency at which the object is vibrating. This is a very important point - the pitch does NOT depend on the force used to set the object into motion. The amount of force used controls the amplitude, or loudness, of the sound, not the pitch. The pitches we use in our musical culture have evolved from years of experimentation and argument. It was not until a conference was held in London in 1939 with the blessing of the International Standards Association that the pitch of the note "A" which is used today was fixed at 440 cycles per second. This replaced the prior internationally agreed upon "A" of 435 cycles per second which had been in place since 1859. The methods of computing the pitches in our 12 tone musical scale have a complex but interesting history which I'll explain after we talk about harmonics, overtones, and intervals.
Harmonics and Overtones
When a string is set into motion the frequency or "pitch" we usually hear is referred to as the "fundamental" pitch. That pitch is also called the "first harmonic." The two points which attach the strings are the "nodes" of the fundamental vibration. What is not obvious is that each string also vibrates simultaneously at many other frequencies, each progressively higher in pitch and lower in amplitude (volume). When the fundamental pitch is created by the vibration of a string of length x, additional frequencies, called overtones or higher harmonics begin vibrating with effective lengths of x/2, x/3, x/4, x/5, etc. Each of these additional frequencies vibrate between additional imaginary nodes that are located exactly at the points on the string which are integer divisibles of the string length. The basic physical law which governs vibrating objects states that the frequency of a vibrating string is inversely proportional to the string length. What that means is if a string of length x vibrates at frequency f, than that same string if cut to length x/2 will vibrate at 2f. Since strings vibrate simultaneously at all integer subdivisions of the string length (effectively creating shorter and shorter string lengths), the vibrating string will produce additional pitches, called higher harmonics or overtones, at frequencies of 2f, 3f, 4f, 5f, and higher.
Intervals or distance between pitches in our musical scale are very important. An interval has nothing to do with the pitch of the note. As a matter of fact, the biggest problem we had to overcome before finally coming to an agreed upon system of calculating intervals was to make sure that the system we chose was NOT dependent on the starting pitch. No matter what pitch we start at there must be a consistent set of intervals between all of the tones in all musical scales that start from that tone. There is an esthetic urge to establish a method of calculating intervals between pitches in our musical alphabet that is consistent with the "natural" harmonic series. Ideally, all notes in our musical scale would be found in the natural harmonic series and perfectly tuned instruments would all complement each other's harmonic palette, swelling to a beautiful consonance as more and more notes were added to the heavenly voice. H e l l o...this is the real world calling...Sometimes reality has a way of getting into the way of idealism. Let's look at what really happens.
Basic definition of intervals
The Interval, or distance, between two pitches is defined as the ratio of the higher to the lower of the two frequencies. It is NOT the numeric difference between the two frequencies. Intervals are added (to find the interval between non-adjacent pitches) by multiplying the fractional representation of each successive interval between the first pitch and the final pitch. For example, the interval between the first harmonic (1f) and the second harmonic (2f) is found by dividing the frequency of the second harmonic by the frequency of the first harmonic: 2f divided by 1f = 2. This tells us that the ratio between the first harmonic and the second is equal to 2, or, that the second harmonic has a frequency that is twice that of the first harmonic. Any frequency that is an exact multiple or divisible by 2 of any other frequency is defined as an Octave of the original frequency (higher or lower octave). The second harmonic is the second loudest component of the various frequencies produced by a vibrating string. Since it is actually the first octave above the fundamental, it reinforces the mis-perception that only the fundamental pitch is being produced by the string.
Computing the intervals between tones in the "natural" harmonic series
The interval between the second harmonic and the third harmonic is found by dividing the frequency of the third harmonic (3f) by the frequency of the second harmonic (2f) = 3/2. This tells us that the ratio between the second harmonic and the third harmonic is equal to 3/2, or, that the third harmonic has a frequency which is 1.5 (1.5 = 3/2) times that of the second harmonic. In our system of music, a note that is 3/2 times the previous note is defined as the "5th" tone of the musical scale starting from the pitch of the previous note. If the first note was "C", the second harmonic would be "C'" (the octave), and the third harmonic would be "G" (the fifth tone of the C scale). The actual pitch of the third harmonic is the fifth tone of the musical scale above the first Octave (called the 12th).
Each harmonic that is an exact divisible or multiple of 2 (1/8, 1/4, 1/2, 2, 4, 8, 16, etc.) is some octave above or below the fundamental. After the forth harmonic (4f = the second octave), the next frequency, 5f, produces an interval from the 2nd octave of 5/4. An interval that is in the ratio of 5:4 is defined as the Major Third in our musical scale. By continuing in this manner from n = 1 to n = infinity, you produce an infinite harmonic series of tones which take the form: (n+1)/n, with progressively smaller and smaller intervals between each succeeding harmonic. Only a few of these harmonics have any correlation to the pitches in our 12 tone musical scale.
Pure Tones, Just Intonation, Pythagorean and Even-Tempered Tuning
The tones of the harmonic series are called "pure" tones. They are exact multiples of the fundamental frequency. The only interval that is pure in our (Even-Tempered) system of tuning is the octave (factors of 2). Tones that are factors of 3 (perfect 5th) are used in the Pythagorean scale, factors of 3 or 5 (Perfect 5th and Major 3rd) are contained in the system of "Just Intonation"; all other tones in the harmonic series do not equate to any of the tones in our 12 tone system of tuning. Neither Just Intonation nor the Pythagorean System of intonation are adequate for modern music because they each have problems with chromatic pitches and problems associated with modulations (changing from one "key" or "tonal center" to another). Our system of "equal-temperment" is based on 1) the acknowledgment that we cannot create a perfect system of intonation, and 2) the decision that it is best to divide the inevitable errors evenly over the space of the entire 12 tones in our musical scale.
The Mathematics of an Equal Temper
If you don't want to get into the mathematics of our equal-tempered tuning system, don't raise your temper, just take a few breaths, drink a beer or two (if you're old enough), and go back to lesson 5.
On the other Hand:
To understand Equal temperament let's look at what we're trying to do. We need to find a way to divide the difference in frequency between two octaves into 12 equal sub-divisions. In algebra we would have set up an equation like this:
Let x = the ratio of intervals that is constant between all pitches in a 12 tone scale
For simplicity, we'll choose 1 as the first frequency, which makes 2 the octave.
Let c = 1, Let c'(the first octave) = 2
Remember that earlier in the "Basic Definition of Intervals" we had said that intervals are added by multiplying the fractional representation of the interval between each sub-interval which lies between the intervals of interest. The two intervals of interest in this case are the fundamental and the octave, and we have decided to divide the scale into 12 equal components. What we want is to find some number that we can multiply by itself 12 times to bring the frequency of 1 up to the frequency of 2. Expressing this in a mathematical equation we get:
x*x*x*x*x*x*x*x*x*x*x*x = 2
There is a simpler way to express that:
x**12 = 2
Ok, we have the equation, now we have to solve it for x and here is where it gets a little more involved. There are two ways we can solve this. The first way is to take the 12th root of 2 to find x, and the second way is to use logarithms. If you have a scientific calculator you can follow along, otherwise you'll have to just trust me.
First using the 12th root of 2:
x = 2**(1/12) = 1.0595
We can check this result by multiplying 1.0595 by itself 12 times and see if we get 2:
1.0595 * 1.0595 = 1.1225
1.1225 * 1.0595 = 1.1892
1.1892 * 1.0595 = 1.2599
etc.. = 1.3348, 1.4142, 1.4983, 1.5874, 1.6818, 1.1718, 1.8877, 2.0000
we did it!
Next, using Logarithms:
12 * (log x) = (log 2)
(log x) = (log 2) / 12 = 0.0251
x = 10**0.0251 = 1.0595 ... the same result we got before.
How is it done in the real world?
Because logarithms have the neat property that adding them and then taking the "anti-log" (raising 10 to the resultant power) is exactly the same as taking x and multiplying it by itself by the number of times needed to get to any of the 12 frequencies in the scale, the basic factor used in calculating equal temperament is the log of x. This means that a frequency can be calculated by multiplying the log of x by the desired interval and the fundamental frequency. One more innovation has been added to the simplicity of this system. By multiplying the log of the interval by the factor: (1200/(log 2)) we can create a numeric value called a "cent" which turns out to be 1/100 of the distance between each semi-tone. That means that 100 cents equals one-semitone in the equal-tempered tuning system.
For example, let's calculate the frequency ratio in cents of the even-tempered 5th (g) of the musical scale "c" as follows:
1) compute how many semi-tones are between the fundamental and the fifth
c, c#, d, d#, e, f, f#, g -> that's 7 semitones
2) multiply the log of x (x=0.0251) by 7
7 * 0.0251 = 0.1756
3) multiply by the factor (1200/(log 2))
3986 * 0.1756 = 700
Of course, we could have eliminated steps 2 and 3 because we know the mathematics of this system has been devised such that there are 100 cents in one semi-tone, therefore there are 700 cents in 7 semi-tones. But, what if we want to find out the difference between the 5th in the pure harmonic series and the equal-tempered 5th? This can be calculated easily by recognizing that the factor x above was just the log of the interval in question. Here is a way to compute the difference in "cents":
1) Remember that the interval of the perfect 5th was 3/2 so find the log of 3/2.
log 3/2 = 0.1761
2) multiply by the factor (1200/(log 2))
3986 * 0.1761 = 702
We see that the equal-tempered 5th is 2 cents lower than the perfect fifth. That small a difference is very difficult to hear and can usually be ignored when tuning a guitar (although it is taken into account when tuning a piano by first tuning the 5th to no beats and then lowering the 5th to where there is about 1 beat per second). Now let's compute the difference between the pure Major 3rd, the Pythagorean Major 3rd, and the equal-tempered Major 3rd:
1) for the equal-tempered case, compute the number of semi-tones in the Major 3rd:
c, c#, d, d#, e = 4 semi-tones
That makes the frequency ratio of the equal-tempered Major 3rd equal to 400 cents.
2) Remember that the interval of the pure Major 3rd was 5/4 so find the log of 5/4.
log 5/4 = 0.0969
2) multiply by the factor (1200/(log 2))
3986 * 0.0969 = 386
We see that the equal-tempered Major 3rd is 14 cents higher than the pure Major 3rd. This difference is very noticeable and it must be considered when tuning an equal-tempered instrument such as the guitar and the piano.
3) From the Pythagorean System of intonation we see that the Major 3rd ("e" in step 4 of that document) has a ratio of 81/64, or 1.266 from the fundamental. Find the log of that number:
Log 81/64 = 0.1023
4) multiply by the factor (1200/(log 2))
3986 * 0.1023 = 408
We see that the equal-tempered Major 3rd is 8 cents lower than the Pythagorean Major 3rd.
We have seen that vibrating
strings contain a very complex mix of pitches. This mix will be used later
to add interest and "life" to our guitar playing. We have also seen how
the acoustics of music have forced us to create a system of intonation
that compensates for inevitable errors in tuning by accepting a slight
error in every interval. Although it is not necessary to understand this
level of detail in order to play the guitar, I believe doing so will help
you understand and appreciate the guitar more fully.
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