Here is an example of something in nature which is truly simple and clearly mathematical. The Pythagoreans seem to have read huge significance into it. It is a fact about music that consonant intervals -- musical pitches that sound harmonious when you hear them together -- are simply related to ratios of integers, like 2/1, 3/2, and 4/3, as you can demonstrate on a guitar. Tune two strings so that they have the same pitch. (If the guitar starts out tuned in the usual way, lower the pitch of a higher string to match a lower one, so as not to risk breaking the string). Then place your finger firmly down on one of the two unison strings to make it effectively shorter, and the pitch higher, and listen to the two-note chord you get by playing both strings together. If you place your finger in the middle of one string, for example, dividing it into two equal parts, and play both strings, you hear the octave. The lengths of the two strings, once you put your finger down, are in the ratio 2:1, and the octave interval sounds harmonious: it is consonant.

If you place your finger 1/3 of the way along one string, then what is left is 2/3 of the string, and the ratio of the longer string to the shorter is 3/2. When you play it, you hear another consonant interval, the fifth. (It is called that because the higher pitch is the fifth in the major scale, do-re-mi-fa-so-la-ti-do. That is, `so,' if the lower pitch is `do.')

If you place your finger 1/4 of the way along one string, then what is left is 3/4 of the string and the ratio of the longer string to the shorter is 4/3. When you play it, you hear another consonant interval, the fourth. (The higher pitch is the fourth in the major scale, `fa'.)

You might be thinking that this is not really something in nature, but just a conventional human choice. After all, what is so special about the major scale? And who is to say what is harmonious? What is consonant to me may be dissonant to you. Maybe it is just a convenience for guitar makers to place the frets at these positions. But no! Nature really does choose these positions as special.

You can demonstrate these special positions on a guitar with the phenomenon of harmonics. If you play the violin, you are familiar with harmonics already. On a guitar, using the lowest string (so that the harmonics won't be too high in pitch), place your left index finger lightly at the midpoint of the string, and pluck the string. You hear a pitch one octave above the open string, but this is not surprising, because that is also what you would hear if you put your finger down hard. Now, plucking the string repeatedly with your right hand, gradually slide your left index finger, still lightly on the string, toward the end of the guitar neck. You will hear a sort of muffled `plunk' sound with each pluck, and the pitch will go steadily down as you move your finger, because the string is, in effect, becoming longer. But when you reach the special position which is 1/3 of the way along the string, you will suddenly hear a high, ringing pitch mixed with the low muffled one. This is the third harmonic of the string. If you are playing the low E string, the pitch is that of the open B string (second from the top string of the guitar), as you can quickly check by plucking it. Continuing to move your finger lower, you hear higher harmonics. The fourth harmonic, which sounds when your finger is 1/4 of the way along the string, is two octaves above the low E string, and should agree with the high E string (top string). Get a guitar and try to identify these and the next two harmonics.

The surprising phenomenon of harmonics is something nature just does. Harmonics occur, whether we want them to or not, when the index finger is at positions along the string which are associated with the integers, so there is obviously something mathematical going on. These positions include the positions previously identified as `harmonious,' giving notes of the major scale.

To the Pythagoreans, this observation was deeply significant, apparently, and modern physicists cannot help being a little awestruck at their intuition. Significant indeed! With only slight reinterpretations, to fit the topic at hand, this observation about harmonics has very broad significance in continuum mechanics, electromagnetic theory, quantum mechanics, statistical mechanics -- basically all of physics.

Here is how we would describe the vibrating string in modern terms. The figure below shows various ways a string, fixed at its two ends, can vibrate, called normal modes of the string. Notice that each normal mode is distinguished by evenly spaced nodes, places where the string doesn't actually move. An arrow points to one of the nodes of the rightmost normal mode. Each normal mode is labeled underneath by the node-to-node distance in the pattern, taking the length of the string to be L. As you see, the node-to-node distances form a regular progression: L, L/2, L/3, L/4,... as you move from one normal mode to the next. The figure shows only the spatial pattern of the vibration. It cannot show the frequency of the vibration, but that is also indicated underneath. The leftmost mode, called the fundamental mode, vibrates with the lowest frequency, or fundamental frequency, called f1 here. It depends on the tension of the string: you tune the string by adjusting the tension. As you increase the tension, the fundamental frequency goes up, but suppose we have it where we want it, and we don't touch the tuning peg. According to the figure, the second normal mode vibrates with the frequency 2f1, i.e., the frequency is twice the fundamental, the third normal mode vibrates with the frequency 3f1, i.e, three times the fundamental, etc. So the freqencies also form a regular progression: f1, 2f1, 3f1, 4f1, ... as you move from one normal mode to the next.

Now what does all this mean in an example? You might tune the string to low E, if it isthe lowest string on a guitar. This means that the fundamental frequency is about f1 = 165 Hz (read `Hertz'), or 165 cycles per second, i.e., the string moves back and forth 165 times every second. The fundamental frequency determines the musical pitch that you hear. If the pitch sounds flat (too low), you should increase the tension on the string, because the frequency must come up. Of course you have no direct way of knowing that it is vibrating at 165 Hz, but if it sounds like low E, then that is what it must be doing. Now the second normal mode would be at twice that frequency, namely 330 Hz, if you could hear it -- and you can hear it. That is just what happens when you put your finger lightly on the midpoint of the string. By not allowing the string to move at the midpoint, you are requiring the vibration to have a node there, which rules out the fundamental mode but still allows the second mode (or second harmonic). We have already noticed that the second harmonic sounds an octave above the fundamental, but now we know from the theory of normal modes that this is what happens when you go from f1 to 2f1. So going up an octave in pitch means doubling the frequency.

There is another observation we can make about the second harmonic. If you put your finger down hard on the midpoint of the string, thereby shortening the string length to L/2, then the fundamental mode of the string of length L/2 looks just like half of the second normal mode of the string of length L, namely the part on one side of the node, as shown in the figure above. And we have already noticed that in either case the frequency is the same: an octave above the fundamental frequency of the long string, namely 2f1. The physical vibration is different in the two cases, because when the finger is down hard, only half the string vibrates, but when it is down lightly to select the second harmonic, the whole string vibrates. It seems not to matter, though. Apparently you can clamp a node without changing the frequency of the part left to vibrate. Since the node doesn't move anyway, it doesn't matter if it is clamped or not.

Let us use these ideas to understand the third harmonic. Its frequency is 3 x 165 Hz = 495 Hz, and we select it by putting a finger lightly at the distance L/3 from the end of the string. That is the same place where we hear the fifth above E, namely B, if we put a finger down hard. But the third normal mode of E, if you look only at 2/3 of it, is exactly the second normal mode of B, which is B an octave higher. So twice the frequency of B is three times the frequency of E, 3f1, and thus the frequency of B is (3/2)f1. We have found what it means to go up a musical fifth: it means to increase the frequency by the factor 3/2. This is just the ratio that Pythagoras associated with the musical fifth!

Problem: analyze the fourth harmonic in the same way as the preceding paragraph to show that going up a musical fourth means increasing the frequency by the factor 4/3. Write out the argument, with sketches of the relevant modes.

Problem: a musical fourth above E is A. What is the frequency of the lowest A you can play on a guitar?

Problem: the fifth harmonic has a node at the position where you play a major third, A flat on the E string, for example. By what factor does the frequency increase when you go up a major third? Make sketches of the relevant modes and write a careful argument.

We are seeing that the ratios Pythagoras associated with a musical interval by looking at two lengths are just the ratios of the two fundamental frequencies, with this difference: that when the length of the string goes down, its fundamental freqency goes up (by the same factor). Thus the product, fundamental frequency times length, does not change. This is a very simple way to say what happens when you shorten the string by fingering it. When the length becomes (2/3)L, the fundamental frequency becomes (3/2)f1, and the product is Lf1. When the length becomes (3/4)L, the fundamental frequency becomes (4/3)f1, and the product is still Lf1. This is just how you play a musical fifth, or a musical fourth on a guitar string. Shorten the string, raise the fundamental frequency.

We are also beginning to see why certain intervals are naturally harmonious. If you play two strings together which differ in pitch by a musical fifth, then their frequencies are f1 and (3/2)f1, where f1 refers to the lower string. That means they have harmonics which literally agree: the third harmonic of f1 is identical to the second harmonic of (3/2)f1. Why this agreement in the harmonics sounds good to us is still a question of how we perceive and interpret sound, but at least there is something objective and natural about it. And, in fact, those harmonics are truly present in the sound. The normal modes are just certain special ways a string could vibrate, but when you pluck a guitar string or bow a violin string, you generally get the string moving in a more complicated way. These more general motions of the string are combinations of all the normal modes, and if you could listen selectively for different frequencies, you would hear that all the harmonic frequencies are present, although the contribution of the fundamental frequency is the loudest (unless you get rid of it by some trick, like forcing a node to occur). The particular combination of harmonics which you hear gives the instrument its characteristic quality of sound. By plucking or bowing the string differently, a musician can change this quality, even while leaving the pitch the same.

An interesting application of these ideas is used in tuning the viola da gamba (a fretted instrument which looks like a cello, popular in the Renaissance, and still played today because such wonderful music was written for it). The viol has six strings, and the middle two are supposed to differ by the interval of a third. That is, if the lower one is f1, the higher one is (5/4)f1. But for reasons that will emerge soon, the viol is not tuned to this so-called `perfect' third. It should be `off' by a certain definite amount. The gambist achieves this in the following way. If the strings were really a perfect third apart, then the 5th harmonic of the lower one and the 4th harmonic of the upper one would agree. If they differ slightly, then the sounds `beat:' there is a kind of difference frequency. The gambist bows across both strings together, and while everyone else hears the fundamental frequencies of the two strings, she is listening to a wah-wah-wah buried in the sound and about two octaves higher! She adjusts the wah-wah-wah to be about 1 Hz, i.e. one `wah' per second. Now it is tuned the way she wants it! Piano tuners use similar tricks.

Why is a viola da gamba deliberately tuned `wrong'? It turns out that most instruments are. Guitars and pianos are too (but not violins). The `perfect' intervals cannot be used consistently! Compromises have to be made. The reason follows from things we have said already, but is a little bit subtle. One place where it simply does not work to compromise is on octaves. Notes that are an octave apart sound so nearly the same that they are even given the same name. We say `middle C' and `high C', but they are both `C'. That is probably because the second harmonic, which is the octave, is strongly present when you pluck a string. The C above middle C must be at exactly twice the frequency. Octaves must be perfect octaves, or else they sound terrible.

We can think of the notes of the major scale, do-re-mi-fa-so-la-ti-do, as lying on a circle, because the scale ends where it began, on `do', which has the same name, even if it is an octave higher. This circle is shown below, where it includes all the notes on a piano keyboard, white keys and black keys, 12 distinct notes in all. An example of a major scale is C-D-E-F-G-A-B-C. These are exactly the white notes on the piano, starting with C. This scale skips the five black keys, which are not in the scale, but are still available as `accidentals', indicated by a sharp or flat. Each step in the circle from a note to an adjacent note is called a half step. Thus from C to C sharp is a half step. Two half steps make a whole step. Thus from C to D is a whole step. A major scale consists of whole steps and half steps in a definite order: starting anywhere, go up by whole-whole-half-whole-whole-whole-half. You can easily check that C-D-E-F-G-A-B-C follows this pattern, and hence is a major scale. Another one is F-G-A-B flat-C-D-E-F. If you played it on a piano, it would use one black note.

Problem: Write the notes of the major scale which begins on A (it is conventional to indicate the black notes with sharps in this case).

There are other scales in use, of course. The chromatic scale is all half steps, and just goes around the circle using every note. On the piano, it just means playing each successively higher note, regardless of whether it is white or black. A minor scale goes up by whole-half-whole-whole-half-whole-whole. An example is the white notes starting on A.

A fifth is 7 half steps, and a fourth is 5 half steps. A major third is 4 half steps, and a major second is 2 half steps (this is just repeating in a different way what are some of the intervals in the major scale). This seems to offer a way to assign frequencies to intervals. For example, if we start from C, letting it have some convenient frequency f1, then going up a perfect fifth to G, we assign it (3/2)f1. Going up a perfect fifth from there, to D, means multiplying that by 3/2, giving (9/4)f1. This is D in the next octave, so we bring it back down to D above our original C by dividing by 2. This means D is (9/8)f1. But C to D is a whole step, so we have found that going up a whole step means multiplying the frequency by 9/8. Now the trouble begins! To find the frequency of E, we just raise D by a whole step, which means multiply by 9/8, giving (81/64)f1. But in the problem on the fifth harmonic, you should have found that a major third, as for example C to E, meant multiplying by 5/4, which is 80/64, not 81/64. Using the natural,`perfect' intervals leads to inconsistencies. We have two different frequencies, each claiming to be the `perfect' one for E.

In fact neither of the two major thirds found above is quite right, given the unavoidable requirement that octaves must be perfect. Going up three major thirds (each 4 half steps) should be the same as going up an octave (12 half steps). But if we use perfect major thirds (i.e, a factor of 5/4 in frequency), then multiplying three times gives (5/4)(5/4)(5/4)=125/64, which is less than an octave, which would be 128/64 (or 2, as we usually say). Thus a perfect third is actually too small! On the other hand, (81/64)(81/64)(81/64) is about 2.027, which is larger than 2, so the other third, based on perfect whole steps, is too large! The whole step, which gave us this third, is therefore also too large, and the perfect fifth, which gave us the whole step, is also too large. Therefore not all fifths can be perfect fifths -- that would be inconsistent with perfect octaves!

This problem has been dealt with in various ways. Through the Renaissance, keyboard instruments like organs and harpsichords were tuned so that some of the fifths were perfect and other fifths were smaller. There were various systems for doing this. They all had more than one kind of whole step, for example 9/8 for some whole steps and 10/9 for others. Now if you are playing in the key of C, you will often be hearing fifths like C-G and F-C, and C is a key which is frequently used, so it would be natural to make those fifths perfect. Not all fifths can be perfect, but fifths like F sharp-C sharp don't have to be very good ones -- we are never going to hear them anyway! Once the instrument is tuned, therefore, certain strange intervals are built into it. Now if, for some reason, a piece is written in the key of F sharp, say, we will be hearing those strange intervals all the time. It will sound quite different than it would if it were transposed into the key of C. So in these tuning systems each different major scale has its own peculiar character, different from each other one. There are different kinds of whole steps, and depending on the key, they come at different places in the scale. Some of these keys are quite dissonant. Mozart, in his operatic music, exploits these differences to convey various emotions!

One rarely gets to hear such music played in the original tunings, however, because nowadays keyboard instruments are well tempered. They are tuned so that each half step is the same factor in frequency as each other half step. Since 12 half steps make an octave, the appropriate factor x must satisfy x to the twelfth power is 2, i.e., x is the twelfth root of 2, or, in approximate decimal terms, x = 1.0595 .... In the well tempered scale, the fifth, which is 7 half steps, is x^7 = 1.4983.... Well, we knew the perfect fifth, 1.500, was a little too large, so this discrepancy is not too surprising. The well tempered fourth is x^5 = 1.3348..., so apparently the perfect fourth, 1.333..., was a little too small. (This also follows because a perfect fifth followed by a perfect fourth is a perfect octave: (3/2)(4/3)=2. If one factor becomes a little smaller, the other must become a little larger, to keep the octave perfect.) An instrument which is well-tempered can play in any key, and they all sound the same, just moved up or down in pitch. The interval relationships are the same in every key. This tuning for keyboard instruments became popular in the early 18th century: J.S. Bach's Well Tempered Clavier illustrates the versatility of well tempering in pieces which would sound terrible in older tunings. Well tempering frees the composer from the tyranny of a few particular keys. The price is that the special peculiarities of the older keys are lost.

It sounds as if well tempering was an innovation in the 18th century, but actually, fretted string instruments, like guitars and gambas, were always well tempered! The reason is that to play these instruments, you place your fingers on frets, built-in ridges in the fingerboard, to shorten the string. The frets go straight across, so the factor by which a given fret shortens the string is the same for every string. From one fret to the next is a half step -- and it is the same distance, hence the same factor in frequency, on every string. To put it more simply, all half steps are alike -- that is well tempering. That is why, for example, a gambist does not tune the C and E strings a perfect third apart. The trick described above is to make it a well tempered third. Exactly the same remarks apply to a guitar. [I didn't know, when I wrote the above paragraph, that well-tempered tuning of lutes was controversial in the 16th century, and that Galileo's father, Vincenzo Galilei, was its most outspoken advocate.]

Problem: The six strings of a guitar are (going up) E-A-D-G-B-E, with the high E two octaves above the low E. The successive intervals between one string and the next are fourth-fourth-fourth-third-fourth. Call the fundamental frequency of the low E string f1. Now suppose we adopt the following tuning strategy, because we want to build in some perfect intervals. (a) Tune the top E string a perfect two octaves above the low E string. What is its frequency? (b) Make all the fourths between the open strings perfect. What are the frequencies of the middle four strings? (c) Is the third between G and B perfect? What is the ratio of frequencies? (d) Realize that all half steps played with frets are well tempered. What are the frequencies of F sharp and G sharp played on the low E string? (Note: this is just going up by well-tempered whole steps.) (e) By what factor does the frequency go up as you go up the half step from G sharp to the open A string? Is this like the fretted half steps? (f) Find the frequencies of other notes that you could play, and in particular find some octaves which are not perfect.

Project: Once you can hear harmonics of strings, you can actually tune a guitar this way. To tune the high E string, make it match the fourth harmonic of the low E string (this is a good idea anyway). To make the fourths perfect, get relevant third and fourth harmonics to agree. Now listen to the intervals suggested in the above problem, especially the imperfect octaves. If you know how to play a few chords, try them out. It seemed to me that an E chord sounded very good, but G sounded awful. (Why?) This illustrates that in tunings which force certain intervals to be perfect, the major scales are not equivalent. Just so there is no misunderstanding: although you can tune a guitar very accurately by this method, it is not a good tuning.

You may have wondered if the small differences we have noted, between a perfect fifth and a well tempered fifth, for example, really matter. If you do this project, you will know: they do matter.

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