Learn to Hear the Dissonances of Equal Temperament

The Physics of Harmonics

Chapter previously titled

"A Musical Sound Consists of Simultaneous Vibration Modes"

So far, we took as example the simple motion of a moving string. It can be a guitar string, a violin, piano, sitar or banjo string, excited into movement by plucking, striking or bowing it. It makes no difference, because the acoustic laws I demonstrate through current examples are the same for all instruments, no matter their type or principles.

Later on in this chapter I will show how the laws that apply to string instruments also apply to other type of musical instruments. For now, let's take a look at a bowed violin string.

The next video has no sound. A sound played at such reduced speed is below the human hearing range.

Why does the violin string move like that, and not how one would imagine? And why is this sound considered musical?

The answer to both questions is: harmonics. Harmonics are different ways of movement, happening at the same time. This is so important in understanding what music really is, that I'm going to repeat it.

Harmonics are different ways of movement (inside the same body), happening at the same time (instantly combining as a mathematical sum of the parts). The string moves like that, because inside it, different types of movement occur at the same time.

The first type of movement is the simple motion I took as an example when showing that all the sounds around you travel to your ears as beautiful holographic bubbles^{9}. This simple movement, up-and-down like a jump-rope over the entire length of the string, is the first harmonic.

The second type of movement divides the string in two identical but opposite parts, each oscillating like the first harmonic. When one of the halves is up, the other is down and viceversa. This is the second harmonic.

The third simple motion divides the string in three equal parts.

I'll stop at three for now, keeping in mind that the number of harmonics of a musical sound could, theoretically, extend to infinity. In the next section you have a more detailed description of these different types of movement - the harmonics.

The process through which different ways of movement are happening at the same time inside one body is that of addition. The string of the violin oscillates at many of its resonant frequencies (harmonics) simultaneously. By simply adding them, the resulting shape will be that of the bowed string.

The summing implies the simple (algebraic) addition of every point along the lines. If all shapes are above the dotted, neutral line, they add up. Whenever they are on the neutral line, their value is zero. If one (or all) of them is under, they get subtracted.

Try it! Find for yourself what music really is. All you need is Java installed on your machine, and the “Harmonic Sums Applet” developed by the P&M Development Team from University of California, Berkeley.

Help is needed to create an animated version of the images above

by programing in Wolfram Mathematica — **or even better,
using only HTML5 SVG, CSS & JavaScript.**

So far half the model is completed;

see the video presentation further down this page.

Contact me: bo @ this website.

In case you were wondering, the sine wave of a vibrating string reflects its geometry.

Please keep in mind that the harmonic summing depicted above uses drawings of the actual string, which are not mathematical representations of sound (sine waves).

I'm not going to repeat the process of harmonic summing with sine waves, but you can take a look at the following picture, taken from the book "On the Sensations of Tone as a Physiological Basis for the Theory of Music" written by Hermann L. F. Helmholtz in 1877.

This is a sine wave of a sound, composed from the addition of the first three harmonics.

In his example, Helmoltz used only the first two harmonics.

*The heights of the portions of the section of curve e ε have to be [algebraically] added to heights of the section of curve d _{0}δ, and similarly for the sections ε ε_{1}, and δ δ_{1}. The result of this addition is shown in the curve C. The dotted line is a duplicate of the section d_{0}δ in the curve A. Its object is to make the composition of the two sections immediately evident to the eye. It is easily seen that the curve C in every place rises as much above or sinks as much below the curve A, as the curve B respectively rises above or sinks beneath the horizontal line. The heights of the curve C are consequently, in accordance with the rule for compounding vibrations, equal to the [algebraical] sum of the corresponding heights of A and B.*

*Thus the perpendicular c _{1} in C is the sum of the perpendiculars a_{1} and b_{1} in A and B; the lower part of this perpendicular c_{1}, from the straight line up to the dotted curve, is equal to the perpendicular a_{1}, and the upper part, from the dotted to the continuous curve, is equal to the perpendicular b_{1}. On the other hand, the height of the perpendicular c_{2} is equal to the height a_{2} diminished by the depth of the fall b_{2}. And in the same way all other points in the curve C are found.*

Hermann L. F. Helmholtz - "On the Sensations of Tone as a Physiological Basis for the Theory of Music"

"Standing Waves" Are Really Rotating Vortices Projected Down to 2D

Are you a talented programmer passionate about physics and music?

Consider helping completeing the interactive model from the video above

by programing in Wolfram Mathematica — **or even better,
using only HTML5 SVG, CSS & JavaScript.**

Contact me: bo @ this website.

When you oscillate a string you're generating a "string wave" which sits there oscillating up and down in place. This motion is actually a constrainment of a wider, fuller motion that rotates the (equal parts of the) string in 3 dimensions.

When this motion takes place inside a medium which can facilitate transmission like air, the vacuum created behind the oscillating string causes a chain reaction started by air particles falling into the vacuum. Every air particle is then involved in the process of passing on this motion by bumping into its nearest neighbor, making the sonic energy spread in all directions simultaneously, spherically (see next chapter, “Cymatics: The Science of Dance”).

Although this motion is 3D, its mathematical representation on a graph will look like a 2D sinusoidal wave. Thus the term "traveling wave". I'm going to keep using citation marks to emphasize that this is just a perceived form of movement, and not the movement itself. This also applies to "standing wave" - the shape taken by the string, appearing to stand.

The obvious difference between "standing waves" and "travelling waves" is that the first type of motion is confined to oscillations between points of stilness, whether fixed ends or harmonic nodes, while the second travels in all directions. The simple act of travelling causes "travelling waves" to lose energy and die.

**Do not confuse standing waves in strings with the mathematical representation of sound movement in the form of sine waves.**

As usual, modern science settles into looking at the observable effects of periodic motion, disregarding the WHAT, WHY an HOW responsible for the phenomenon. Still, by using this picture, we can make an idea about a basic manifestation of the principles we are discussing.

Chapter previously titled

"Harmonics are Vibration Modes of Exact Multiples"

All objects that are irregular constituted, like a door or a chair, are not musical, because their natural frequencies are not of exact multiples, and do not resonate. They are called "inharmonic", which generally means having a stretched harmonic series, or irregularly distributed overtones.

If the source of a sound is of a consistent structure, it will emit regular oscillations and vibrations. Any sound producing source, such as a violin string, vibrates not only as the whole string, but also in fractional segments of that string. It's as though the string were infinetely divisible into equal parts of itself: two halves, three thirds, four quarters and so on. These harmonics (also called overtones) are all sounding together with the fundamental sound of the full string^{11}.

These statements are so important in understanding harmony and what music really is, that with the risk of sounding redundant, I'm going to repeat them all over, using different words.

Whenever you hear a musical note, you are simultaneously hearing a whole series of higher tones that are sounding at the same time. These are arranged in an order preordained by nature and ruled by universal laws^{11}.

Movement becomes music when the source of sound is of a consistent structure, and all its components are locked in harmonic (of whole number multiple) proportion. A moving string describes a complex motion and generates a complex sound, that can be broken into the sum of its own series of harmonic overtones. These overtones are like waves of motion that fit perfectly into each other, because they are made from exact divisions, or sub-multiples, of the entire string.

If the body generating the sound is made from parts unequally divided, that are nonharmonically related, but which despite this is artificially driven into (constant) oscillation, it will produce a different kind of musical tone, called nonharmonic. Bells and gongs are a good example.

The first harmonic of the harmonic series is also known as the fundamental, and is often erroneously associated with the sound of the open string. The other harmonics are naturally less aparent to the ear; they sound more and more faintly as they go higher.

The second harmonic divides the string in two identical segments. The smaller the segments, the faster they move. Each segment of the second harmonic oscillates twice as fast as the first. This new tone is literally a doubling of the source tone, and even though it is twice as high, it sounds almost exactly like it. There is a feeling of maximal similarity between them, they seem to be “the same note, but different”.

The doubling of a tone leads to the experience of coming back to an already known quality, on a new level. Yet this is a new tone, having it's own musical identity. The second harmonic's segments are half as long, so they makes two oscillations by the time the first harmonic finishes one. This causes the sounds to reinforce each other once every repetition of the larger cycle.

The third harmonic is made of three identical segments, each one third of the entire length of the string, each moving thrice as fast as the first. The third harmonic performs three oscillations for every one of the fundamental, developing a periodicity recognized instantly by our ears. The tripling of a tone leads to the experience of coming back to an already known quality, on a new, even higher level.

The fourth, fifth, sixth and so on harmonics increase the number of vibrations, creating every time a new tone which is nothing but a reiteration of the fundamental, every time on an even higher level.

*As a general rule, the farther the overtone is from the fundamental, the less is its intensity. Taking the sound from a well-bowed violin as a model of tone, Helmholtz has given the following approximation to the relative intensities of its partials, the intensity of the fundamental being taken as 1. ^{7}*

The intensity (or volume, or loudness) of the harmonics are given in the table as exponents, ratios and percentages. The resulting numbers are the same (bottom numbers are all 1). Notice the connection between the frequency of harmonics and inverse powers of 2. For example, harmonic number 3 moves three times as fast as the fundamental, producing a sound three times as high, but nine times as lower in intensity (9 times quieter).

Overtone intensity table inspired from: Thomas Frederick Harris -

"Handbook of Acoustics for the Use of Musical Students (8th ed)"

Images in this chapter don't reflect harmonic intensity accurately. If that be the case, the wavy shape of harmonic overtones above the second would be just a line. The amplitude has been exagerrated to allow a better visual interpretation. The colors used are arbitrary, and do not specifically convey any link between sound and color.

In order to make an idea about the reality behind the numbers, let's hear the first 6 overtones. Each of the following sounds represents a simple motion, a pure tone component or simply, an isolated harmonic. Their loudness intensities are those from the tableabove, not from the picture to the left. For best results, use bigger speakers or headphones. Laptop speakers will compromise the acoustics.

Isn't it somehow strange, that the sounds seem to have almost the same loudness? If I would use scientific talk, I would say that their intensities are inversely proportional with the square root of their frequency. Isn't science fun?

Let's take a look at the "sine waves" of these sounds. Keep in mind that what you're looking at is just the side view of a spinning helix, a mathematical construct: a 2-dimensional graphical projection of a 3-dimensional real-life phenomenon.

The image is accurate in terms of intensity proportions, although only a small part of the total duration of each overtone is depicted.

Together, the six harmonics sound like this:

At this point, you might be wondering why this sound is different from that of a violin. If the theory is correct, then the sum of harmonics should sound like a note played on a real instrument.

The answer is quite simple; it reinforces the theory, and it will be expounded in the next section Why the Same Tone Sounds Different on Different Musical Instruments.

Try it! Find for yourself what music really is. You will need an instrument with at least one string. It doesn't have to be in great shape or even tuned; any old guitar lying around will do. Also, a measuring tool like a folding ruler or measuring tape and if you really suck at math like me, a calculator.

Specific harmonics can be isolated by limiting the string's range of motion. This can be achieved by creating a point of stilness at specific locations along the string, that will silence other harmonics by forcing the movement into only one shape. Here's what I mean by that:

In order to isolate the second harmonic, rest your finger gently on the string, right at the centre. Pluck, strike or bow one of the sides. Notice how the string moves, generating a sound twice as high as the fundamental: the second harmonic.

My favourite way of doing it is to pluck the string and touch it precisely in the middle, while still oscillating. The result: all harmonics are canceled, except the second. This gives me a great insight about where the second harmonic is located in the complete spectrum of the string, and its intensity. The string reveals this harmonic, proving that its whole tone is actually a composite that human ears accept as a unity.

This can be taken further by repeating the process for the third harmonic. Rest your finger gently on the string, at a point one third along the way (measure and divide by three). It doesn't matter which end you measure it from; as you will see, when you pluck, strike or bow the third of the string, one more point will magically form, dividing it thus into three symmetrical parts. What you're hearing is the third harmonic.

You can continue like that until the physical laws governing the thickness of your fingers won't allow you to clearly isolate harmonics.

Let's consider another approach to generating harmonics: instead of dividing the string, multiply it. Instead of looking for the small parts that make up the whole, look for a whole made up from small parts, one of which is the string.

Any sound producing source, such as a violin string, bowed with precise control, could oscillate as many different larger segments, multiples of that string - one at a time. It's as though the string were infinetely multipliable with equal parts of itself: two doubles, three triples, four quadruples and so on. These harmonics are called undertones or subharmonics.

Every musical sound is part of a whole series of lower tones that represent symmetrical multiple portions of itself. These undertones are like waves of motion in which larger parts of the entire string fit perfectly, because they are made from exact multiples of that string.

Acoustically, the undertone series is like a mirror image of the overtone series; it is opposite in every way.

The first harmonic of the (sub)harmonic series is the same fundamental we have hitherto discussed, the same sound erroneously associated with the open string.

The second harmonic represents twice as many equal parts than the first. In other words, the first harmonic represents two equal parts from the second. At the same time, the first harmonic represents (only one of the) three equal parts from the third, and four equal parts from the fourth - so the fourth is four times larger, or deeper, or lower in frequency than the fundamental.

For this reason, the undertones do not sound together simultaneously. In order to emphasize each undertone, we are required to create exact multiples of the the string's length by enlarging it. Everytime the string gets larger, a new and deeper sound is created and this can only happen asynchronously.

The picture above is helpful in seeing the relation between fundamental and undertones, by sacrifing a correct representation of the intensities. The amplitudes have been greatly diminished in order to allow a better visual interpretation. An accurate image of the undertones would depict them overlaping. We'll come back to this in a moment.

The second subharmonic multiplies the string length of the fundamental by two. The larger the string, the slower it moves, and so it oscillates two times slower in the same unit of time. By moving two times slower, it generates a sound two times deeper. This new tone is literally a halving of the source tone, and even though it is twice as deep, it sounds almost exactly like it. There is a feeling of maximal similarity between them, they seem to be “the same note, but different”.

The halving of a tone's pitch leads to the experience of coming back to an already known quality, on a deeper level. Yet this is a new tone, having it's own musical identity. The second harmonic is twice as long, so it makes one oscillation by the time the first harmonic finishes two. This causes the sounds to reinforce each other once every two repetitions of the smaller cycle.

The third harmonic consists of three identical segments, each as long as the fundamental. Being thrice as large, it moves thrice as slow, generating a sound three times as deep. The third harmonic performs one oscillation for every three of the fundamental, developing a periodicity recognized instantly by our ears. Trisecting a tone's frequency leads to the experience of coming back to an already known quality, on a new, even deeper level.

The fourth, fifth, sixth and so on subharmonics decrease the number of vibrations, creating every time a new tone which is nothing but a reiteration of the fundamental, every time on an even deeper level.

Each of the 6 undertones you're about to hear represents a simple motion, a pure tone component or simply, an isolated harmonic. Their loudness intensities are those from the table. In order to hear them, you will have to use big speakers, with a subwoofer attached where possible. On laptop speakers the sounds will be unhearable.

The table on the right is an inversion of the "overtone harmonics" table above. It is currently theoretical, as no serious study of undertones has taken place so far. Though the sound example seems to confirm this theory.

A correct acoustic representation of the subharmonic series would imply starting from a point in time and procedeing in the opposite direction on the time axis. Since we can only move forwards through time, the undertone series will have to be represented accordingly, using the "from left to right" convention.

Depicted below are the "sine waves" of these sounds. Keep in mind that what you're looking at is just the side view of a spinning helix, a mathematical construct: a 2-dimensional graphical projection of a 3-dimensional real-life phenomenon.

The image is accurate in terms of intensity proportions, although only a small part of the total duration of each undertone is drawn.

Try it! Find for yourself what music really is. You will need a stringed instrument - preferably a big, long stringed, fretless one. For this experiment I'm using my custom fretless classic guitar (I pulled out the frets with a knife). For this experiment, you will find frets to be frustrating; a violoncello (cello) or contrabass (double bass) will do better. You'll also need measuring tape and a calculator.

Decide how deep you want to go. It is important that you state from the beginning up to which level you want to descend on the harmonic series. I will have to say three, because of the limitated length of my guitar's fingerboard.

Measure the string and divide it in three equal parts. Press the string against the fingerboard with your finger at one third of its length measured from the bridge. Pluck, strike or bow this portion between your finger and the bridge. The sound you hear is the fundamental. (It also contains its own overtone harmonic series.)

Repeat the process with a length of two thirds. This segment will be twice as long as the fundamental and its sound twice as deep. This is the second harmonic of the subharmonic series (which also includes its own overtone harmonic series).

The third can be heard on the open string (also with its own overtones), now three times longer than the fundamental.

But what if we want the fundamental to be the same as when we demonstrated the ascending series of harmonics? Well, we would need to multiply the string length way beyond the nut of the guitar, and this would be obviously impractical.

Even though the above image represents a fact and its limitations, it is possible to play subharmonics of an open string, without multiplying the string length physically.

Mari Kimura is the first violonist of which I know of achieving this. She uses a special technique of bowing the violin which makes possible a phenomenon thought to be impossible until now. When listening to the deep sounds produced, it is hard to believe that they come from a violin and not from a larger instrument like a viola or cello.

*What I'm doing is inventing a way to play notes below the open G (the lowest note on the violin) without changing the tuning. This is done by the precise control of the pressure and the speed of the bow. It is quite difficult, and if you can't do it successfully it will sound awful :) The new generation of violinists are more adventurous and I believe this will be a standard technique in the future.* - Mari Kimura

Ever wondered why some of the music on your computer sounds like life was taken out of it? Because most likely, those files are in MP3 format, and they miss harmonics.

MP3's are smaller in size than the original audio file. This reduction in size is done at the expense of harmonic content. Basically, compression is achieved by removing the harmonics that are inaudible to the human ear. The higher the compression rate, the larger the number of harmonics removed. Thus the specific hollow, lo-fi sound.

The process used for MP3 encoding is called FFT, or Fast Fourier Transform, after Joseph Fourier, the french mathematician who discovered it.

*Fourier’s Theorem says that any periodic motion can be represented as a sum of simple harmonic motions. “Fourier analysis” or “harmonic analysis” refers to the decomposition of a vibration (periodic function) into its harmonic components (sine functions).
Fourier analysis permeates all of science and engineering. This far-reaching mathematical tool is used in applications ranging from building a music (Moog) synthesizer, to designing circuits, to analyzing waveforms, to processing signals in electronics, to processing light in optical systems, and to calculating a quantum jump in atomic systems.*

The process known as "Fourier" decomposes any signal - in our example, the tone generated by a violin's string - into its harmonic components.

This crucial detail is usually overlooked, or shortly mentioned in physics literature. The most important thing to remeber about what Fourier really means, is harmonics: 1, 2, 3, 4, 5, 6, and so on.

Any periodic signal is composed of many other signals, that are in harmonic relation to each other. These can never be irrational numbers, like the square root of two (1,4142...) but only harmonic numbers or integers - that is, values that fit perfectly inside each other.

Chapter previously titled

"Why the Same Tone Sounds Different

on Different Musical Instruments"

"The same tone" is not the same at all. We usually think of a tone as the frequency of one sound: the fundamental frequency common to all instruments exemplified here, but every tone is in fact a complex entity - a chord if you will - consisting of many other simple tones.

These simple tones are reiterations of the same energy, on different levels. They are harmonically related to each other (do not confuse true harmony with the simultaneous playing of equal tempered tones). The simple mathematical law governing the harmonic series is depicted here: the fundamental frequency at 128 Hz is harmonic 1, or simply the fundamental. The next harmonic oscillates exactly 2 times faster at 256 Hz; harmonic 3 precisely 3 times giving 384 Hz, and so on.

This progression 1 2 3 4 5 ... generating aurally unequal tones continues to infinity. Physically, a harmonically rich timbre like that of the sitar or accordion goes beyond the 120th harmonic - visible on the graphic. Musically, the Harmonic Series is nature's perfect tuning system, generating three-dimensional spiraling vortexes having perfectly consonant harmonies and evolving fractal beatings (as opposed to the stiff two-dimensional sound waves of equal temperament, spinning in closed circles with their constant and annoying beatings and phaser-like dissonances).

Many tunings based on rational numbers like Just Intonation and the tuning systems of different ethnic and indigenous cultures are in true harmonic alignment with the nature of sounds. True harmonic music is the miniature of energy laws throughout the Universe.

*Sound, tone, and note each have a specific meaning, even though they may refer to the same auditory phenomenon. Each represents a different response to a musical event — a different way of feeling and thinking about what has been heard.
Sound simply refers to the transmission of vibratory motion and its perception by the auditory center in the brain after the various parts of the ears have resonated to it.
A tone is a sound that conveys significant information to the hearer, because it transmits the nature and character of the source of the sound. Thus a tone is a meaning carrying sound.
A musical note, on the other hand, has no meaning in itself. It has meaning only in relation to other notes. The same note may be played by several instruments producing very different actual sounds. *

Timbre Is the Quality of Sound that Makes it Distinctive.

The sound of a bow made from wood and horse hair sliding on a violin string, together with the resonant body of the violin is different from the sound of a wooden hammer hitting a piano's string, even if they are tuned alike.

Even if their fundamental harmonic resonates at the same frequency, other harmonics may have different amplitudes or could miss entirely; thus their sum and the sound produced by their added motion will be different.

The tone produced by a musical instrument has two main components: the oscillating body (in our example a streched string) and the vibrating body (like the body of a guitar or violin). These resonate together, each adding its specific characteristic to the tone.

HARMONIC OVERTONES FOR AIR COLUMNS (FLUTES AND REEDS, OPEN AT BOTH ENDS):

If the string of a guitar oscillates at a certain frequency, the body of that guitar will vibrate accordingly. Together, they will develop a periodic movement representing the sum of certain harmonics. It is these harmonics, their number and intensity that determines the specific sound of a guitar. A piano will have harmonics arranged in another manner, and consequently will sound different, even though the oscillation speed of its string is the same with that of guitar.

HARMONIC OVERTONES FOR CERTAIN INSTRUMENTS AND OTHER AIR COLUMNS (FIXED AT ONE END, OPEN AT THE OTHER - THUMB PIANO, PANPIPE):

TO DO: Fix CSS vertical spacing problem after images

The physical aspect of consonance and dissonance is closely related to the perceptual. A definition of these is given in the Main Components section of the HarmoniComb chapter.

Consonance and Dissonance - The Main Theories:

http://www.music-cog.ohio-state.edu/Music829B/main.theories.html

**COMING SOON!!!**

CONSONANCE AND DISSONANCE

through summing

OVER- and UNDER-HARMONICS as

PITCH and RHYTHM

governed by PERCEPTION

applied only to SIMULTANEOUS PLAYING

and extrapolated to CONSECUTIVE PLAYING.

TO DO: change text (make understandable), optimize

KEY: every what-you-see-is-how-you-hear number line (whether vertical or horizontal) has both ascending and descending series, marked by the operators × and ÷. These number lines represent the way we hear pitches and rhythms, while the linear drawings of physical lengths on the side of diagrams represent the way we measure them.

Hold tight, working on it!

Hold tight, working on it!

Last updated: 17 may 2015