ToneGAD
Tone Generator and Display

A Guided Tour and Some Information About Harmonics

The Controls

It is suggested that you begin the tour with a fresh copy of the program. You can press Reload or Refresh, if needed.

In the "Tone" mode, a tone will start to immediately (assuming your speakers are on). The tone can be controlled by the other controls. To turn the tone off, switch to either the "Off" or "Piano" mode. In the "Piano" mode you will need to press certain keys in order to get a tone to play. "Harmonics" is an advanced mode that will be discussed later.

Volume

Things are more interesting, if you click the "Tone" box. You can vary the volume with the "Volume" control either using the buttons whose notations are those used in music or with the slider. "pp" means very soft while "ff" means very load. The volume can range from 0 to 100.

The amount of tone you hear at various frequencies will depend on the quality of your sound generation system and speakers. With my not very HiFi earphones, I can't hear frequencies below about 35 Hz and above about 7000 Hz. Quality systems might do much better. Really good humans ears might hear frequencies from 20 Hz up to about 20,000 Hz if the sound system will produce them.

Advanced: If the "Allow inversion" box is checked, the volume slider range changes. Normally, the range is from 0 to 100. When checked, the range becomes -100 to 100. When the volume is slider is set so the volume is negative, the curve is inverted - that is, turned upside down. (Mathematically, the wave is being subtracted instead of added.) Because pure sine waves are very symmetric, you may not notice the difference unless you look very closely with the update rate set to 1. But it is much more obvious when sawtooth waves are selected. Observe the slopes change from going uphill to the right to downhill.

Frequency

Next move to the "Frequency" control which provides 3 different ways to select the frequency or pitch. The higher the frequency, the higher pitch the sound seems to us. The top 4 buttons:

• "100 Hz". Hz stands for "Hertz" or "cycles per second". 100 Hz was selected just because it is a convenient number.
• "Middle C". This the pitch is Middle C on the piano. Its frequency shows in the display.
• "A": Over the years, several different frequencies have been used for the key of A above middle C. In 1926, the American music industry agreed to use 440 Hz for A. It was adapted an international standard in 1955 although not every one agrees. Orchestras is the U.S. and U.K. normally tune their instruments to A pitched at 440 Hz.
• "B flat": Most concert, marching, and brass bands tune on "B flat" which just a little higher than "A". The reason: many band instruments are pitched in B flat.

Just below the those buttons is a slider that allows adjusting the frequency anywhere between 50Hz and 2200Hz.

Below them are 10 multiplier buttons which multiply the selected frequency by the indicated number. Later we learn the multipliers 1 to 10 are also called harmonics. (While initially exploring these buttons, it strongly suggested that you select "100 Hz" button.) The default is "1". "2" multiplies the selected frequency by 2, "3" multiplies it by 3 and so on. The selected button is shown in "red". In music, doubling the frequency increases the tone by 1 octave, reducing the tone by 1/2, lowers by 1 octave. This makes the multipliers 1/4, 1/2, 2, 4, and 8 especially interesting as they can reduce the pitch by 2 octaves or 1 octave or increase it by 1, 2 or 3 octaves.

The following table show the musical notes produced by the harmonics of the 3 special frequencies available by clicking a button.

Multiplier	C Fundamental	A fundamental	A sharp/B flat Fundamental
1	Middle C	A, above middle C	B flat, above middle C
2	C, 1 octave higher	A, one octave above	B flat, one octave above
3	F, above that C	E, 2nd octave above	F, 2nd octave above
4	C, 2 octives higher	A, 2nd octave above	B flat, 2nd octave above
5	D, 2 octives higher	C sharp, 3rd octave above	D, 3rd octave above
6	F, 2 octives higher	E, 3rd octave above	F, 3rd octave above
8	C 3 octave higher	A flat, 3rd octave above	B flat, 3rd octave above

(Multiplier 7 was omitted from the above chart because the corresponding notes (A sharp, G, and A flat) are flat and are avoided musically but appear naturally.)

FFT and the two displays

In case you are wondering, "FFT" stands for "Fast Fourier Transform", a mathematical way of analyzing periodic (regularly repeating) functions. But don't worry. You will not need to do any math. But fortunately we can take advantage of its results. (I was able to advantage of somebody else's coding and didn't have to do the code the mathematics!)

The upper display shows an image of the wave. If the display seems to be too bouncy to understand, try clicking on one of the "1 time/sec" button. Then the upper display will show a snap shot of the wave which is updated the indicated number of times each second. Assuming that the wave form is still a sine wave, you should see a nice image of the common sine wave if the frequency is somewhere between 100 Hz and 3000 Hz. Below that, you may see only part of one cycle, above that you may see a smear.

You will the notice the width of a set of humps in the curve gets smaller as the frequency increases.

If you are still displaying a sine wave, you will see a spike at the specified frequency in the lower display. The spike was produced by some rather complicated mathematics that "listens" to the music and determines what frequencies are in the tone being produced. Right now there is a pure tone and only 1 frequency is being detected. Later in our discussion we will listen to more complex tones and the FFT analysis will pick up multiple frequencies in the sound. In any case, the analysis figures out what frequencies are in the tone by sampling ("listening" to) the tone. It didn't pay attention to the button you used to select the tone.

(We will discuss this display again when we look at other wave forms. It will be much more interesting then and the "Show peaks" check box may be useful at that point.)

Wave Control

You are probably tried of listening to the sine wave. So lets try some of the other options. Looking at the wave forms in the upper display, you can easily understand the names for the triangle, sawtooth and square wave. The later is the easiest to understand. A square wave could be produced by just turning on and off an electrical circuit in a periodic fashion. I am not sure how this program produces the wave because again I am taking advantage someone else's coding. One could assume that the triangle and square wave could be produced by turning a voltage control up and down periodically but again I can't say how this program does it.

The lower display becomes quite interesting with these wave forms. It will be easiest understand them if you set the frequency to 100 Hz and the multiplier to 10 giving a 1000 Hz tone and turn on the "Show peaks" option in the FFT display. If you look at the triangle or square wave forms, you will see peaks at 1000 Hz, 3000 Hz, 5000 Hz .... . One the other hand, if you look at the sawtooth wave, you will see peaks at 1000 Hz, 2000 Hz, 3000 Hz, ... . What gives? You picked 1000 Hz but the FFT analysis says that tone contains several multiples of the specified frequency. It is those extra frequencies that make these alternate wave forms sound different the pure sine wave.

Comment: Don't be disturbed that the frequencies in the "Show peak" display in the upper right corner of the lower display are not exactly those numbers. The reason is that the analysis divides all the frequencies up to about 25,000 Hz into 1024 "bins". Each "bin" is about 24 Hz wide and the number report in the peak display is more or less an average for those frequencies. So if the peak is reported by 1006 Hz, it means that the analysis saw some frequencies near 1006 Hz which would certainly include the 1000 Hz we expected see. At high frequencies, having 24 Hz wide bins is not a major problem but it is when the frequencies are lower and the reported peaks will not be very accurate.

Consider the sawtooth wave. FFT reports that the frequncies 1000 Hz, 2000 Hz, 3000 Hz, ..., all present in the sound that we hear. Notice these frequencies are all multiples of the base frequency 1000 Hz. The height of the peak gives you some idea of the amplitude needed although some may actually be positive and some negative. On the other hand, the triangle and square waves only use the odd (1,3, 5, 7, ...) multiples of 1000 Hz. The even (2,4, 6, 8, ...) multiples are not needed because of a certain symmetry in the wave form. It turns out that the sound from some musical instruments contains only the odd multiples (or harmonics) but both even and odd harmonics may be produced by other instruments.

The peak frequencies that you see with these wave forms are sometimes called harmonics or overtones. In the harmonics section we will discus why the musical notes we hear are affected by the harmonics the wave form generates even though nice musical tones rarely have the extreme shape of the three none-sine wave options.

The Piano mode

The piano mode allow you to individual play individual notes using keyboard keys To see the keys that work, just click the "Piano Note Keys" button. You will see there are two ways. You can use the note names like "a", "b", ..., in lower case while capping a letter to gets you its sharp (for those keys that have sharps). In some cases, it is more convenient to use the top row of keys on a standard keyboard which include the number keys. The notes sound as long as you hold the key down. Their (approximate) frequency shows in the Frequency control. The Volume and Wave Form controls work as normal. The multipliers are reset to 1 when one enters the "piano" mode and before and after every note. But while a note is playing, you can use multipliers in their normal fashion.

In attempt to mimic many instruments, notes are accented at lower volume levels when the notes starts and they decay slowly when the key is released.

Advanced: If you hold a key down while playing a note in the piano mode, you can change the volume using the Volume control. However, changing the volume with a slider in the Harmonics control does not change the current volume. This mimics the response of a piano where it is impossible to change the volume of note while it holding its key down.

The Harmonics mode

Until you select the "Harmonics" mode, you will always hear the single pitch as shown in the frequency display. That pitch is called the "first harmonic" or "fundamental" It is the pitch you heard when the multiplier buttons are set at "1". After selecting the "Harmonic" mode, you can play a note together with some of its harmonics.

When you select "Harmonic" a new control opens below the other controls. It will stay open even when you select another mode. The other thing that happens is that the Wave will be changed to "sine" because normally one studies harmonics with "sine" waves. However, you can select another wave type, if desired.

After the harmonic control is displayed, the "Harmonics" and "Tone" modes are identical.

We will continue this discussion assuming the "sine" is being used. It may be helpful if you set the frequency slide bar for a frequency about 1000Hz. You will not be able to get 1000 exactly, just something close to it. If you click some of the harmonic checkboxes, that multiple will be added to the sound you hear. The wave form will no long be a simple "sine" wave. The harmonics will show up in the lower display. You can adjust the harmonic's slide bar to adjust the intensity of the harmonic. In a later section, you will learn that most musical instruments produce harmonics which give the instrument it characteristic tonal quality. Those harmonics give the instrument its distinctive tone when a note is being held. In the "Experiments" section of this paper, we study how harmonics can be used to produce other wave forms. For the time being, you may enjoy playing with the different harmonics at various intensity levels.

What about the "Set volume ..." buttons? They will be useful in the Experiments section.

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Pianos, organs and woodwind instruments have lots of keys that allow the musician to play lots of different notes. String instruments may have only 4 or 6 strings, but the player can press the string at various points to pick the desired frequency. But have you ever noticed that most brass instruments like trumpets and tubas typically only have three valves? They can be used in combination, but for practical purposes, there are only 7 different combinations. How in the world can brass players produce lots of different notes? Typically, the length of the tubing determines the frequency of sound and the purpose of the valves is to change the length of the tubing. The valves allow for 7 different tube lengths. (Trombone players effectively only have the same 7 options but they are able to adjust the slide in between those options.) Again, how can brass players produce lots of different notes?

The answer is they use their lips to produce different harmonics. It is something like selecting one of the multiplier buttons in the Frequency control. Most players use there lips to produce harmonics (or multipliers) of 2, 3, ..., 6. An excellent player (depending on the instrument) can pick harmonics from 1 to 8 or possibly even higher. Harmonic 1, called the fundamental or petal tone, is often hard to produce. Harmonics higher than 6 take lots of practice to control in a useful manner.

Lets take a closer look at the trombone, in part because I try to play one. (The other part is the trumpet players actually play B flat when they call the note being played C which may be confusing.) Recall that we said the concert bands typically tune on B flat because most band instruments are pitched in B flat. Trombones are pitched in that key. If the trombone players leave the slide all the way closed, they can play harmonics based on B flat. (Trumpets can get the same effect if they leave all their valves up only they are an octave higher because their tubing is only half as long. But for reasons that musicians can only understand, they call it C.) Lets look carefully at the harmonics of a trombone with the slide closed.

• 1st harmonic, multiplier 1: a low B flat two measures below Middle C on the piano but remember it normally is very hard to play. It is sometimes called the fundamental or petal note.
• 2nd harmonic, multiplier 2: Remember that multiplier 2 produces a tone one octave higher, so it is also B flat, one octave below middle C.
• 3rd harmonic, multiplier 3: F below middle C
• 4th harmonic, multiplier 4: As we know, this is 1 octave above the 2nd harmonic so it is the B flat just below middle C.
• 5th harmonic, multiplier 5: D just above middle C.
• 6th harmonic, multiplier 6: 6 is twice 3, so this another F, this time above middle C.
• 7th harmonic, multiplier 7: A rather flat A flat so it is not normally used on a trombone and almost never on other brass instruments with valves.
• 8th harmonic, multiplier 8: 8 is twice 4. So can you guess that it is also a B flat, an octave above middle C. Typically only expert players can produce this harmonic well enough to play it a concert.

Exactly the same concepts apply to baritones and euphoniums. while a Double B flat tubas plays one octave lower because their tubing is twice as long. Trumpet players play the same harmonics but because their tubing is only half as long, the notes are an octave higher. (Remember that for musical reasons understood by trumpet players, the would call the harmonics C, C, G, C, E, G, a flat B flat that would almost never be used, and C. The French horn, alto horn (also called tenor horn, E flat horn, or althorn), and the E flat tuba are pitched differently and have different fundamentals but the harmonics work the same way.

We have discussed these harmonics on brass instruments, but they also apply to stringed instruments like a guitar. Have you ever looked closely at a thicker, heavier, lower note string vibrate? If you have, you will notice that it vibrates with the shape similar to the top half of a sine wave. To play a note one octave higher, guitar players press string at the half way point and only half the string vibrates. But if the one very gently touches the string at the half point, the first harmonic is suppressed and one may see both halves of the string vibrate - the string is now vibrating in two sections - the second harmonic. In theory, we could try gently touching string 1/3 of the way down. This would mean neither the first or 2 second harmonic could sound and the string would be vibrating in 3 sections producing the third harmonic. Actually the harmonics were being produced when we thought we were just playing the fundamental. What appeared to us as string vibrating with the first harmonic, is actually much more complex. The string was also vibrating in ways that produced some of the higher harmonics. Similar things can happen on a violin. Composers sometimes ask a player to touch a string and produce harmonic in order to give the instrument a different sound.

You may wonder how the harmonics work in brass instruments. The following is a very simple minded explanation and not really 100% accurate. Playing harmonic 1 means that one period of the sine wave uses the entire length of the pipe. When brass players users use their lips to play the 2nd harmonic, they are effectively getting the instrument to vibrate in two sections in much the same manner as we were able to produce the second harmonic on the guitar. If they play the 3rd harmonic, they are getting instrument to vibrate in three sections and so on. Fortunately, their harmonics can be played usefully and loudly on brass instruments although they would be quite soft on the guitar.

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In the previous section we looked at how harmonics can be used to produce different pitches especially in brass instruments. In this section we will look at how harmonics, often referred to as overtones in this context, enhance almost all music. Harmonics (or overtones) help explain why a trumpet sounds different than a guitar or flute even they are holding on the same note.

Did you find that sine waves sound boring? They are just too pure. If you played much in the piano mode on this program, you might have found yourself preferring to use one of the other wave forms even though they are not really nice sounding. Even the human voice is enriched by harmonics.

In the previous section, we spent a little time discussing how harmonics could be produced on a guitar. Actually, even when a player is playing a certain note higher harmonics are occurring at the same time. Even though we may think we are only hearing the first harmonic or fundamental, it is the harmonics that help the note to sound pleasing to us instead of being a boring sine wave.

Trumpets, cornets, and flugalhorns

As the first example, we are going to look at trumpets, cornets, and flugalhorns. If you are not familiar with these instruments, you might want to look at the video https://www.youtube.com/watch?v=ZhfGM19bsmM where the presenter compares these instruments and their sound. The typical versions of these instruments are all pitched in B flat and have the same valves. Players can move from one of the instruments to the other without much difficulty although a player may specialize in just one of them. One the other hand, perhaps you may have seen trumpet players switch to the flugalhorn in the middle of the piece. Why would they do that? The answer is they have a different sound. But why do they sound different even when they play the same note? The answer is they produce different harmonics or overtones. The tubes in a trumpet are straighter and have a smaller diameter. That means that the higher harmonics are more pronounced giving the trumpet a more brilliant, piecing tone. On the other hand, the rounder, somewhat larger diameter tubes in the flugalhorn give it a mellower tone with less of the higher harmonics. The cornet might be considered an compromise between those two extremes.

Pipe organs

As seen in the picture at the right, larger pipe organs often have several different kinds of pipes. There are multiple pipes of each style as they are needed produce different pitches. This organ has both square and triangular shaped wood pipes as well metal pipes of different shapes. Why? The different materials and shapes produce sound with differing amounts of harmonics or overtones. As a result, the sound coming from the different style of pipes has a different tonal quality even when they are playing the same note.

As mentioned, it is the harmonic spectrum of different instruments that make them sound different. The picture on the left comes from a video where the presenter plays the same note on different instruments to help illustrate the importance of harmonics. Towards the end, he shows the harmonic spectrum of several instruments including this picture which shows the spectrum of a piano. The image was produced by using a microphone to capture the sound of the piano and then having a FFT analyzer sample the sound and produce a chart showing the frequencies it found in that sound. Notice the regularly spaced peaks for the harmonics.

Below are the FFT spectrum analysis for cello and a guitar. Again note the regularly spaced harmonics which mean the cello and guitar sound different even when they play the same note.
 

A final note: If you search on-line for "musical harmonic spectrum" or something similar, you will find some spectrum images similar to those presented above where the peaks are spaced regularly. However, some may be confusing at first because the peaks get closer together on the right. The reason is the frequencies are shown on what is called a log scale instead of the equally spaced scale that we have been using. On these charts, octives are equally spaced. That is, the octave from 100 Hz to 200 Hz has the same width as the octave from 200 Hz to 400 Hz and the octave from 400 Hz to 800 Hz, and so on. In some respects that is a better way to show sound spectrums because that is the way we hear them. Higher harmonics sound close together. However despite that fact, that I like to show equally spaced harmonics.

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A simple experiment with inversion.

With the "Allow inversion" option turned on, turn off all the harmonics except for the 1st harmonic and adjust its volume to 100, if needed. Observe the wave form which should be a perfect sine wave then turn it off. Adjust the volume 6th harmonic to a positive value of approximately .4. Turn on that harmonic and observe its wave form. Notice it is vibrating 6 times as fast. Turn it off. Turn on the 1st harmonic and then turn the 6th harmonic back on. Can you see how the two harmonics are added? Now slowly decrease the volume of the 6th harmonic to about -.4 and observe the wave form as you go. Can you see that the inverted harmonic is being subtracted from the 1st harmonic?

You may want to experiment with other harmonics and volumes.

The 2nd experiment: Creating complex wave forms.

This experiment does not always work properly. Most recently the problem has occurred with the new version of Bing but even with that browser, it has sometimes worked as expected. Perhaps the problem is the oscillators associated with the various harmonics are not synchronized to each other properly. If you do not see a nice pattern in the wave forms in this experiment it is suggested that you try a different browser, if possible. In all honesty, I expected this problem before I began testing the experiment and am surprised that it normally works.

Discussion: In this experiment, we will show that it reasonable to think that sawtooth and square waves can be approximated by using harmonics of the fundamental frequency.

Perhaps you were wondering why by default the sliders in the Harmonic control were positioned the way they were and why there are special buttons to set the harmonic volumes to 1/n and 1/n². These volumes were chosen not random. Consider the default harmonic volumes (and those set by the 1/n button). The value for harmonic 1 is 100/1 = 100, the value for harmonic 2 is 100/2 = 50., the value for harmonic 3 is 100/3 = 33.3 (at least to the nearest tenth) and so on. These values were picked because they lead to some very interesting results.

The experiment: Refresh or reload the program to make sure the values are as expected and select the Harmonics mode. Select a convenient frequency like "A" or 1000 (or something near it). You will see a nice sine wave. Turn on the 2nd harmonic and observe the wave in the upper display. Then turn on the 3rd harmonic and turn on observe the wave again. Do you see something interesting? Continue turning on the next harmonic and observing until you are using all 8 of the harmonics. Now just suppose that you were able to add in the 9th, 10th, 11th, ... harmonics. If we could do this a hundred times, what do you think we would get?

Repeat this experiment. Turn off all the harmonics. Repeat the experiment but this time only use the 1st, 3rd, 5th, and 7th harmonics. Leave the other harmonics off. What do you think the wave would look like if we could use 100 or more odd numbered harmonics?

Now check the "Allow inversion" check box. Turn off harmonics 2 to 8 and click the "Set volume: ± 1/n² " button. Turn on the 3rd, 5th, and 7th harmonics one at a time, observing the wave form each time. Once again, if we had 100 odd numbered harmonics available, what do you think we would get?

If you would like to see what the results could be if you could add lots of harmonics, check out GraphingSeries.html which allows you to add lots of harmonics so you can see what happens as more and more harmonics are added at the right intensity. It shows how the square wave, sawtooth and triangle waves can be approximated by using lots of harmonics. Unfortunately, generating the triangle wave requires "subtracting" some harmonics which is the reason for that ± was added to the 1/n² button when the "Allow inversion" option was turned on. Mathematically inversion (when the volume of the harmonic is negative) is equivalent to subtracting the wave form instead of adding it.

(In case you are wondering about the "1/n²", it means the volume was selected by dividing 100 by the square of the harmonic number. For harmonic 1, that means 100/1² = 1/1 = 1. For harmonic 2, we get 100/2² = 1/4 = 25 and so on. The plus and minus signs were selected just to get the desired wave form for this experiment.)

Why does it sound bad to play two notes very close to each other?

This program doesn't allow you to play two notes next to each other on the piano but we can come close using the 7th and 8th harmonics. For example, if we start out in C frequencies, we know the 8th harmonic is also a C only 3 octaves higher. The 7th harmonic would a somewhat flat B flat just below it. Those notes are pretty close and we can begin to see the problem.

You could use just about any note as the fundamental (1st harmonic) but it is pretty easy to see the problem if you use 100 Hz as the fundamental. In the harmonic control, set the sliders for harmonics 7 and 8 both at 50 (or very close to it). Then turn off harmonics 1 to 6. Turn on harmonics 7 and 8 so you are seeing only the 7th and 8th harmonics. (You may want to do this with the sound turned off if the sound is too bad.) Now click the 1/4 multiplier button. Watch the upper display to see the wave form that is generated. Then click the 1/2 multiplier, the 1st multiplier, and so on still watching the wave display. Some where along the line you should begin to something interesting. It is very obvious with the number 10 multiplier if you use 100 Hz as the fundamental frequency. (If you use a higher fundamental like B flat, you may have the best results with multipliers in the range of 3 to 5.) You see a relatively high frequency wave bounded by a low frequency wave.

You can imagine what happens when someone plays to keys immediately next to each. Even worse is if players play the same note but one of the players is a little out of tune.

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Version 2.1.b5, 3/13/2020. This is a draft version and improvements are expected. 2.0 introduced the harmonic control and the separate help page. 2.1 allows volumes to be negative and the waves to be inverted.

The source files:
toneGadMic.pjs The main program.
modeControl.pjs Code for mode control and keyboard keys.
Controller for the wave selection.
Controller for volume.
Code for frequency selection.
Controller for the FFT section and the two displays.
Controller for the harmonic section.
The web page for toneGAD.
The image files:
Images/PipeOrganPipes.png
Images/trumpetVariations.png
Images/VibratingString.png
Images/celloSpectrum.png
Images/guitarSpectrum.pif
Images/piano2.png

+/- Downloading considerations.

Comments, suggestions, corrections: James Brink

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