Essays on Pitch, Tuning And the Physics of Musical Tone

Written by Gilbert Hoek van Dijke, Rotterdam
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Edited for these pages with the author's permission, by Wendy Morrison. For the most part, all I have done is correct typos and punctuation for clarity, broken up the paragraphs, and Americanized the English a little bit.
These essays were originally posted to the accordion and concertina mailing list (now the newsgroup rec.music.makers.squeezebox) in response to questions to the list concerning degree of reed tremolo, tuning (just vs. equal temperament) and some other related subjects.

Basic Physics of Sound and Tuning

Gilbert's introduction:

As far as I saw, all questions were answered. However, the relationship between different phenomena is not shown by the separate answers. I can imagine people still struggle with this stuff. To clarify relationships, I will try to explain some basic physical principles of sound and tuning, without going into detail.

I'd like to emphasize that this story will not end as a guide on how to tune your instrument. The theory itself will show that theory is not enough to tune an instrument. The purpose of my writings is that players should understand some basic principles so the physical aspects are no longer an unknown. Additionally, this knowledge can help you to recognize special sounds or problems, and pinpoint tuning wishes, so you can discuss them with instrument sellers, repairpersons, and tuners.

Basic Physics of Sound and Tuning

• What is sound, pitch, volume?
• How are beats (tremolo, wetness) generated?
• Overtones and intervals
• Dilemmas in tuning strategy
• Theory vs. practice of tuning

What is sound, pitch and volume?
Sound has to do with vibrations. A vibrating object (a reed of an accordion, a string) causes vibrations (periodical compression and decompression) in the surrounding air. These vibrations propagate in all directions. When they reach our ear, they cause the eardrum to vibrate. We experience this as sound.

Pitch: a high frequency (many vibrations per second) gives a high tone, a low frequency gives a low tone.
Volume: a big amplitude (deflection) produces a loud sound, a small amplitude produces a soft sound.
The latter can be seen in a guitar string: after striking a string, we can see the string vibrating. Both the amplitude of the string and the "loudness" diminish, while the frequency (pitch) is constant. The frequencies cannot be seen without technical equipment, since they are too fast. Children can hear frequencies from 20 to 20000 Hz (vibrations/second). As people get older, the sensitivity to pitch, especially to high frequencies, diminishes.

How are beats (tremolo, vibrato, wetness) generated?

Beats are produced by at least two reeds (or strings, etc.) that are tuned just a little bit different. In an accordion this is sometimes called musette, vibrato, or tremolo. How this works can be seen in slow-motion in the swings on a children's playground. The frequency of the swing depends on its dimensions. When two swings are exactly the same, they will swing with the same frequency: their movements are the same. The same goes for two accordion reeds: when two reeds are the same, they produce the same "dry" sound without tremolo. The sound is intensified with respect to only one reed, and this is what you expect (two reeds produce more sound than one reed).

Now, suppose that one swing makes a cycle every second (its frequency is 1 Herz), thus 10 cycles in 10 seconds. The other swing makes 11 swings in the same 10 seconds. What happens now? When the swings start in the same direction (in phase), they move together forward and backward. 5 seconds later, the first swing has completed 5 swings. The second swing, however, has competed 5 and a half swings! So, when the first swing is moving forward, the second swing is moving backward. The first swing is compressing air, the second swing is decompressing air. They are no longer intensifying each other, they are counteracting each other and the produced sound will be softer. Again 5 seconds later, the first swing has completed 10 cycles, the second swing 11 cycles. They are in phase again, intensifying each other. The produced sound will be louder and softer periodically; it has a tremolo.

Back to the accordion reeds: in the analogy of the swings, two reeds that are tuned a little different will intensify and counteract each other periodically, producing a tone with vibrato. The more the reeds are out of tune, the faster the vibrato will be (a wetter sound, more beats per second). The amount of beats per second is the difference in frequency: two reeds, tuned at 440 Hz and 442 Hz, will give 2 beats per second: starting in phase, half a second later the first reed has competed 220 cycles and the second reed 221. They are in phase again, after producing one beat. Again half a second later, they will have produced two beats, which is the difference in frequency.

Next part:
Why is the sound of a violin different from the sound of a saxophone, even if they are playing the same tone at the same loudness?

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Overtones and intervals

• Overtones: how do we recognize the sound of various instruments?
• Intervals.

Overtones: how do we recognize the sound of various instruments?

When two different instruments, say a flute and a saxophone, play the same note continually, we can hear which one is the flute and which one is the saxophone. How is this possible? It is not the frequency, since this determines the pitch, which is equal. It is not the amplitude, since this determines the volume. A loudly playing flute never becomes a saxophone.

We recognize the instruments because a tone is not made by one single frequency. Only in a laboratory is it possible to produce a tone that exists of (almost) one single frequency. Nobody would like such an instrument, since this sound has a very clean, cold, soulless timbre. A tone of an instrument always contains several frequencies. These frequencies are related to each other. Unrelated frequencies produce noise. You can hear this in waves at the seaside or in the wind in the trees: all kinds of unrelated frequencies are produced and we hear noise, not the most suitable sound to make music.

The relationship between frequencies of pleasant tones appears to be simple: in addition to the base frequency, there is the double frequency, a frequency three times the base frequency, four times the base, etc. These additional frequencies are called overtones or harmonics. The contribution (volume) of the separate overtones determines the timbre of the tone and is different for each instrument. For example, in a given instrument the first overtone can be loud and the second one almost absent. This instrument produces another sound than an instrument where each overtone is half as loud as the previous one. The overview of the contribution of each separate frequency is called the frequency spectrum.

The existence of overtones can be demonstrated in a guitar string. For every overtone, the string has another moving pattern. The moving pattern of the base frequency (the most dominant one) is: the endpoints are not moving (of course, these are fixed to the instrument), the biggest amplitude is in the middle. The moving pattern of the first overtone (twice the base frequency) is: endpoints not moving, also the middle not moving(!), half strings at both side of the middle vibrating in opposite directions.

After playing the string, we can eliminate the base frequency by carefully touching the string for one moment in the middle, creating a flageolet tone. What we hear is a soft tone, one octave higher than the base frequency. This is the first overtone, vibrating twice as fast as the base. This tone is not eliminated by our finger, since this tone does not move the middle of the string. More flageolet tones show more overtones: touching at one third of the string shows the second overtone (three times the base, the string vibrates in three parts), at one fourth four times the base, etc.

Intervals

Overtones and intervals are related to each other. Suppose we have two objects (strings, reeds, whatever) A and B that both produce a tone. When the frequencies are not related, the two tones we hear are not related as well. But now, reed B has twice the frequency as reed A. Say reed A = 100 Hz, with overtones 200, 300, 400 Hz etc.; reed B = 200 Hz (overtones 400, 600, 800 Hz etc.). Then, the first overtone of reed A coincides with the base tone of reed B. And so does the third overtone of A coincide with the first of B, and all other overtones of B with some overtones of A, as in this table:

Overtone #	Reed A	Reed B	Overtone #
base	100
1	200	200	base
2	300
3	400	400	1
4	500
5	600	600	2

The result is that, when both reeds are played, the two separate tones fuse into one new tone, with a new frequency spectrum. However, no new frequencies are added by the high tuned reed, only the contribution of separate overtones is changed. Now we can understand that we recognize the relation between these reeds as a nice interval: the octave. Also, we can understand that the ear is sensitive for proportions: if reed B was tuned at 105 Hz instead of 100 Hz, reed A should be tuned at 2 times 105 = 210 Hz and not to 105 + 100 = 205 Hz to insure that the overtones will coincide.

We have seen that the interval 'octave' belongs to two frequencies in the proportion of 1 to 2 (notation: 1:2). In a similar way, we can find the interval 'fifth' (in just tuning, later more about tuning) as two frequencies in the proportion 2:3, as in this table:

Overtone #	Reed A	Reed B	Overtone #
base	100
		150	base
1	200
2	300	300	1
3	400
		450	2
4	500
5	600	600	3

Again, many overtones of the high reed coincide with overtones of the low reed, but now not every one. Besides, the first coinciding tones are the second overtone of A together with the first overtone of B (no base tone). Again, we recognize this interval 'fifth' as nice, but it is somewhat more difficult.

The same goes for other intervals, each interval corresponds to a specific proportion of frequencies. When the proportion between two frequencies consist of whole numbers (1:2, 2:3, 3:5 etc.), overtones coincide and we experience that as a nice interval. When these proportions become more complex, only higher overtones coincide and this will be more difficult to hear.

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Basic physics of sound and tuning, Part 3

• Tuning by ear
• Conflicts in rules for tuning

Tuning by ear

The previous episode explained that intervals are based on simple proportions of frequencies. The most simple proportion (1:2) corresponds with the octave. How can we hear whether an octave is tuned correct?
Until now, the theory has dealt only with correct tuned intervals. In that case, overtones coincide and nice tones are produced. When an octave is a little bit out of tune, overtones no longer coincide. For example, a low reed is tuned at 220 Hz. The high octave reed should be tuned at 440 Hz (twice the base frequency), equal to the first overtone of the low reed (which is also twice the base frequency).

Suppose, the octave reed is tuned at 442 Hz. Then, this frequency deviates from the first overtone (440 Hz) of the low reed. This produces beats, here beats of 2 Hz (the difference between these frequencies). The origination of these beats is the same as the origination of beats in two reeds at about the same pitch, but a little bit out of tune. This process is discussed in the first episode.

The same happens in other intervals, for example the fifth. This interval corresponds with a proportion 2:3 of frequencies, for example 220 Hz (base) and 330 Hz (fifth). Now the second overtone of the low reed coincides with the first overtone of the high reed: both 660 Hz. These overtones will no longer coincide when the fifth is out of tune, then these overtones will produce beats.

Tuning by ear is a matter of listening to beats (and practical skill, of course). The more complex the proportion of frequencies is, the higher the numbers of coinciding overtones will be. In general, the contribution of higher overtones is less, and it requires more training to hear the beats.

Conflicts in rules for tuning

From the sections so far, some simple conclusions can be drawn:

• intervals are based on overtones
• the proportions between base frequency and overtones are simple numbers
• the ratio between base frequency and octave is 1:2
• the ratio between base frequency and fifth is 2:3

In spite of this plainness, there is a conflict in these rules. This can be explained by an example. Let's tune an imaginary instrument and let's see where it goes wrong.

For the first tone, we use a reference, for example a tuning fork. Tuning an A of 440 Hz is common, but for reasons of clarity it is better to use simple numbers. Suppose we tune the A at 400 Hz. Listening to the beats, we can tune the A one octave higher at 800 Hz, one octave below at 200 Hz, and in the same manner all A's at:

100, 200, 400, 800, 1600, 3200, 6400 and 12800 Hz
(frequency of every A twice the frequency of the previous A).

Then, we tune the fifths (E's): one-and-a-half times the frequency of the A below. So we find for the E's:

150, 300, 600, 1200, 2400, 4800, 9600 and 19200 Hz.
(Notice that you can find each frequency either by multiplying the frequency of each foregoing A by one-and-a-half or by multiplying the frequency of each foregoing E by two).

Starting from E, we can tune the fifth of E (=B) by multiplying the frequencies of E by one-and-a-half: we find:

150 x 1.5 = 225
300 x 1.5 = 450
and so forth, up to
19200 x 1.5 = 28800 Hz.

Then the next fifth ( = F#) etc. This series of fifths is called 'circle of fifths'. Successively, we find C# ( = the fifth of F#), Ab, Eb, Bb, F, C, G, D and A again. Now the conflict appears: the A, which was the starting point, turns out to be not the fifth of D! What has happened?

We found the frequency of the highest A (12800 Hz) by multiplying 7 times the frequency of the lowest A (100 Hz) by two. We also found the frequency of the same A by multiplying the frequency of the lowest A by 1.5 12 times. The first calculation dictates the high A to be 128 times the frequency of the lowest A, the second calculation dictates the high A to be 129.7 times the frequency of the low A.

So if octaves are tuned right, not every fifth can be tuned right and the other way around. Since an incorrect tuned octave is terrible (the octave is the most simple interval to hear), octaves have to be tuned correct. So something has to be done with the fifths.

Equal temperament and just tuning.

Tuning in fifths gives higher pitches with respect to pitches found by tuning in octaves. One solution is to tune the fifths a little bit flat: every fifth is found by multiplying the base frequency by 1.498 instead of 1.500. If this is repeated 12 times, it will appear that the result for the high A is the same as found by multiplying the base frequency 7 times by 2. The conflict in tuning fifths and octaves is eliminated by spreading the differences equally over each fifth. There is no preference for any key, so you can play in any key you want. (Cynics state you can't play in any key, since every interval except the octave is out of tune).

Another solution is to give prevalence to some specific intervals. Intervals that are important for a specific key are tuned 'just', this is according the theory above. However, this will cause other intervals to be more out of tune with respect to equal temperament. Suppose the fifth from A to E is tuned just (2:3). Then, there are fewer intervals left for spreading the difference between 'octave and fifth tuning'. This will increase the deviation from just tuning in the remaining intervals, hampering playing in other keys. For other intervals a similar train of thought can be made.

However, this will not clear up the theory. The aim of this writing is to show the dilemmas in tuning, not to be a guide to tune an instrument. There are different strategies for choosing a pitch for each interval. I don't know all the details, so I shall leave the matter here. More about the practice of tuning in the next episode.

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• theory vs. practice of tuning
• epilogue

Theory vs. practice of tuning

In the previous episodes I discussed some theory on sound and tuning. Practice, however, differs from theory at several points. This section deals with some aspects of tuning practice.

In the second episode I asserted that the frequency of overtones can be found by multiplying the base frequency by whole numbers. However, in reality musical instruments produce overtones with frequencies that deviate a little bit from these ideal overtones. This has to do with physical properties of the vibrating parts (reeds, strings etc.).

The moving pattern of a vibrating string, for example, is always represented as if the endpoints can rotate freely. However, a string has some bending stiffness, like a rod. The real vibrating length is somewhat smaller then the total length of the string. This affects the various overtones differently, causing these overtones to be not exactly in tune. Also a rod, like a leg of a tuning fork, produces overtones that are not in tune.

This can be calculated from elasticity theory. So, even in a tuning fork the overtones are out of tune! Fortunately, these deviations affect mainly higher overtones. These have small contributions and mute after a short time. But undoubtedly they are there, and the ideal tuning with coinciding overtones is just theory. A tuner has to search for compromises, so that all intervals sound acceptable. This search for compromises has to be done by an experienced tuner.

Besides these skills concerning hearing intervals, an accordion tuner needs skill in how to tune a reed. Basically, a reed is tuned by removing material from the end of the reed (to raise the pitch) or from the base of the reed (to lower the pitch). This influences not only the base frequency, but also the overtones. Of course, it is important to know how to remove material so that base frequency and overtone frequencies stay in correct proportions. If the reed is treated wrong, it can become permanently damaged. On this point, a reed differs from a string: a string is tuned by altering its tension, this can be repeated over and over again (as long as the tension is not raised excessively), until the string breaks from metal fatigue at the fixed ends.

A tuner will never use the strategy described in the previous episode. This was described simply to illustrate the theory. Since no interval can be tuned faultlessly, every fifth is tuned a little bit wrong. The described strategy gives no feedback in between, so it would always end with an additional sum of errors. Therefore, during the tuning process a tuner listens to other intervals, so he can adjust the tuning between times.

Epilogue

In my writings, I have tried to explain some basic principles of sound and tuning. As I stated before, the aim is helping you to understand what is happening in your instrument, not to teach how to tune your instrument. This would be impossible, since tuning can only be learned from an experienced tuner who shows you in practice how to tune. For me, it was difficult to explain everything in a foreign language and without the use of figures. Feel free to ask me when something that you want to know is unclear. But realize I am just an interested player, I do not tune, build or repair instruments, so I can't give you much more details than I have written already in my postings. Finally, I want to emphasize that it's nice to know something about the physics of sound and tuning, but it does not help you to play better. So keep practicing!

Gilbert Hoek van Dijke
Rotterdam

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the Just Intonation Network
Microtonal bibliography
Scales
Mugwumps article on stringed instrument tone