Physics

Artificial Harmonics on the Violin

Violin Harmonics 的图像结果

Abstract

This report presents the results of research into artificial harmonics on the violin. In the first experiment, the ratio of string lengths between fingers was measured to determine their effect on the pitch of the artificial harmonic. The results of the first experiment indicated that the fourth finger was always placed a quarter of the way up the string, removing all harmonics but the fourth and its multiples. In the second experiment, the harmonic content of the same note played with five different techniques (one was played as an artificial harmonic) was analyzed to find out why artificial harmonics have a ghostly timbre. The artificial harmonics had a harmonic content dominated by the fundamental and second harmonic, thus resulting in a purer tone which sounds less bright and rich than normal.

INTRODUCTIONdiagram of bowing harmonics

The aim of these experiments was to find out why artificial harmonics sound two octaves higher than the note they are based on and to find the reasons for their unique timbre.

How Violins Produce Sound

Violins produce sound when the bow is drawn across the string. Rosin (crystallised pine sap) is rubbed on the bow hair to increase friction by creating a rougher surface. This is an example of dry friction, which can be further split into static friction (between 2 surfaces not moving in relation to each other) and kinetic friction (between 2 moving surfaces).[1] Typically, static friction is greater than kinetic friction as it usually takes a smaller force to keep an object moving at a constant velocity than to accelerate it.[2]

F=μR

where F is the force of friction, R is the normal contact force and µ is the coefficient of either static or kinetic friction depending on whether the 2 surfaces are moving relative to each other.[3] The application of rosin to the bow hair increases µ and also the difference between the coefficient of static and kinetic friction by making the bow hair rougher. Image result for slip stick bow motion

As a result, the coefficient of static friction is much greater than that of kinetic friction. The string is pulled along with the bow during the ‘stick” phase due to this high static friction as the string and bow move in the same direction with similar speeds. This creates a wave that travels from the finger position (1⇒2 or “stick phase in Figure 1a), returning to the contact point where the string’s tension pulls it back the other way easily in the “slide” or “slip” phase due to low kinetic friction as the bow and string move in opposite directions. During this phase, the string moves opposite to the bowing direction and the wave reflects at the bridge. When it reaches the contact point once more, the string is moving in the same direction and at about the same speed as the bow. Therefore, there is static friction and this cycle of sticks and slips repeats. This is analogous to the string being plucked every time the kink in the string is moving in the same direction as the bow.[4]

As well as the fundamental with 1 kink, the second, third, fourth etc. harmonics are also produced with two, three, four etc. kinks, respectively[5] (see Figure 1b). As the waves of these fundamentals travel and reflect up and down the string, they interfere with themselves and each other when they occupy the same space on the string, resulting in superposition of the various harmonics to form one complex standing wave.

On a violin, this complex wave is usually saw-toothed in shape due to relative amplitude of the various harmonics (Figure 2) being about 1, ½, 1/3, ¼ etc.[6]

As the “stick” phase occurs whenever the string and bow move in the same direction and the “slip” occurs when their motion occurs in opposing directions, the period/frequency of the catch-release cycle is the same as that of the vibration of the string. violin sawtooth wave 的图像结果

However, this oscillation of the string by itself is not sufficient to produce any audible sound: too little air is moved. About 40% of the string’s tension is directed downwards over the bridge[7]. It increases in the direction of bow movement during the “stick” and in the opposite direction during the “slip”. The force on the bridge oscillates with the same frequency and waveform of the waves in the string (Fig. 3) with a frequency ranging from 196 Hz (open G string – the lowest note) up to around the mid-2000s (E7, the practical limit for orchestral parts, is 2637 Hz[8]).

sawtooth wave of violin 的图像结果

This wave is transmitted to the violin body, whose vibrations move enough air to create audible sound.[9]

Harmonics

For a regular note, the harmonics (second, third, fourth etc.) are produced as well as the fundamental (see Figure 1b for catch-release cycle of second harmonic). Each consecutive harmonic has an additional node and antinode[10], and therefore an additional half-wavelength in the string[11].

Harmonic No. of Waves in String No. of Nodes No. of Antinodes Wavelength
1 1/2 2 1 2L
2 1 3 2 L
3 3/2 4 3 (2/3)L
4 2 5 4 (1/2)L
5 5/2 6 5 (2/5)L
6 3 7 6 (1/3)L

 

https://lh6.googleusercontent.com/V85F6WqvujK9mwnLIKvEtKUMALsOedWH39ft-ZHtQDwu3iFmFH7sjQ4pmoW_yl0Fn4aSqi5lcGBFXsNCd2KhnekNHGRF9sVCfTCNXC3Hg_JHGxgsPqUARIH9IQbjCOb8TUwdEOZm

 

Wavelength = 2L/n for the nth harmonic where L is the length of the string.

We can determine the frequencies of the harmonics of A5 which has a fundamental frequency of 880 Hz.

where v is the speed the wave travels, f is the frequency and λ is the wavelength.[12]

v is dependent only on the properties of the medium[13] (in this case the violin string) and not the properties of the wave itself so the frequency of the second harmonic can be determined as follows:

Therefore, the second harmonic has twice the frequency of the fundamental.

For the nth harmonic of a note with fundamental frequency F:

Therefore, the nth harmonic has n times the frequency of the fundamental.

Each note played creates a series of standing waves as shown in the first graph below. The fundamental has twice the amplitude of the second harmonic and thrice that of the third.[14] Therefore the sound you hear is a blend of these harmonics and the fundamental (Fig. 2) which can be obtained by adding the wave equations[15] (see right) due to the principle of superposition.[16]

Image result for violin fingers

Types of harmonics

When a violinist plays a natural harmonic, they lightly touch the string ½, ⅓, ¼ etc. of the way along the string. This creates a node at that position and isolates a specific harmonic. Only harmonics with a node at that position form, and other harmonics are silenced.

This is different to how an artificial harmonic is produced. The violinist stops the desired note with their first finger and touches the string a perfect fourth (the fourth note up the scale) above. Figure 5 shows how the fingers are numbered for violinists. This produces a note two octaves higher than the stopped note itself. This must be the fourth harmonic as a note two octaves higher must have 4 times its fundamental frequency. The fourth finger artificially creates a node at its position, forcing only the standing waves of harmonics which have that point as a node to form and therefore be heard.

Timbre

Timbre describes characteristics of sound that allow the human ear to distinguish between sounds of equal pitch (dependent on frequency) and loudness (dependent on amplitude). Timbre is determined by harmonic content and the attack-decay-sustain-release (ADSR) envelope[17] (see Figure 6) of the sound as well as other dynamic characteristics such as vibrato[18]. Image result for adsr envelope

For sustained tones, only the sustain portion is relevant; so harmonic content (relative intensity of the different harmonics present in the sound) is the most important contributing factor of timbre[19]. A sustained tone is a repeating continuous function and, as such, can be reproduced by [20]Fourier analysis as a sum of sine waves with frequencies that are integer multiples of the fundamental frequency. Fourier synthesis allows the graph of the sustained note to be obtained by adding the various sine waves previously obtained through Fourier analysis.

The attack is the initial action that causes the instrument to produce the tone (in the case of the violin, it is the action of drawing the bow across the string). The sound rises to its maximum amplitude and decreases over time in the decay. The human ear can differentiate between different attacks and decays though the difference in timbre due to attack and decay is less noticeable during long, sustained notes as the harmonics produced during the attack are very important.

Any sound shorter than 4 microseconds is perceived as an atonal click and the ear needs at least 60 microseconds to identify the timbre of a note[21].

Vibrato occurs when the violinist rolls his finger forwards and backwards on the stopped note which causes periodic changes in pitch as the length of stopped string repeatedly lengthens and shortens.

At the date when this was written, there was no specific research into the timbre of artificial harmonics though there have been studies regarding how violins produce sound and the timbre of the violin in general.

Why do Artificial Harmonics Work?

APPARATUS

  • Fig. 7a: Violin & bow to produce artificial harmonics
  • Fig. 7b: 30cm rule to measure string lengths to nearest mm
  • Fig. 7c: Rosin to increase coefficient of friction between bow and string
  • Fig. 7d: Tuner to determine the pitch of the stopped note and the harmonic.   

Image result for Violin rosinImage result for Rule measuringImage result for violin tunerImage result for Violin

Image result for Violin guarneri

Method

This Experiment aims to find out the ratio of the string length between the first and fourth fingers and between the fourth finger and the bridge for consecutive artificial harmonics on the A string (see Figure 8).

 

  1. Mark points on the sides of the first and fourth finger so that each measurement is between the same 2 points.
  2. Produce first artificial harmonic on A string and compare the pitches and frequencies of the stopped note and the harmonic itself.
  3. Measure ratio of distances between marked points on fingers (D1) and between the bridge and fourth finger (D2) with the ruler
  4. Repeat steps 2 & 3 for a total of 3 times and take mean values for D1 to obtain D1av and D2 to obtain D2av as well as the frequency of the notes both stopped and heard.
  5. Repeat steps 2-4 for as many harmonics up string as practical

To calculate the mean fraction of the way up the stopped string that the fourth finger is placed (Fav), divide (D1av + D2av) by D1av.

Results

Table 1: ratio of string lengths

Stopped note [frequency /Hz] Heard note [frequency / Hz] D1 /cm D2 /cm Ratio (Fav)
B4 flat [466] B6 flat [1865] 7.9 22.3 0.26
B4 [494] B6 [1976] 7.4 23.4 0.24
C5 [523] C7 [2093] 7.1 21.1 0.25
C5 sharp [554] C7 sharp [2217] 6.4 20.0 0.24
D5 [587] D7 [2349] 6.2 18.7 0.25
D5 sharp [622] D7 sharp [2489] 5.8 17.7 0.25
E5 [659] E7 [2637] 5.7 16.7 0.25
F5 [698] F7 [2794] 5.4 15.8 0.25

Analysis

Thus the 4th finger touches the stopped string at one quarters of its remaining length. This forces a node at a position such that only the fourth, 8th, 12th etc. harmonics form. This is equivalent to the harmonic spectrum of a note with four times the frequency. Every time the frequency doubles, the note is higher by an octave and hence a note which is 2 octaves higher is heard.

Uncertainty analysis:

D1: Greatest absolute uncertainty = 0.5 * range = 0.5*0.2 = 0.1cm. Therefore maximum uncertainty = (0.1/5.4)*100 ≈ 1.9%

D2: Greatest absolute uncertainty = 0.5 * range = 0.5*0.2 = 0.1cm. Therefore maximum uncertainty = (0.1/15.8)*100 ≈ 0.6%

Fav: 1.9 + 0.6 + 0.6 = 3.1% = +/- 0.00775 ≈ +/- 0.01 (2 decimal places)

It was difficult to determine exactly where the fingers touched the string. The markings on the ends of the sides of the first and fourth fingers helped to measure the distance between the same two points each time. This introduced a systematic error into the measurements.

In order to reduce parallax error, the measurements were taken after the notes had been played. I endeavored to keep my fingers still after playing and before measuring distances. I also tried to keep my fingers still and hence did not use any vibrato. A possible improvement to reduce experimental inaccuracies would be to have one person play the notes and another to simultaneously measure the distance between the fingers.

Timbre of Artificial Harmonics

Aim: to find out why artificial harmonics produce a ‘ghostly’ sound by comparing 5 different ways of playing A5 on the violin.

Apparatus

  • Violin & bow for playing notes
  • Visual Audio application to record played notes and perform fast Fourier transforms (FFT)
  • Excel to record and modify data
  • Teraplot application for Fourier synthesis of waveforms of the notes

Method

Different methods of playing (see figure 9)

  • Type I (Normal on E): stop string fully on A5 on the E string (yellow A5)
  • Type II (Normal on A): stop string fully on A5 on the A string (blue A5)
  • Type III (Normal on G): stop string fully on A5 on the G string (pink A5)
  • Type IV (Natural harmonic on A): play A5 as a natural harmonic on the A string (lightly touch blue A5)
  • Type V (Artificial Harmonic on G): play A5 as an artificial (touch fourth) harmonic on the G string (stop green A3 fully and touch lightly the green D3 with fourth finger)

Recording

Controls & precautions:

  • The notes were recorded in a quiet, soundproofed room in order to minimize the level of extraneous sound which could alter the data.
  • The notes were recorded in the middle of long bow strokes to only capture the sustain portion of the ADSR (attack-decay-sustain-release) cycle.
  • Additionally, no vibrato was used in order for the pitch to remain constant.
  • Referencing the sound level detector in the Visual Audio app, the notes were kept at c. 80db throughout and for each measurement as I tried to play each note as loud as the preceding one and discarded any recordings of notes which were not around 80db.

Procedure:

  1. Download and open the Visual Audio app from the Google Play store.
  2. In the app, set a delay timer of 5 seconds before the app starts to record and a timing interval of 2 seconds. The app records at a data sampling rate of 44,100Hz (44,100 samples per second) using the microphone and converts it into a sound pressure level (SPL) in decibels (dB).
  3. Warm up the Normal E during the delay period by playing a few initial strokes of the note then sustaining the note at about 80 Db into and past the end of the 2 second recording period.
  4. Repeat step 3 five times in total for each method of playing the note (Normal E, Normal A, Normal G, Natural A and Artificial G).

Fourier Analysis

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Determining the amplitudes of the harmonics:

  1. The Visual Audio app performs Fourier transforms on the sound waves recorded and produces graphs showing their harmonic content (see figure 8):
  2. In the FFT graphs, the peaks show the decibel values of the harmonics: the peak furthest to the left is the fundamental. The decibel values of the first ten harmonics for each of the 25 graphs were entered into Excel.
  3. As the average loudness of each of the 25 recorded notes was not constant, the decibel values of the harmonics were scaled as follows:
  4. The decibel values were converted to amplitudes for the different harmonics:
  5. The amplitudes for the five repeats were averaged for each harmonic of each of the five methods of playing (this produced table 2 of results).
  6. Then the relative abundance of the harmonics were calculated (this produced table 3 of results):

Fourier Synthesis

In order to see how the difference in the relative heights of the harmonics and therefore the harmonic content affected the sound quality, the graphs of the complex sound waves for each of the 5 methods of playing could be obtained by adding the waves of each of the 10 harmonics in the Teraplot app.

  1. Plot each of the sine graphs of the 10 harmonics with the equation shown below:

Where y is the amplitude of the sound wave at a distance x away from the source, A is the maximum amplitude, and f is the frequency.

This produced graphs 1a, 2a, 3a, 4a and 5a for the 5 methods of playing

  1. Add the 10 equations to obtain the complex sound wave for each method of playing:

This produced graphs 1b, 2b, 3b, 4b and 5b, showing the complex waves for the 5 methods of playing

Results

Table 2: Amplitude of harmonics

Harmonic Frequency /Hz Type I /amps Type II /amps Type III /amps Type IV /amps Type V /amps
Fundamental 880 3615 4382 3070 4557 3107
2 1760 1111 1499 1443 1849 1737
3 2640 2081 825 732 387 341
4 3520 853 695 583 493 276
5 4400 1769 958 788 652 267
6 5280 600 376 128 416 56
7 6160 714 317 70 332 24
8 7040 603 204 54 152 23
9 7920 250 271 79 157 31
10 8800 98 124 33 52 9

Table 3: Harmonic content

Harmonic Type I Type II Type III Type IV Type V
Fundamental 31% 45% 44% 50% 53%
2 9% 16% 21% 20% 30%
3 18% 9% 10% 4% 6%
4 7% 7% 8% 5% 5%
5 15% 10% 11% 7% 5%
6 5% 4% 2% 5% 1%
7 6% 3% 1% 4% 0%
8 5% 2% 1% 2% 0%
9 2% 3% 1% 2% 1%
10 1% 1% 0% 1% 0%

Graph 1a: shows individual waves of the first ten harmonics for Type I

Graph 1b: Type I complex wave

Graph 2a: shows individual waves of the first ten harmonics for Type II

Graph 2b: Type II complex wave

Graph 3a: shows individual waves of the first ten harmonics for Type III

Graph 3b: Type III complex wave

Graph 4a: shows individual waves of the first ten harmonics for Type IV

Graph 4b: Type IV complex wave

Graph 5a: shows individual waves of the first ten harmonics for Type V

Graph 5b: Type V complex wave

Analysis

As no vibrato was used and the recordings were taken in the middle of a long smooth stroke, harmonic content had the greatest effect on the timbre.

Harmonic content

Table 3 shows the relative amplitudes of the first ten harmonics for each of the methods of playing A5. In comparison to the rest, the harmonics (Type IV and Type V) have greatly reduced amplitudes beyond the second harmonic. As such, their complex waves are smoother and more similar in shape to a graph of only the first two harmonics (graph 6).

This result can be shown by pie charts of the percentage harmonic contents from Table 3:

In both of the harmonics, the first and second harmonics (red and blue sections, respectively) make up a very large (c.75%) proportion of the harmonic content of the sound. This results in a complex wave which is very similar to the complex wave only containing the first two harmonics. Therefore the timbre of the harmonics is similar to that of a note with only the first two harmonics. As the artificial harmonic has an even greater proportion of fundamental (53% vs 50%) and second harmonic (30% vs 20%) in its harmonic content, it sounds even more like a tone consisting of just the first two harmonics. Notes consisting mainly of only one or two harmonics have a very pure sound, as opposed to the brighter and richer sound of notes with a greater spread of harmonic content (see below).

This can be shown by pie charts of the harmonic content of the notes which were played normally: Type I, II & III

The normally played notes have a much higher percentage of their harmonic content distributed to the third fourth and fifth harmonics (green, yellow and purple, respectively). They also have a smaller proportion of the fundamental and second harmonics. This is perhaps most noticeable for Type I where the first and second harmonics are greatly reduced (though still prominent) and both the third and fifth harmonics are greater in proportion than the second.The result is the ‘bright’ nature of notes played on the E string as there are a greater proportion of higher harmonics.

In comparison to the notes played normally, the artificial harmonics have a much purer sound and are consequently ghost-like in timbre.

Uncertainty analysis

The Fourier transform application provided data correct to the nearest 0.1 dB. Therefore the uncertainty for the majority of calculations was so the percentages in the pie charts are correct to the nearest hundredth of a percent for the smallest sections to the nearest percent for the largest ones.

To improve the reliability of my results, I repeated my experiment 5 times for each note and took their averages. I also removed anomalies and discounted and repeated recordings where any background noise was present or where the note was played incorrectly (for example, not sustaining for the whole two seconds)

In order to reduce the impact of background noise on my results, I recorded the notes in a quiet room with the phone’s microphone positioned close to the violin on a music stand. By marking out a position on the floor, I ensured that the violin was the same distance away from the phone for each recording. Additionally, as the recording app showed the average dB values in real time, I was able to sustain notes at about 80dB.

However, I only performed this experiment for my own violin with a set of Dominant string, so this experiment has no data regarding other violins and different types of strings (for example gut strings or metal strings from a different manufacturer).

Conclusion

An artificial harmonic plays a note two octaves higher than the stopped note because the fourth finger creates a node at a quarter of the length of the stopped string. Only the fourth harmonic and its multiples are produced as others are silenced. The fourth harmonic has a frequency which is 4 times greater. As notes sound an octave higher every time the frequency doubles, artificial harmonic therefore sounds two octaves higher. These artificial harmonics sound ghostly compared to notes played normally on the violin because they have a very pure harmonic content comprised of mainly the fundamental (c. 50%) and the second harmonic (30%).

An extension of this project could be to analyse the harmonic content of different artificial harmonics. The results may differ across or up strings. To improve the reliability of the experiment, it could be repeated on different violins and an average taken as different violins also have slightly different timbres. Alternatively, one could perform an experiment to determine why the perfect fourth interval is always a quarter of the remaining string length by measuring how consecutive semitones up one string get closer.

References

Adams, Steve, and Jonathan Allday. “Friction.” In Advanced Physics 2nd Edition, by Steve Adams and Jonathan Allday, 94. Oxford: Oxford University Press, 2000.

Adams, Steve, and Jonathan Allday. “Standing waves.” In Advanced Physics 2nd Edition, by Steve Adams and Jonathan Allday, 264-269. Oxford: Oxford University Press, 2000.

Adams, Steve, and Jonathan Allday. “Superposition.” In Advanced Physics 2nd Edition, by Steve Adams and Jonathan Allday, 248-249. Oxford: Oxford University Press, 2000.

Adams, Steve, and Jonathan Allday. “Waves.” In Advanced Physics 2nd Edition, by Steve Adams and Jonathan Allday, 236-237. Oxford: Oxford University Press, 2000.

American Physics Society. Fiddle Physics. 2013-2016. http://www.physicscentral.com/explore/action/fiddle.cfm (accessed 5 1, 2017).

Baek, Sangyeol. “Artificial harmonics.” 2015.

Editors of Encyclopaedia Britannica. “Envelope (sound).” In Encyclopedia Britannica, by Editors of Encyclopaedia Britannica. Encyclopaedia Britannica, Inc., 2016.

J.H. Smith, J. Woodhouse. The Tribology of Rosin. Cambridge: Journal of Mechanics and Physics of Solids, 1999.

Katznelson, Yitzhak. An introduction to harmonic analysis. Cambridge: Cambridge University Press, 2004.

Kneubuhl, Fritz Kurt. “Oscillations and Waves.” In OScillations and Waves, by Fritz Kurt Kneubuhl, 365. Springer, 1997.

Nave, R. Timbre. 2007. http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/timbre.html (accessed May 2017).

Piston, Walter. “Orchestration.” In Orchestration, by Walter Piston, 45. 1955.

Sheppard, Sheri D, and Benson H Tongue. “Statics: Analysis and Design of Systems in Equilibrium.” In Statics: Analysis and Design of Systems in Equilibrium, by Sheri D Sheppard and Benson H Tongue, p. 618. Wiley, 2005.

Wolfe, Joe. Bows and Strings. 2006. http://newt.phys.unsw.edu.au/jw/Bows.html (accessed May 12, 2017).

Zukovsky, Paul. “On Violin Harmonics.” In Perspectives of New Music. Princeston University Press, 1968.

  1. (Adams and Allday, Friction 2000)
  2. (Sheppard and Tongue 2005)
  3. (Adams and Allday, Friction 2000)
  4. (Wolfe 2005)
  5. (American Physics Society 2013-2016)
  6. (American Physics Society 2013-2016)
  7. (American Physics Society 2013-2016)
  8. (Piston 1955)
  9. (American Physics Society 2013-2016)
  10. (Adams and Allday, Standing waves 2000)
  11. (Zukovsky 1968)
  12. (Adams and Allday, Waves 2000)
  13. (Adams and Allday, Waves 2000)
  14. (American Physics Society 2013-2016)
  15. (Kneubuhl 1997)
  16. (Adams and Allday, Superposition 2000)
  17. (Editors of Encyclopaedia Britannica 2016)
  18. (Nave 2007)
  19. (Baek 2015)
  20. (Katznelson 2004)
  21. (Nave 2007)

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