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Precision Strobe Tuners

The Equal Tempered Scale and Some Peculiarities of Piano Tuning

Jim Campbell


The first skill to be learned by the musician sometimes involves tuning the instrument. For most of the band instruments this requires a single adjustment to a reference pitch, and is not really critical if the instrument is being played solo. The multiple string and fretted instruments however, must have the strings tuned to each other frequently. When tuning a fretted instrument, each string can be tuned to the adjacent string, by fretting one string appropriately and matching the open string to the fretted string. Unfretted instruments are usually tuned using a separate pitch reference for each string. Piano tuning presents several problems and peculiarities. With more than 200 separate strings, this task is usually handled by a professional. The process of setting the correct spacing between the notes is known as setting the temperament. The equal tempered scale used forms the basis for modern western music and will be described in more depth. A further peculiarity in piano tuning is the practice of stretching the upper and lower octaves. This refers to progressively sharpening the notes of the upper octaves from a perfect equal tempered scale. The reason for this has to do with what is known as the inharmonicity of ringing strings. This phenomena will also be described in detail. Understanding the equal tempered scale and the rational for the stretching of the upper octaves in piano tuning involves an understanding of something about both the physics of sound, and the history of music.

The Phenomena of Sound and the Harmonic Series

Sound is the result of periodic fluctuations in air pressure propagating through the atmosphere. When these pressure fluctuations pass by the eardrum, we here sound. Differences in the rate of fluctuation are perceived as differences in pitch. Differences in the shape of the pressure wave forms are perceived as differences in tone, or timbre. When these pressure fluctuations strike a microphone, they are converted to electrical signals, and an oscilloscope can then be used to observe how these pressure wave forms vary as a function of time. A pure tone, such as from a flute, will look like a nearly perfect sine wave. Most musical wave forms are more complex than this. A tone produced by a reed instrument, may appear to be shaped similar to a square wave. You can imagine how the reed periodically opening and closing would produce such a wave form. The wave form of a bowed violin may resemble a saw tooth. This results from the way in which the string is grabbed and released periodically by the bow. Each of these tones may be of the same fundamental frequency, but the ear and brain perceive distinct differences in timbre.

The harmonic series is an important concept in the study of sound. It allows us to describe characteristics of sound in terms of what our ears hear, rather than what our eyes see when it is viewed on an oscilloscope. It also provides insight into how different instruments produce the different sounds we hear. The harmonic series consists of a set of frequencies which are multiples of a fundamental. The frequency of any component of the harmonic series can be figured using:

f(N) = N * fo

where N is the harmonic number, and fo is the frequency of the fundamental. For example, if fo = 110 Hz and N = 1, then the resultant frequency is 110 Hz, which is the fundamental. If N = 2, the resultant frequency will be the second harmonic, or 220 Hz. If N=3, the resultant frequency will be the third harmonic, or 330 Hz. And so on. The fact that the first overtone above the fundamental is called the second harmonic can lead to confusion. Some references will refer to it as the first harmonic. Therefore, whenever these terms are used, it is important to realize whether it is the fundamental, or the first overtone, which is being referred to as the first harmonic.

It can be shown, using a branch of mathematics known as Fourier analysis, that any periodic wave form is composed of a summation of frequencies from the harmonic series. If only the odd harmonics, that is odd values of N, are summed in the proper proportions, the resulting wave form will resemble a square wave. Likewise, a saw tooth wave form can be generated by summing both odd and even harmonics in the proper proportions:


It is possible to create any periodic wave form by a summation of weighted elements of the harmonic series. This has direct application in music synthesis. It is also useful to decompose a sound wave form into its component harmonics. A spectrum analyzer is used for this purpose. The spectrum analyzer samples the sound wave forms, and displays a graph of magnitude as a function of frequency. When the spectrum of a periodic wave form is viewed on such an instrument, there will appear a peak at the fundamental, and a series of peaks at each harmonic present in the wave form:


Real sound wave forms however are more complex. They change with time or may even have overtones which are not perfect multiples of the fundamental. It is interesting to note that wave forms with mirror symmetry are composed of odd harmonics only. This is why distortion which results from the clipping of the top of the wave form, such as when over driving an amplifier, is sometimes referred to as "odd harmonic" distortion. The ability to describe a wave form in terms of its frequency components is also useful because this resembles the way the ear hears. The function of the ear is to detect the relative strength of each of these frequencies, while the brain interprets this information as both pitch and timbre. The difference in sound we perceive between a saxophone, a bowed violin string or a flute can be explained in terms of the harmonic content of the sound wave forms.

Beats

Another important phenomena encountered in the study of sound is the beat. When two notes, which are slightly different in pitch, are played simultaneously, a resultant modulation can be perceived as a slow periodic change in volume or timbre. This is known as the beat. This beat rate slows as the pitches are brought closer and stops when they are exactly matched. The cause of this is that as both notes are combined by the eardrum or microphone, the frequency difference results in a steady shift in the phases of the two notes. As the phase difference shifts from constructive to destructive interference, a slight difference in volume and timbre is perceived.


The rate at which this occurs is equal to the difference in frequency. If a 440 Hz tone and a 441 Hz tone are sounded together, a beat rate of 1 Hz will be heard. By sounding two notes close in pitch, and adjusting one so that the beat rate slows until it is stopped, a process known a zero beating, two notes can be tuned extremely accurately. Even someone who would consider themselves tone deaf can do this, if they know what to listen for.

Beats are also perceived between a tone and its natural harmonics. For example if a tone of 440 Hz is sounded with a tone of 660 Hz ( (3/2)440, or a natural fifth) the resulting tone will be beatless. If however, the 660 Hz tone is shifted to 659.2551 Hz, a beat rate of .745 Hz will be heard. This corresponds to 3 beats every 4 seconds, or a metronome setting of 45 beats per minute. If you play A5 and E5 together on a properly tuned piano this is exactly what will happen. This is a result of the equal temperament used in modern music.

The Physics of Ringing Strings

Sounds containing periodic wave forms are produced in many ways. Initially primitive music was based on rhythm. An object with elastic properties is struck, and it vibrates, radiating pressure fluctuations which rapidly decay. Methods were discovered of generating sustained tones. One involved the plucking of stretched strings. Another the resonating of columns of air in tubes. The air column could be resonated by blowing across the end, like the flute, or buzzing ones lips in the end, like the horn. With the realization that several pitches could be produced by varying ones lip technique, the natural harmonic was discovered.

A stretched string, plucked carefully, will ring with pure tone. Due to it's elastic properties, the string oscillates back and forth between two gentle curves. The condition is known as the fundamental mode of oscillation. If the length is cut in half, say by pressing the string down to a fret, it will ring in a similar manner at twice the frequency, or an interval of one octave above the fundamental. This property of a stretched string was first identified by the Greeks.

A string can be induced to ring in modes other than the fundamental. By gently touching the string at its midpoint while plucking it, a mode is induced with two gentle curves, crossing in the middle. The point at which the curves cross is called a node, and each section between nodes rings exactly as though it were a string of that length. In this case the pitch produced is 2 times the fundamental pitch which is an interval of one octave above the fundamental. By touching the string at the 1/3 point (above the 7th fret), a mode is induced with two nodes and three ringing sections. The pitch produced is 3 times the fundamental, which is an interval of one octave plus a major fifth. By touching the string at the 1/4 point (above the 5th fret), a mode is induced with 3 nodes and four ringing lengths. The pitch produced is 4 times the fundamental, an interval of two octaves. By touching the string at the 1/5 point(above a spot between the 3rd and 4th frets), a mode is induced with 4 nodes and 5 ringing lengths. The pitch produced is 5 times the fundamental, an interval of two octaves plus a third. By touching the string at the 1/6 point, a mode is induced with 5 nodes and 6 ringing sections. The pitch produced is 6 times the fundamental, which is an interval of two octaves and a fifth. This process continues:



When a string is plucked, it produces the pitch of the fundamental, as well as a series of overtones harmonically related. This is due to the fact that the string is excited in several modes of oscillation. As the string rings, the higher modes will decay, leaving predominately the fundamental mode. This combination of decaying overtones results in the distinctive sound of a ringing string. Careful measurements will show that the overtones of the string are not necessarily perfect harmonics. To differentiate between the ideal harmonics and the pitches actually produced, the actual overtones produced by the string are called partials. Although the partials are very close to the ideal harmonic series, they can differ. This effect becomes important in the art of piano tuning.

The Evolution of the Equal Tempered Scale

Early musical scales and harmonies were based on intervals from the natural harmonic series. The natural third and fifth intervals can be combined to form pentatonic scales (with five intervals dividing the octave, such as the pentatonic blues scale). With additional natural third and fifth intervals diatonic scales can be formed. A diatonic scale divides the octave into eight intervals. These eight intervals consist of 5 large steps and 2 small steps. Each of these large steps is referred to as a whole tone interval, and each small step is referred to as a semi-tone interval. A semi-tone interval is half a whole tone interval. By rearranging the pattern of whole tones and semi-tones, scales can be formed with different tonal characteristics. The major and minor scales are examples of diatonic scales. By splitting each whole tone, a twelve tone chromatic scale can be formed. A scale developed like this from pure harmonic intervals is known as a just toned scale. Modern musical terminology developed from the early use of these types of diatonic scales. The interval from the root note to the eighth note of the scale is referred to as an octave. The interval from the root note to the fifth note in a major scale is referred to as a major fifth. The interval from the root note to the third note in a major scale is referred to as a major third. It works out that the fifth interval is obtained by multiplying by 3/2 (or 2/3 to descend a fifth), and the third interval is obtained by multiplying by 5/4. So, the numerical ratio of frequencies of the fifth, has nothing to do with 5, and the ratio of frequencies of the third, has nothing to do with 3.

A twelve tone chromatic scale can also be developed as a natural consequence of the cycle of fifths. The Chinese discovered the cycle of fifths more than 5000 years ago. This is how it works: Begin with a root note, and play the note a fifth above it. Then you play the note an interval of fifth above this note. If this process is repeated a total of twelve times, you will end up with a note very close to the root note you started with, only seven octaves higher. If each of these notes is translated down by octaves, a scale can be formed dividing the octave into twelve intervals.

An interesting thing happens if you actually play a cycle of fifths using only intervals which are perfect natural fifths, that is where each interval is exactly (3/2)fo. If you start with fo = A0 = 27.5 Hz, then the twelfth interval will result in f = 3568.02 Hz. If on the other hand, you start at A0, and play perfect octaves up to A7, you will get f = 3520 Hz. This can be quickly verified on a hand calculator; try it. This difference in pitch between the twelfth note of a natural fifth series, and a natural octave series of 48.02 Hz results in a pitch error of 24 cents, almost a quarter of a semi tone. If on the other hand, you play a cycle of fifths using intervals which are each slightly flat by just the right amount, when you reach the twelfth interval, you will end with f =3520 Hz! If each fifth in the series is flattened by the same amount and the adjustment distributed evenly throughout the cycle, it works out that each fifth should be played flat by 2 cents. When the resulting twelve notes are all shifted, by octaves, into ascending order, the result is an equal tempered chromatic scale. The spacing between the notes of this scale will have intervals which are all equal.

The advantage of this equal tempered scale over a just toned scale may not be apparent at first. In fact, harmonies played on a just tone scale may sound purer because of the total absence of any beats produced by the natural intervals. But it becomes clear when harmonies are shifted to other keys. The intervals between the notes of a just toned scale are not constant. When the patterns of intervals making up a chord are shifted in key in a just toned scale, the intervals within the chord are shifted very slightly, resulting in some chords with beats that can clash quite badly. In an equal tempered scale, chords can be shifted at will without harmful effect.

Two developments occurred in music technology which necessitated changes from the just toned temperament. With the development of the fretted instruments, a problem occurs when setting the frets for just tuning, that octaves played across two strings around the neck would produce impure octaves. Likewise, an organ set to a just tuning scale would reveal chords with unpleasant properties. A compromise to this situation was the development of the mean toned scale. In this system several of the intervals were adjusted to increase the number of usable keys. With the evolution of composition technique in the 18th century increasing the use of harmonic modulation a change was advocated to the equal tempered scale. Among these advocates was J. S. Bach who published two entire works entitled; "The Well-tempered Clavier". Each of these works contain 24 fugues written in each of twelve major and twelve minor keys and demonstrated that using an equal tempered scale, music could be written in, and shifted to any key.


Piano Tuning

The initial task of a piano tuner is known as setting the temperament, that is, to tune a section in the middle of the keyboard to an equal tuned temperament. Once this is done, the rest of the strings may be set by tuning octave intervals up and down the keyboard. The trick then, is in adjusting the spacing between notes over a two octave range, starting with a single C tuning fork. One might suspect this could only be done if one had perfect sense of pitch, but this is not necessary. There are systematic methods of setting the temperament which depend on setting intervals to natural harmonics, and then adjusting them sharp or flat by listening to, and adjusting for a specific beat rate. Beat rates can be adjusted to the speed of a metronome, and measured quite accurately. A tuner, armed with a good sense of time, and an ear that knows what to listen for, can very accurately set the temperament from a single pitch reference.

The are many systems of temperament setting, The Fischer method is not generally used, but has an advantage of simplicity. First, a strip of felt is looped between each group of three strings over a two octave interval to damp out the outside strings in each group and allow only the middle string to ring when struck. Middle C is tuned directly to a fork. Then the octaves above and below are tuned against middle C until the intervals sound beatless. The next step is to actually set the temperament. When finished, the two octaves between the upper and lower C already tuned, will be set to equal temperament. The interval from the C to G in the lower octave is tuned to a beatless natural fifth, then flattened slightly to a beat rate of .7 beats per second forming an equal tempered fifth. Then the G in the second octave is tuned beatlessly to the lower G. Next, the D in the second octave is tuned to the lower G , 2 cents flat of a natural fifth, with a slightly faster beat rate. This procedure continues on up through the cycle of fifths until both octaves are tuned. The catch is, that the beat rate corresponding to 2 cents on the highest fifth interval used is 1.4 beats per second, rather than .7. As each interval is adjusted flat, the beat rate must steadily increase from .7 to 1.4 beats as the interval moves up the keyboard. Using a metronome, one can measure the beat rate accurately and develop a good sense of timing to be able to make this adjustment smoothly. The final fifth of the series ends back on C, and must end up with the proper beat rate. Once the two temperament octaves are completed, the damper strip is removed and the outside strings in each group are tuned beatlessly to the middle string. This is known as tuning the unisons. Wedges are used to damp each edge string while tuning the other. Finally the remaining strings are tuned to the temperament octaves using beatless octave intervals up and down the keyboard.

When one measures a piano that has been tuned by a skilled tuner, something interesting will be discovered. The middle of the piano will be close to perfect, while somewhere about the second octave above middle C, the notes will gradually become sharper, until at the high C this sharpening may be as high as 20 to 30 cents! Also there will be some flattening of the notes in the lower octaves. Shown is a plot of actual measurements of a piano tuned by a skilled tuner, along with a curve resulting from the average of many such measurements:

This affect is known as the "stretching" of the upper and lower octaves. It is due to the fact that a string does not necessarily vibrate at perfect natural harmonics. The harmonics of a vibrating string tend to be sharp of the natural harmonic series. This becomes more pronounced as the string is made shorter or thicker. When the tuner is tuning octave intervals up the keyboard, they tune for the best sound, compromising the beat between the string fundamentals and the beat between the octave partial of the lower note to the fundamental of the upper. This produces a slightly sharp octave interval between the two strings. This slight adjust accumulates towards the high and low end. If a piano is tuned to a perfect scale top to bottom, the beats produced between partials of the high strings clash. A piano tuned with the upper and lower octaves stretched just sounds better.

The shape and degree of the stretch is dependent on the string configuration of the instrument. Pianos with longer bass string sections require less stretch in the lower end. In principal then, a piano could be fairly well tuned using a tuning indicator device, such as a Precision Strobe Tuner, and adjusting the device calibration accordingly across the keyboard. A further refinement of this method might be to develop and use different tuning curves for pianos of different construction. A simplification of the above method would be to tune the piano correctly up the keyboard to about E5. From this point up, progressively sharpen each note about .5 cent ending at C7 sharp about 30 cents. This two line curve fit can be readily set using a chromatic tuning indicator device and seems to result in a reasonably well sounding tuning. This is probably a good starting point for the beginning and amateur tuner. There are algorithms and software available which will calculate a temperament for a given piano based on inharmonicity measurements. There are also people who will say this can be done properly only by human ear. Unfortunately, this is a skill that seems to be falling into decline. Perhaps the best method for the amateur tuner of finding the best temperament of a given piano is to locate a skilled tuner who tunes by ear and have them tune the piano. The stretch could then be measured and recorded for future use. Of course, a student with patience and perseverance can learn to tune a piano properly using a single pitch reference and their ears. It is said this is a skill that can take years to develop. If the number of skilled piano technicians continues to decline, it is a skill that could become more and more valuable.

The goal of musical instrument adjustment and tuning is an instrument that sounds good. The qualitative definition of what sounds good has been being developed by musicians and instrument makers for thousands of years. Modern technology has provided the tools to make very accurate quantitative measurements of these qualities. But these tools are only tuning aids, alone they do not guarantee the goal of adjusting an instrument so that it sounds good. As we have seen, a piano tuned perfectly to an equal tempered chromatic scale will not sound as good as one that has been stretched somewhat in the upper and lower octaves. However, when these tools are used along with a bit of judgment based on knowledge of music, good results can be obtained.


Appendix A, The Equation of Pitch of the Equal Tempered Scale.

The octave is divided into twelve intervals to form the chromatic scale. Each of these intervals is further divided into 100 cents. Cents are used to describe small differences in pitch in terms of percentage of a semi tone.

Intervals of pitch are described in terms of the ratios of the frequencies, not absolute difference in frequencies. An octave interval is always twice, or half the frequency, of the first note. In the equal tempered scale, the twelve intervals are spread evenly between the octaves. If the ratio of each semi-tone is the inverse of the twelfth root of two (1.059463), this condition will be met. The frequency of each note in the scale can be figured by multiplying each successive note by this number to get the next. The frequency of any note can also be figured from:

f(N) = 27.5*2^(N/12)

where N is an index into the chromatic scale notes starting with 0 for A0, the lowest note on the keyboard. N increases by 1 for each note on the keyboard. Note that the actual key on the keyboard is N + 1. Table 1 shows the index and corresponding frequencies for all of the keyboard notes based on A4 = 440 Hz. To find the frequency of a note 12 cents sharp from A4, a value of n = 48.12 would be used. Sometimes it is useful to convert a frequency into N, solving for N:

N(f) = (12/ln(2))*ln(f/27.5)

This formula can be used to determine the note corresponding to a given frequency. Once N is figured, the integer value closest to it can be looked up in the following table. This equation also tells you the cents error from the corresponding note by taking the difference from it to the closest integer and multiplying by 100.
Oct
0
1
2
3
4
5
6
7
Note
N
f
N
f
N
f
N
f
N
f
N
f
N
f
N
f
A
0
27.5000
12
55.0000
24
110.0000
36
220.0000
48
440.0000
60
880.0000
72
1760.000
84
3520.000
Bb
1
29.1352
13
58.2705
25
116.5409
37
233.0819
49
466.1638
61
932.3275
73
1864.655
85
3729.310
B
2
30.8677
14
61.7354
26
123.4708
38
246.9417
50
493.8833
62
987.7666
74
1975.533
86
3951.066
C
3
32.7032
15
65.4064
27
130.8128
39
261.6256
51
523.2511
63
1046.502
75
2093.005
87
4186.009
Db
4
34.6478
16
69.2957
28
138.5913
40
277.1826
52
554.3653
64
1108.731
76
2217.461
88
4434.922
D
5
36.7081
17
73.4162
29
146.8324
41
293.6648
53
587.3295
65
1174.659
77
2349.318
89
4698.636
Eb
6
38.8909
18
77.7817
30
155.5635
42
311.1270
54
622.2540
66
1244.508
78
2489.016
90
4978.032
E
7
41.2034
19
82.4069
31
164.8138
43
329.6276
55
659.2551
67
1318.510
79
2637.020
91
5274.041
F
8
43.6535
20
87.3071
32
174.6141
44
349.2282
56
698.4565
68
1396.913
80
2793.826
92
5587.652
Gb
9
46.2493
21
92.4986
33
184.9972
45
369.9944
57
739.9888
69
1479.978
81
2959.955
93
5919.911
G
10
48.9994
22
97.9989
34
195.9977
46
391.9954
58
783.9909
70
1567.982
82
3135.963
94
6271.927
Ab
11
51.9131
23
103.8262
35
207.6523
47
415.3047
59
830.6094
71
1661.219
83
3322.438
95
6644.875


Appendix B. Further Reading:

The Science of Sound
Thomas D. Rossing (1982), Addison-Wesley Publishing

The Acoustical Foundations of Music
John Backus (1977), W W Norton and Company

Piano Servicing Tuning & Rebuilding
Arthur A. Reblitz (1993), The Vestal Press

PIANO TUNING, A Simple and Accurate Method for Amateurs
J. Cree Fischer (1907)

THE SEVENTH DRAGON: The Riddle of Equal Temperament
Anita T. Sullivan (1985), Metamorphous Press.

"The Endangered Piano Technician"
Essay, James Boyk, Scientific American December 1995


Copyright 1997 James A. Campbell
All Rights Reserved