Automatic Piano Tuning

Transcription

1 AMERICAN UNIVERSITY OF BEIRUT FACULTY OF ENGINEERING AND ARCHITECTURE MECHANICAL ENGINEERING DEPARTMENT Final Year Project Report Automatic Piano Tuning Prepared By: Project Supervisor: Matossian, Garo Tilbian, Joseph Dr. A. Smaili June 3,2002

2 Abstract The most popular and loved musical instrument is the piano. However the Tuning of this instrument requires tremendous effort and a good knowledge. Piano tuning technicians are the only qualified people to this task. The purpose of our project is to design a device, which can be easily used by anyone who doesn t have the background knowledge about piano tuning and whishes to tune his piano by himself. After a brief introduction, we will introduce the art of piano tuning and try to define the different variables involved in the process of tuning a piano. Next, we will try to design and put together a system capable of doing what the technician does. Finally, we will discuss the results delivered by our system and propose ways to upgrade this system. 2

3 Table of Contents Introduction and Problem Statement 4 Literature Review 5 The Nature of Sound and Theory of Music 5 The Harmonic Series 10 The Equal Temperament 13 The Piano 17 Tuning the Temperament Octave 22 Tuning instruments 27 Design Considerations 29 Proposed Solution 31 The Microphone and The Interface Circuit 32 The Micro Controller 35 Sequence of Events 35 Capture and Compare Module 36 Tuning Gun Control 45 LCD Control 46 The Tuning Gun 47 The Torque 47 The Rotational Speed 48 The Motor 51 Worm Gearing 52 The Tuning Tool 57 Results and Discussion 62 Packaging 62 How to Use the Instrument 64 Using the Tuning Gun 65 Conclusion 67 References 68 Appendices 72 3

4 Introduction and Problem Statement The Piano can be considered as the most popular musical instrument, that nearly every well-to-do household is furnished with one. To supply this demand for pianos companies manufacture some three hundred thousand pianos yearly, in the United States alone. These pianos must be tuned many times (at least 10) in the factory before they are shipped to the salesroom; there they must be kept on tune until sold. And, finally, they take up their permanent place in the homes of the purchasers, where they should be tuned at least twice every year. Piano tuning is considered a profession. The tuner has to be well qualified. He needs to have a good musical ear, some mechanical ability, and some knowledge about music theory and harmony. Basically, what the tuner does is that he sets the piano s temperament. By controlling the tension in the strings of the piano with the tuning hammer, he can achieve the desired frequency for a desired key. To set all the keys (88), the tuner usually needs three to five hours of continuous work. To achieve a good tuning the tuner has to set a good temperament, which can hold for all scales. Until today the only instrument which can help a tuner in accomplishing his task is the Strobe Tuner. This equipment has a mount on stroboscope that responds to the acoustic frequencies it receives through a microphone. Our Final Year Project involves the construction of a precision instrument Automatic Piano Tuner that does the job of a qualified tuner, in a simple and easy way. Anyone with no prior knowledge about tuning will be able to set the temperament of a piano by simply following instructions, clearly explained in a user manual booklet. The basic idea behind such an instrument is to somehow capture the acoustic signal generated by a single string of the piano, process and analyze this signal, compare it to some preset value, and from the result of this comparison, control a tool placed on the pin relative to the string being tuned, capable of changing the acoustic frequency of the string, until the desired frequency is achieved. The final product will be composed of a small rectangular box and a tuning gun. The box houses the electric as well as electronic components, a build in microphone, an Input and an output jacks for external microphone, a liquid crystal display, an off/on switch, as well as two control buttons. The tuning gun houses a DC motor, a gearbox, and a control button. This instrument has two markets. It can either be sold to a piano manufacturing company or to any household that owns a piano. As we mentioned before, tuning a piano can take up to five hours of hard work. We believe that this machine will be able to accomplish its task in about fifteen to thirty minutes time. 4

5 This machine, if adopted by a manufacturing company, can save both time and money. If the final product is marketed at a reasonable price and becomes available to every household, people will be able to tune their own pianos in no time, and for only the price of the equipment. Literature Review Piano tuning is an art. To be able to design such a precision instrument, which would rely on scientific methods and procedures, we have to understand and study this art thoroughly. During this study, we will first discuss the nature of sound and the laws of physics, which apply to the musical sound, as well as some basic theory of music and the tempered scale. Second, we will introduce the piano as an instrument and briefly talk about the seven essential parts that make up a piano. Third, we will introduce the method of fifths and fourths used in setting the temperament of a piano used by professional piano tuners, by relying on their ears. Fourth, we will talk about a few instruments currently available on the market used in piano tuning, and try to point out some of the shortcomings of these instruments. And finally, we will point out all the variables that can be identified from this discussion and try to relate these variables to come up with a design for this precision instrument. The Nature of Sound and Theory of Music (This section is taken from Reference 3. Only slight changes have been made to the text.) We can describe sound as a physical phenomenon and state that sound is the vibration of a solid, liquid or a gas. Sound, my be perceived as either musical in nature or as noise, depending somewhat on the view of the listener. Noise tends to be irregular and often harsh, while musical sounds like those produced by a songbird or a musical instrument tend to be more regular and pleasing to the ear. The concept of music is a matter of perception rather then a matter of physics. However laws of physics do apply to musical sounds. The knowledge of how these laws apply to music and piano is necessary to understand why we tune a piano the way we tune it. 5

6 The laws of physics that govern the vibration of the piano string are: The first law states that by keeping the tension, diameter, and density of a string constant, the frequency of a vibrating string is inversely proportional to its length. (The shorter the string, the higher the pitch) The second law states that by keeping the length, diameter and density of the string constant, the frequency of vibration of the string is proportional to the square root of the tension. (The greater the tension, the higher the pitch) The third law sates that by keeping the length, tension, and density of the string constant, the frequency varies inversely as the diameter of the string.(the greater the diameter, the lower the pitch) The fourth law states that by keeping the length, tension, and diameter constant, the frequency is inversely proportional to the square root of the density of the string. (If a gut string replaces a metal string, the frequency will be lower) When we come to define music we must first start by defining the musical scale, the octave, the intervals, the frequency ratios and the harmonic series. The musical scale is a series of musical tones, each bearing some mathematical relation with the other. The modern scale is divided into 12 separate tones called an octave. Seven of these notes are given names from A to G and are found by playing the white key on the piano. The white keys are also called naturals. The black keys are either called sharps or flats in music, but in piano tuning they are referred to as sharps. A sharp gets its name from the natural key to its immediate left. The symbol # is used for the word sharp. Therefore the black key directly to the right of the G key is referred to as G #. Since we have 7 octaves in a piano, at least seven keys have the same name. One way to identify them is to number the keys from the leftmost of the piano to the far right and refer to them with their numbers for example A49 which refers to the A note in the fourth octave counting from the left. The second way is to refer to them by the number of the octave they are found in. Considering an octave starts at a C and ends at the following C, we can refer to all the notes starting from C4 till C16 by including C4 and excluding C16. That is the A found between C40 and C52 is referred to as 4 A. An interval is defined as the frequency distance between any two notes of the scale. This frequency distance can be expressed as a ratio or fraction. The simplest interval in music is the Unison. This interval is the result of two tones from two separate sources that are at the same frequency. An example is the 6

7 three strings in a piano corresponding to the same note. These three strings are tuned to produce the same frequency when hit by a hammer. Because of this they are called unison strings. The mathematical relation between tones of Unison is expressed as the ratio 1:1. The next simplest interval and the one on which the modern scale is designed is called the Octave. The tones of the octave have a frequency ratio of 1:2. The octave derives its name from the eight steps in the scale between the lowest note in the interval and the highest. For Example, C40 and C52. The interval having a ratio 2:3 is called the fifth. It encompasses five steps in the scale. For example, if a scale is played at C40, G47 is the fifth note encountered. The interval of the fifth can be used to find all the notes of the scale, the Western man has used in his music, called the diatonic scale. Begin at any C on the piano and play a fifth downward. Then go to the same note and start playing a series of notes up the scale, each a fifth higher than the scale preceding it. These notes spread over five octaves comprise the notes of the diatonic scale as begun on C. If these notes are compressed into the same octave they produce the scale used today. The mathematical relationship between these tones is shown in Table 1. These ratios apply to the frequency distances of the intervals. The ratio will be the same, no matter what note is used as the starting point. For example the frequency of 1 C is 266 Hz then the frequency of 1 F is equal to 266 * 4/3 = Hz. Interval Name Note Frequency Ratio From C First (Unison) C1 1/1 Second D 9/8 Third E 5/4 Fourth F 4/3 Fifth G 3/2 Sixth A 5/3 Seventh B 15/8 Eight (Octave) C2 2/1 Table 1: Frequency Ratios of the intervals of the Diatonic Major Scale [3, p. 31] The ratio of the frequencies of any two adjacent notes in the scale can be found by dividing their ratios relative to the first note in the scale with each other. For example you want to find the frequencies of E and F in Table 1. This ratio is found by dividing 5/4 by 4/3 which equals 15/16. The frequency ratios of two adjacent notes for an octave are presented in Table 2. 7

8 Ratio Notes Between Steps C D 8/9 D E 9/10 E F 15/16 F G 8/9 G A 9/10 A B 8/9 B C 15/16 Table 2: Ratios between adjacent tones in a diatonic scale [3, p. 31] Table 2 shows that there are three different ratios between the adjacent tones of the scale. Two of these are fairly close together, namely 8:9 and 9:10. At two places in the scale, the frequency difference is 15:16, which is smaller. If the scale is played, and the difference between each tone listened to carefully, it can be perceived that the sound interval between the second and third note and between the seventh and eighth note is less than elsewhere in the scale. This is what should be expected considering the smaller proportional difference that is found to exist mathematically at these places. Because the ratios 8:9 and 9:10 are very close, the diatonic scale can be considered to be made of steps that have two basic frequency distances. Most of the steps, having the wider distance of either 8:9 or 9:10, are called whole steps or whole-tones. The smaller intervals are called half steps or semi-tones. The diatonic scale has eight steps, with a semi-tone difference between the third and fourth notes as well as between the seventh and eighth notes. The rest of the steps are a whole-tone apart. A look at the piano keyboard shows that its design accommodates this scale exactly when it begins on C. There are no sharp keys between the E and F or the B and C, where the semi-tone steps should fall. If C1 in Table 1 is assigned the frequency of 264 Hz, which is close to that of middle C, the frequencies of the other notes can be found by multiplying 264 by the appropriate ratios, which are given in the table. These are shown in Table 3. The frequency of D is found to be 297 Hz. 8

9 Frequency Note (Hz) C1 264 D 297 E 330 F 352 G 396 A 440 B 495 C2 528 Table 3: Frequencies of the tones in the diatonic major scale derived from the ratios of Table 1, with C1 assigned a frequency of 264 Hz [3, p. 32] Suppose now this table is used and the scale begun again but this time starting on D. The second note in the scale is now E. Using the same ratios given in Table 1, its frequency is found to be 334 Hz, which is different, but close to the frequency assigned to it when played in the key of C. Therefore, if the piano had been tuned using C as the starting point, the E which serves as the Third from C could also serve fairly well as the Second from D. Now, using D as the starting point at 297 Hz, and multiplying this by 5/4, the frequency of the Third from D can be found to be 371 Hz. The piano, which was tuned with C as the starting point, has no note tuned to this frequency. It falls between the F and G. To accommodate for this, the piano has a black key between the F and G, called here the F sharp. The other sharps, which are found on the keyboard and in the musical scale, are there so that a scale may be played on an instrument when starting with any note. No matter what the starting point, a scale can be played while maintaining the diatonic relationship of wholetone and semi-tone steps by using a combination of white keys (naturals) and black keys (sharps). The early Greeks did not have this flexibility. Because they had not fully developed the scale, they used a system called modes. Each mode had eight steps, and the placement of the whole-tone and semi-tone steps was specified as well as the pitch on which the mode was to begin. This must have presented some difficulty when changing from one mode to another because all the instruments would have to be re-tuned. The pattern of Fifths, which was used to find the eight notes of the scale beginning on C, can be extended to find the sharps that are needed to play the diatonic scale in other keys. Continuing from there, it is found that the Fifth above B is F#. A Fifth above that is C# and so on for the remaining G#, D#, and A#. The Fifth above A# turns out to be E#. There is, however, no E# key on the piano. Because a sharp is a semi-tone above a natural, and F is a semi-tone above E, F 9

10 is played for E#. Because F was the starting point of the pattern, the cycle is completed. So far, the scale that has been considered is the diatonic major scale. There are also diatonic minor scales. The minor scales are similar to the major scale, but the semi-tone steps fall in different places. Playing music in the minor scales results in a sound that is more moody or haunting. The uses and derivations of the minor scales are not really important to the piano tuner. He should, however, be able to find the minor Third interval because it is used in tuning. The minor Third falls a semi-tone below the Major Third. It is important to realize that the two notes of a given interval are always spaced the same distance apart on the keyboard, regardless of where in the scale they are located. The piano tuner can always find the second note of an interval by counting the correct number of semi-tones from his starting point. The intervals most commonly used by piano tuners are the Octave, Major Sixth, Fifth, Fourth, Major Third, and minor Third. Table 4 shows the number of semi-tones found within each of these intervals. Interval Number of Semi-Tones Octave 13 Major Third 10 Fifth 8 Fourth 6 Major Third 5 Minor Third 4 Table 4: The number of semi-tones within each interval in piano tuning [3, p. 34] To use this table, it is necessary to count both the key on which the series begins and the one on which it ends, as well as all the sharps and naturals in between. As an example, suppose the Fifth above F # is to be found. The table shows that there are eight semi-tones in the interval of a Fifth. By beginning to count on F #, you find the eighth semi-tone above it to be C #. C # is then the Fifth above F #. The Harmonic Series (This section is taken from Reference 3. Only slight changes have been made to the text.) Because sound is defined as the vibration of an object, much can be learned about why a piano string (actually a piece of wire) sounds as it does by studying the nature of its vibrations. If a string is stretched tightly between two fixed points and is plucked or otherwise induced to vibrate, it will begin to swing back and 10

11 forth along its entire length, as shown in Figure 1. This is the first mode of vibration, and it produces a tone called the fundamental, which is recognized as a note in the musical scale. Figure 1: The first mode of vibration of a piano string [3, p. 34] At the same time, the string will divide and vibrate in halves as shown in Figure 2. Since the string is now vibrating in halves also, a second tone will be produced from this same string. The tone produced by this second mode of vibration will have twice the frequency of the sound produced by the first mode of vibration because the length of each segment is half that of the entire string. For example, if the string in Figure 1 produces a tone with 440 Hz, the tone produced by the second mode of vibration will be 880 Hz. This tone is called the second harmonic of the string; it is much quieter than the fundamental. It is so quiet that the casual listener does not normally hear it. Figure 2: The second mode of vibration of a piano string [3, p. 35] In addition to dividing and vibrating in halves, the string will also vibrate in three equal segments as shown in Figure 3. This will happen at the same time that the two other modes are occurring. The string divides and vibrates in increasingly shorter sections, with each mode producing a separate distinct tone. All these tones occur simultaneously. 11

12 Figure 3: The Third mode of vibration of a piano string [3, p. 34] The harmonic series is the term used to describe this group of tones. The fundamental, caused by the string vibrating along its entire length, is called the first harmonic. The next highest tone, caused by the string vibrating in halves, has already been identified as the second harmonic. The next highest, caused by the string vibrating in thirds, is called the third harmonic. The fourth harmonic is caused by the string dividing in four equal segments, and so on. Suppose for example that the string in Figure 1 is vibrating and producing a fundamental tone with a frequency of Hz, which is the frequency of C16. The second harmonic is always twice the frequency of the fundamental, so in this example it would have a frequency of Hz, which is the same frequency as the fundamental of key C28. Table 5 shows the harmonic series that results from C16. Each of these tones will correspond to the fundamental of some note on the keyboard. By playing these notes on the keyboard, the interval distances between the successive harmonics can be seen easily. These same interval distances will apply to the harmonic series of every note of the scale. Harmonic Coincides with Note Interval Distance from preceding harmonic Eight C52 Second Seventh A#50 Minor Third Sixth G47 Minor Third Fifth E44 Major Third Fourth C40 Fourth Third G35 Fifth Second C28 Octave First C Table 5: The interval distance between successive frequencies in the harmonic series [3, p. 36] 12

13 The harmonic series of two tones will have some frequencies that will coincide. Consider the harmonic series of C16 and G24. The second harmonic of G24 should have the same frequency as the third harmonic of C16. By listening to the frequencies that should coincide in the harmonic series of two notes, a piano tuner can accurately tune a piano. To do this, he must listen to the beating that occurs when the notes are tuned so that the sound waves of their coincidental harmonics are slightly out of phase. Consider again what happens when C16 and G24 are played at the same time. The harmonic series of each should overlap at certain frequencies. If C16 and G24 are tuned to their mathematically correct frequencies, then what is called a pure or just interval results. If these two notes are tuned to a pure Fifth. then the second harmonic of G24 will be Hz, and the third harmonic of Cl6 will also be Hz. The sound that results when both are played simultaneously will be smooth; almost as if made by one string. If, however, the G is tuned so that its second harmonic is 2 Hz low, the sound the two produce together will not be smooth. Instead it will produce the alternating loud-soft pulsation of beats. Because the coincidental harmonics are 2 Hz out of phase, the sound will beat two times in each second. Consider now the interval of a Unison. The note A49, in the middle of the piano, has three strings. If two of the strings are plucked, each should vibrate with a fundamental of 440 Hz. If one string is slightly low, producing for example a fundamental of 437 Hz, the sound the strings produce together will beat three times in each second. If one string is too high by 3 Hz, the resulting sound will still beat three times per second. Although beating can be used to determine if a string is out of tune, it cannot be used to tell automatically if it is sharp or flat (i. e., too high in frequency or too low). It is enough to realize that the piano tuner does not have to play each string separately and then guess if it is at the same pitch as another. Two strings can be tuned, comparing one to the other, until the beats finally disappear. This is how pianos are tuned. The Equal Temperament (This section is taken from Reference 3. Only slight changes have been made to the text.) The term temper means to modify or regulate. When tuning the piano, it is necessary to temper the diatonic scale because of some difficulties arising from the relationships of tones in it. The problem can be illustrated by tuning the piano beginning at A1 in two different ways. First, A1 is tuned to its proper frequency of Hz. The rest of the A s are then tuned by Octaves from this starting point. Because the ratio of the Octave is known to be 2:1, the correct frequencies for the other A's can be figured quite 13

14 easily. A1 is tuned to Hz, so this is multiplied by two to show that the frequency of A13 is 55 Hz. This is then multiplied by two in order to find the frequency of A25, and so on. Upon reaching A85, which is the seventh in the series to be tuned, the correct frequency is found to be 3520 Hz. Beginning again at A1 having 27.5 Hz, but this time using the ratio of successively higher Fifths, the tuning will once again arrive at A85. This time, calculations based on the ratio of 3:2 for Fifths indicate that A85 should have a frequency of Hz. This means that the frequency of A85 would be about 48 Hz higher if the tuning were done by Fifths instead of by Octaves. This discrepancy is known as the comma of Pythagoras. The term comma is defined as a small differentiation in pitch. The nature of how this discrepancy affects the scale within one octave can be shown by comparing a scale designed with the frequencies figured from C and one figured from D. Earlier, when investigating the nature of scales, it was shown that sharps were needed in order to be able to play in any key. It was discovered that if C40 were arbitrarily set at 264 Hz, then E44, which is a Third above C40, would have a frequency of 330 Hz. If the scale is now started on D, the Second above it should be E44, a whole-tone. Calculations done using the ratio of 9/8 for the interval of a Second, show that E44 should now have a frequency of Hz. If these calculations are continued for the remaining intervals, the frequencies for the octave from D42 to D54 can be found. Table 6 shows the frequencies for the notes of this octave as figured from C40 at 246 Hz, and as figured from D42 at 297 Hz. In the second case, D42 is assigned the frequency of 297 Hz because this is found by multiplying the frequency of C40 by 9/8. This is the mathematical ratio that should exist between these two when calculating the scale from C40. Scale Beginning on D42 as calculated from C40 at 264 Hz Scale Beginning on D42 as calculated from C42 at 297 Hz Note Frequency(Hz) Note Frequency (Hz) D D C# C# B B A A G G F# F# E E D D Table 6: The frequencies derived form the diatonic ratios for the Octave D42-D54 as calculated from C40 and D42 [3, p. 39] 14

15 Even though both sets of figures are determined by the perfectly legitimate means of multiplying by the intervals' ratios, the answers can be seen to vary, depending upon what note is used as a base for the calculations. It is evident then that no diatonic scale can be tuned on the piano by using-only pure intervals. Because the piano locks the player into using combinations of 12 separate tones in each octave, the tuning must be altered from pure intervals, so that the instrument can be played in any key. This characteristic of the keyboard causes a temperament to be required. Instruments in the violin family, for example, have no such limitations. The violinist can make minute changes in tuning as he plays by changing slightly the position of his fingers. A singer can do the same by adjusting the tension of his vocal chords. When tuning keyboard instruments, it is necessary to make some kind of compromise in the spacing of the intervals. Although this will result in some distortion, the result is an acceptable intonation with which any possible interval or group of intervals can be played and still sound pleasant to the ear. The inaccuracies that result from the necessary compromises are small, and, through time, Western culture has become accustomed to hearing this altered scale. It has become so accepted that it is considered by the modern listener to be accurate, and a scale built on pure intervals would sound odd to him. Even the violinist, who could alter his fingering to play pure intervals, tends to use the tempered ones, because they are what he is used to hearing. The various systems that have been used through time to modify the diatonic scale have been called temperaments. An adjective is used to describe what type of temperament it is. Modern music is played in equal temperament, which was popularized by J. S. Bach. His studies on the well-tempered "clavier" are written to demonstrate the flexibility of this type of tuning. To understand how an equally tempered scale is designed, consider again the ratios between the steps of the diatonic major scale as shown in Table 2. The ratio between whole-tone varies slightly, depending upon the location in the scale. Because a whole-tone is comprised of two semi-tones, the ratios of these will also vary throughout the scale. Equal temperament is used to eliminate this variation and to make the proportional distances between all semi-tones equal. The end result is a scale in which the Fifths are all tuned a little bit narrow of a pure Fifth, but not so much as to be objectionable. The Fourths all end up being slightly wide of pure. All the intervals are affected; some tuned wide and others, narrow. None are so radically altered as to be dissonant, at least to modern man. The only exceptions are the Unisons and the Octaves. Unisons, being two tones of the same frequency, will always be pure no matter how the scale is tempered. Because the Octave is the basis for Western music and because the tempering that must be done takes place within this framework, the Octave remains pure. 15

16 To understand the theory behind equal temperament, remember that the frequency of the highest tone in the Octave is twice that of the lowest. Remember also that the frequency of one note is multiplied by the proper ratio to find the frequency for the other notes. Because the Octave contains 12 semi-tones and because equal temperament requires that the ratios between all 12 be equal, the proper ratio between adjacent semi-tones will be that number that equals two when multiplied by itself 12 times. This number is the ratio that can be expressed as the decimal Using the ratio and the accepted convention that A49 should have a frequency if 440 Hz, it is possible to assign frequencies to all the notes of the piano. Table 7 shows the frequencies that result from using this ratio for the Octave C40-C52. Also shown in Table 7 are the frequencies for these same notes when figured from the ratios of the diatonic scale, assuming that A49 is at 440 Hz. Table 7 shows that the frequencies of the equally tempered scale are different from those derived solely from the diatonic ratios. Frequencies in equally tempered scale from C40-C52. Based on Standard Pitch of A49 at 440 Hz Frequencies of Diatonic scale using the ratios given in Table 3. Based on Standard Pitch of A49 at 440 Hz Note Frequency(Hz) Note Frequency (Hz) D D B B A A G G F F E E D D C C Table 7: The frequencies of an equally tempered octave, compared with those derived from the diatonic ratios. Both are calculated from A49 at 440Hz [3, p. 41] Piano tuners have established logical and relatively easy systems for tuning one Octave of the scale so that it is equally tempered. This Octave is then used as the starting point for tuning all the remaining notes of the piano. 16

17 The Piano Literally piano means soft and the word forte means loud. The combination of both words Piano Forte, describes the wide range of sound and volume which are the main characteristics of the instrument, which is keyed and stringed and simply known as the Piano. The first pianofortes were developed in southern Italy about three hundred years ago. Any piano has seven essential parts, which are the case, the keyboard, the action, the plate or metal frame, the soundboard, the pin block, and finally the strings. The case is made of hardwood veneer, such as oak, walnut, rosewood or mahogany. There are four types of case The grand piano, which is usually used in concert halls and large studios. These pianos have the longest strings and produce a very deep tone. The upright piano, which is called so because its action, its case, and its plate are vertical. It has a very simple action and the hammer is forward. The studio piano, which was made to accommodate the smaller homes and apartments. Its strings are somewhat shorter then those of the grand and the upright, to conform to the reduced size of the case, and the result are somewhat less depths of tone. The spinet or console piano, which has a drop action, that is an action which is inverted and placed in such a position in the piano that the action is actually lower than the keyboard. These pianos are smaller than the vertical or studio upright. The keyboard consists of 52 white keys, which represent tones of the absolute pitch, and 36 black ones, which are referred to as sharps. The black and white keys are laid in a symmetrical pattern so that every 12 of them forming an octave are identical. The black keys are in a group of two s and three s, and retain this arrangement in every octave throughout the piano. The width of each key is less than an inch, and the overall width of the octave is seven and five eighths inches. 17

18 Figure 4: The Grand Case [2, p. 94] Figure 5: The Keyboard [2, p. 80] The action of the piano is made of over eighty separate parts. The purpose of the action is to cause the string to start vibrating when the key is depressed, and to stop the vibration of the string when the key is released. When a key is pressed, the back end of the key rises, pushing the sticker, which in turn lifts the whippen on its pivot. The jack flange is glued on one end of the whippen. As the whippen rises the jack pushes the hammer butt, and the hammer strikes the string. Just before the hammer strikes the string, the jack is tripped by the jack 18

19 regulating button, and the hammer moves onward to the string by its own momentum. At the same time the hammer moves on its way to hit the string, the spoon lifts the damper off the string and it remains off until the key is released. When the hammer hits the string, it is rebounded and caught by backcheck. All this is done simultaneously, in one operation. When the key is released everything goes back to its original position, the rest position. The plate is made of cast iron or bell metal. Piano manufacturers devote hours of work on the construction of the plate since it has to be both acoustically and mathematically correct. The plate is the piano is many aspects. The holes and open spaces, more or less, are the resonance chambers. At the top of the plate are the holes through which the tuning pins must go, and at the bottom, driven into the holes of the frame are the taped hitch pins to which one part of the string is to be secured. These plates are cast and later carefully machined. The plate has to be manufactured according to specifications, and has to be strong enough to withstand the forty thousand pounds load that result at producing a 440Hz sound. Figure 6: The Plate [2, p. 86] The soundboard is made of thin wood. It has a crown in the center (convex). The crown is maintained and secured by means of the ribs, the underside of which has been beveled and pinned to the soundboard with dowels and glue. The soundboard not only acts as an amplifier but also actually vibrates in the same manner as a drumhead. 19

20 Figure 7: The Vertical Piano Action [2, p. 83] 20

21 The pin block consists of several layers, or laminations of hard rock maple, which are cemented, grain against grain. The wood is cross-grained so that the pins are locked in and this determines whether the tuning will hold or not. After the pin block is fitted on the plate, the tuning pinholes are drilled and the pins are either screwed in by hand or driven in at the time of the stringing. Figure 8: The Soundboard [2, p. 87] Figure 9: The Pin Block [2, p. 89] The strings of a piano are made of fine tempered steel. They will not vary in pitch to any great degree over a period of time. If atmospheric conditions are favorable, piano strings should remain about the same tension during a period of approximately sixty to ninety days, or from season to season. The diameter of the piano string increases gradually as the scale descends, and decreases as the scale ascends. The bass strings and the treble stings are made of the same material. However, bass strings are wrapped with either copper or steel wires. 21

22 Figure 10: Cross section view of the pin block [2, p. 89] The inside of the bass string is called the core and the outside is known as the loading. Some of the bass strings are double wrapped to weight to the string, which is directly related to the depth of the sound it produces. Each note on the piano has a number of strings corresponding to it. The number of strings for a certain note is relative to the position of the note on the scale. The bass notes on the piano usually have one or two strings corresponding to each note. The middle part of the piano as well as the treble part has three strings corresponding to each note. The distribution of the number of strings varies from one piano to the other. Tuning the Temperament Octave by Fourths and Fifths (This section is taken from Reference 3. Only slight changes have been made to the text.) To start tuning the piano, a technician should first open the front cover of the case on the piano, so that he can easily access the pin block. Next, the technician has to set the strip mute. Since we mentioned earlier that most of the notes on the piano have three strings corresponding to each note, the technician tunes a single string at a time. Therefore, before starting tuning the technician should somehow mute two strings out of three, and this is accomplished by placing a strip mute. If the technician wants to tune the middle string he should place the strip mute as shown in Figure 11. Next, the tuner has to place the tuning hammer on the pin corresponding to the string he is about to tune, as shown in Figure 12. By Rotating the hammer clockwise the technician will add tension to the string thus increasing the pitch (frequency) of the sound, and turning it counter-clockwise would decrease this pitch. The goal of the tuner is to set the exact pitch of the string. 22

23 Figure 11: A strip mute used to silence all but the middle string of a series of unison group [3, p. 67] One method of tuning a temperament octave involves tuning a series of Fifths, which are tuned up the scale, and Fourths, which are tuned down the scale. Because Fifths are narrowed in equal temperament and Fourths are widened, all notes tuned in this system will be tuned so that they are slightly flat of pure. The only exception is the first interval, which is a Fifth tuned down the scale. Step 1: Tuning C40 Tuned Note: Tuning Fork C Note To Be Tuned: C40 Beat Speed: Test: 0 (pure) Roughly tuned G#63 so that it beats between 3 and 7 times per second with C40. Compare the beat speed of G#36-C40 with that produced when G#36 is played while the tuning fork is sounding. The beat speeds should be identical. Step 2: Down a Fifth Tuned Note: C40 Note To Be Tuned: F33 Reference Note: C52 Beat Speed: 3 times in 5 seconds-sharp Test: None Step 3: Down a fourth Tuned Note: C40 Note To Be Tuned: G35 Reference Note: G59 Beat Speed: Once each second-flat Test: None 23

24 Figure 12: A tuning hammer placed on the pin corresponding tot he 1 st string of the note being tuned [3, p. 55] Step 4: Up a Fifth Tuned Note: G35 Note To Be Tuned: D42 Reference Note: D54 Beat Speed: 3 times in 5 seconds-flat Test: None Step 5: Down a Fifth Tuned Note: D42 Note To Be Tuned: A37 Reference Note: A61 Beat Speed: Once each second-flat 24

25 Test: Major Third F33-A37 should beat about 6 times in each second. Step 6: Up a Fifth Tuned Note: A37 Note To Be Tuned: E44 Reference Note: E56 Beat Speed: 3 times in 5 seconds-flat Test: Major Third C40-E44 should beat very rapidly, about 12 times in each second. Compare the beat speeds of the Major Third C40-E44 and the Major Sixth G35E44. They should beat at about the same speed. Step 7: Down a Fourth Tuned Note: E44 Note To Be Tuned: B39 Reference Note: B63 Beat Speed: Once each second-ftat Test: Major Third G35-B39 should beat faster than the Major Third F33-A37. G35-B39 should beat about 7-8 times per second. Step 8: Down a Fourth Tuned Note: B39 Note To Be Tuned: F#34 Reference Note: F#58 Beat Speed: Once each second-flat Test: None Step 9: Up a Fifth Tuned Note: F#34 Note To Be Tuned: C #41 Reference Note: C#53 Beat Speed: 3 times in 5 seconds -flat Test: Major Third A37-C#41 should beat faster than G35B39. A37- C#41 should beat about 8-9 times per second. Step 10: Down a Fourth Tuned Note: C#41 Note To Be Tuned: G#36 Reference Note: G#60 Beat Speed: Once each second-flat Test: Major Third G#36-C40 should beat slightly faster than G35- B39 and slightly slower than A37-C#41. 25

26 Step 11: Up a Fifth Tuned Note: G#36 Note To Be Tuned: D#43 Reference Note: D#55 Beat Speed: 3 times in 5 seconds-flat Test: Major Third B39-D#43 should beat slightly slower than C40- E44. Compare the beat speed of the Major Third B39-D#43 to the beat speed of the Major Sixth F#34-D#43. They should beat about the same speed. Step 12: Down a Fourth Tuned Note: D#43 Note To Be Tuned: A#38 Reference Note: A#62 Beat Speed: Once each second-flat Test: Major Third A#38-D42 should beat slightly faster than A37- C#41 and slightly slower than B39-D#43. Compare the beat speeds of the Major Third A#38-D42 and the Major Sixth F33-D42. They should beat at about the same speed. Fourth F33-A#38 should beat once each second. Step 13: Up a Fifth Tuned Note: A#38 Note To Be Tuned: F45 Reference Note: F57 Beat Speed: 3 times in 5 seconds-flat Test: Major Third C#41-F45 should beat slightly faster than C40- E44. The octave F33-F45 should be pure. Fourth a C40-F45 should beat once each second. Compare the beat speeds of the Major Third C#41-F45 and the Major Sixth G#36-F45. They should beat at about the same speed. A properly tuned temperament octave should exhibit the following qualities that can be used to check the overall tuning. All Fifths within the octave should beat three times in five seconds. All Fourths should beat once each second. As the Major Thirds are played up the scale, they should slowly increase in beat speed, with each successively Major Third beating slightly faster than the one immediately below it. The same pattern of increasing beat speeds applies to ascending Major Sixths. After checking the overall tuning, the technician can now tune the two other strings corresponding to each note by Unison, and move on to tuning the rest of the piano based on the tuning of the temperament octave. 26

27 Tuning Instruments There are several precision instruments currently available on the market, However, these instruments are designed to assist qualified tuners. One of these instruments is the chromatic tuner and another is the strobotuner. The KORG Auto Chromatic Tuners are precision solid state electronic instruments which produce tones of the equally tempered chromatic scale to a very high degree of accuracy. No other audible tuner matches their accuracy, versatility, or range. Some of the characteristics of these instruments are: Most versatile and widely used audible tuning instruments Superb accuracy, within 1/3 of 1/100 of a semitone All solid state circuitry for long term reliability Full range pitch control accurately calibrated in hundredths of a semitone Lightweight and compact Full seven octave range No warm up, ready to use instantly No calibration required either before or during use Figure 13: KORG Chromatic Tuner CA-30 [8] They are used extensively in the tuning of pianos, band and orchestra instruments. They are an excellent aid for use in music education. The sevenoctave range, the pitch control and powerful tone will satisfy the most difficult tuning requirements. 27

28 Figure 14: KORG Digital Tuner DT-3 [8] The Strobotuner [9] is a highly sensitive and accurate instrument enables the tuner to use his eyes and his ears in tuning the piano. The strobotuner is a precision electronic device, which detects and compares musical frequencies with a fixed reference. The instrument visually displays this comparison using the stroboscopic principle via the scanning disc on the Face Plate of the unit. The fixed reference is the scanning disc itself, which revolves at twelve different speeds determined by the setting of the Selector Knob on the front panel. There are twelve possible settings of the Selector Knob corresponding to the twelve notes in the equally tempered. If the job of tuning a piano were simply a matter of using the strobotuner and making its bands "stand still" for 88 different notes, even a person totally deaf could accurately tune. Alas, such is not the case - for to musical ears, such tuning would be aural chaos. Figure 15: Peterson Strobotuner Model

29 A new generation of virtual strobe tuners is currently available on the market. The mechanical system which rotates the face plate for visual reference in the old models has been replaced with an LCD in these new models and consequently their size has considerably decreased. Figure16: The Peterson Virtual Strobe Tuner VS-1 [10] Design Considerations and Variables From our discussion, so far, we can infer that piano tuning is an art, which only qualified technicians can handle. However, throughout the rest of this report we will try to design a precision instrument, which will play the role of the technician and can be used by anyone, who doesn t have a prior knowledge about this art, and would like to tune his own piano on his own. The technician s job as we have seen in a previous section, is to listen to the pitch generated by a string related to a certain note, use his hammer, and set the frequency of the string to a desired pitch, by increasing or decreasing the tension in the string. To be able to design an instrument that does the job of the technician, our instrument needs to be equipped with an ear, a brain, and a hand with a tuning hammer. 29

30 The microphone is the simplest device that can do the job of the ear. It can capture any acoustic frequency in the air, within its range. A micro controller will do the job of the brain, in analyzing the captured acoustic signal and controlling a custom made tuning gun, which will play the role of the technician s hand controlling the tension in the string, with his hammer. Any microphone can be used to capture the acoustic signal. However there is one very important consideration. The microphone should have a working range of 25Hz to 4200Hz, the acoustic frequency range of any piano. The micro controller will be the brain of our system. The micro controller to be selected for our application should have the following characteristics: High operating speed High capacity program memory Power on reset High sink / source current A 16 bit free running timer A capture module 3 input / output ports As for the tuning gun, it should do the job of the hand of the technician and the tuning hammer. This implies that, the tuning gun should have an end tool, which can fit on the pins of the piano pin board, and a motor and a gearbox to rotate this tool. There are two considerations to take into account in designing the tuning gun. First, the tool should rotate with a low angular velocity to achieve a high tuning resolution. Second, the motor should be able to deliver a torque enough to rotate the pin supporting a string. Our system should also have a user interface, which would guide as well as allow the user to navigate through and select different notes tune. A LCD (Liquid Crystal Display) will do the job. Therefore, a microphone will capture the signal. The signal will then be amplified and processed by an interface electronic circuit, which will deliver a signal that can be processed and analyzed by the micro controller. The micro controller will capture the signal and compare the frequency of this signal to a preset value in memory. After a successful capture and comparison, the micro controller will turn on and control the direction of the motor of the tuning gun. When the acoustic frequency generated by the string equals the preset value in memory, the motor stops, and the string is in tune! 30

31 Proposed Solution Procedure The system we are about to design is a control system, since we are causing a system variable (acoustic frequency generated by a string) to conform to some desired value, the reference value (frequency value saved in controller memory). The system has feedback characteristics, since we are measuring the controlled variable (acoustic frequency of the string) and using that to influence the controlled variable (acoustic frequency of the string). Our system is a closed loop control system, since it measures its output (acoustic frequency of string) and adjusts its input accordingly (rotational direction of tuning gun) by using feedback. In our control system, the sensor is the microphone, the actuator is the tuning gun, the controller and compactor is the micro controller. Figure 17: Closed Loop Feedback Control System The main variable in our system is the reference frequency. To be able to calculate the reference frequency of every note, first we have to select a reference pitch and then multiply this reference pitch by Since, an octave contains 12 semi-tones and because the equal temperament requires that the ratios between all 12 be equal, The proper ratio between adjacent semi-tones is

32 We are going to select the standard pitch. (A49 at 440Hz) For the sake of simplicity we are going to choose an octave to base our study on. The octave is A49 A61. We will also include the two peripheral notes on the piano, which are A1 and C88. Since A49 is vibrating at 440 Hz, to calculate the vibrating frequency of the next note in the scale, we multiply 440Hz by Therefore the Frequency of vibration of A#50 is Hz. The rest of the frequencies are shown in Table 8. Acoustic Note Frequency (Hz) A A # B C C # D D # E F F # G G # A A C Table 8: Frequencies in equally tempered scale from A49 A61 as well as the notes A1 and C88, based on standard pitch, A49 at 440 Hz The Microphone and Interface Circuit For our system we chose a piezoelectric microphone, with a working range wider than that of the piano. The raw signal coming out of the piezoelectric microphone, which is of a sinusoidal nature, needs to be processed (Figure 18). 32

33 The processing of the signal should somehow transform this noisy sinusoidal signal to a clear signal with clear rising and falling edges and within the 0 to +5 V dc working range of any micro controller. The signal is filtered and amplified by a low noise dual pre-amp integrated circuit, LM387A (see appendix A for data sheet). The schematic diagram of the processing circuit is shown in Appendix D. The raw signal is first fed into the non-inverting end of the first pre-amp. It is filtered, amplified, and shifted to a higher voltage level (Figure 19). The signal is next fed into the inverting end of the second pre-amp. It is filtered, amplified and inverted (Figure 20). The signal is then clipped and now has a peak to peak 0 to 8 volts range (figure21). Finally, the signal is fed through a 5V low signal zener diode and now the peak to peak voltage ranges between 0 to 5 Volts (Figure 22). Figure 18: The raw noisy sinusoidal signal coming out of the piezoelectric microphone Figure 19: The signal after passing through the first pre-amp circuit 33

34 Figure 20: The signal after passing through the second pre-amp circuit Figure 21: The signal is clipped 0 to 8 volts peak to peak Figure 22: The signal 0 to 5 volts peak to peak 34

35 The Micro Controller For our application we chose the PIC16F877 manufactured by Microchip (see Appendix A for data sheet) as our micro controller. This micro controller satisfies all the design criteria that we mentioned in a previous section. The PIC16F877 operates at a 20 MHz by a crystal oscillator. However, every 4 oscillations make up a working cycle. Therefore, the clock speed of the micro controller is 5 MHz. In our application, the micro controller has four tasks to perform. The first is to control the sequence of events. Second, capturing a signal and compare its period to a preset value in memory. Third, control the tuning gun based on the comparison. And fourth, control the LCD. Sequence of Events The micro controller is the core of our instrument and the sequence of events that happen while using the instrument is the result of the program written on it. Our program should support the following sequence of events: 1. The user turns on the instrument and a welcome message is displayed. 2. Next, the name of the first note to be tuned is displayed on the LCD and the user is alerted that the controller is ready to capture a signal by a second message on the LCD. 3. The user at this point should have 2 options. Either place the tuning gun on the pin of the note whose name appear on the LCD and starts tuning, or he should be able to choose another note and tune it. 4. If the user chooses to tune the note whose name is displayed on the LCD, first he has to place the tuning gun on the pin of the string corresponding to the note to be tuned. Second, he has to press the key of the same note and third press a button mounted on the gun to start tuning. Once the tuning process starts a message is displayed on the LCD. Once the string is tuned the user is alerted. The program should automatically jump to display the name of the adjacent note on the screen. 5. If the user decides to select another note, he should be able to do so by pressing one of the two buttons on the instrument. One button will take him forward in the scale and the other backward. The procedure explained in 4 is executed. 35

36 From this discussion we can deduce that the user needs three control buttons. The first, to scroll forward through the notes. The second, to scroll backwards. Finally, the third on the tuning gun to start tuning a note. Capture and Compare Module After capturing the acoustic frequency by the microphone and processing it by the microphone to micro controller interface circuitry, we are now ready to measure the period of the processed signal and compare it to a preset reference value in the memory of the micro controller. To measure the period of the processed signal we need to capture the value of a free running timer at two consecutive falling edges, subtract them from each other and the result is the period of the processed signal. Before we proceed in explaining the capture process on the micro controller we need to calculate the reference values that would be stored in the micro controller s memory. So far we calculated the frequencies of our working range A49 A61. Now that we know the cycle speed of our micro controller we can calculate the number of counts by which the free running timer increases for every frequency in our scale. A49 is vibrating at 440 Hz and the micro controller timer is running at 5 MHz. The number of counts by which the counter/timer increments between two falling edges of the 440 Hz signal is calculated by dividing 5 MHz by 440 Hz. The result is However this number should be rounded and presented as a whole number equal to The hex value corresponding to is 2C64. The values for the rest of the sample scale are given in Table 9. Since we rounded the number of clock counts, we should calculate the error. The number of counts corresponding to A49 was rounded to The rounded number corresponds to the frequency that can be calculated by dividing 5 MHz by The result is The error is equal to the difference of and which is Hz. The error for the rest of the sample scale is given in Table 10. To be able to estimate the intensity of this error, we have to define the word cent used in piano tuning. If we take two adjacent semi-tones and divide the frequency range between these two to hundred equal parts, each division will be called a cent. To illustrate this, let s take A49 at 440Hz and its two adjacent semitone A#50 at Hz and G#48 at A cent between A49 and A#50 is equal to / 100 = Hz A cent between A49 and G#48 is equal to / 100 = Hz 36

37 Note Acoustic Frequency (Hz) Period (sec) Controller Frequency (Hz) Number of Clock Counts Rounded Number of Clock Counts Hex Value A E E C 64 A # E E E6 B E E C C E E C # E E B D E E D # E E F 63 E E E D A0 F E E B F7 F # E E A 65 G E E EA G # E E A E E A E E C6 3A C E E AA Table 9: Number of counts by which the free running timer increases for every frequency in our scale Note Acoustic Frequency (Hz) Rounded Number of Clock Counts Frequency Corresponding to Rounded Number Error A A # B C C # D D # E F F # G G # A A C Table 10: Rounding Error of the sample scale 37

38 Table 11: Values of +/- 5 cents about the frequencies of the sample scale Note Acoustic Frequency (Hz) 1 Cent to Right Semi-tone Error Ratio A A # B C C # D D # E F F # G G # A A C Table 12: Ratio of 1 cent to right semi-tone and error 38

39 Table 11 shows the frequencies of our sample scale and the peripheral notes to +/- 5 cents. Anything within the margin of +/- 1 cents from the desired frequency is undetectable by the human ear. As long as our error is within this range, our error is acceptable. The error we calculated for rounding was which is about 18 times smaller than the one-cent values we calculated above. Therefore our rounding error can be neglected. The comparison for notes of the sample scale is shown in Table 12. Coming back to the micro controller, the PIC16F877 has two capture modules with16 bit capture registers. Each module has 3 registers. In our program we are going to use the first module, which is connected to pin CCP1 and therefore the three capture registers associated with it are: CCP1CON : CCP Control Register CCPR1H : CCP High Byte CCPR1L : CCP Lower Byte The capture modules use Timer1 as the timer resource. The Timer1 module is a 16-bit timer/counter consisting of two 8-bit registers (TMR1H and TMR1L). The TMR1 Register pair (TMR1H:TMR1L) increments from 0000h to FFFFh and rolls over to 0000h. The TMR1 Interrupt, if enabled, is generated on overflow, which is latched in interrupt flag bit TMR1IF (PIR1<0>). This interrupt can be enabled/disabled by setting/clearing TMR1 interrupt enable bit TMR1IE (PIE1<0>). In Capture mode, CCPR1H:CCPR1L captures the 16-bit value of the TMR1 register when an event occurs on pin CCP1. When a capture is made, the interrupt request flag bit CCP1IF (PIR1<2>) is set. The interrupt flag must be cleared in software. If another capture occurs before the value in register CCPR1 is read, the old captured value is over-written by the new value. The values written into register CCPR1 should be copied to a user defined memory address, before they are over-written by new values. Since Timer1 is a free running timer, we should check if an overflow occurred between two consecutive captures. Therefore the register PIR1, which contains the Timer1 overflow bit TMR1IF, should also be copied to a user defined memory address. As to our program, during initialization all interrupts are disabled, Timer1 is stopped, the higher and lower bytes of Timer1 are cleared, capture module is turned off, CCP1 pin is made an input pin, all peripheral interrupts are cleared, all 39

40 interrupt flags are cleared, Timer1 is set to 1:1 prescaler, capture mode is turned on to capture at every falling edge, and finally Timer1 is turned on. CLRF INTCON CLRF T1CON CLRF TMR1H CLRF TMR1L CLRF CCP1CON ; Disable interrupts ; Stop timer1 ; Clear timer1 high bite ; Clear timer1 low bits ; CCP module is off BSF STATUS, RP0 ; Select bank 1 BSF TRISC, CCP1 ; Make CCP1 pin input CLRF PIE1 ; Disable peripheral interrupts BCF STATUS, RP0 ; Select bank 0 CLRF PIR1 ; Clear interrupt flag MOVLW 0x000 MOVWF T1CON ; Timer1 stopped and prescaler set to 1:1 MOVLW 0x04 MOVWF CCP1CON ; Capture mode on, every falling edge BSF T1CON, TMR1ON ; Timer1 is activated Now, we need to allocate 6 memory locations. CCPAL CCPAH TMA CCPBL CCPBH TMB : Location to where CCPR1L will be copied to after 1 st capture : Location to where CCPR1H will be copied to after 1 st capture : Location to where PIR1 will be copied to after 1 st capture : Location to where CCPR1L will be copied to after 2 nd capture : Location to where CCPR1H will be copied to after 2 nd capture : Location to where PIR1 will be copied to after 2 nd capture To capture the first value we first need to clear CCPR1, the CCP interrupt flag, and the Timer1 overflow flag. CLRF CCPR1H CLRF CCPR1L BCF PIR1, CCP1IF BCF PIR1, TMR1IF ; Clear CCP1 high bit ; Clear CCP1 low bit ; Clear CCP interrupt flag ; Clear Timer1 overflow flag Next we have to loop until the CCP interrupt flag is set. Once the flag sets we copy the CCPR1 register and the PIR1 register to the allocated memory addresses for the first capture. 40

41 Clear CCP Registers Clear Capture Flag Clear Timer Overflow Flag Capture Flag Set? No! Yes! CCPAL = CCPR1L CCPAH = CCPR1H TMA = PIR1 Clear CCP Registers Clear Capture Flag Capture Flag Set? No! Yes! CCPBL = CCPR1L CCPBH = CCPR1H TMB = PIR1 Figure 23: 1 st and 2 nd capture flow chart CL1 BTFSS PIR1,CCP1IF GOTO CL1 MOVF PIR1,W MOVWF TMA MOVF CCPR1H,W MOVWF CCPAH MOVF CCPR1L,W MOVWF CCPAL ; Move first int. register to TMA ; Move first capture high byte to CCPAH ; Move first capture low byte to CCPAL 41

42 Now we need to clear the CCP register and the interrupt flag and wait for the next capture to occur. Once the flag sets we copy the CCPR1 register and the PIR1 register to the allocated memory addresses for the second capture. BCF PIR1,CCP1IF CLRF CCPR1H CLRF CCPR1L ; Clear CCP interrupt flag ; Clear CCP1 high bit ; Clear CCP1 low bit CL2 BTFSS PIR1,CCP1IF GOTO CL2 MOVF PIR1, W MOVWF TMB MOVF CCPR1H, W MOVWF CCPBH MOVF CCPR1L, MOVWF CCPBL ; Move second int. register to TMA ; Move second capture high byte to CCPAH W ; Move second capture low byte to CCPAL Now is the time to subtract the two captured values to determine the period of the processed signal. The result is placed in the user allocated memory address CCPRESL for the lower byte and CCPRESH for the higher byte. However, before we do this we should first check if an overflow occurred in Timer1. We have three cases: No overflow occurred between the two captures An overflow occurred before the first capture An overflow occurred between the two captures In the first two cases we have to subtract the 2 nd capture value from the 1 st capture value. The result is the period of the signal. In the third case, since an overflow has occurred between the two captures, we first need to subtract the 1 st capture from FFFFh and then add the result to the 2 nd capture value. To check for the three cases we do an exclusive OR operation on TMA and TMB. If the result of the exclusive OR is 0 then the one of the first two cases have occurred. If the result is 1, then the third case has occurred. MOVF TMB,W XORWF TMA,F BTFSS TMA,0 GOTO CALC1 GOTO CALC2 ; Move TMB to w register ; Exclusive OR TMA with W Result in TMA. ; IF TMA<0> is set goto CALC2 else CALC1. 42

43 TMA<0> TMB<0> XOR Is the Result 1? Yes! No! CCPAL = FF CCPAL CCPAH = FF CCPAH CCPRESL = CCPBL CCPAL CCPRESH = CCPBH CCPAH CCPRESL = CCPAL + CCPBL CCPRESH = CCPAH + CCPBH Figure 24: Calculation of the period of the signal flow chart For the first two cases, CALC1 CALC1A CALC1B MOVF CCPAL, W SUBWF CCPBL, W MOVWF CCPRESL BTFSS STATUS, C GOTO CALC1A GOTO CALC1B DECF CCPBH, F MOVF CCPAH, W SUBWF CCPBH, W MOVWF CCPRESH ; CCPRESL = CCPBL - CCPAL ; Check if borrow has occurred ; Decrement CCPBH IF Borrow Else ; CCPRESH = CCPBH - CCPAH And for the third case, CALC2 MOVF CCPAL, W 43

44 SUBLW 0xFF MOVWF CCPRESL MOVF CCPAH, W SUBLW 0xFF MOVWF CCPRESH MOVF CCPBL, W ADDWF CCPRESL, F ; CCPRESL = FF - CCPAL ; CCPRESH = FF - CCPAH ; CCPRESL = CCPRESL + CCPBL MOVF CCPBH, W BTFSC STATUS, C ADDLW D'1' ; IF Carry add 1 ADDWF CCPRESH, F ; CCPRESH = CCPRESH + CCPBH Now we need to compare the Results saved in CCPRESH:CCPRESL with the preset values in memory. The higher byte of the preset memory is set in the user allocated memory address CCPVALH and the lower byte in CCPVALL. The objective of this comparison is to know whether the period of the processed signal falls between the range of the preset values. We first start by comparing the higher bytes. If CCPRESH < CCPVALH, this implies that the frequency of the processed signal is higher than that of the preset value. The controller should rotate the motor counter-clockwise to decrease the frequency of the string. If CCPRESH > CCPVALH, this implies that the frequency of the processed signal is lower than that of the preset value. The controller should rotate the motor clockwise to increase the frequency of the string. If CCPRESH = CCPVALH, this implies that we have to compare the lower bytes. Now we have to test the lower bytes. If CCPRESL < CCPVALL, this implies that the frequency of the processed signal is higher than that of the preset value. The controller should rotate the motor counter-clockwise to decrease the frequency of the string. If CCPRESL > CCPVALL, this implies that the frequency of the processed signal is lower than that of the preset value. The controller should rotate the motor clockwise to increase the frequency of the string. If CCPRESL <= CCPVALL, this implies that the string has been tuned to the desired frequency, and we can move to tuning another string. 44

45 MOVF CCPVALH, W SUBWF CCPRESH, W BTFSS STATUS, C GOTO CCWROT MOVF CCPRESH, W SUBWF CCPVALH, W BTFSS STATUS, C GOTO CWROT MOVF CCPVALL, W SUBWF CCPRESL, W BTFSS STATUS, C GOTO CCWROT MOVF CCPRESL, W SUBWF CCPVALL, W BTFSS STATUS, C GOTO CWROT GOTO MOTSTOP ; W = CCPRESH - CCPVALH ; IF CCPRESH < CCPVALH ; IF CCPRESH >= CCPVALH ; W = CCPVALH - CCPRESH ; IF CCPRESH > CCPVALH ; IF CCPRESH = CCPVALH ; W = CCPRESL - CCPVALL ; IF CCPRESL < CCPVALL ; IF CCPRESA >= CCPVALL ; W = CCPVALL - CCPRESL ; IF CCPRESL > CCPVALL ; IF CCPRESL = CCPVALL The preset values as we saw are all given in 16 bit data, represented by Hex numbers, since the captured values are captured from a 16 bit timer. However the 16-bit data space is not enough to hold the preset value of the strings found on the left periphery of the piano. From Table 9 we can see that the number of counts for the note A0, which is 181,818, exceeds the address space, which is FFFF in Hex corresponding to 65,635. Adding a prescaler value to the timer can easily solve this problem. By forcing the timer to increment every 4 cycles instead of every single we can put back the count number within the range of the of the timer. We have to find the note from where on when moving to the left we should use the prescaler. 65,635 counts correspond to the frequency Hz that is the frequency of D#19 with 37 cents offset. To be on the safe side we will add the prescaler to all the notes whose counts exceed 60,000. This value corresponds to the note E20. Tuning Gun Control 3 output pins of the PIC16F877 micro controller control the motor in the tuning gun. The motor is designed to rotate both clockwise and counter-clockwise and stop whenever needed. This control is achieved by the use of a Dual Full Bridge Driver integrated circuit L298N (see Appendix A for data sheet). 45

46 The full bridge driver needs 3 pins to control the motor connected between its output pins. The first pin is the Enable pin. The second and the third are the Direction1 and Direction2 pins. Setting the enable pin to high activates the motor and setting it to low stops the motor. To turn the motor in the clockwise direction Direction1 pin is set to high and the Direction2 pin is set to low. To reverse the direction of the motor, to turn in counter-clockwise, direction Direction1 pin is set to low and the Direction2 pin is set to High. These pins are set to high and low depending upon the results of the capture and compare module explained in the previous section. The code in our program to control the motor is given in the following lines. CWROT BSF PORTB,DIR1 ; DIR1 = 1 BCF PORTB,DIR2 ; DIR2 = 0 BSF PORTB,EN ; EN = 1 CCWROT BCF PORTB,DIR1 ; DIR = 0 BSF PORTB,DIR2 ; DIR1 = 1 BSF PORTB,EN ; EN = 1 MOTSTOP BCF PORTB,EN ; EN = 0 LCD Control The LCD in our system is the user interface with the instrument. The messages displayed on the LCD guide the user through the tuning process, allow him to navigate and select the note he wants to tune. The LCD we selected for our application is a Hitachi based LCD with 2 Lines of 20 characters each (see Appendix B for data sheet). When the instrument is activated, the first message the LCD displays on its first line is PIANO TUNER 2002, and the second line displays the message Initializing which blinks three times for about one second. The display is then cleared and the following message is displayed on the first line NOTE: 4 A and on the second line PRESS TO START! 46

47 If the user presses the button on the tuning gun, the first line will keep displaying the name of the note that is being tuned and the second line will display the message TUNING Once the string is tuned the first line will keep displaying the name of the note which has been tuned and the second line will display the message TUNED!. This message will blink three times to alert the user and afterwards it will displays the name of the note to the right of the note that was tuned on the first line and on the second line PRESS TO START!. This sequence of events is repeated over and over until the piano is tuned and the user turns of the instrument. The Tuning Gun Before we start designing the tuning gun, there are two variables to be calculated. The first is the torque and the second is the speed of rotation of the motor. The Torque The tuning pins carry the tension in the string and therefore are the mean through which the tension can be altered, by turning them in either sense. Hence, they carry torque, by being driven into holes bored in a slab of wood. We needed an accurate measure of the maximum torque, so that we could design the tuning gun to accommodate for that tension without any mechanical problems. It is important to mention at this stage, that the torque needed to start turning the pin is higher than the torque needed to maintain turning it. So, we measure the starting torque and base our calculations on it. To measure this torque, we used a standard torque wrench, with the proper tuning pin- socket assembly installed to its square drive. The procedure for measuring the torque on all the pins is as follows: 1. Set the graduation on the handle to the minimum amount of torque. 2. Fit the tuning pin socket assembly to the pin on the piano. 3. Balance the wrench with your hand and pull the handle. 4. Feeling and/or hearing a wrench click during the pull means that the actual tension in the pin is higher than the set value. In this case, set the graduation to the next highest value. Repeat until the click disappears and you can 47

48 actually turn the pin by the wrench. This value is an indication of the tension in the pin. Note Measured Torque ( in lbs ) Torque ( N-m ) A A # B C C # D D # E F F # G G # A A C Table 13: Reading from the torque wrench The values experimentally calculated for the sample scale that we selected and the peripheral notes are tabulated in Table 13. The maximum torque value from the table is around 25 N.m. For safety, we will design the tuning gun to deliver 30 N.m torque. Now that we have an estimate of the torque the tuning gun must deliver, we present a detailed design procedure for the different components and the overall assembly of the device. The Rotational Speed The tool of the gun will be designed to rotate at 2.4 degrees per second. However we should calculate the error generated from this value. Before we start calculating the error induced from the speed of rotation of the tool we have to gather some experimental data about the strings of the piano. To be able to calculate the above-mentioned error, we need to measure the change in 48

49 the frequency of the string when the pin corresponding to it is rotated 180 degrees. We collected this data by arbitrarily choosing a piano and measuring the values. First each string corresponding to a note in our sample scale was tuned to the frequencies tabulated in Table 9. Each string was then rotated 180 degrees clockwise and the new frequency of each string was measured. The values are tabulated in Table 14. The values are within +/- 5 cents error. Let us assume that the frequency increases linearly with the 180 degrees rotation. Let us take A49 and use it for sample calculation. The time it takes to loop around our closed loop feedback control system is the number of micro controller instructions multiplied by the instruction cycle time in the loop plus twice the value of the period of the note being tuned. The longest path the program can take to capture and compare is 33 instruction. The execution time of each instruction is 200 nanoseconds. Therefore the time needed for the execution of the 33 instructions is 6.6 microseconds. Note Original Frequency (Hz) Frequency After 180 Degree Clockwise Rotation Change in Frequency (Hz) A A # B C C # D D # E F F # G G # A A C Table 14: Change in Frequency of strings after 180 degree of clockwise rotation of pin 49

50 A49 has a frequency of 440Hz and a period of seconds. To capture two values of the timer the maximum time needed, is twice the period of the frequency being measured, which equals to Therefore, to move around the closed loop while tuning A49 takes us seconds. In seconds the gun rotates by seconds x 2.4 degrees per second which equals to degrees. In degrees rotation the frequency of our string changes by (63.6 Hz / 180 degrees) x = Hz This value is far below the 1-cent error that we calculated for A49, which is The calculation for the rest of the sample scale is tabulated in Table 15 and Table 16. Note Time it Takes to Complete The Closed Loop (sec) Change in Frequency After 180 Degree Clockwise Rotation Change in Frequency of String in 1 sec (Hz) Change in Frequency to Complete Loop A E E-03 A # E E-03 B E E-03 C E E-03 C # E E-03 D E E-03 D # E E-03 E E E-03 F E E-03 F # E E-03 G E E-03 G # E E-03 A E E-03 A E E-03 C E E-03 Table 15: Change in frequency of the string while completing the closed loop of our system 50

51 Note Change in Frequency to Complete Loop 1 Cent to Right Semi-tone Ratio A E A # E B E C E C # E D E D # E E E F E F # E G E G # E A E A E C E Table 16: The ratio of 1-cent error to the change in frequency to complete one loop From these calculations we can state that the speed of rotation of the motor generates en error that can be neglected. The Motor The selection of the motor is based on speed, horsepower, torque, and inertia requirements. The tuning pin must rotate at a constant steady speed, so our design does not call for a motor with more than one speed or a range of speeds. The motor has to operate on dc power source. The load to be carried by the motor is assumed constant, so the motor must deliver the peak horsepower at the necessary operating speed. The peak horsepower determines the maximum torque required by the driven pin and the motor must have a maximum running torque in excess of this value. The rootmean-square-average of the peak horsepower also indicates the proper motor rating from a heating standpoint. 51

52 The inertia effect of the rotating parts of a driven machine will, if large, appreciably affect the starting of the motor and the amount of heating in it. Therefore, all axes of rotating parts must be precisely aligned. Besides all these design criteria, another is the aesthetics of the motor. It has to be light, easy to handle, flexible, and easily integrated in the assembly. For our application, we found a 50 VDC, 0.88 Amp motor, with a gear-case of 1:220 speed reduction. The unit (motor + gear-case) is manufactured by Globe Industries at Dayton, O PN 102A415, USA, with US PAT. 2,935,785. The motor armature shaft rotates 3520 revolution per minute when the supply voltage is 28 volts. The motor has a small gearbox connected to it. The gearing inside the gearbox is of the planetary type. This type of gearing permits a large speed reduction with few parts. Hence, it is well adapted for motor units where economy and compactness are essential. The slow-speed output shaft of the gearbox is in line with the armature shaft of the motor. With the 28 volts supply, the output shaft rotates at 16 rpm. To be able to reduce the rotational speed of the shat further we need a second gearbox, capable of both reducing the speed and withstanding the high torque. After a thorough study we decided to use a worm gear. The calculations for the selection of the worm gear are presented below. Worm Gearing But, we need the tuning pin to rotate less than 3 degrees per second. If the tuning pin is made to rotate at 0.4 rpm, this will correspond to 2.4 degrees per second rotation. Hence, to reduce the 16 rpm to 0.4 rpm, we need a 40:1 ratio gear reducer! And, since our assembly requires the tuning head (slow-speed shaft) be placed at right angles to the motor armature, we selected a worm gearing type of reduction that is quiet in operation. Figure 25 illustrates a typical worm and worm gear set. In our case the worm has a single thread. The geometry of the worm is similar to that of a power screw. The rotation of the worm simulates a linearly advancing involute rack. The geometry of a worm gear is similar to that of a helical gear, except that the teeth are curved to envelop the worm. This figure shows the usual 90 degrees angle between the nonintersecting shafts. Next, we present a detailed design analysis of the worm gearing. We will determine appropriate values of d w, d g, N w, N g, p, maximum lead angle (lambda), 52

53 and pressure angel (phi n ). We will also determine whether the worm can be bored for mounting on a separate shaft. Given: The worm will be driven at 16 rpm. The combination of large loads and high sliding velocities encountered with worm gear sets requires the worm to be of hardened steel and the worm gear of chillcast bronze. Bronze is a material capable to wear-in and increases the contact area. Figure 25: Typical warm gear [4] Decisions: 1. The reducer will not be fan cooled, instead, grease will be used for cooling and lubrication. 2. The center distance must not be more than 25 mm. We chose 21 mm, due to standard bearing and shaft size. Assumptions: 1. The worm and gear are mounted and aligned to mesh properly at mutually perpendicular axes. 2. The entire tooth load is transmitted at the pitch point and in the mid-plane of the gears. 53

54 Design Analysis: 1. Since the pitch diameter of a worm is not a function of its number of threads Nw, the velocity ratio of worm gear set is W w / W g = N g / N w. In our case, this ration must be equal to 40. Also, from design specifications, N w + N g must be greater than 40. Therefore, we choose N g = 40 and N w = 1 (single thread). 2. For high efficiency, the Figure 26 indicates that lambda should be as large as possible, hopefully close to From Table 17, we select phi n = For maximum power-transmitting capacity, the pitch diameter of the worm should normally be related to the shaft center distance by: c /3.0 <= d w <= c /1.7 To obtain the large value for lambda (35 degrees), d w must be small. For c = in, d w permitted by above equation is: <= d w <= Figure 26: Efficiency of ACME screw threads when collar friction is negligible [4] 54

55 Table 17: Maximum worm lead angle and worm gear and Lewis Form Factor for various pressure angles [4] Taking the minimum, d w = 0.29 in, dg = 2c d w = (2*0.825) 0.29 = 1.36 in. The circular pitch for the gear is p = d g *pi/n g = 1.36*pi/40 = in. We select as standard pitch of p = 0.1 in. 4. p = 0.1 in, d g becomes 0.1*40/pi = in. i.e. the change in p results in a smaller gear and requires us to choose between making worm slightly larger than the normally recommended, or, decrease the center distance for contact between gear and worm be established. We chose changing the center distance. D g = in. Using d w = 0.29, c = 0.29/ /2 = We choose c = 0.8 in. Therefore, dw = in. This value turns out to be slightly larger than the 0.29 in minimum value. Therefore, d w = in d g = in. c = 8 in. 5. The smallest worm diameter normally suitable for boring (to fit over a separate shaft) is: d w = 2.4p (in) Here, d w = (2.4*0.1) = 1.34 in. Obviously, the chosen worm diameter requires that the worm be cut directly on the shaft. 6. Tan (lambda) = N w *p / pi*d w = 7*0.1 / pi* = Therefore, lambda = 34.3 degrees. 55

56 7. To estimate the efficiency of the gearbox, we must first estimate the friction coefficient. For this we must find V s, which requires knowledge of Vg. V g = pi*d g *n g = pi*( /12)*(16/40) = fpm. V s = V g / sin(lambda) = / sin(30.31) = fpm. From Figure 27, f = Figure 27: Worm gear coefficient of friction [4] E = [cos(phi n ) f*tan(lamda)] / [cos(phi n ) + f*cot(lamda)] = = %. The above results are checked on Figure 26. Bearing, shaft seal, and oil-churning losses would reduce this figure slightly to about 68 %. 8. Gear face width should be close to but not greater than half the worm outside diameter. The worm outside diameter is (2*p/pi) + d w = 0.4. Therefore, the gear face width is: 0.2 in. On the market we were able to find a gearbox that has close characteristics to the gearbox we designed After connecting the gearbox to the output shaft of the motor connected by an adapter the gearbox rotational speed was reduced to 0.4 rpm, which is equivalent to 2.4 degrees per second rotation. 56

57 The Tuning Tool An adapter made of aluminum connects the tuning tool to the gearbox. The tool rotates with the same speed as the output shaft of the gearbox, which is 2.4 degrees per second. The parts and assembly drawings of the motor, the gearbox, the tool, and the adapters are presented in Appendix C. Pictures of these components are presented in the figures below. Figure 28: The Tuning Gun (front view) Figure 29: The Tuning Gun (rear view) 57

58 Figure 30: Exploded View of Tuning Gun 58

59 Figure 31: The Gearbox Figure 32: Exploded View of The Gearbox 59

60 Figure 33: Exploded View of The Motor Figure 34: The Motor Figure 35: The Motor Gearbox 60

61 Figure 36: The Worm Figure 37: Exploded View of The Worm Figure 38: The Worm Gear Figure 39: Exploded View of The Worm Gear 61

62 Results and Discussion The design was built as a simulation and tested. After testing the simulation the results turned out to be as expected. Next, the design was packaged. In the following sections we will talk about the packaging of the prototype, and the way to use this prototype to tune a piano. Packaging We packaged the instrument we designed into a 28x16x17 cm black transparent box. The box houses two transformers and 3 electronic boards. An off/on switch, 2 control buttons, a LCD, a piezoelectric microphone, and 2 input/output jacks are placed on the front panel of the box. The front panel as well as the box cover are made of black, semi-transparent 3mm plexy-glass. The base of the box is made of 2-cm thick wooden plate. The first electronic circuit is that of the piezoelectric microphone and the input/output jacks. This circuit operates at 12-Volts DC and 10-mA current. The output of this circuit is the processed signal, which is connected to the micro controller circuit. This circuited is supported on the front panel. The second electronic circuit is that of the micro controller. The circuit has 3 inputs connected to the 3 control buttons of our system, 2 on the front panel and one on the tuning gun. The circuit has another input connected to the first electronic circuit. This input is the processed signal from the microphone circuit. On this circuit we also have 14 bi-directional pins used to control the LCD. The output pins on this circuit are 3. They are connected to the third circuit through opto-isolators. These outputs control the motor of the tuning gun. The micro controller as well as the LCD work on 5-Volts DC and a 40-mA current. The micro controller circuit is supported by four vertical threaded axes, which are fastened to the base wooden block. A transformer supplies the power to the first and the second circuits. The transformer has an input of volts AC and an output of Volts AC with a 0.4 Ampere rating. The 7.5 outputs are fed into a full bridge rectifier for 15 volts AC input. The output of the bridge rectifier is connected to a 30 Volts 470 u-farad anti ripple capacitor. The output of the capacitor is then connected to the inputs of LM7805 and LM7812 voltage regulator integrated circuits (see Appendix A for data sheet). The output of the LM7805 IC supplies a 62

63 Figure 40: The packaged design without a cover steady 5 Volts DC to the micro controller and the LCD. The LM7812 IC supplies a steady 12 Volts DC to the first electric circuit which it the microphone circuit. The electric supply circuit we described is fitted on the micro controller board. The third electronic circuit is that of the H-bridge driver L298N. The circuit has 3 inputs, which are the control pins to the H-bridge and two output pins connected to the motor. The three inputs coming from the micro controller circuit are connected to opto-isolators instead of direct connection with to H-bridge. The opto-isolators are used to separate the power of the two circuits in order to avoid possible grounding problems. This circuit is also supported by the same threaded 4 axes which support the micro controller circuit. The L295N IC needs two DC power sources. One source to provide power to drive the motor and the second to provide a 5 volts logic level voltage. The motor in our system operates at 28 Volts DC and a current that reaches 1 Ampere for a few milliseconds and then falls to 0.2 Ampere. A second transformer is used to supply the power needed for the third circuit. The transformer has an input of volts AC and an output of Volts AC with a 3 Ampere rating. The 15-Volt outputs are connected to a full bridge rectifier for 30 Volts AC input. The output of the bridge rectifier is connected to a 36V 1000 u-farad anti ripple capacitor. The output of the capacitor is then connected to the inputs of LM7805 and LM350 voltage regulator integrated circuits (see Appendix A for data sheet). 63

64 Figure 41: The final design of the box with cover opened The LM 7805 IC supplies a steady 5 Volts DC. The LM350 is regulated by a variable resistance, and supplies a steady 28 Volts DC to run the motor. A spiral chord, which can extend to 3m, connects the box to the tuning gun. The power supply to the instrument is 220 Volts AC. The box is powered connected to the 220-power source. The schematic diagrams of all three circuits are shown in Appendix D. How to Use the Instrument The instrument we designed is very simple to use. All the user has to do is follow certain set of instructions listed below. Hook the instrument to a 220 Volts Ac power source. Open the front cover of the piano to get access to the tuning pins Place the mute strip to silence two of the three strings of each note Turn on the instrument by pressing the Off / On switch Wait 2 seconds for the instrument to initialize The name of the first note to be tuned is displayed on the LCD. At this point the user can either start tuning the piano from the note whose name currently appears on the display or scroll the notes and select another note by pressing the Forward / Backward control buttons. To tune a string fit the tuning gun s head on the pin corresponding to the string being tuned 64