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Chapter 1-1 CHAPTER ONE MUSICAL KNOWLEDGE REQUIRED FOR PIANO TUNERS NOTE: The illustrations referred to in this book are not presented because they include graphics which cannot be shown on all computers. They are available directly from the author. However, they are not necessary to understand the text. ALSO: If you request the audio tapes that are offered, you will find that I occasionally refer to page numbers in the printed text. After reformatting the text from a commercial product to a disk presentation, these page numbers may not coincide. However, It will be no problem to find exactly where I want you to look. In order to properly tune a piano, I recommend you learn a "little" about music terminology, acoustics, how a string vibrates, how the musical scale is organized, a little about the mathematics of the musical scale, and the theory surrounding the art of tuning. Although I can teach you to tune a piano without requiring much knowledge in these areas, the more you know, the more confidence you will have. I believe the more you can learn about the complete subject of "TUNING", the better tuner you will become. This sounds like I am going to ask you to become a music major rather than a tuner. Nothing could be farther from the truth. You will find the musical knowledge re- quired to tune a piano can be learned in a very short time. A piano is tuned by listening for beats (explained later) and adjusting the tension of the strings to either eliminate or set the speed of these beats. A good ear is necessary, but a good musical ear is not. NOISE AND MUSIC Webster's dictionary defines noise as "something that lacks agreeable musical quality or is noticeably unpleasant." A musical tone is defined as a "sound of defi- nite pitch and vibration." When a piano string is struck, a musical tone is heard, and when you hear the sound of a jack-hammer, you are hear- ing noise. You probably have learned elsewhere that in order for a sound to exist, it must be heard. If a sound vibrates at a certain rate and causes your ear to vibrate at the same rate, you are hearing a musical sound. Conversely, if a sound vibrates in an unorganized fashion causing your ear to vibrate the same way, you are hearing noise. Chapter 1-2 THE VIBRATING PIANO STRING If you secure a length of piano wire on both ends and pluck it with your fingernail, you will hear a musical sound. The sound (pitch) you hear is determined by 1) the thickness of the wire; 2) the length of the wire; 3) the tension put on the wire; and 4) how stiff the wire is. It is not necessary to try this experiment at this point - just remember the characteristics of a vibrating string. NOTE: If you are not familiar with basic musical notation, please refer to appendix D. When a string is struck, it vibrates in many different ways. First, and foremost, the sound you hear will be the FUNDA- MENTAL. Secondly, the string produces a series of PARTIALS by dividing itself into halves, thirds, quarters etc. This phenomenon occurs simultaneously (see illustration 1-1). | When you enroll as a student and receive | | your pack of illustrations, attach them | | in the empty spaces throughout this book.| - / \ \ / illustration 1-1 (The above illustration simply shows you how a string vi- brates and produces partials when it is struck in different places along its length). The first eight PARTIALS produced by striking an indi- vidual string are shown in illus. 1-2 built on the fundamen- tal note C-28 (explained later). / \ \ / illustration 1-2 Chapter 1-3 The partials shown above (over the FUNDAMENTAL C-28) are C- 40, G-47, C-52, E-56, G-59, A# (or Bb - explained later) and C-64. A little later, after I have explained these numbers attached to the notes (pitches), I will ask you to play them on the piano. PRODUCING BEATS If one piano wire is adjusted to sound exactly the same as another wire, they are "in tune" with each other. On the other hand, If one wire is just a little "flat" or "sharp" to the other, they will produce a softer tone when sounded together and you will hear a VIBRATION. This VIBRATION will either be fast or slow, depending on how far sharp or flat one wire is to the other. For example, if one wire is tuned to sound at 440 C.P.S. (cycles or beats per second) and the other wire is tuned to sound at 441 C.P.S. you will hear ONE beat per second. You will hear this because the faster vibrating string will overtake the slower vibrating string ONCE per second. Every time you hear the sound getting louder and then softer, you are hearing ONE beat (cycle). Therefore one string (or partial) vibrating at a specific C.P.S. will cause you to hear beats if it is sounded with another string vibrating an a different C.P.S. Please don't give up yet. This subject will be pre- sented in more detail later on. I am just filling your head with facts that will magically make sense as you progress. I promise! THE PIANO KEYBOARD Now, I am going to introduce you to your piano in a way you may not have experienced before. FIRST: Sit down in front of the piano - say "HI! I am going to tickle your ivories and make you feel and sound great". If you are sitting in front of a full size piano, you will be looking at 88 individual keys. The key at the far left of the key- board will be a white key and it will be given the name of A-1. The key at the far right of the keyboard is also a white key and will be given the name of C-88. SECOND: Observe that there are 52 white keys, and 36 black keys (which we will call SHARPS). If you do not know the names of all the keys you will now learn them very easily. I will take you up the keyboard as you are SITTING IN FRONT OF THE PIANO. Chapter 1-4 The keys (for identification) are numbered from left to right 1 thru 88. LEARN this sequence: A-B-C-D-E-F-G-A. This is the way the scale progresses from A-1 up to C-88 ON THE WHITE KEYS. TRY IT. Start on A-1 and play every white key all the way up to the top. You just played 52 keys, NOT 88. The other 36 keys are the black ones. As you progress up the keyboard on the white keys, and come to a black key between two white keys, give it the name of the key you just left and add the name SHARP. In other words, the first black key you come to will be called A- sharp (written usually as A#). The second black key you come to will be called C#. The third black key will be called D#. SO NOW, you have the ability to name all the keys from A-1 to C88. I'm sure you are familiar with the word "FLAT" as pertains to musical sound. When most people hear this term, I imagine they think of a tone (note or pitch) that sounds a little "off". This is correct, but another way tuners and musicians use the term FLAT is to identify musical pitches. If you start at the top of the piano on pitch C-88 and come DOWN, you will find that the black keys are in exactly the same place. Brilliant? I thought you would think so. As you come down the keyboard the first black key you come to is just BELOW B-87. Since it is BELOW the note we are going to call it B-flat (normally written Bb). Simply put, when you are going UP the keyboard, the black key takes the name of the white key BELOW it and adds the term SHARP (or #). When your are coming DOWN the key- board the black key takes the name of the white key ABOVE it and adds the term FLAT or (b). At this point, make sure you understand that C# is the same as Db; D# is the same as Eb; etc... One other point to make - Please note that between the notes E and F; and B and C, there are no black keys. This merely means that E# can also be called F and B# can be called C. Also Fb is the same as E, and Cb is the same as B. Please do not let this confuse you. Just accept it for now . IMPORTANT: Tuners, for the most part, call all black keys SHARPS. Musicians use both SHARP AND FLAT. For the pur- poses of this course we will use the term SHARP exclusively in the printed text and illustrations. I just wanted you to understand why you may hear C# called Db - A# being called Bb etc... Chapter 1-5 On the audio tapes you will hear me occasionally refer to both Sharps and Flats. This is so you will be able to better understand the terms and feel comfortable with either one. Take a break - have a cup or glass of your favorite beverage, think about it until just before you get a head- ache and then proceed reading. Believe me, it WILL eventual- ly make sense. Earlier, when you learned the sequence of notes as you go up and down the keyboard, you saw that the notes start repeating after 12 have been hit. Start on A-1 the first note on the left side of the keyboard and go up note by note and the 13th note you hit will be A-13. REMEMBER THIS: The distance between one note and another one with the same letter name (higher or lower) is called and OCTAVE. Now, start with A-1 and go up counting the A's and determine that there are 7 more - plus 3 more notes. This tells you that the complete piano scale contains 7 OCTAVES plus three notes. When you start at the bottom of the piano and ascend by playing each note (white and black) one after the other, you will be going up the keyboard CHROMATICALLY. Practice going up and down the keyboard in this manner and saying aloud the notes as you play them. / \ / \ Illus. 1-3 Chromatic scale (from C-40 up to C-52) INTERVALS An INTERVAL is a unit of harmony, resulting from sound- ing two tones (notes) simultaneously. For our purposes we will think of an interval as the DISTANCE between two notes measured by their differences in pitch. If the two notes are played one after the other, it is referred to as a MELODIC interval. If they are played together, it is re- ferred to as a HARMONIC interval. Chapter 1-6 The distance from any note to the next note, higher (to the right) OR lower (to the left) is defined as a HALF-TONE or HALF-STEP. This is the SMALLEST interval. Recall now that the LETTER NAMES of the notes are A-B-C-D-E-F-G-... Now if you want to find out the GENERAL name of any inter- val, you simply start counting on the first note of the interval and continue up or down to the second note of the interval. EXAMPLES: If the first note of the interval is C-28 (the 28th note from the bottom of the piano) and the second note of the interval is D-30, you would count 1-2. The interval would be called a SECOND; C-28 up to E-32 is a THIRD; C-28 up to F-33 is a FOURTH and so on until you reach the 8th which is called the OCTAVE (C-28 to C-40). / \ / \ Illus. 1-4 (Various intervals within the octave) Since sharps (#) and flats (b) take their LETTER NAMES from the adjacent white keys, they are not considered when you are determining the size of an interval. A to C is a THIRD and A to C# is also a third. This brings us to another term called the UNISON. Look at the strings on the piano and you will find that when you strike them by pressing the keys, the higher notes will have three strings per note. The notes to the immediate of these will have two strings per note and the bottom 10 or so will have only one string per note. When the strings struck by one hammer are tuned to each other the are said to be in UNISON. This is also referred to as the interval of a perfect PRIME. We must now learn to identify the intervals by counting HALF-STEPS. A half step is the distance from on note up or down to an adjacent note (black or white). The chart below will show you how to construct the intervals. You then need to be able to start on any note and play any interval neces- sary. Chapter 1-7 FROM TO HALF-STEPS INTERVAL NAME _____________________________________________________ C-28 E-32 4 MAJOR THIRD C-28 D#-31 3 minor third C-28 F-33 5 PERFECT FOURTH C-28 G-35 7 PERFECT FIFTH C-28 A-37 9 MAJOR SIXTH C-28 G#-36 8 minor sixth C-28 C-40 12 PERFECT OCTAVE _____________________________________________________ Some new terms were introduced in the chart - MAJOR, minor and PERFECT. As you have probably have figured out by now, if C-E is a third and C-D# (Eb) is also a third, we need some way to label the difference since they will not sound the same when played together. So, a third will be MAJOR if there are 4 half steps between the two notes and it will be minor if there are only 3 half-steps. Practice identifying intervals starting on various notes until you are able to start on ANY note and play the intervals of the MAJOR & minor thirds and sixths and the Perfect fourths and fifths. You will use these intervals over and over while learn- ing to tune and in every tuning you perform in the future. The importance of learning the keyboard cannot be over- emphasized. You certainly do not have to know how to play a piano to tune it and an auto mechanic does not have to know how to drive, but you wouldn't take your car to a mechanic if he/she didn't know a spark plug from a carburetor. A "LITTLE" MATH There are numerous books you can find that will delve deeply into the mathematics or mechanics of the musical scale. My purpose in this book is to present just enough (hopefully) but not too much of the technical aspect of tuning. Once you grasp the information herein you may find your appetite has been whetted sufficiently and you can expand your knowledge. As in all professions, there is always more to learn. The rest of this chapter is fairly technical, but no course on tuning would be complete without at least includ- ing this information. I recommend you at least read the rest of the chapter because there are many non-technical bits of information you should know. Don't worry that you will not be able to tune Chapter 1-8 without knowing everything I will present. I tuned pianos professionally for a few years without knowing MOST of the information on the next few pages. If you are really seri- ous about entering this profession, you will refer to and learn the theory of tuning eventually. So, speed read the following info and proceed to chap- ter two. If you understand it all - great, if not - don't worry. EQUAL TEMPERAMENT As you sit in front of your piano and play the notes up and down, it is apparent that they all sound at a different pitch or frequency. You learned that a string, when struck, vibrates at a certain rate causing your ear to vibrate at the same rate. Since there are 88 different pitches on most pianos, there has to be a way to space these pitches one to another so that the piano will be in tune. For instance, we know that within any octave there are 13 separate sounds. These sounds must be arranged so there is exactly the same distance between each note as we go up or down. There are 13 separate sounds, but only 12 half- steps. In order to obtain the frequency of a tone one half- step higher than another and have 12 equal half-steps from the lower note of an octave to the upper note it is neces- sary to multiply the frequency of the tone by the 12th root of the octave ratio, which is 1:2. The 12th root of 2 is 1.0594631 for those of you who understand this terminology. More simply, the note A-49 vibrates at 440 cycles per second (C.P.S.). If you multiply 440 by 1.0594631 you will get 466.163764 which is the number of C.P.S. of A#-50. If you multiply 466.163764 by the 12th root of 2, you will get the frequency of B-51. You could do this from the bottom of the piano all the way to the top and you would go from A-1 with a frequency of 27.5 to C-88 with a frequency of 4186.009. OR you could just refer to appendix B from the table of contents (main menu) and find that I have provided this information for you. When 12 successive half-steps (comprising one octave) are EQUALIZED by the method explained above, the result is called and EQUAL TEMPERED octave. A smaller unit of measurement was introduced by A.J. Ellis called the CENT. Ellis divided the equal tempered octave into 1200 units called "CENTS" with each half-step being exactly 100 cents distance from the next, regardless what octave you are in. The cent is too short a distance to be heard by the ear, but a trained ear will hear a distance of 2 cents and the average person can hear a distance of 3-4 cents. Chapter 1-9 Now that we know how the EQUAL TEMPERED octave was created, it is a simple matter to "equally temper the entire keyboard." For Example, the lowest note on the piano is A-1 which beats at 27.5 C.P.S. To obtain the beats of A-13 an octave higher) we multiply 27.5 by two and get 55.00 C.P.S. We then could multiply 55.00 by two and get the C.P.S. of A-25 (110.00). If we proceed by multiplying each frequency by successive of 2 we will reach A-85 at a frequency of 3520. Again, please refer to Appendix B for clarification. At the beginning of this section, I told you that a tuner tunes a piano by listening for beats. You are surely wondering how you are supposed to hear 440 or whatever cycles per second. YOU DON'T HAVE TO. Since it is impossi- ble to hear those frequencies, we will use a system of tuning based on COINCIDENT partials. Don't let this new term frustrate you. You will understand soon enough. Recall that we learned when a string vibrates it pro- duces a series of PARTIALS. When two strings are sounded together forming an INTERVAL, you will find (explained later) that there is a common partial sounding at close to the same frequency. So instead of comparing the extremely high frequencies of the FUNDAMENTALS, we will be comparing the closely related frequencies of the coincident PARTIALS. SERIES OF PARTIALS In order to follow the discussion of partials, it will help to have the chart on pitch frequencies (Appendix B) in front of you Just return to the Table of Contents and highlight the topic "Theoretical Fundamental Pitches of All Notes. Press ENTER, and when Appendix appears, turn on you printer and press P. It is only two pages long. Chart (1) gives you the cycles per second that every note on the piano sounds when struck. Chart (2) starts on C-28 (the 3rd C from the bottom of the piano). Locate C-28 on the piano. Beneath The word NOTE on Chart 2, the notes from C-28 up to C-40 are listed and the first column to the right will give you the C.P.S. of these pitches. When you play C-28 on the piano the FUNDAMENTAL will be sounding at 130.81 C.P.S. Since the string produces PAR- TIALS, I will give you the first eight partials that will be produced. Remember, the FUNDAMENTAL is actually the FIRST partial. Chapter 1-10 PARTIAL NOTE C.P.S. INTERVAL 1st C-28 130.81 FUNDAMENTAL 2nd C-40 261.63 OCTAVE up from C-28 3rd G-47 392.44 FIFTH up from C-40 4th C-52 523.25 FOURTH up from G-47 5th E-56 654.07 MAJOR third up from C-52 6th G-59 784.88 minor third up from E-56 7th A#-62 915.69 minor third up from G-59 8th C-64 1046.50 ONE OCTAVE up from C-52 and TWO OCTAVES up from C-28 Now, start on C-28 and while holding the RIGHT pedal on the piano down, play the partials one after the other from C-28 up to C-64. As you play each note try to learn the intervals listed above. Listen to the sounds of the inter- vals. Since our goal is to tune the piano by listening for beats or vibrations as one note is sounded against another, I will now show you how we get these beats down from the hundreds of cycles per second to the range in which we can distinguish them. For this exercise, we are going to assume that the note C-28 is perfectly in tune. How to do this will be explained later, but for now we already have it in tune. We are going to tune E-32 to C-28 so we will have two notes on the piano in tune. Look at chart TWO in Appendix B which lists the fre- quencies of the first eight partials of each note in the temperament octave. By the way, the TEMPERAMENT octave is the octave we will use later when we begin tuning the piano. Locate C-28 under the column labeled NOTE. Follow this to the right until you come to the 5th partial. The 5th partial of C-28 produces 654.07 C.P.S. NOW, E-32 in the same column. Follow this to the right until you come to the 4th partial. You will find the 4th partial of E-32 produces 659.26 C.P.S. We subtract 654.07 from 659.26 and find that when C-28 and E-32 are tuned we will hear approximately 5 C.P.S. You will be able to hear 5 C.P.S. easily once your ear is trained (later). For now just try to follow the mathematics all well as you can. It will gradually (believe it or not) become easy. The simple fact is, that when we sound any note with another, somewhere in the series of partials of each note we can find a partial of one series that beats very close to the other. Above, we found that the 5th partial of C-28 beats very close to the 4th partial of E-32. Therefore, we can conclude that the RATIO of C-28 to E-32 (which is the interval of a MAJOR third) is 5:4. Chapter 1-11 I will now give you the ratios of the intervals we will be using later so you will be able to find the COINCIDENT PARTIALS by using the chart. If you didn't have the chart, you could find the C.P.S. of any partial by finding the multiple of the fundamental. For example, if you wanted to know what the C.P.S. of the sixth partial of C-28 is, you merely multiply the fundamental (130.81) by six. You will find it to be 784.86, which you can find under the column labeled 6th in the chart. The cycles have been rounded off to two decimal places. You can find the C.P.S. of any partial of any fundamental by the same method. Simple - Right? RATIOS INTERVAL RATIO Unison 1:1 Octave 2:1 Perfect Fifth 3:2 Perfect Fourth 4:3 MAJOR Third 5:4 minor third 6:5 MAJOR Sixth 5:3 minor sixth 8:5 REMEMBER to multiply the lower note in the interval by the larger number in the ratio and the upper note by the smaller. ONE MORE EXAMPLE and then you must spend some time working on this procedure until you feel comfortable with it. We just tuned E-32 to C-28. Now we will tune G#-36 to E-32. We will then have three notes in tune - C-28, E-32 and G#-36. First, determine that the interval from E-32 up to G#- 36 is a MAJOR third. Then find the ratio of a MAJOR third from the above chart. Since the ratio is 5:4 we know that the 5th partial of E-32 will sound very close to the 4th partial of G#-36. Locate the C.P.S. of the two notes. E-32 beats at 164.81 C.P.S. and G#-36 beats at 207.65. Multiply 164.81 by 5 to obtain the C.P.S. of the 5th partial and get 824.05. Then multiply 207.65 by 4 and get 830.56. Subtract and come up with approximately 6.5 (6 1/2) C.P.S. So we would then tune G#-36 to E-32 until we hear 6.5 C.P.S. You could also just have looked up the notes on the chart and saved the hassle of multiplying. Chapter 1-12 IN THIS CHAPTER YOU LEARNED: 1. The difference between noise and musical sound 2. How a piano wire vibrates 3. What partials are and how they are used in tuning 4. Identification of keys on the piano keyboard 5. What intervals are and how to identify/construct them 6. What "equal temperament" means 7. What coincident partials are 8. The ratios of intervals and how they are applied Press P to print out this chapter or ESC for the menu