

                                                         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.|
                                     -
        
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          \                                                     /
                               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
        
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