Abstract:
A method and apparatus that enables tuning a piano to exact equal temperament includes beat time template generator. The beat time template generator is provided a fundamental key describing the fixed pitch the reference key is tuned and a secondary key to be tuned. The beat time template generator determines all the harmonics of the fundamental key and the secondary key. A difference beat frequency for all the harmonics is calculated for the fundamental key and secondary key and one beat time is selected as a reference beat time template signal. The reference beat time template signal is then transferred to a speaker for reproduction. A piano tuner then adjusts the strings of the secondary key such that when the fundamental key and the secondary key are struck simultaneously, the beat time of the struck keys is identical to the reference beat time template signal from the beat time template generator.

Description:
TECHNICAL FIELD 
     This disclosure relates generally to methods and apparatus for tuning musical instruments. More particularly, this disclosure relates to methods and apparatus for equal temperament tuning of pianos and other similar instruments. 
     BACKGROUND 
     As is known, the keys of a piano are divided into seven groups or octaves. Each octave has twelve keys that are tuned with equal temperament. Equal temperament tuning is a method of tuning each of the notes within each octave such that every pair of adjacent notes has an identical frequency ratio. Pitch is perceived roughly as the logarithm of frequency by the human ear and thus the perceived “distance” from every note to its nearest neighbor is equal to a ratio of the logarithms of frequencies of the adjacent keys. Each frequency for each key of a piano is determined by the equation: 
     
       
         
           
             
               f 
               ⁡ 
               
                 ( 
                 N 
                 ) 
               
             
             = 
             
               
                 f 
                 ⁡ 
                 
                   ( 
                   M 
                   ) 
                 
               
               * 
               
                 ( 
                 
                   2 
                   
                     ( 
                     
                       
                         N 
                         - 
                         M 
                       
                       12 
                     
                     ) 
                   
                 
                 ) 
               
             
           
         
       
         
         
           
             Where: 
             f(N) is the frequency of the N th  key of the piano keyboard. 
             N is the numbering of the keys of the piano with the farthest left key being N=0. 
             f(M) is the frequency of a key tuned to a standard pitch. 
             M is the key number of the key tuned to the standard pitch. Usually the standard pitch or frequency is the A key above middle C or A 4 . M=49 for the key A 4 . 
           
         
       
    
     The history of the evolution of equal temperament is divided into two parts, before and after invention of equal temperament formula. First recorded date of the development of the concept of equal temperament is 1584 by Chu Tsai-Yu of the Ming Dynasty. Then European mathematicians Simon Stevin (1585) and Marin Mersenne (1636) developed their versions of equal temperament. Before equal temperament, it was a struggle to find some kind of universal system for tuning a musical instrument After introduction of the equal temperament concept, it was about 300 years before it was finally accepted for use in a few selected countries. It was assumed that J. S. Bach had intended equal temperament in his “Well-Temperd Clavier”. However, there is a difference between “well tempered” and “equal temperament”. The evolution of equal temperament still took decades for it to be permanently adopted by the musicians. In 1939, the standard pitch or frequency of the A key above middle C or A 4  as 440 Hz was adopted. 
       FIG. 1  is a diagram of a piano keyboard showing the designations for each of the keys of the keyboard. To tune a piano minute adjustments are made to the tensions of the strings of a piano to properly align the intervals between their tones so that the instrument is in tune. In the context of piano tuning, being in tune is not simply setting the tensions of the strings of the piano to a set of assigned frequencies or pitches. A piano tuner must assess the interaction between the notes. The tuner will tune one of the keys to a standard pitch or frequency such as the “A” key above middle “C” or “A 4 ” that is tuned to a frequency of 440 Hz. The remaining keys are tuned in relationship to the chosen fixed pitch (i.e. “A 4 ”=440 Hz). 
     SUMMARY 
     An object of this disclosure is to provide an exact equal temperament tuning method and apparatus. 
     To accomplish at least this object, a beat time template generator is provided a fundamental key describing the fixed pitch that a key is tuned and a secondary key that is to be tuned. The beat time template generator determines all the harmonics of the fundamental key and the secondary key. A difference beat time is determined for all the harmonics calculated for the fundamental key and secondary key and one beat time is selected as a reference beat time template signal. The reference beat time template signal is then transferred to a speaker for reproduction. A piano tuner then adjusts the strings of the secondary key such that when the fundamental key and the secondary key are struck simultaneously, the beat time of the struck keys is identical to the reference beat time template signal from the beat time template generator. 
     The beat time template generator includes a control interface device such as a display and a keyboard, a touch sensitive display, or a panel including indicator lights and switches. The beat time template generator has a fundamental note selector for indicating which note of the piano is tuned to an accurate pitch or frequency. A secondary note selector indicates the note of the piano that is to be tuned relative to the fundamental note. The control interface device has a fundamental note register and a secondary note register to respectively receive and retain a digital code indicative of the frequency or pitch of the fundamental note and the secondary note. 
     The beat time template generator has a fundamental note harmonic generator connected to the fundamental note register to receive the digital code of the fundamental note and to generate a set of digital codes indicative of a set of the harmonics of the frequency of the fundamental note. The secondary note register is connected to a secondary note harmonic generator to receive the digital code of the secondary note and to generate a set of digital codes indicative of a set of the harmonics of the frequency of the secondary note. In some embodiments, there are six harmonic frequencies included in each set of harmonics of the fundamental note and the secondary note. 
     The harmonic codes for the fundamental note and the harmonic codes for the secondary note are transferred to a beat time calculator. The beat time calculator determines all the combinations of beat times as a difference between the fundamental note and its harmonics and the secondary note and its harmonics to generate a set of all the beat time codes. In various embodiments, the beat time is transferred to the control interface device where the numerical frequencies represented by the beat time codes are displayed. 
     A beat time selector is connected to the beat time calculator to receive the beat time codes. In some embodiments, the piano tuner selects the desired beat time and the beat time select signal chooses the selected beat time code. In other embodiments, the beat time code that is an inverse of a number of beats per second (BPS) is chosen automatically. 
     A digital to analog converter is connected to the beat time selector to receive the digital code representing the selected beat time template between the fundamental note and the secondary note. The digital to analog converter generates an electrical signal that is transferred to a speaker for reproduction of the beat time template. The piano tuner then strikes the fundamental note and the secondary note simultaneously. The piano tuner detects the difference between the beat time of the simultaneously struck fundamental note and the secondary note and the reproduced beat time template from the speaker of the beat time template generator. The piano tuner then adjusts the string of the secondary note until beat time of the simultaneously struck fundamental note and the secondary note is identical to the beat time template of the beat time template generator. 
     In various embodiments, a method for tuning a piano begins with tuning a fundamental note such as the “A” key above middle “C” or “A 4 ”=440 Hz. A secondary note is chosen for tuning and the harmonics for the fundamental note and the secondary note are determined. In some embodiments, six harmonics are determined for the fundamental note and the secondary note. The beat times are determined for all combinations of the harmonics of the fundamental frequency and the secondary frequency. The desired beat time is chosen. In other embodiments, the beat time is chosen automatically. The chosen beat time is generated and reproduced through a speaker. 
     The piano tuner then strikes the fundamental note and the secondary note simultaneously. The piano tuner detects the difference between the beat time of the simultaneously struck fundamental note and the secondary note and the reproduced beat time from the speaker. The piano tuner then adjusts the string of the secondary note until beat time of the simultaneously struck fundamental note and the secondary note is identical to the beat time template of the beat time template generator. 
     The piano tuner then selects another secondary note for tuning, determines the beat time rate and generates the beat time template for reproduction. The piano tuner then adjusts the tension of the string of the secondary note until the beat time of the simultaneously struck fundamental note and the secondary note is identical to the beat time template of the beat time template generator. When all notes of the piano are tuned the method is complete. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a diagram of a piano keyboard showing the designations for each of the keys of the keyboard. 
         FIG. 2  is a chart of the harmonic frequencies for the notes of the octave from the second “F” Key (F 2 ) to third F” Key (F 3 ). 
         FIG. 3   a  is a chart of the beat frequencies between the pitch or frequencies of any two notes of the octave from the second “F” Key (F 2 ) to third F” Key (F 3 ) in beats per second. 
         FIG. 3   b  is a chart of the beat frequencies between the pitch or frequencies of any two notes of the octave from the second “F” Key (F 2 ) to third F” Key (F 3 ) in beats per second and beats per minute. 
         FIG. 4  is a chart of the beat frequencies between the pitch or frequencies of any two notes of the octave from the third “F” Key (F 3 ) to fourth F” Key (F 4 ) in beats per second. 
         FIG. 5   a  is a waveform illustrating a beat per second for an audio signal. 
         FIGS. 5   b - 5   d  is the musical symbology for the beat structure for harmonic frequencies of a metronome. 
         FIG. 6  is a block diagram of a beat time template generator embodying the principles of this disclosure. 
         FIG. 7  is a flowchart of a method for tuning a piano embodying the principles of this disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     In tuning a piano, a key is tuned to a fundamental frequency. In most cases, the initial key that is tuned is the “A” key above middle “C” or “A 4 ” is tuned to a pitch or frequency of 440 Hz. This is identified as a fixed pitch and provides the fundamental frequency. The secondary key(s) to be tuned is tuned relative to the pitch of the A 4  Key. The A 4  key and the secondary key that is being tuned are struck simultaneously. The notes will sound with the fundamental notes and their harmonics. Each or the harmonics will additively or subtractively combine to form a beat frequency. The second key is adjusted such that the relationships of the frequency between the fixed pitch key with its harmonics and the second key with its harmonics have the correct beat frequencies. If the beat frequencies are not correct, the piano is not tuned to be in equal temperament. 
       FIG. 2  is a chart of the harmonic frequencies for the notes of the octave from the second “F” Key (F 2 ) to third F” Key (F 3 ). It can be shown that each of the keys have an equal logarithmic between each of their frequencies.  FIG. 3   a  is a chart of the beat frequencies between the pitch or frequencies of any two notes of the octave from the second “F” Key (F 2 ) to third “F” Key (F 3 ) in beats per second. The chart is read by picking a frequency on one horizontal row and picking a second frequency on one vertical column. The beat frequency is chosen as the smallest difference between the pitch of the first key and its harmonics and the pitch of the second key and its harmonics. For example, the beat frequency between the second “F” Key (F 2 ) and the second “A” key A 2  is 3.465 BPS. The beat frequency is the result of combining the 4 th  harmonic of the A 2  key and the 5 th  harmonic of the F 2  key. 
       FIG. 3   b  is a chart of the beat frequencies between the pitch or frequencies of any two notes of the octave from the second “F” Key (F 2 ) to third F” Key (F 3 ) in beats per second and beats per minute. As is known, metronomic notation is standardized in beats per minute. Therefore, the beats per second for the beat frequencies are multiplied by 60 to designate the beats per minute. For example, the beat frequency between the F 2  key and the A 2  key is 3.465 BPS, as noted above. That translates to 104 beats per minute (BPM). 
       FIG. 4  is a chart of the beat frequencies between the pitch or frequencies of any two notes of the octave from the third “F” Key (F 3 ) to fourth F” Key (F 4 ) in beats per second. As can be seen, when  FIGS. 3   a  and  4  are compared, the beat frequencies in the octave between the F 3  key and the F 4  key are double the beat frequencies of the octave between the F 2  key and the F 3 . For example, the frequency of beat frequency discussed above between the F 2  key and the A 2  key is 3.465 BPS and the beat frequency between the F 3  key and the A 3  key is 6.930 BPS. 
       FIG. 5   a  is a waveform illustrating a one beat per second wave for an audio signal and  FIGS. 5   b - 5   d  is the musical symbology for the beat structure for harmonic frequencies of a metronome. As is known, the fundamental wave structure or a pure tone for sound is a sine wave. If the time from one peak of a sine wave to a second peak of the sine wave is one second, the wave is said to have a frequency of 1 Hz. A metronome will make a sound at the time of each peak or one beat per second that is a setting of sixty beats per minute for the metronome. The universal symbol for a metronome beat is the quarter note (           ) and indicative of the number of notes played or beats of the metronome. To replicate the beat frequency of the between the F 3  key and the A 3  key that is 6.93 BPS or 416 BPM, the metronome is set such that quarter note (         ) equals 416 BPM as shown in  FIG. 5   b . Alternately, the metronome may be set to 208 BPM. Then the beat frequency of the between the F 3  key and the A 3  key must occur twice during the metronome period as shown in  FIG. 5   c . In  FIG. 5   d , the metronome is set to a frequency of 104 BPM. In this case the beat frequency of the between the F 3  key and the A 3  key occurs four times in the period of the metronome beat.
       FIG. 6  is a block diagram of a beat time template generator embodying the principles of this disclosure. The beat time template generator has a control/display panel  100  that includes start switch  105  and stop switch  110  for activating and deactivating the beat time template generator. The beat time template generator has a fundamental note selector  115  that in some embodiments is a selector switch with a display to show indicate the fundamental note. Similarly, the beat time template generator has a secondary note selector  115  that in some embodiments is a selector switch with a display to show indicate the secondary note to be tuned. 
     In some embodiments, a beats per minute display  125  is a media display device that is a light emitting diode (LED) or liquid crystal (LCD) matrix or seven segment display for displaying the calculated beat frequency for the fundamental note and the secondary note. In other embodiments, the control display panel  100  is a media display device such as a touch screen or computer monitor. In embodiments, where the control display panel  100  is a media display device, the start switch  105 , the stop switch  110 , the fundamental note selector  115  and the secondary note selector  120  are incorporated in the media display device. The beat frequency chart  130  shows a set of the keys that are to be tuned. In the illustration, the beat frequency chart  130  has the notes between the F 2  key and the A 3  key. The fundamental note selector  115  and the secondary note selector  120  are incorporated in the displayed beat frequency chart  130 . In a touch screen or a screen with a mouse/cursor selection mechanism as in a computer system, the notes between the F 2  key and the A 3  key are sensitive to selection and function as the fundamental note selector  115  and the secondary note selector  120 . 
     The control/display panel  100  is connected to the fundamental note register  135  and the secondary note register  140 . The fundamental note register  135  and the secondary note register  140  respectively retain a digital code representing the frequency of the selected the fundamental note and the secondary note. The fundamental note register  135  is connected to the fundamental note harmonic generator  145 . In some embodiments, the fundamental note harmonic generator  145  is a digital multiplier that multiplies the code for the frequency of the fundamental note by a set of digits from 2 to an upper range of the harmonics (i.e. 6 or 7) to generate the fundamental harmonic codes  155 . 
     Similarly, the secondary note register  140  is connected to the secondary note harmonic generator  150 . In some embodiments, the second note harmonic generator  150  is a digital multiplier that multiplies the code for the frequency of the second note by the set of digits from 2 to an upper range of the harmonics (i.e. 6 or 7) to generate the secondary harmonic codes  160 . 
     The fundamental harmonic codes  155  and the secondary harmonic codes  160  are transferred to the beat time calculator  165 . The beat time calculator  165  additively and subtractively combines the fundamental harmonic codes  155  and the secondary harmonic codes  160  to generate the set of beat time codes  170  for all the combinations of the harmonics of the fundamental note and the secondary note. In various embodiments, the beat time codes  170   a  are transferred to the control/display panel  100  for display and selection by a user. A user selects a desired beat time and a beat time select signal  190  is transferred to the beat time selector  175  for selecting the beat time. 
     In other embodiments, the beat time codes  170   b  are transferred to the beat time selector and the desired time frequency is chosen automatically based on the parameters such as the ability for a user to discriminate the beats and determine if the beat time of the fundamental note and the secondary note of the piano are equal to the generated beat time template. The selected beat time code is transferred to a digital-to-analog converter  195  that converts the selected beat time code to an analog signal that is applied to a speaker  200 . 
     In some embodiments, it is desirable for the user to be able to have reference beat signals such a 60 beats per minute, 120 beats per minute, or 240 beats per minute to become attuned to these beat patterns to distinguish the beat times of the fundamental note and the secondary note from the piano to establish the desired beat time for generation by the beat time template generator. The display panel  100  has 60 beats per minute, 120 beats per minute, and 240 beats per minute selectors. The selectors generate 60 beats per minute, 120 beats per minute, or 240 beats per minute selector signals  180   a ,  180   b , and  180   c  that are applied to a beat register  185 . The beat register  185  stores the necessary beat codes for the 60 beats per minute, 120 beats per minute, and 240 beats per minute signals. The selected code is applied to the digital-to-analog converter  195  that converts the selected beat time code as the beat time template signal to an analog signal that is applied to a speaker  200 . 
       FIG. 7  is a flowchart of a method for tuning a piano embodying the principles of this disclosure. The method begins with selecting and tuning (Box  205 ) a fundamental note on the piano. In most embodiments, a tuning fork for the fundamental note is used as an aid to adjust the fundamental note. For example, if the fundamental note is the fourth “A” Key (A 4 ), the tuning fork will provide a frequency of 440 Hz and the A 4  key will be adjusted to produce the 440 Hz. A secondary note will then be selected (Box  210 ) and the desired beat rate between the fundamental note and the secondary note will be determined (Box  215 ). As described above in  FIG. 6 , a beat time template generator may determine all the possible beat frequencies and the user may select the appropriated beat time. Alternately, the beat time template generator will choose the beat time based on the parameters of ease of discrimination and ability to match the adjustment of the secondary note to match the beat time of the beat time template generator. The selected beat time is then generated (Box  220 ). 
     The fundamental note and the secondary note are struck simultaneously (Box  225 ) and the user determines (Box  230 .) the difference between the generated beat time template and the beat time of the fundamental note and the secondary note of the struck keys. The user further determines (Box  235 ) whether the beat times are aligned. If they are not aligned, the user adjusts (Box  240 ) the tension of the string of the secondary note. 
     The user then strikes (Box  225 ) the fundamental note and the secondary note simultaneously and the user determines (Box  230 .) the difference between the generated beat time template and the beat time of the struck fundamental notes and the secondary note of the struck keys. The user further determines (Box  235 ) whether the beats are aligned. If they are not aligned, the user adjusts (Box  240 ) the tension of the string of the secondary note and repeats the process. If the beat time of the struck fundamental note and the secondary is aligned with the beat time template of the beat time template generator, the user determines (Box  245 ) if all the notes are tuned. If all the notes of the piano are not tuned, the next secondary note is selected (Box  210 ), the beat time is determined (Box  215 ), and the beat time template is generated (Box  220 ). The user strikes (Box  225 ) the keys of the fundamental note and the secondary note and determines (Box  230 ) any difference between the beat time of the struck keys and the beat time template of the beat time template generator. The user then determines (Box  235 ) if the beats are aligned. If the beats are not aligned the user adjusts (Box  240 ) the tension of the selected key. This is repeated until the beat times are aligned and all the keys have been tuned and the process is ended. 
     The structure of the beat time template generator as shown is  FIG. 6  is functional. The beat time template generator may be a computer-based system with the computer performing the functions of the described elements. The fundamental note register  135  maintains the frequency of the fundamental note and the secondary note register  140  maintains the frequencies of remaining notes. Alternately, the frequencies of the remaining notes may be calculated using the equation detailed above and stored in the secondary note register  140 . If the piano is structured for tuning in other temperament structures rather than twelve keys per octave, the equation for the tuning frequencies will be calculated based on that structure. 
     The computer processor may be programmed to function as the fundamental harmonic generator  145  and the secondary harmonic generator  150 . The processor then calculates the beat frequencies for all the harmonics of the fundamental note and the secondary note to perform the function of the beat time calculator  165 . The computer processor may be programmed with the necessary parameter to determine the desired beat time as performed by the beat time template. Many modern computer systems have a digital-to-analog converter  195  included in the structure for the conversion of the beat time template to an analog signal for application to an included speaker  200 . 
     While this disclosure has been particularly shown and described with reference to the preferred embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made without departing from the spirit and scope of the disclosure.