ELASTIC STRING ENERGY.
Getting back to Tonal Sustain and Elastic String Energy E, In Fig 8 page 3, the value of E for the 1st overtone (2nd harmonic) is seen as E/4, if we take a relative value of E for the fundamental f to be 100%, then in comparison the available Energy E produced by the 2nd harmonic is only = to 25% of the fundamental Energy level and for the 3rd harmonic only 11.1% of the fundamental. (The vibrating string segments for the overtones of Fig 8 have been drawn out of proportion, for clarity). A quick glance at the E levels shown for the 1st, 2nd, 3rd 4th, and 5th...n (number) of harmonics, in the series indicates a dramatic fall rate in Energy levels, respectively 100%, 25%, 11.1%, 6.25%, 4%. Mathematical Fourier sine wave formulation has been used to calculate E for each harmonic, for further harmonics in the series, the 6th=E/36, 7th=E/49 and 8th=E/64. Fortunately only the first few harmonic overtones are needed to build a rich full tone. In fact if we were able to hear the higher overtones, the tone would actually sound somewhat grating and unpleasant. Of importance are the 3rd and 5th harmonics, adding a more musical tone to the predominant fundamental.
OVERTONES - that sound Harmonious or Discordant.
When you play an acoustic guitar that is out of tune a harsh sounding noise is created due to a lack of harmony, this is discord. At first you may re tune the guitar but find that the tone is still in discord, for other reason unbeknown. With displeasure you may conclude it does not have a very good tone and so put the guitar down. Possibly if you are looking to purchase an acoustic guitar you will then consequently try another? As a musician at one time or another you may have taken your own guitar to a luthier, to have the string length (intonation) readjusted, in order to make the guitar play in tune. Applying normal finger pressure to various fretted string positions, you may check out the intonation job carried out with a chromatic guitar tuner. Then you may find the guitar played in tune by say + or - 0 to 3 cents depending on what notes were fretted as indicated by the tuner meter; ( As to be expected from a fretted instrument, this is normal and generally higher readings are common, 100cents = 1 semitone = 1 fret). But you may still have been dissatisfied with the tonal quality of your acoustic guitar whilst playing it, simply because there still seems to be some kind of discord when you play it. You may even notice the louder you play the guitar the worse the discord becomes.
NODAL SHORTENING due to a Fixed, Mass Loaded String
The above discussion brings us back to the tonal quality of the acoustic guitar. Fig 9 may help to explain what is happening. Fig 9 is similar to Fig 8, it shows in particular an enlarged small section of the string support structure, the large corner section coloured in dark blue. The first harmonic wave or fundamental frequency is seen as the two curving black lines connected to the boundary end of
the support structure. Inside the fundamental we can also see the 1st overtone coloured in dark blue. You may notice that the nodal area for the vibrating string is moving away from the string length boundary, for the fundamental and with a further increase in distance (dx) for the 1st overtone in the direction of the dark blue arrowhead. The nodal positions continue to move further away from the boundary for each overtone that forms in turn, within the series of the fundamental frequency. This particular vibrating string action occurs only if in one instant any part of the support structure moves or wavers, by a greater or lesser degree as indicated in Fig 9 by the double arrow "S" curve lines. Consequently, for the fundamental but more so for its series of overtone standing waves as seen in Fig 8, do no longer fit perfectly within the boundaries or importantly within the fundamental. What we end up hearing due to inner nodal migration from an inadequate wavering support structure, is discord.
Why is this discord, you may remember from the discussion of sine waves and referring to Fig 4; if the wave length (λ) of the frequency is shortened then the frequency is increased. Each standing wave pattern of each overtone will still look the same as those seen in Fig 8, but because their string length boundary nodes are forced to move away from the boundary, the waves become compressed. With the standing wave lengths now closer together, their frequency changes, i.e. increases. Let's investigate the situation; take A=110Hz, its first overtone = 2 x 110=220Hz, a perfect octave. Now say the wavering motion of the support structure causes a 5% compression of the wave length, so that's 220/100% = 2.2Hz=1%, and 5x2.2Hz=11Hz, summing-up 220+11= 231Hz is therefore no longer a true 1st overtone octave, the sound we hear is out by 5 cents or 11 cycles.
As stated the nodal shift dx also increases for each consecutive overtone within the series, only to create further discord, where the above mild case of 5% wave compressions are increased further. This situation is made worse if you like simply by playing the guitar strings more forcefully and especially so for an industry standard X-braced acoustic guitar! Why, because the X- brace is not put in place to somehow directly or even indirectly support the fixed end of the string length, at the bridge. The X-brace arms' end on the very thin side walls of the guitar body, a flimsy situation to say the least and so, allows the bridge to initially rock back and forth greatly in the direction of the double arrow "S" curve as shown in fig 9. Reference as to why the x-brace is used as a compromise, has been discussed in the overview pages.
The x-brace is used simply to stop the soundboard from buckling up too much, under the weighted mass of ~74kg of the string load tension. X-braced guitars are notorious for allowing bridges to be lifted clean off the face of a soundboard, and the braking of the x-brace arms especially forward of the string load tension is also a common occurrence.
When a string is made to vibrate on the x-braced guitar, the bridge undertakes a rocking motion, as indicated by the direction of the curved arrows' from 0⁰ towards either side of the ±90⁰ shown central of the boundary in Fig 9. Where the above description of nodal shorting wave compression, is greatly enhanced and takes place! This rocking motion is in excess and tends to slacken the string, this allows the Elastic string Energy to quickly dissipate, due to the lack of direct support for this particular system of a Fixed, Mass Loaded String length. Forceful playing of the guitar strings only increases nodal migration and harmonic discord. If a slow softer melody is played, the rocking motion of the bridge will still occur. In this case the string slacking, results in less amplitude and less sustain for the higher harmonics which quickly diminish in sound level, normally refered to as dead notes. Not only do we end up with in harmonic discord (poor tonal quality), but we also lose the ability to have tonal sustain!
The ability to have tonal sustain comes from keeping the string vibrating, this can only occur if the bridge does not have a rocking motion in the direction of the string length, and secondly if the soundboard is able to vibrate, from an initial stress free state. Losing Elastic string Energy into the soundboard, is short lived. While you may have an initial loud sound or more precisely a booming sound for a heavier bass string as an example, the string will quickly settle and the singing tonal voice of the instrument will be lost.
You may be thinking well what other motion does the bridge undertake, it must also be rocking side to side. Yes it does, and it is also vibrating up and down. If we consider this to be so then there are three dimensions' of movement. We could label the direction of string length as x, the direction perpendicular to the soundboard (up and down) as y, and the direction of side to side of the bridge as z. Now there must also be Energy entering the soundboard in all three planes: x, y and z. The energy associated with the x direction is our all important natural string vibration, and needs to be maintained as our all important source. We could use a rigid support in order to produce tonal sustain. While in the planes of y and z , the vibrating motion of the bridge could be allowed since it would not greatly affect the well supported vibrating string length direction of x.
Getting back to Tonal Sustain and Elastic String Energy E, In Fig 8 page 3, the value of E for the 1st overtone (2nd harmonic) is seen as E/4, if we take a relative value of E for the fundamental f to be 100%, then in comparison the available Energy E produced by the 2nd harmonic is only = to 25% of the fundamental Energy level and for the 3rd harmonic only 11.1% of the fundamental. (The vibrating string segments for the overtones of Fig 8 have been drawn out of proportion, for clarity). A quick glance at the E levels shown for the 1st, 2nd, 3rd 4th, and 5th...n (number) of harmonics, in the series indicates a dramatic fall rate in Energy levels, respectively 100%, 25%, 11.1%, 6.25%, 4%. Mathematical Fourier sine wave formulation has been used to calculate E for each harmonic, for further harmonics in the series, the 6th=E/36, 7th=E/49 and 8th=E/64. Fortunately only the first few harmonic overtones are needed to build a rich full tone. In fact if we were able to hear the higher overtones, the tone would actually sound somewhat grating and unpleasant. Of importance are the 3rd and 5th harmonics, adding a more musical tone to the predominant fundamental.
OVERTONES - that sound Harmonious or Discordant.
When you play an acoustic guitar that is out of tune a harsh sounding noise is created due to a lack of harmony, this is discord. At first you may re tune the guitar but find that the tone is still in discord, for other reason unbeknown. With displeasure you may conclude it does not have a very good tone and so put the guitar down. Possibly if you are looking to purchase an acoustic guitar you will then consequently try another? As a musician at one time or another you may have taken your own guitar to a luthier, to have the string length (intonation) readjusted, in order to make the guitar play in tune. Applying normal finger pressure to various fretted string positions, you may check out the intonation job carried out with a chromatic guitar tuner. Then you may find the guitar played in tune by say + or - 0 to 3 cents depending on what notes were fretted as indicated by the tuner meter; ( As to be expected from a fretted instrument, this is normal and generally higher readings are common, 100cents = 1 semitone = 1 fret). But you may still have been dissatisfied with the tonal quality of your acoustic guitar whilst playing it, simply because there still seems to be some kind of discord when you play it. You may even notice the louder you play the guitar the worse the discord becomes.
NODAL SHORTENING due to a Fixed, Mass Loaded String
The above discussion brings us back to the tonal quality of the acoustic guitar. Fig 9 may help to explain what is happening. Fig 9 is similar to Fig 8, it shows in particular an enlarged small section of the string support structure, the large corner section coloured in dark blue. The first harmonic wave or fundamental frequency is seen as the two curving black lines connected to the boundary end of
the support structure. Inside the fundamental we can also see the 1st overtone coloured in dark blue. You may notice that the nodal area for the vibrating string is moving away from the string length boundary, for the fundamental and with a further increase in distance (dx) for the 1st overtone in the direction of the dark blue arrowhead. The nodal positions continue to move further away from the boundary for each overtone that forms in turn, within the series of the fundamental frequency. This particular vibrating string action occurs only if in one instant any part of the support structure moves or wavers, by a greater or lesser degree as indicated in Fig 9 by the double arrow "S" curve lines. Consequently, for the fundamental but more so for its series of overtone standing waves as seen in Fig 8, do no longer fit perfectly within the boundaries or importantly within the fundamental. What we end up hearing due to inner nodal migration from an inadequate wavering support structure, is discord.Why is this discord, you may remember from the discussion of sine waves and referring to Fig 4; if the wave length (λ) of the frequency is shortened then the frequency is increased. Each standing wave pattern of each overtone will still look the same as those seen in Fig 8, but because their string length boundary nodes are forced to move away from the boundary, the waves become compressed. With the standing wave lengths now closer together, their frequency changes, i.e. increases. Let's investigate the situation; take A=110Hz, its first overtone = 2 x 110=220Hz, a perfect octave. Now say the wavering motion of the support structure causes a 5% compression of the wave length, so that's 220/100% = 2.2Hz=1%, and 5x2.2Hz=11Hz, summing-up 220+11= 231Hz is therefore no longer a true 1st overtone octave, the sound we hear is out by 5 cents or 11 cycles.
As stated the nodal shift dx also increases for each consecutive overtone within the series, only to create further discord, where the above mild case of 5% wave compressions are increased further. This situation is made worse if you like simply by playing the guitar strings more forcefully and especially so for an industry standard X-braced acoustic guitar! Why, because the X- brace is not put in place to somehow directly or even indirectly support the fixed end of the string length, at the bridge. The X-brace arms' end on the very thin side walls of the guitar body, a flimsy situation to say the least and so, allows the bridge to initially rock back and forth greatly in the direction of the double arrow "S" curve as shown in fig 9. Reference as to why the x-brace is used as a compromise, has been discussed in the overview pages.
The x-brace is used simply to stop the soundboard from buckling up too much, under the weighted mass of ~74kg of the string load tension. X-braced guitars are notorious for allowing bridges to be lifted clean off the face of a soundboard, and the braking of the x-brace arms especially forward of the string load tension is also a common occurrence.
When a string is made to vibrate on the x-braced guitar, the bridge undertakes a rocking motion, as indicated by the direction of the curved arrows' from 0⁰ towards either side of the ±90⁰ shown central of the boundary in Fig 9. Where the above description of nodal shorting wave compression, is greatly enhanced and takes place! This rocking motion is in excess and tends to slacken the string, this allows the Elastic string Energy to quickly dissipate, due to the lack of direct support for this particular system of a Fixed, Mass Loaded String length. Forceful playing of the guitar strings only increases nodal migration and harmonic discord. If a slow softer melody is played, the rocking motion of the bridge will still occur. In this case the string slacking, results in less amplitude and less sustain for the higher harmonics which quickly diminish in sound level, normally refered to as dead notes. Not only do we end up with in harmonic discord (poor tonal quality), but we also lose the ability to have tonal sustain!
The ability to have tonal sustain comes from keeping the string vibrating, this can only occur if the bridge does not have a rocking motion in the direction of the string length, and secondly if the soundboard is able to vibrate, from an initial stress free state. Losing Elastic string Energy into the soundboard, is short lived. While you may have an initial loud sound or more precisely a booming sound for a heavier bass string as an example, the string will quickly settle and the singing tonal voice of the instrument will be lost.
You may be thinking well what other motion does the bridge undertake, it must also be rocking side to side. Yes it does, and it is also vibrating up and down. If we consider this to be so then there are three dimensions' of movement. We could label the direction of string length as x, the direction perpendicular to the soundboard (up and down) as y, and the direction of side to side of the bridge as z. Now there must also be Energy entering the soundboard in all three planes: x, y and z. The energy associated with the x direction is our all important natural string vibration, and needs to be maintained as our all important source. We could use a rigid support in order to produce tonal sustain. While in the planes of y and z , the vibrating motion of the bridge could be allowed since it would not greatly affect the well supported vibrating string length direction of x.