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From: paisley@exodus.austin.wireline.slb.com (Gary Paisley)
Subject: Guitar Resonances - tap tones

I've had a couple of responses about how to determine the main
resonances associated with the body, top and back plates of an
acoustic guitar. The specific frequencies of these resonances and the
relation of these frequencies to each other have a lot to do with how
"good" a guitar sounds. In a later note, I'll go a bit into why this 
is. For now, I'm putting some references at the tail of this
article.

This first note goes into how to determine what they are. I have
measured them on some of my guitars and would like to compile and
compare data from a lot of guitars. Hopefully this will stimulate
some of you do do this.

The study of body and plate resonances is not particularly new, as 
it was undoubtably at least intuitively understood by early 
16th century violin makers. These resonances can be determined
by several techniques, ranging from the simple one described here,
to the Chladni (sp) technique , or to using laser interferometry.

This technique requires some means of determining the frequency
of a note induced by tapping the plate with your fingers. 
You have to listen quite carefully, as the tapped note is often
a muffled thump, but with practice you can determine ( by comparing
it to a note on a guitar string, or to a piano ) what it is. 
You can also match it with your voice, and sing it into 
a guitar tuner. You can tap the top with strings on , but I've
found that it leads to irreproducible results, it seems that the
strings add even more resonances...


Quantification of guitar resonant frequency modes.

Main Body Resonance.

	The guitar body comprises a cavity with a fundamental 
resonant frequency determined by the cavity volume and sound hole
diameter. This frequency can be determined by singing a sweeping note
into the sound hole and noting the frequency at which a resonance is
heard ( or felt). 


Top Plate Resonances.
	
	A guitar top plate has many resonant frequency modes, of which
I'll describe the first four in increasing frequency. These are for a 
guitar with a top of solid wood. ( For guitars with tops of laminated or 
composite materials modes 2 and 3 will likely be reversed. This is because
solid top guitars are stiffer ( anisotropic ) in the grain direction
as compared to the cross grain direction, and laminated tops are likely
to have equal stiffness in all directions ). Of course bracing patterns
will tend to move things around a bit as well. 

A note on the ASCII art. The little plus (+) and minus (-) signs
are to indicate the region of maximum movement. "O" is the sound hole.


 Mode 1 : The maximum excursion of the top is centered on the bridge - 
	(the top movement is as a whole, as if the bridge moved in
	and out as a unit). The top is essentially hinged at the bindings,
	This resonance can be determined by tapping the 
        center of the bridge and listening for the resonant note.
     ______   ___
    /      \_/   \             
   |              \___________%%%
   |  ([+]) O      ___________   ]
   |              /           %%%
    \______/ \___/


  Mode 2 : There are two points of maximum excursion, at each end of the
           bridge - (The top rocks with the center of the bridge as
	   a fulcrum point, each end of the bridge alternately moving
	   up and down) . This resonance can be determined by tapping the 
	   guitar top at a point about an inch or so from the end of the 
	   bridge while holding a finger tip at the center of the bridge.
     ______   ___
    /      \_/   \             
   |    (+)       \___________%%%
   |     I  O      ___________   ]
   |    (-)       /           %%%
    \______/ \___/


   Mode 3 : Again there are two points of maximum excursion, now on
	either side of the bridge, the entire bridge is a fulcrum .
	This frequency can be determined by tapping the guitar body at
	a point  about 3 inches from the bridge towards the tail block
	while putting several fingers on the bridge to hold it
	immobile.


     ______   ___
    /      \_/   \             
   |              \___________%%%
   | (+)I(-)O      ___________   ]
   |              /           %%%
    \______/ \___/

   
   Mode 4 : There are 3 points of maximum excursion, the bridge as a 
	whole, and the points off either end of the bridge...
	This frequency can be found by tapping about midway between the
	end of the bridge and the side of the body, and holding
	a finger at about the end of the bridge itself to suppress
	mode 2.


     ______   ___
    /      \_/   \             
   |   (+)        \___________%%%
   |   [-]  O      ___________   ]
   |   (+)        /           %%%
    \______/ \___/



Back Plate Resonances

	The back plate resonances are similar to the top
plate resonances, expect that they are much weaker. Consequently
the mode 1 resonance is the only one we are concerned with.


	Mode 1 back plate. This resonance has maximal excursion
	approximately under the bridge. This resonance can be found
	by holding a finger on the back plate directly under the sound 
	hole, and tapping the back plate under the bridge. 
     ______   ___
    /      \_/   \             
   |              \___________&&&
   |   [+]    (-)  ___________   ]
   |              /           &&&
    \______/ \___/


References:

1) First of all, the bulk of this posting is derived from chapter
9. pp 133-135 of "Fundamentals of Musical Acoustics" by Arthur H. Benade. I
have the Dover edition published in 1990. This is a good
book, flawed in its ommision of formulae.

2)" A Study of Acoustical and Hologram Interferometric Measurements
of the Top Plate Vibrations of a Guitar"
Acustica 25 (1971): 95 -100


3) I have another larger list of references, which I cannot find
just now. They are articles which have appeared over the last 
50 years or so in the Journal of the Acoustical Society of America,
predominantly by members of the Catgut Acoustical Society with respect
to their research on the acoustics of violins. Scientific American
has also run a few of these articles over the years. Within the
last two years, Physics Today had an exceptionally good article on
the vibration nodes of drums.

                             General Reference


                     Notes and corresponding frequencies
(thanks here to Frank Bounds bounds@gulaam.glv.com )

    C2   65.406         C3   130.81         C4   261.63         C5   523.25
    C#2  69.296         C#3  138.59         C#4  227.18         C#5  554.37
    D2   73.416         D3   146.83         D4   293.66         D5   587.33
    D#2  77.782         D#3  155.56         D#4  311.13         D#5  622.25
    E2   82.407         E3   164.81         E4   329.63         E5   659.26
    F2   87.307         F3   174.61         F4   349.23         F5   698.46
    F#2  92.499         F#3  185.00         F#4  369.99         F#5  739.99
    G2   97.999         G3   196.00         G4   392.00         G5   783.99
    G#2  103.83         G#3  207.65         G#4  415.30         G#5  830.61
    A2   110.00         A3   220.00         A4   440.00         A5   880.00
    A#2  116.54         A#3  233.08         A#4  466.16         A#5  932.33
    B2   123.47         B3   246.94         B4   493.88         B5   987.77

 Gary Paisley
--
Gary Paisley  Senior Engineer   512-331-3271 
Austin Systems Center / Schlumberger Well Services / Austin, Texas 78734



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