This site has something for you if you are either making a violin or you want to improve a low cost violin or viola.

By tuning the top & back plates you can get a good instrument that responds well to the bow and that can sound like a £1500 instrument.

Making or modifying a violin for good tone

 In making or modifying a violin or viola for good or better tone you can choose one of a number of methods. These are, in historical sequence:-

• Use craftsmanship! Get help, get trained, get experience, and make your first dozen violins. There is no substitute for knowledge, craftsmanship and for experience, not least in choosing your wood.
• If you do choose to use wood tap-tone tuning methods, then adjust the traditional Tune Mode 5 (Ring tone) with the back Mode 5 sounding F#, and the front at F (as a ‘raw’ plate). This ‘F’ later becomes F# to G with ff’s bassbar fitted and varnished . The tap tone must be a full, true ring, the very best you can get. Mode 2 frequency ‘floats’, and the plate’s weight is ignored. Typically, allied with good practice, very good wood and experience it works very well**.

  This gives no clues about any cases that require the back to be tuned above or below the front’s final Mode 5 ring tone**. Harry Wake describes tuning the plates in his book, but tap tuning with mic and computer is much quicker, as described on the page about plate tuning!

• use the ‘CAS’ method: Tune Mode 5 and Mode 2 to be an octave apart (Mode 5 = 2 x Mode 2), and ignore plate weight. This is Carleen Maley Hutchinson’s (or the CAS***) method. It works well if you are choosing the best wood with low or standard density and therefore plate weight. Modes 2 & 5 frequencies of front and back should match exactly.              Or you can ..........

• use Dr. Harris’s method: Tune Modes 5 and Mode 2 and take the plate’s weight into account by making the ‘plate stiffness’ proportional to the plate’s weight. It works well with good nearly standard density spruce or maple. Unfortunately Dr. Harris’ method or theory it does not take into account the effects of the plates’ edges being attached to the stiff and relatively massive bouts.                                      Or you can .......

• use a modified version of Dr. Harris’s method. Here you Tune Modes 5 and Mode 2 but only drop the tap tone frequencies a little if any plate has turned out rather heavy. I, and number of others who have contacted me via this web site have found that Dr. Harris’ method gives plates that turn out too thin. Only about half the ‘extra’ weight of plate needs to be taken into account. So this is half way between the two methods shown above - Carleen’s CAS method, and Dr. Harris’s. It takes account of the plate edges, and plate weight and grain stiffness as significant factors in determining the resonant frequencies of front and back plates when in an actual violin.

  Looking at the weighs of plates given below you may see just how much plates vary in weight: e.g. a belly can be about 54 grams right the way to 101 grams - nearly a 100% range, so, as Joseph Curtins says, we really do need to take weight into account in choosing tap tone frequencies.

Dr. Nigel Harris’s figures.

 In his paper Dr. Harris gives Stiffness Figures * of

• 7,255,000 for backs, and
• 4,247,000 for fronts, with the back unvarnished and the front ‘raw’, i.e. with no ‘ff’ holes, bassbar or varnish. I personally have found this belly figure to be rather high, and a stiffness as low as 3,800,000 is fine if you don’t intend to to ‘final’ thinning later when the ff’s and bass bar are in place. Dr. Harris’ violins are presumably intended for ‘solo’ use, so I would expect them to be rather thick.

 This figure, Dr. Harris’ “Stiffness Figure” is a simple product. It is the average of the Mode 2 and 5 frequencies (in Hz) squared, multiplied by the plate’s weight (in grams), and is explained too in Joseph Curtin’s Strad article.

 The table below (Table 1) shows the way in which the plate Stiffness Figures above are derived, and shows that these figures are all directly derived from Carleen Hutchin’s work with CAS, combined with Dr. Harris’ Stiffness figures: the spreadsheet itself can be seen here. They are given below for each ‘tone’: you can choose which - Student, Orchestral or Soloist. I have introduced “Stiffness Factors here too, as they are easier to work with, and are described next.

Plate Stiffness Reference.xls			15 Feb '10		Table 1
			Stiffness Figures are for fully finished plates with varnish
	Belly ref. Weight (gms.)	64.7	Belly Stiffness Figure	4,500,000
	Back ref. weight (gms.)	109.3	Back Stiffness Figure	7,580,000
	Weight (gms.)	Mode 2 (Hz)	Mode 5 (Hz)	Stiffness Factor	Carleen Hutchin's tone definition
Belly	64.7	165	330	0.88	Soft bowing/student tone
Back	109.3	165	330	0.88
	Derived from Jo Curtin's Strad violin data		330 Hz is E natural	relative to Stiffness Figures above
	Weight (gms.)	Mode 2 (Hz)	Mode 5 (Hz)	Stiffness Factor
Belly	65.62	170	340	0.94	Chamber/ orchestra/ teachers
Back	111.4	170	340	0.95
			350 Hz is F natural	relative to Stiffness Figures above
	Weight (gms.)	Mode 2 (Hz)	Mode 5 (Hz)	Stiffness Factor
Belly	67.7	180	360	1.07	Soloist/exceptional bowing
Back	116.05	180	360	1.08
				relative to Stiffness Figures above
	Weight (gms.)	Mode 2 (Hz)	Mode 5 (Hz)	Stiffness Factor
Belly	68.7	185	370	1.14	Soloist/ very exceptional bowing
Back	118.3	185	370	1.16
Note that belly & back are heavier for higher tap tone frequencies			370 Hz is F#.	relative to Stiffness Figures above

   To simplify things a little and to also take account of the inaccuracy in Dr. Harris’ Stiffness Figure, I define the “Stiffness Factor” as the ratio of f² x corrected plate weight, relative to the Stiffness Figure. This is shown in the formula below, and shows the specific relationship that needs to be in place between the tap tones and the plate weights for a good violin, once the kind of tone you want has been chosen:-

   It may be interesting to note that a ‘good’ violin, with ‘viable back and front’ needs its plates to have a Stiffness Figure within the +10% / -15% of the Reference Figures given here. In terms of plate weight, because removing wood changes weight, Mode 2 and Mode 5 all downwards, then the plate’s weight needs to be within the small band of -5%/ +4% of its ‘proper’ weight. In fact the plates’ weights need to match within ~ 2 or 3% for a good instrument. This explains why factory fiddles have plates that are so wrong (and mostly too heavy) most of the time: it just doesn’t happen by accident.

Time for Examples

     So for instance, a good belly or front plate of spruce might have a Mode 2 of 172 Hz, a Mode 5 of 327.8 Hz, and a weight of 71.3 grams in the raw with no ff’s, bassbar or varnish. So the average frequency is (172 + 327.8) / 2 = 249.9 Hz. Squared (multiplied by itself) gives 62,450, multiplied by the corrected plate weight as follows: 71.3 actual wt. + 64.7 ref. weight [from table 2 below] / 2, = 68 grams, giving a Stiffness Figure of 4,247,000. So this belly has a Stiffness Factor of 1.0, and will give an ‘Orchestral’ tone, though some final thinning is assumed on the finished belly.   There is a Wiki page giving the relationship between frequency (Hz) and modern pitch here.

     Similarly, a back, unvarnished, with Mode 2 of 171 Hz, a Mode 5 of 345 Hz and weighing 109.3 grams would have stiffness factor of (171 + 345)/2 = 258 Hz, squared = 66,564, x 109.3 grams = 7,255,000. This too has a Stiffness Factor of 1.0, for an Orchestral tone. This example has the plate exactly equal to the reference weight (Table 1), so there is no weight correction.

   If you’re not fond of maths, then here is a chart (left) I have created for the ‘Stiffness Factor’ to be read off for both front and back plates before the ff’s are cut, the bassbar added, or plates varnished. It uses a [very] high (Harris’s) Stiffness Figure of 4,247,000, and makes sure the belly wont be too thin.

This simple chart allows the Stiffness Factor to be read off directly from

1. the average tap tone, i.e. the average of Modes 2 & 5, typically = 262.5 Hz, and
2. the plate’s final weight: 61.6 grams for a belly before the ff’s are cut and bass bar added, and 105 grams for a back.

Plate Reference Weights    Table 2 Back Length mm. Back Length ins. Ref. Front or Belly wt. (gms.) Ref. Back wt. (gms.) 357 mm 14” 64.7 109.3 394 mm 15 1/2” 77.7 131.2 407 mm 16” 90 145 419 mm 16 1/2” 105.2 162.9

Table 2 right shows the Reference weights of violin (first line), and the larger Viola plates for various sizes. It takes into account the increased width of a viola compared with a violin, especially in the 419 mm (16 1/2”) size. (These weights are a little high for raw plates, as they actually include about 2 gms of varnish!)

Chart 1 above has log graduations on the scales^#. To save and print it , just right click it, select ‘Save Link As...’ and save to ‘My Pictures’. Then open it in any picture/photo program and print it out as an A4 page from there.

To use it, just use a ruler’s edge to connect the average frequency of Modes 2 & 5 at the bottom, and the plate’s weight at the top. Read off the Stiffness Factor on the middle line scale. A Stiffness Factor of 1.0 on the middle line is equivalent or is scaled to 4.25 x 10E6 (4,250,000) for a belly, and 7.25 x 10E6 (7,250,000) for a back. 

* Just where this formula is derived can be found in Dr. Harris’s paper here.

**Interestingly, F.J.Fetis’ book “A Notice of Anthony Stradivari”, on page 81, is the only one to say that the back, (actually Villaume’s real Strads and Guarnarii violins of 1850, obviously with ff’s bassbar & varnish) should be tuned a tone lower than the belly.  Heron-Allen in his “Violin Making, as it was and is”, page 132 says the back should be a tone higher than the belly. Other sources say they should match .

*** CAS = Catgut Acoustical Society, now part of the Violin Society of America, the VSA.

****  I had a good violin with a finished front of 101 grams! This was the third fiddle I ever worked on and though heavy, it produced an excellent tone. It’s an oddity that set me on the road to tap tones.... why should such a belly with a Mode 5 at only 316 Hz (final, with ff’s, varnish) make such a good sounding fiddle ? A standard front is nearer to 65 - 70 grams with ff’s, bassbar and varnished, and even as low as 57 grams.

 David Langsather gives low plate weights of 54 gm front with bass bar, and an incredible 86 grams back. Stradivarius’ bellies, with bass bar & varnish, are also typically low at 58 - 70 gm. Dr. Harris uses heavier plates, typically 65 gm front and 109 gm. back before ff’s and varnish.

  A Charles Davis has been in touch to tell me (Jun08) about Joseph Curtin’s Strad magazine article on Stradivarius front plates: found at the Strad magazine website. Based on these (obviously) finished Italian violin bellies I suggest the front’s target ‘stiffness figure’ is 4.5x10E6 (i.e. just 5% or nearly 1 semitone above raw plates of Chart 1) for a finished belly or violin front plate. Chart 2 (see right) incorporates these figures.

Matching the plates: Matching Stiffness Factors!

It’s important that the “Stiffness Factors” of the front and back plates match for a good instrument. The general rule seems to be that the front ‘s stiffness factor should be equal to or not more than 15% less that of the back.

Finished plates: the front or belly with ff holes bassbar & varnish, and the back varnished.

Based on the good old Italian fronts of Joseph Curtin’s Strad article then final violin fronts should have a ‘Stiffness Factor’ some 5% or nearly 1 semitone higher than the raw plate of Chart 1, and this is shown in Chart 2 below.

 Cutting the ff holes lowers Modes 2 & 5 frequencies. Then adding the bassbar and varnish raises them again. If there were no “final tuning”, i.e. thinning the belly a little from the ‘raw plate’ tuning, then typically the plate stiffness would be 15% - 20% higher than the ‘raw plate, implying that the Stiffness Factor of a raw belly plate (no ff’s, bassbar or varnish) can be as low as 3,800,000.

Final tuning will reduce either or both Modes 2 & 5. Varnish will then increase Modes 2 & 5 by about 5 Hz each, and also increase the weight slightly. Varnish will harden too over the first year an more, raising Modes 2 & 5 slightly with time. The back is only slightly affected by varnish.

As before for the ‘raw’ plates before the ff’s are cut and bassbar put in, a similar relationship applies:-

The average of the Modes 2 and Mode 5 tap tones (in Hz) still needs to be modified somewhat to take account of the plate’s weight, and what sort of ‘tone’ you want from the instrument.

So for an ’orchestral’ tone (see below) the final plates need to have a stiffness factor of :-

• about 4,500,000 for the front, with its ff’s, bassbar & varnish, calculated, or a Stiffness Factor of 1.0, and
• 7,580,000 for a good maple back plate that is varnished, calculated as above.: a Stiffness Factor of 1.0 again.

  To avoid the maths, use the chart (left) that allows the ‘Stiffness Factor’ for both a front (belly) and back plates in final varnished state to be derived directly from:- 

1. the average tap tone, i.e. the average of Modes 2 & 5, typically 260 Hz,  and
2. the plate’s final weight: e.g. 112.1 grams for a back, and e.g. 66.6 grams for a belly with bassbar, ff’s and varnish.

I use a pocket calculator to derive the average plate freq. from the Modes 2 and 5 figures, and the average (corrected) plate weight using the plate weight data of Table 1 left.  The two Mode frequencies can be read off the one tap tone recording and its FFT, made with mic & computer as shown here.

The chart above has log graduations on the scales^#. To save and print it , just right click it, select ‘Save Link As...’ and save to ‘My Pictures’. Then open it in any picture/photo program and print it out from there.

As before, this chart above has log graduations on the scales^#. To use it, just use a ruler’s edge to connect the average freq. (Mode 2 + Mode 5)/2 at the bottom scale, and the plate’s weight at the top, to read off the stiffness factor relative to 1.0 off the middle scale.

The figure 1.0 on the middle line is equivalent to or is scaled to 4.5 x 10E6 (4,500,000) for a belly in finished form, and 7.58 x 10E6 (7,580,000) for a back.

Different ‘Stiffness Factors” for different tones.

 To summarise Carleen Maley Hutchinson’s figures for higher or lower tap tones for a range of violin tones, she found that both front and back plates with Modes 2 & 5 of 170 & 340 Hz gave a ‘Student or Chamber instrument tone’ that is easy to bow, but doesn’t carry all that well. She also says how to make ‘Solo’ instruments with tap tones of 190 & 380 Hz shown in Table 1 left. Other tap tone frequencies in between gave ‘amateur, ‘Orchestral or teacher’ violin tone.

 So start with the ‘Stiffness factor’ of the back. This should be at about 1.1 for a ‘Solo’, 1.0 for an ‘Orchestral’ and about 0.9 for a Student’ tone. The relative ‘Stiffness factor’ is read off Charts 1 or 2 above or found with a spreadsheet.

 I’ve found it better that the stiffness factor of the front should match the back or be below it: this a similar to what Carleen (CAS) recommends. Never take too much wood off the back plate - it’s better to leave it with a Stiffness Factor not less than 0.95, even for a student instrument.!

You can choose to take the front’s ‘stiffness factor’ anything up to 10% below that of the back for a good tone, but I would not generally recommend more then 5% - 8% below the back. This uses the data from the Strad fronts Joseph Curtin wrote about in his Strad Magazine article. I have modified some instruments with final belly Stiffness Factors of as low as 0.85 (3,800,000), (and indeed many Strads etc. have bellies with a stiffness factor as low as this), and the modified instruments are an absolute delight to play under the bow, but need top quality spruce to have good solo or carrying power as well.

 I have found that in a dozen violins so far a Stiffness Factor between 90% and 100% of the figures above (0.9 to 1.0 on the charts) for both front and back plates makes a good violin with Student, Chamber to Orchestral tone.

Plotting the Stiffness Factor for a plate as you go along.

 The method I use is as follows: I keep a constant watch on the calculated 'Stiffness Factors' of front and back all the time as I evenly thin the plates from the inside. Back & front have to match subject to the rules above, and don’t take the back too thin!

 The wood you have may limit your choice of ‘tone’ perhaps, but you can put in a rather high new bassbar to raise the stiffness factor of the front, and graft (i.e. glue) in a round or oval maple patch (~45 x ~50 mm across) into the middle of the back to raise its stiffness factor.

 It took me a week or two to get used to the method, but now I don't really have to think about it: it's just habit.

I have do calculations at every step (!), but they are simple: find the average of the plate’s Mode 2 and 5 frequencies, and watch the weight of the plate too.

Spreadsheets.

Record the details and the stiffness factor at each step for each plate. I mostly use a simple spreadsheet to do all the calculation on the same PC I'm measuring the Mode frequencies, but you might prefer the charts I’ve given above if you don’t like spreadsheets. Here is an example I created for a JTL ‘Steiner’ violin with finished, varnished plates to show how easy it is. The screen looks like this:

Ssheet Steiner violin SFs V1.0.xls 8 Jan '10 Table 3     Stiffness Figures are for fully finished plates with varnish Belly Reference Weight (gms.) 64.7 Back Stiffness Figure for Reference 4,500,000 Belly Reference Weight (gms.) 109.3 Back Stiffness Figure for Reference 7,580,000             Weight  (gms.) Mode 2 (Hz) Mode 5 (Hz) Stiffness Factor Belly 78 162.5 351 1.05 Back 100 164.5 374 1.00   Italics = estimated figures relative to Stiffness Figures above

   Only the cells shaded yellow need to have your measured figures entered, and the Stiffness Factor can then be read out from cells E7 and E8.   It is easy to create a similar spreadsheet that will do all the calculation for you. Use OpenOffice for a good free spreadsheet.

A note on humidity

  It is worth noting that the Stiffness Factor of plates and the final violin changes with humidity. What I have noticed however is that as the weight of a plate increases with water content (high humidity) then the tap tones decrease to compensate, so the effects of changing humidity are significantly reduced if you derive the Stiffness Factor for a plate and use it rather than just the tap tones, Modes 2 & 5.

^# I created it using using the ‘SmallBasic’ language on a PC.