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 Last updated:                28th. March. 2015  Copyright (C)     platetuning.org

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.
• There are many plate thickness maps available, and there are many available free on the web. They kind of work, but most platee need to be thicker or thinner than the map, as every bit of wood has different elasticity. Tune the pates too!
• 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**.
• use the ‘CAS' method: Tune Mode 5 and Mode 2 to be an octave apart (Mode 5 = 2 x Mode 2). This is Carleen Maley Hutchins' (or the CAS***) method. It works well if you are choosing the best wood.  Modes 2 & 5 frequencies of front and back should match.

Update: this is a major revision March/April 2012.

    After reading Patrick Kreit's book “The Sound of Stradivari” in February 2011 it seems to me now that all of this work on plate Stiffnesses  is interesting, but really all we need to do is make the Mode 5 frequencies of the plates right and we can get the tone we want from a violin without always focusing on the Stiffness Figures for the plates. What to set the Mode 5 tap tones of the plates for the various choices of tone is explained on the page ‘Platetuning 4 dummies'.

   The basics are: make or alter the violin plates so that the Mode 5 frequencies fall within the area in yellow, light green or dark green in the diagram shown right.

  Click on it, and then right click that screen to save it as a .jpg to print.

    If the plates are of good light, stiff wood then the belly Mode 5 should be either equal to, or up to a semitone below the back plate's Mode 5 in the range 315 Hz to 370 Hz, just as is traditional! But if either plate turns out heavy it will need a rather lower Mode 5 frequency than that shown.

That is when the plate's Moisture Content (MC) is at 6%, for a heated workshop in winter, or 15 Hz less if done in the summer months with the wood at 12% MC.

       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 a quarter of the ‘extra' weight of plate needs to be taken into account. So this is in 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 Curtin 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 of 4,000,000 is fine if you don't intend to do any ‘final' thinning later. 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 the “Stiffness Factor” here too, as it is much easier to work with, and is described next.

To simplify things a little and to also take account of an inaccuracy in Dr. Harris' Stiffness Figure (an error due to the effects of the stiffness and weight of the edges of the plates), 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 : Belly plate

         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 or rathyer larger than the 4,000,000 of  a ‘raw' belly plate. So this belly has a Stiffness Factor of 1.06, and will give a bright ‘Orchestral' tone.  There is a Wiki page giving the relationship between frequency (Hz) and modern pitch here.

       It is worth noting that roughly speaking, a belly plate ring tone or Mode 5 will be increased by about 0.7 of a semitone, or be raised by 14 to 15 Hz when the ff holes are cut, the bassbar added (which itself increases Mode 5 by 2 to 3 semitones), and the plate varnished. This is equivalent to a Stiffness Figure increase of ~12.5%.

Back Plate

Plate Reference Weights: violin, viola & 4/4 ‘cello    Table 2     updated 20 May ‘10 Back Length mm Back Length ins. Ref. Front or Belly wt. (gms.) Ref. Back wt (gms.) 300 mm (1/2 size) 11.8” 41.7 73.1 328 mm (3/4 size) 12.9” 53.1 91.1 357 mm (4/4 size) 14” 64.7 109.3 387 mm viola 15 1/4” 79.4 131.7 394 mm viola 15 1/2” 83.1 137.3 407 mm viola 16” 89.6 147.2 419 mm viola 16 1/2” 97.1 158.3 4/4 ‘Cello ~30” (760 mm) 438.2 627.8

          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 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.

Table 2 right shows the Reference weights of violin (first 3 rows), and then the larger Viola plates for various sizes. 

  Note: These weights are a little high for raw plates, as they actually include about 2 gms of varnish!]

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 * This formula or relationship 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 12.5% or ~0.7 of a semitone  above raw plates) for a finished belly or violin front plate. Chart 2 (see below) 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 varnished violin fronts should have a ‘Stiffness Factor' some 12.5% or about 0.7 of a semitone higher than the raw plate, and this is shown in Chart 2 below.

 Cutting the ff holes in a raw belly plate lowers Modes 2 & 5 frequencies. Then adding the bassbar  and varnish raises them again. If there is no “final tuning”, i.e. thinning the belly a little from the ‘raw plate' tuning, then typically the  plate stiffness would be 10% to 15% higher than the ‘raw plate, tying up with a Stiffness Factor of a raw belly plate (no ff's, bassbar or varnish) of 4,000,000. 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. 109.3 grams for a back, and  e.g. 64.7 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 Hutchins' 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 Chart 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 chart) for both front and back plates makes a good violin with Student, Chamber to Orchestral tone.

Plates for ‘Cellos

  It has been possible to use data that Carleen Hutchins, Dr. Nigel Harris and particularly John Osnes Violins (Anchorage, Alaska) have provided to estimate the Stiffness Figures required for ‘Cello plates too. Carleen strongly recommends that the plates have matching Mode 2's, and they are ‘octave' plates (Mode 5 = 2 x Mode 2) in the range: Mode 2 = 60 to 65 Hz, and Mode 5  120 to 130 Hz.

Not surprisingly the ‘cello Stiffness Figures are very similar to those for violin and viola plates. For normal ‘cello ‘Orchestral Tone' the belly unvarnished but with ff's and bassbar should  have a Stiffness Figure of 4.03E6 (4,030,000 ) and the back unvarnished 6.0E6. Varnished this rises to 4.44E6 for the ‘cello belly (note it is almost identical to the violin figure) and 6.16E6 for the ‘cello back, which is 19% less than for violins.

 You should change the Stiffness Figures say 5% lower to 15% lower for a ‘student' tone (3.78E6 and 5.24E6) for easy bowing, or increase it by up to 15% for a ‘Solo' tone (5.1E6 and 7.08E6). Note that plates still need to have matched Stiffness Figures, as for violins! This is summarised in this table: click on it ....

  In the CAS Journal III there is a particularly interesting example of a ‘factory' Mittenwald ‘cello that has been rethicknessed. Thomas King did the work under Carleen's supervision, and the article is here (I strongly recommend you get the full set of CAS articles). The ‘cello has a low but matching Mode 2 in front and back plates (57 Hz): the back has a standard ‘orchestral' Stiffness Figure, but the belly is 20% low, assuming standard plate weights. The tone apparently is outstanding and is easy to bow.

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 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 chart I've given above if you don't like  spreadsheets. Here is an example spreadsheet 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 tap tones change with the moisture content (MC) of the  wood. So in a heated workshop in winter wood will have an MC of ~6% and the figure given on this website will apply. However in summer the wood MC will rise to ~ 12%, and the Mode 5 tap tone frequencies given on this web site will need to be  reduced by about 15 Hz! The plate weights also increase with higher MC too.

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