Making or modifying a violin for good tone.
I will now show how to get an estimate of the key body frequencies of a finished violin from the weights and tap tone frequencies of the plates that go to into making it.
First a warning: a note on humidity
It is worth noting that tap tones change all the time (!) with the moisture content (MC) of the wood. So in a heated workshop in winter wood will have an MC of ~6% and the figures given on this website will apply. However in summer the wood MC will rise to ~ 12%, and the Mode 5 tap tone frequencies I give will need to be reduced by as much as 15 Hz! The plate weights also increase slightly with higher MC too.
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 some available free on the web. They kind of work, but most plates need to be thicker or thinner than the map, as every bit of wood has different elasticity so I recommend ‘tuning the plates’ too! Dominic Excell’s articles on a First Violin does not use plate tuning (but I’m talking to him!).
- 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 before f-holes are cut. This ‘F’ later becomes F# and even G with f-holes, bassbar fitted and varnished . The tap tone must be a full, true ring, the 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 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 be equal.
- Dr. Nigel Harris’s came up with a method Tune Modes 5 and Mode 2 and take the violin plate’s weight into account by making the ‘plate stiffness’ proportional to the plate’s weight. Dr. Harris may achieve tonal repeatability using his ‘stiffness’ method but my data shows that heavy plates need to be made with significantly higher not lower tap tones!
- I did suggest using his ‘stiffness figure’ method for years on this site. I apologise profusely to anyone who used it, it is simply wrong and unnecessary.
- An excellent book by Patrick Kreit “The Sound of Stradivari” came to my notice in 2011. Dr.Harris’ work on plate Stiffnesses is not compatible with Patrick Kreit’s work. I have found that light plates are much better, and so far as I know Patrick’s book explains the only method to produce the lightest plate from any given piece of wood.
A ‘model’ that uses weights and tap tones to predict key body resonances1.
Otto Moeckel published a book “Die Kunst des Geigenbaus” in 1930 which had data on the tap tones of the plates of 30 good Italian violins including 7 Stradivarius violins .... [with thanks to Anders Buen and Maestronet]. It is shown on the right.
The 30 violins have belly plates with a Mode 5 averaging about 355 Hz +/- 45 Hz, and back plates with an average of about 366 +/- 59 Hz.
You can see it is difficult to discern a pattern from this data, but it does set a target or test for any ‘model’ that attempts to predict the key body resonances of violin starting from the belly and back plate tap tones.
By concentrating on the very important B1- , B1+ and A1 violin body resonances I have derived a model to predict the B1- and B1+ violin body resonances and hence help you make what is likely to be a fine-sounding violin.
With regard to accuracy of the model, the prediction for the frequency of B1+ has a Standard Deviation (sigma) of 10 Hz.
So there is a 68% likelihood that the actual ‘B1+’ frequency will fall within +/- sigma (+/- 10 Hz) of the predicted frequency.
Otto Moeckel’s data on ‘old Italian violins’ can be fitted to the model with reasonable assumptions that are shown on the diagram. On the diagram there are 13 violins outside outside the model's coloured ‘viable’ zone. These have very high back Mode 5 frequencies, but their Mode 2 frequencies are not recorded.
It turns out from the data that it is the back plate's Mode 2 frequency that is most important. The back plate's Mode 2 is kept high by leaving the area between the C-bouts particularly thick. The table below shows what effects the back plate's Mode 2 and the belly's weight have on the B1- and B1+ body mode frequencies.
The model to predict B1- and B1+ from the plate Mode 5 frequencies is derived by extending the work suggested in a paper by Alonso Moral (1984) outlining these relationships. My data looks for (assumed linear) relationships between B1-, B1+ and 7 aspects of the violin plates: Modes 2 and 5 in belly and back, their weights and the bass-bar height.
My results however show different figures from those of Moral, but then he did not measure Mode 2. I have found that the back's Mode 2 frequency dominates the resulting key body resonances.
By taking 25 violin examples where the plate Mode 2’s and Mode 5’s and the final violin B1- and B1+ mode frequencies are known I used a method of seeking multiplier values that give the lowest standard deviations for B1- and B1+ by successive iteration. This is a technique used elsewhere in Finance to minimise risk in investment portfolios.
The details of just how each factor changes the B1-, B1+ frequencies is given in then table above.
Estimating B1-. Accuracy: sigma ~12 Hz.
1) The data shows that the most important factor in what frequency B1- will be is the back plate's Mode 2 frequency. In any violin this is likely to affect B1- by about 10 Hz.
2) The next most important things are:-
a) The back plate Mode 5 frequency.
b) the back plate weight and
c) the belly plate weight.
These (a), (b) and (c) may affect the B1- frequency by about 5 Hz each. A heavy back and belly both reduce B1-.
3) The least important things are the belly plate's Mode 2 and Mode 5 (which is surprising!), and the bass-bar height, although that does affect the belly Mode 5.
>> Ideally ‘B1-’ needs to be matched or close to the A1, the lengthwise internal air resonance. It should ideally be within 0 to 16 Hz of B1-, and A1 is typically 462 Hz and in the range 450 to 485 Hz.
Estimating B1+. Accuracy: sigma ~10 Hz.
>> Ideally ‘B1+’ is 60 - 90 Hz above A1.
1) The data shows the most important factor for the B1+ frequency is
a) again the back plate's Mode 2 frequency. A high Mode 2 frequency is likely to affect B1+ by up to 40 Hz, and
b) the belly plate weight. A heavy plate reduces B1+ by as much as 13 Hz.
2) The next most important things are:-
a) the back plate Mode 5 frequency,
b) the belly plate Mode 5 frequency, and
c) the back plate weight.
These (a), (b) and (c) may affect the B1+ frequency by about 5 Hz each. A heavy back plate reduces B1+.
3) The least important things are the belly plate Mode 2 and Mode 5, and also the bass-bar height, although that does affect the belly Mode 5.
Estimating the frequency difference between B1+ and B1-. Accuracy: sigma ~13 Hz.
The data shows that the most important factor in what the difference will be is
1) a) the back plate’s Mode 2 frequency. This may affect it by ~12 Hz.
b) the belly plate weight. A heavy plate reduces it by ~10 Hz.
2) The next most important things are:-
a) The back plate weight, and a heavy back plate reduces the difference.
b) the belly plate Mode 2 frequency, and
c) the belly Mode 5 frequency.
These (a), (b) and (c) may affect this frequency difference by only about 2 Hz each.
It looks as if Sr. Stradivarius set the key body resonances after completing the violin, as he only finished the edge-work and inserted purfling after gluing on the belly!
The key violin body resonances are very sensitive to re-thicknessing the plates on the outside of the plates rather than the inside: so beware!
A few scrapes on the outside of one or both plates can drop the body resonances quickly, and they can't be raised again!
The ‘ideal’ body resonances.
There is data available on the key body resonances of important Italian and other violins in The Strad and other magazines. The data I found is that is most accessible is an article called “Measurements of 10 Preference-Rated violins”, a paper presented at ACOUSTICS 2012 written by by C. Saitis, C. Fritz, B.L. Giordano, & G.P. Scavone: it can be found here.
If you look at the page Resonances of the Violin Body I have extracted and enlarged the data on the best 3 violins of the 10 tested for sound quality to show the kind of body resonances you would be looking for.
A Charles Davis was in touch to tell me about Joseph Curtin’s Strad magazine article on Stradivarius front plates: found at the Strad magazine website. Jo Curtin has some useful tap tone frequencies there.
- Dr. Harris’s paper can be found here.
- 1 This data also includes the data published in Patrick Kreit’s book and website that gives the ‘deltas’ between these (A0), A1, B1-, B1+ body resonances we should look for in a first rate violin. You may need to get Patrick Kreit’s book to get details.
- ** 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”, on 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 was an oddity that set me on the road to tap tones .... a belly with a Mode 5 at only 316 Hz (final, with ff’s, varnish) make such a good sounding fiddle ? Odd.