Equal Tension (ET) stringing is an historically informed idea which has been gaining in popularity among the followers of HIP since over 15 years by now. Several sources such as Mersene, De Colco, Mozart, Speer and others talk about equal-tension stringing in terms of equal kilograms exerted on the string longitudinally. Dowland, Mace and several other well known sources however talk about Equal Feel, that is not tension per se but how the strings "feel" when downward pressure of the bow or the fingers is exerted on them.
EF stringing can be established empirically, by trial and error, with reliance on the known historical gauges from the 17th to the 19th centuries and one's feeling. There is but little chance for error.
ET stringing is however based on some sort of calculus: modern HIP players inspired by the idea of ET and lead by Mozart&Co rely on an algebraic formula rather then equal weights.
Tension = ((Hz2 x lstring2 x ⌀2) x 1.3) / 3183
Naturally, their practical results are identical with those historical. The problem is that the sources on ET rather than EF are wrong from the viewpoint of EF! Here is what happens.
ET stringing on instruments with rounded bridges almost inevitably result in unequal pressure. In turn, unequal pressure results in an unequal feel (EF) and this is what the more practical Baroque sources warned about. They say: the strings must feel the same. We shall analyze the downward string-pressure on instruments with rounded bridges in detail below, but first let us make clear what we mean by the word "feel".
When we play violins and the like instruments, we press against the strings - with the fingers and the bow - in a downward fashion. The strings naturally resist to the pressure. If there is not enough resistance, then we have strings that are too slack. When there is too much resistance, then the strings are too tight. Longitudinal tension of strings expressed in Kgs and their resistance to the downward pressure are related, but when you are playing, you are interacting with the strings in the downward fashion, not in the longitudinal.Therefore, the downward resistance of the strings is much more important than their longitudinal tension.
There are several reasons for the confusion between ET and EF, and the modern HIP players are not to be blamed for it because the confusion apparently originates in the 17th century:
1. Terminology: what we mean by tension, feel, pressure and how these are related, the types of strings etc.
2. Baroque artists' addiction to the proportions because the God, in their opinion, created the world according to the musical proportions, and much of European artworks are created in accordance with such proportions. This is why string-gauges are expected to have the relation of 2:3 on instruments tuned in pure fifths, or, as in De Colco-Mozart, the strings should form pure fifths (2:3) if equal weights attached to them. This is ET, and the physical realization of their much cherished vision of the Universe - Harmonia Mundi.
3. The principle of EF was first mentioned in relation to the Lute - an instrument with a flat bridge. Indeed, if there is no breaking angle over the bridge, the resistance of strings to the pressure of the fingers will be theoretically equal - as long as the longitudinal tension is equal*.
It is unclear how could the lute principle be directly applied to the instruments with rounded bridges (viols, violins) but that's what De Colco and Mozart explicitly do: see the picture from De Colco (Serafino Di Colco: "Lettera. prima...", Venice 1690.)
With all the respect to the sources and venerable Leopold Mozart, I feel urged to explain, that ET does not result in EF on violins and instruments with rounded bridges, and even less on instruments with more than four strings! Rounded bridges are always higher in the middle, hence the breaking angles are not the same but more acute on the middle strings. This difference in angles is further aggravated, apparently on purpose, by the use of mismatching baroque tailpieces: they are usually quite flat in comparison to the bridges.
Now, let us leave the "feeling" alone and see what the math tells.
Pressure = Tension x sin(α1 x π/180) + Tension x sin(α2 x π/180).
This simple trigonometric calculation shows that the difference in breaking angles can be tiny, but their influence on the pressure/feel can be dramatic. What we see on historical flat tailpieces is that the difference in angles is very big.
Now we can go back to the sources on string gauges actually in use (see works by Barbieri-Peruffo for the details).
The 18th and 19th century sources on string-gauges indicate heavy e" and g (wound g made from an a'-string instead of e''-string as is the case with the modern strings), and often surprisingly light a' and d'. Why?
Obviously, the choice of seemingly too thin middle strings is directly related with the idea of EF (see the mathematical expression above). Let us make it clear: there is no ET in case of historical gauges mentioned by a large number of players specially from the 19th century. Yet the notion of EF was still very much in vogue. Here we are dealing with a very fine balance between string-gauges and the flatness of historical tailpieces.
Observe the flatness of the tailpiece attributed to Andrea Amati's viola. Given that the bridge is rounder than the tailpiece, it inevitably results in an acuter angle on the middle strings. This in turn allows to reduce the gauges of the middle strings - a' and d' - putting our choice of strings in line with historical gauges.
Proportioning the strings as 2:3 is the worst method because it does not consider that the thinner strings loose up to 6% of it's original gauge after they have been stretched and played-in. To avoid this error one must choose top strings at least 6% thicker than the "calculated" diameters. For example, if your calculated diameter is 0.69, you should opt for 0.73-75. Thicker strings, such as d' and g-strings on violins do not thin down quite as much as the e''-strings do, at least not in a few days or even weeks.
Some HIP players who uses ET-stringing choose the first string much thinner than the gauge commonly used in the 18th-19th century (ca.70+). 17th century sources on gauges are virtually nonexistent, but Mersenne indicated that the violin e''-string has the same size as the Lute's fourth, which is roughly similar to its 18th century counterpart (though it could go up to as thick as ca.80, and I have tested 0.82 in a concert. N.B! An e''-string of 0.82 after two days of stretching at A-465Hz becomes ca0.76 and stays as such for over a month).
One of the main points of this article is clear by now: finding out the correct diameters for ET strings may result a well balanced instrument in terms of longitudinal tension but not in terms of downward pressure, that is not in terms of EF!
Let us examine what happens on your instrument when the tension is equal and when it is unequal.
The above ET results in the following downward pressure on the bridge:
e'' - 1.93Kgs
d' - 2.61Kgs
g - 1,87Kgs
This is very far from anything that one can feel or call an equal resistance! Naturally, it can be be called EF. On my own violin the differences are not as big as usually found on the violins of others due to the minute variations in the setup.
Now let us bring the string gauges in line with the equal downward pressure, that is EF:
Now the downward pressure of each string on the bridge is 1.93 Kgs, while the longitudinal tension is unequal. This is what one can call EF stringing. These above gauges look more like the historical ones, except that the historical e'' is usually above 0.70. Typically I use e'' 70+, and adjust the gauges depending on circumstances.
The effect of the angles have been described first by Higgins, and then, more recently by Mimmo Peruffo of Aquila Corde. )WILLIAM HUGGINS: "On the function of the sound-post and the proportional thickness of the strings on the violin", Royal Society proceeding, XXXV 1883, pp. 241- 8: 247.)
In order to balance the instrument in both the longitudinal tension and the downward pressure, the middle strings must be thinner and the outer strings must be thicker. If player, however, wants to use thicker middle strings, the tailpiece must be rounder, perhaps as round as the modern tailpiece, nearly matching the curvature of the bridge. This is rarely the case with the surviving tailpieces. Additionally, most of the healthy, robustly built or well preserved instruments are not that capricious and accept little deviations from the "optimal setup" in terms of Kgs - be it tension or pressure.
Though I did my best to be thoroughly informed, this is just a quick note, not a scientific article. Its goal is rather practical: to explain how ET works on instruments with rounded bridges and to help the HIP string-community to understand more clearly what the 17th-18th and 19th century authors had in mind when dealing with stringing issues, where they were correct and where they erred. References to the sources on stringing and historical gauges, surviving tailpieces etc are left out intentionally because they are well known to the HIP players.
* Even this is not entirely correct because if the diameter of the strings is not the same, then there can be no "equal feel" if the downward pressure is equal. This is because the pressure of the bow or of the fingers is being spread on a larger contact-area on the thicker strings (Giordano Riccati ‘Delle Corde…’, Bologna 1767). The contact area also naturally depends on whether you are playing forte or piano. Here we leave this consideration out - for the sake of clarity - and because it is of little practical importance to either luthier or musician.
The two lower pictures are downloaded from aquilacorde.com, the above picture is one of my technical drawings.