Lutherie Myth/Science:
Human Perception of String Tension and Compliance in Stringed Musical Instruments

Tension and Compliance

As mentioned in the introduction above, string tension is completely defined by three factors: the pitch the string is tuned to, its vibrating length, and its mass (weight) per unit of length. Those interested in the formula by which these factors are related to pitch should take a look at page on this website about String Tension. Conventional wisdom has it that a number of other factors affect string tension. Some of these are (in no particular order):

The length of the string between the string anchor and the saddle of the bridge;

The length of the string between the tuning machine post or peg and the nut;

The breakover angle of the string at the bridge and/or the nut;

Once again, looking at the formula it is clear that none of these have any effect whatsoever on string tension. But it is both possible and in some cases likely that some or all of these can affect the longitudinal stiffness of the string and this can affect just how taut the string feels to the player. It should be noted that the term tension has a specific meaning in physics and that it is a more restrictive meaning than that of the more vernacular use of the word. I don't want to be too geeky here, but I would be remiss if I didn't point out to both players and technical types that most of the disconnect between the known physics of string tension and the perceived string tension of musicians is simply a matter of semantics. But just for the record, the quality which players generally refer to as string tension is really (mostly) longitudinal stiffness, which is discussed below. Now, there are other factors which probably affect string feel, like the bending stiffness of the string, the height of the frets, and the general flexibility of the instrument. So I'd like to introduce a term here which wraps up everything that players often refer to as perceived tension. I will use the term compliance for this.

Probably the best way to envision the relationship of string tension and longitudinal stiffness in the context of musical instrument strings is to consider a schematic representation of a string stretched between nut and bridge.

Note that in this schematic there is no extra string length from the bridge to a string anchor – the string is anchored at the bridge. And there is also no extra string length between the nut and the tuning machine. In the schematic the string is anchored at the nut. This is obviously not at all how strings are attached to real instruments. In real instruments there is always some distance between the bridge saddle and the point where the string anchors. The amount depends on the style of instrument. On classical and steel string flattop guitars this extra length is pretty small, considered as a ratio of the “speaking length” (nut to bridge) of the open string. But in violin family instruments and archtop guitars and mandolins this extra length is generally quite large.

While on the subject, all instruments without locking nuts present a sizable extra length of string between the nut and the tuning peg or post, at least for some strings. Consider just for example the Fender electric guitar. The extra length of string between the nut and the low E string is pretty small, but the extra length between the nut and the high E string is quite large.

To visualize why these extra lengths of string might have something to do with longitudinal stiffness, consider another schematic of another string. In this one there is considerable extra length of string between the bridge saddle and the string anchor and also between the nut and the tuning post. In addition, both the nut and the bridge saddle are implemented as relatively large diameter friction free rollers. Consider that the scale length, string construction, and pitch of this string are all identical to those of the string in the previous example.

Now, consider what happens when you attempt to stretch both of these strings by pulling them to the side. In the case where the string is firmly anchored at both bridge and nut any stretch you impose on the string has to come from that length of string. But in the other case the stretch you impose on the string comes from the entire length of the string – not just from the speaking length. Combined they are considerably longer than the “speaking length” of the open string alone. Please note that the stretch imposed on the string is imposed on its entire length and not just on the speaking length because the friction free rollers allow the string to stretch uniformly along its entire length. So if the schematics were rendered in real hardware the first string would be less compliant and might feel stiffer to attempts to stretch it than would the second string. In the vernacular the first string might be said to feel like it is under greater tension, but given that the scale length is the same, the strings are made identically, and they are tuned to the same pitch, we know that this is not possible given the physics definition of tension.

Human Perception of Tension and Compliance

Please notice that I didn't say that the first string would positively feel stiffer than the second. It is (positively) stiffer, but it may not feel that way to a musician. It is accurate to say that there have been no comprehensive studies done and published on the extent to which humans can distinguish differences in compliance when actually bending strings. It may be that people are exquisitely sensitive to differences in compliance, but the only study I know of indicates that they are not very sensitive at all. Master archtop guitar maker Bob Benedetto described a couple of informal experiments he made in this area in an article that appeared in American Lutherie #68. Bob built two simple demonstration “necks”. The first had a number of identical strings but with different scale lengths, the scale lengths varying from 23” to 26”. All strings were tuned to the same pitch, so according to the relationship between tension, pitch, mass per unit length, and speaking length, the strings with the longer scale lengths will be under greater tension than the shorter ones. You can take it as fact that this must be so (they are called laws of physics after all) but no one that had this apparatus in their hands could feel any difference in elasticity between any of the strings. Again, let me make this clear. The issue here is not that the longer strings were under greater tension – that is a physical fact. The issue is that people could not sense any difference in the feel of the strings when attempting to bend them.

The second of Bob's test necks had a number of identical strings tuned to the same pitch too, but this time they all had the same scale length. What differed was the amount of extra string length between the bridge saddle and the anchor for each string. With this apparatus we know that the tension has to be the same for each of the strings because the strings are identical and their lengths and pitches are identical as well, and we also know that the strings with the longer extra length behind the bridge saddle should be more compliant. But here again, no one that handled the apparatus could detect any difference in elasticity among the strings.

[Note for those intending to read the original American Lutherie article: In the paper Bob uses the term “tension” in the vernacular sense, referring to perceived tension as tension. Whereas this sin probably would have got a submission of a paper to a physics journal rejected, I encourage you to not get too hung up on the terminology there. Bob is not a physicist. But he designed and performed a very clever and useful experiment and obtained good results.]

Nut and Saddle Friction

One possible reason that the folks that handled Bob Benedetto's second apparatus could not detect any differences in compliance among the strings is that there weren't any. Note that Bob's apparatus differs significantly from the second schematic presented above in that the schematic specified that the bridge saddle and nut were implemented as large diameter friction free rollers, thus guaranteeing that any stretch imposed on the string be uniformly distributed along the entire length of the string. Presumably (there is not enough info in the article to tell) Bob implemented his nut and bridge saddle using more typical materials and dimensions. Assuming such materials and considering that in typical guitars the breakover angle of the strings at both the nut and bridge are typically at least 15 degrees, there could be considerable friction between the nut and string and the bridge saddle and string (see the page Calculating Downforce on the Bridge for more information on the relationship between breakover angle and pressure). Depending on how much the strings were bent during evaluation, it is possible that they were not bent enough to overcome the friction at either of these points, and in fact the act of bending may have induced even more friction there. In that case the apparatus would behave as if there was no additional string length in front of the nut and behind the bridge saddle. That is, it would be just as if the string was anchored at the nut and saddle, as in the first schematic above.

Other Factors (Comparing Apples to Apples)

The primary property of a string involved in the discussion of compliance above is called longitudinal stiffness, and it is a function of the core material of the string and how long the stretched length of the string is. For this reason different materials will feel different in terms of stiffness. For example, the rubber strings of an Ashbory bass will feel less stiff than the Nylon strings of a classical guitar, which will in turn feel less stiff than steel guitar strings, even if all of these are stretched to the same tension. (To further confuse the issue, they are generally not stretched to the same tension in real instruments!) But there are other factors that can influence how stiff a string feels. One is simply how far you bend it. Fretting an instrument with low action feels a whole lot different than doing a deep string bend on the same instrument. Another factor is bending stiffness, which differs with string material, gauge and construction. It is important when performing experiments in this area to keep these other factors constant.

Summary

It should be clear that more research is needed in this area, particularly in measuring typical friction between nut and string and saddle and string and is ascertaining just how much or how little tension difference and compliance difference can be detected by musicians. But from a practical perspective all of this may be purely academic. Bob Benedetto's experiments strongly suggest that modifications such as scale length changes and tailpiece length changes intended to provide some difference in string feel may not be very fruitful.

References and Suggestions for Further Reading

String Tension
This is the page in the technical section of this website on the topic of string tension. The formula for determining string tension and other string tension related information and tools are included here.