The Well-Tapered Cavalier (or Beware Inharmonicity)


I have been using the same strings and bass for about 13 years, though my current instrument is a replacement for one that was damaged recently in an accident. They’re nearly identical except that this one is a prototype model so there are some specific minor differences.

I have always preferred taper-core strings (i.e. the outer wrap diminishes in circumference to the core near the end) because with only the core to go over the saddle much less damping occurs; the string is more resonant and brighter in my opinion. Plus, it allows for easier setup and low action. But there is an inherent danger that comes with tapered strings, one that came to visit recently (and stayed too long). But the explanation and the solution were so interesting I thought it might be worth posting about.

I’m finishing up work on a full-length recording project, one that has demanded a great deal of intense work in terms of writing, rehearsing, and recording over the past 6 months or so. And while everything was going well for the most part, I was having an unusual amount of trouble with my intonation, but only in the lower register (approximately G2 and below). This is a little strange because intonation errors tend to creep up and become more apparent in the middle to high registers, not so much down in the basement.

Being a fretless bassist, my first reflex of course was to think it must be me, and that I just needed to make an adjustment in my fingering (like we always do). But nothing seemed to be working. Every once in a while things would sound alright though, which gave me a false sense that it was just something I needed to tame in my technique—but I was still surprised that it kept happening: while I certainly would never claim to have perfect intonation, this isn’t my first day either. And all this sour rumbling underneath was spoiling everything.

I started to think about the differences found in this bass, especially since I took the bridge from my previous instrument and put it on this one, and they were very different in size and design. But still, even if the bridge placement was off I should still be able to roll my fingertips into the correct pitches.


I stripped off all the strings, put on a new set and did a thorough job of setting the intonation (which to be honest, was nearly spot-on to start with). I don’t know if I had convinced myself that it was better or if it actually was better, but I embarked on re-recording everything I had already finished. Some spots had improved, but in those sections where the texture is thinner (with clean guitars), those low notes were just out—no two ways about it. And now, after so much intensely-focused listening and scrutinizing, it started to sound like everything was a little bit off. I realized then that I needed to take some time away from things and start fresh in a few days.

So I spoke with a friend of mine who has been building string instruments for some 40 years now, and the first thing he said was that it was probably the strings. He asked if they were tapered, I told him they were. He asked if the problem was only happening on those two strings, I told him it was. The problem, it seems, was inharmonicity.

Inharmonicity occurs when the overtones of a vibrating body diverge from being whole-number multiples of the fundamental. Imagine a B string generating a harmonic series that is quarter tone between B and B-flat: you’d hear the fundamental when you plucked the string, but also the out-of-tune overtones mixed in. (Here is a small file that demonstrates the sound; a single note played on the B string where you can hear the unfocused pitch. Change the suffix from .pdf to .zip, etc. Headphones recommended.)

But how was this happening now, on a bass that was nearly identical to the one I’ve been playing for 13 years, along with the same strings, which had never exhibited this problem before? The key here is that they were “nearly” identical, and what follows is a simplified attempt at describing the cause of the problem.

When you use a taper-core string and set it up as shown in example 1 below (where the bridge saddle meets the taper more or less) the effective length of the string (L)—that is to say, the length of the string that contributes to the overall pitch when at proper tension—begins where the taper has reached the full diameter of the string and continues until it is stopped by the nut.

However, if you have things set up (as I have more or less) in figure 2, you might be inclined to think that the length from the saddle to the taper (x) adds to the overall effective length of the string (x + L). This is not correct. The only portion of the string that contributes to its pitch at tension, and therefore subject to intonation, is that which is at full diameter. Instead, this additional length (x) adds a complex component to the string’s overtone series because that portion of the string is not vibrating at pitch like the rest of the string. So, imagine setting the intonation for example 1: divide the string in half from the saddle to the nut, and the octave would be just about at the center of L. If you applied the same thing to example 2, the octave would no longer be at the center of L.


The degree to which this happens (and is noticeable) depends on several factors, including string length, string gauge, taper length, bridge setup, etc. In fact it doesn’t rule out the use of tapered strings; the taper would just need to be shorter. In my case, the differences in the tolerances between the two instruments and the swapped-out bridge were the tipping point—enough to cause a problem where none had occurred before.

The solution? For now, I’ll bridge the gap and go taper-less (but with every intention of soon becoming well-tapered, and a little less cavalier).

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2 Responses to The Well-Tapered Cavalier (or Beware Inharmonicity)

  1. Kenneth Goodenough says:

    Sean, were you the only who noticed this effect during recording, before mentioning it?
    And besides this problem, how is the new bass working out for you?


    • Sean says:

      Kenneth, picking up from your e-mail—you might enjoy the book “How Equal Temperament Ruined Harmony (and Why You Should Care)” by Ross Duffin. It discusses, among other things, topics such as “expressive intonation”— how string players adjust their intonation based on key. For example, a C-sharp in the key of D Major (the leading tone) would be played a little sharper than a C-sharp in, say, B Major (the supertonic). It’s all addressed in a historical context, through the development of equal temperament.


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