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“Why can’t I get this thing to play in tune?”
Lots of banjo players think it, even if they don’t say it. It’s the banjo player’s common lament.The old banjo joke goes something like this: Banjo players spend half their time tuning and the other half playing out of tune. While somewhat true, you can generally substitute a lot of other instruments as the butt of the joke if frets or fixed tuning intervals are involved. The banjo is particularly problematic in this regard due to the complex timbre of the notes produced by the instrument due to the string’s interaction with the stretched membrane head. There’s a reason for all this, as we shall see.
Order in the universe
Way back when I first developed an interest in music as a pre-teen I initially thought that music must be based on an empirically correct order of notes that had perfect mathematical relationships to each other. I had up to that point been generally satisfied that the universe and life within it could be quantified and contained within a strict mathematical structure, so I was quite happy to find that the western scale of music that our ears have become accustomed to was based on an orderly sequence of frequencies that were all separated from each other by the same mathematical distance. I soon learned that this distance was calculated from a mysterious number known as the twelfth root of 2, or approximately 1.05946309. This number represents a value that when multiplied by itself 12 times yields a result of 2. I eagerly punched the number into my hand calculator (at that time itself considered a new technology), selected the multiplication function and poked the “equals” button the required 12 times. Walla! The result was a number very, very close to 2 indeed! Music did indeed conform to the precision of mathematics, so I could go to bed each night and rest a little easier with the assurance that all was right in the cosmos.Not so fast, it’s not that simple.
As I progressed as a musician I would notice that occasionally things just didn’t sound right to me. I chalked it up to an inability to properly tune or perhaps a problem with the mechanics of my chosen instrument. I had selected guitar as my first love as most of the other gazillion new players do. This one came with the requisite high action and cheap construction, so sour notes were part of the territory. I learned to live with them.As my abilities as a player increased I also progressed up the food chain of better instruments but the old “Something’s rotten in Denmark” feeling was still there. I could usually tweak the tuning enough to be more or less satisfied, but I continued with my search for an answer until I eventually learned the root of my tuning dilemma. The problem was a little thing called equal temperament, (itself created as a solution to another problem), but to understand why it’s the problem we need to step back a few centuries for a closer look.
Music and math
It turns out that someone with a little higher understanding of mathematics, Pythagoras, had discovered that there were indeed pure mathematical intervals that we as humans perceived as most pleasing to our ears. These turned out to be simple ratios of frequencies, such as 2:1 (octave) and 3:2 (fifth). There are similar simple ratios to define the rest of the 12 notes in the musical scale, also. If you really want to know, they are 16:15, 9:8, 6:5, 5:4, 4:3, 7:5, 3:2, 8:5, 5:3, 7:4, 15:8, and 2:1. Too much information! Could this be the answer? Well, yes and no. This system is commonly referred to as “just intonation” and the simple harmonies created from various combinations of these ratios sound pleasing to us because they are composed of exact harmonic multiples of the key note which they are created from. The problem with just intonation is that it is only musically pure in the key (based on the beginning note) that the intervals are calculated from. If you wish to play in another key then all of the note values need to be re-calculated from that particular starting note. That’s not very handy at all, is it?Side note:
Occasionally instruments that are key specific such as the diatonic accordion use just intonation to full advantage. By utilizing just tuning the chords produced by the instrument sound sweet and pure. The third note of the scale is tuned 14 cents flat (100 cents is the distance between notes in the 12 note scale) from its equal temperament counterpart, and a few of the other notes are pitch corrected to a lesser degree. Listen to any Cajun band and you can hear an example of this.
“We have a problem, Houston.”
Simply said, as long as an instrument is designed with fixed pitches defining the musical scale notes and it is desired to have the ability to play in more than one key there needs to be an adjustment in the fixed pitch intervals to allow us to play in these desired keys. Various ways of “tempering” the 12 tone scale were used from a period beginning with the 18th century and eventually culminated in the development of the “equally tempered” scale in the mid-nineteenth century. Other temperaments are still in use to a limited extent, (See Side note above) and there are many composers who feel that equal temperament was a grand mistake. These composers and music theorists are very active today, and many work with even more highly refined systems of temperament. That’s beyond to scope of this article, though. Suffice it to say that the system of equal temperament, where all pitch intervals are spaced out equally from each other, became the standard for western music. Equal temperament allows us to play in all keys with all note relationships sounding more or less in tune. You can also say that the note relationships all sound equally out of tune. Such is the compromise that we present ourselves with by adjusting the note pitch relationships to utilize frets or play upon any instrument that features fixed note intervals.”Say it ain’t so, Joe.”
In the process of homogenizing our 12 tone western scale to accommodate easier key changes we created note intervals that are inharmonious to a small degree. Over a long period of time we have become desensitized to the inharmonic intervals within the scales used by the equally tempered twelve tone system. Many people are more pitch sensitive than others and if you’re one of the lucky ones that have difficulty ignoring equal temperament’s disharmony you’ll most likely always experience difficulty with tuning a fixed pitch instrument. (There’s always room for more trombone players, though.)Another side note:
If you wish to test your reaction to the contrast between just tuning and equal temperament tuning listen to the following 15 second long mp3 example that I’ve recorded for your listening displeasure/pleasure HERE .
The first half demonstrates a chord in equal temperament and the second half demonstrates the same chord in just intonation. They are recorded identically with the only change being pitch adjustment. Notice the slight “buzzing” and “warble” caused from note interaction in the equal temperament example. All of these artifacts are not present in the just intonation example. Can’t hear any difference? Congratulations! You’ve been successfully re-programmed by western music and can return to bliss with out any further reading.
Why does the equal temperament chord sound like that?
The “buzzing and warbling” in the equal temperament example is a result of the way that the individual frequencies combine together. When frequencies are combined you also hear new frequencies that are created by their interaction of the original frequencies. These new frequencies are the sum and difference of the original frequencies. These newly produced frequencies also combine to create yet new frequencies that are the sum and difference of them. This process continues, with each newly produced set of sum and difference frequencies becoming lower in amplitude. You can see that the three initially combined notes can get rather complex in a short time. If the initial notes were pure intervals, then the complex resultant sound is harmonious. If the note values are not perfect intervals (as in the equal temperament example) things can turn ugly fast. That’s what you hear.Bottom line, anyone?
The bottom line is that if you’re sensitive to the inharmonic intervals presented by equal temperament then the best thing to do is use an electronic tuner to tune your instrument accurately and very slightly adjust the tuning by ear to compensate for the key in which you choose to play. It’s counterintuitive, but the best thing to do is to manually add inaccuracy to improve the sound of the chords that are produced on a fretted instrument, keeping in mind that retuning will be necessary if another key is selected.Here’s a quote from a fellow Banjo Hangout member that I found particularly succinct: “Simply put, equal temperament allows us to play equally out of tune in any key -- the thirds are not mathematically "pure" intervals, which is why you need to tune your B string slightly flat in G tuning. We are so used to this; it passes unnoticed by all but a few unlucky souls who are cursed with perfect pitch. Unless you want to have interchangeable fingerboards with frets placed for different tunings in different keys or movable frets (like a lute), or no frets at all, there's no getting around it.”
He nailed it.
Still another side note:
I am fully aware of the possibilities and/or likelihood that individuals within other cultural systems can and do often develop their musical preferences in ways that reflect their own particular preferences based upon the relationships within their own culture. Scale systems with different pitch structure and different intonation systems are common while additional sound structures based on rattles, buzzes, or other noises are also often incorporated within the overall structure of many diverse forms of music. What is reflected in my writing here is referring to music structure that is generally considered to be based on the foundations of western music. YMMV!
**************** OPEN BACK BANJO CONSTRUCTION GUIDE ON CD ****************
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Please visit my other website designed to provide information on musical instrument construction. There are free plans as well as construction tips and techniques available at the present time.
Rudy's Sketchbook of Musical Instrument Plans, Ideas, and Inspiration
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