Yes, I promised this a couple weeks ago. But once I started reading about the mathematics of tuning, I realized this would have to be a tad more involved than complaining about electronic tuners (though you’ll find that here, too!). So I’ve been brushing up on integers, wolf intervals and syntonic commas, so that I could sparse it down to some basic principles and questions, and perhaps even a few suggestions, that I could then offer up. Also, I feel I must admit, I’m using this post to vent some pent up frustration acquired in jam situations where tuning is treated with the same abandon as fast food wrappers offered up to city streets.
Tuning – the biggest little thing you can do to sound better. I am far from an accomplished musician, and where I onced neared perfect-pitch (the ability to recognize a pitch by name when hearing the frequency), I now bust out my tuning fork to find the A440. BUT! But,…the mind is infinitely malleable, and pitch discernment is acquired, not inherited. In other words, you can improve your ear. For that, an understanding of the underlying principles of harmony are helpful. That’s what this article is about.
If you play a string instrument in group settings, you are bound to have come across an electronic tuner or probably own one yourself. Popular ones have a clasp so that you can place the tuner on your instrument and use both hands to tune. You than strum or bow your open string, while changing the tension, until the needle on the screen stabilizes in the middle of the spectrum and you are ‘in tune’ (The measures on tuners are usually in cents of Hertzs. More on that later). These tuners are very practical, but present one huge problem: you are tuning with your eyes. What’s going on with your ears?
Your instrument produces sound waves. Waves have amplitudes (which we hear as volume) and frequencies (which we hear as pitch). When you are tuning your instrument, you are looking to make the frequencies of the various strings combine pleasantly. What does this mean? Mathematically, it means that the sound waves of two different notes produce a harmonic interval where the ratio of the waves approaches as close as possible to a simple integer ratio.
Say what!?! A simple example illustrates this: an octave is the interval between two frequencies, of which the higher note is twice the frequency as the lower note. That means that the wave of the higher note cycles through a period in time twice as fast as the lower one. The ratio of the notes is 2:1. Very elegant, mathematically speaking, and pleasant to the ear. When waves combine symmetrically, such as with simple ratios (2:1, the octave; 3:2, the fifth; 4:3, the fourth) there is none of the ‘buzzing’ that your ear would detect if the ratio between the two notes was more complex (1264:636, for example). That buzzing is the interaction of the two sound waves, which are out of step with each other.
Where it gets complicated: Musical tuning was theorized as far back as Antiquity, in China, Egypt and most relevant to old time and bluegrass musicians, in Greece. Pythagoras, whom you’ll remember from grade nine math, made explicit the diatonic scale ratios. So, tuning to just intonation, C is equal to an abstract 1, the value of the C above is 2:1, and the G is 3:2. The other intervals also have nice clean fractions, such as 9:8 for the major third (think of sectioning off strings). But there is one big problem with this elegant math: you can only play in C. If you tune your instrument on the basis of that C, playing in any other key (save A minor), will sound terribly out of tune. And that is because frequencies are logarithmic, not arithmetic.
[I’ll skip over the math and history, because several pages could go into the details of the evolution of the many disciplines that informed the music we make today. But I would like to point out that the line between the inherent qualities of sound waves and the culturally affected experience of music are beautifully blurred. Why the major and minor scale modes predominate in Western music is in fact directly related to Church reformation, for example.]
In other words, if the sound waves combine perfectly in C, they will not combine as symmetrically in other keys. Because of that, Pythagorean tuning is not standard practice in this day and age. The standard system we use is known as equal temperament. A piano tuned in this system would not present perfect octaves, and the greater the distance between notes, the farther off the ratios are. But so that the piano can be played in any key (C, or B, or Gb), the inexactitude between all the intervals is distributed evenly across the board. Let me explain:
“Equal temperament is a musical temperament, or a system of tuning in which every pair of adjacent notes has an identical frequency ratio”. In other words, in just intonation, the interval between C and G is nice and pretty like 3:2 (a fifth), the interval between another fifth such as F and C will be off (instead of a pretty 3:2, it’ll be closer to something like 47:31 – close, but not perfect), and the discrepancy increases the further away from the reference note you go. To make your piano playable in as many keys as possible, equal temperament approximates the intervals between all notes, so that the total difference in cents (increments of Hertz) is distributed across the board. So starting on your middle C, you would tune all the keys based on an equal difference in ratio (think three dimensions instead of two, though mathematicians will scowl at this analogy.) – i.e., all notes will be equally out of tune in relation to each other. But because this difference is distributed to all notes, it is so slight as to be almost entirely undetectable.
Oh my. I suspect I might be confusing the issue more than clarifying it. I refer you to the wikipedia articles on the subject for helpful graphs and tables, and scientific explanation.
What this really is about is tuning your instrument. So, with even only a vague impression of the physics in mind, here’s what to look for when tuning:
1) Use your tuner for one string only! Then close your eyes and listen while tuning your other strings using that first one as the reference. You will hear the sound waves (like a pulsing, throbbing or waving quality) combining. As you adjust the tuning (usually, it’s easier to adjust a higher note in relation to a lower note, than vice versa), the pulsing will speed up as you near ‘perfection’ – and once you hit, it will slow down dramatically. Go higher or lower than that perfect point, and the pulsing speeds up again. You might also hear the sound as fuller; those are the natural harmonics produced by your instrument.
Don’t worry if this all sounds terribly esoteric. It isn’t. Like many things, the explanation is a million times more complicated than the actual experience itself. The more you practice tuning your instrument with you ears, the easier it is. No surprise there.
2) If you play a fretted instrument, just like the piano, tuning it to mathematical perfection will only sound good in the key you tune to (with guitars, that’s usually E). So if you plan on playing in more than one key, play a variety of chords, adjusting the tuning with each one until they also sound pretty damn close (it won’t be ‘perfect’, but “close enough” is usually not close at all).
3) If you are playing a non-fretted instrument, pythagorean tuning by ear is more likely. Symmetry is easier to achieve than planned assymetry (especially in terms of 2 cents of a hertz difference). With the fiddle, for example, tuning in fifths of perfect 3:2 ratios is easy, and because your fingers adjust the pitch on the spot, you modulate from key to key subconsciously.
I’m sorry, I indulged in math on a blog about fiddle tunes. I said “close enough” is not close at all. There are exceptions – Old Time being one of them. Old Time is folk music, which means it’s all about the heart and soul. So you can get away with ‘close enough’ (but, that being said, there is a reason Bruce Molsky is revered in the way he is). But for the most part, playing in tune goes an incredibly long way in making you sound good. It brings out the best in your instrument (letting natural harmonics emerge) and takes away the buzzing sound which our ears detect, even if we are not nessecarily conscious of it. Being poor, I lost my tuner and was left with a tuning fork. I am thanful for my financial condition, because as a consequence of it, I was forced to learn to tune by ear. Don’t get me wrong, electronic tuners are great and they’re a helpful tool, but like anything, they can become a crutch which hinders your progress if you rely on it too much. Also, $40 tuners vs. the human brain – no contest.
I humbly suggest it is better to tune with your ears than with your eyes.
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