This is because of what I've decided to call The Tuner's Dilemma, and it's best understood by placing oneself in the mindset of a tuner. As a practical example, let's think about tuning a guitar. Our task is to tune a six-string guitar in standard tuning: E2-A2-D3-G3-B3-E4.
This means that the musical interval between each of the strings is a perfect fourth, with the exception of the G- and B-string, which form a major third. Thinking about this, let's see what happens if we tune these intervals perfectly.
We won't worry about our starting point. In this example, we'll assume our bottom E-string is perfectly in tune (perhaps when played open, it plays at 82 Hz, but that doesn't really matter).
A true perfect fourth is typically defined as two pitches whose frequencies form a ratio of 4:3. The ratio for a major third is 5:4. There are alternative definitions of these intervals, as we'll see later, but these simple, rational ratios are very commonly accepted. An octave is almost universally understood to form a 2:1 ratio.
Ready, Set, Tune
Let's start. Our low E2 is in tune, so now we tune our A2 so that its pitch is exactly 4/3 higher. If you have a guitar, you can do this yourself -- if your guitar is reasonably in tune, adjust your A-string so that you hear zero beats in the sound wave when E2 and A2 are played simultaneously. Whether you're using math or your ear to come up with the new pitch, this process is known as forming a just interval.Say we now do the same between A2 and D3 and between D3 and G3. Then we tune a just major third from G3 to B3 and a final perfect fourth from B3 to E4. Each interval we've tuned has been completely in tune.
So what's the problem? The problem presents itself when you try playing your two E-strings simultaneously. These notes are two octaves apart and should thus form a 2:1 × 2:1 = 4:1 ratio. Yet your ears will tell you right away that something is amiss.
Let's get a better idea of what happened by examining the numbers. You'll notice from the previous paragraph that I used simple multiplication to come up with a compound interval (i.e. the double octave, 4:1). We can use this technique to find out what the actual interval between our two E-strings actually is:
4:3 × 4:3 × 4:3 × 5:4 × 4:3= 1280:324 = 320:81
320:81 is approximately, but not exactly, equal to 4:1. This is the mathematical view of the dilemma, and the technical term for the 81:80 (4 ÷ 320:81) discrepancy is a comma. There are all kinds of commas, because there are all kinds of tuning systems. Let's stay focused on this "guitar comma" and investigate what we can do about it.
Solving the Tuner's Dilemma
Put succinctly, there is no way to solve The Tuner's Dilemma. However, the problem can be at least somewhat mitigated. For the sake of example, let's implement a naïve solution that fixes our octaves at the cost of making our perfect fourths, well, not so perfect. We'll keep our major third intact.Since our E4 fell flat of where it should have been, common sense tells us that our perfect fourths will all have to become a little sharp. How sharp? We need to distribute the comma evenly between our four perfect fourths. This means that each fourth is widened (multiplied by, mathematically speaking) by a factor of the fourth root of the comma, or 3 / (2 × 4√5). Our resulting definition of a "perfect" fourth is a ratio of 2:4√5. This is getting pretty gnarly. Let's persevere and try it, though. Here's the new tuning math:
2:4√5 × 2:4√5 × 2:4√5 × 5:4 × 2:4√5 = 80:20 = 4:1
That outcome looks better. This isn't a very satisfying solution, though. We've essentially distributed 100% of the comma across our perfect fourths and nowhere else, rendering them noticeably out of tune while other intervals, like the major third between the G- and B-string, remain in tune. In practice, a more balanced approach is used.
Equal Temperament
Equal temperament is the system used to tune the majority of instruments in the Western musical world. It allows instruments to be tuned so that they sound the same in all 12 keys, and that all intervals are acceptably close to their just counterparts, although in equal temperament, only octaves remain justly tuned.This is done by means of a simple function, f, which takes as its input the number of semitones, i, above a given starting pitch and produces the tuning ratio for the interval corresponding to that number of semitones.
f (i) = 2i/12
A fourth can also be defined as being equal to 5 semitones, so let's try using the equal temperament function and seeing how its result compares.
f (5) = 25/12 = 1.334839854
| Tuning system | Value | Intonation |
|---|---|---|
| Just intonation | 4/3 (1.3) | In tune |
| Equal temperament | 1.334839854 | 2 cents sharp |
| Solution above | 1.33748061 | 5 cents sharp |
So you can see that equal temperament produces a fairly acceptable "perfect" fourth. Note that this interval is much closer to being in tune than what we came up with in our naïve solution above (5 cents is often thought to be the minimum distinguishable difference in pitch for most humans). The noticeable improvement in the equal tempered perfect fourth is due to the fact that equal temperament distributes the comma evenly throughout all twelve notes in the scale.
Conclusion
Does equal temperament solve The Tuner's Dilemma? No. However, it does a reasonable job of allowing us to tune our instruments once and be able to play in any key without retuning.Over the centuries, countless tuning systems have been devised to work around The Tuner's Dilemma. Many of them have interesting characteristics that make them worth studying and hearing. Here are some links to more information:
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