Accuracy and resolution of electronic tuners
What's this all about?
Every time I read a discussion about electronic tuners, the question of accuracy arises. Discussions inevitably revolve around two points:
- Which tuners are more accurate than others, and
- How much accuracy is really needed?
The first point is conventionally answered via reference to the manufacturer's specifications. The second point is - rightly so, I believe - open to debate.
What I'm interested in, though, isn't either of those points.
I've been concerned for a while that both manufacturers and players may be citing specs that aren't necessarily indicative of the tuner's actual performance. This article is an attempt to explore the meaning and validity of electronic tuner "accuracy" specifications.
First, some groundwork...
If you ever took an introductory science course, you probably learned something about measurement. In particular, you would have learned that there's a difference between accuracy and resolution.
- Accuracy describes how closely your measured value corresponds to the true value, while
- Resolution describes the amount of precision in your measurement.
For example, consider that you have a ruler with markings every inch. For purposes of this discussion, you're allowed to use this ruler only to measure to the nearest inch; this is a matter of judging whether the measurement falls closer to one marking on the ruler. We say that this particular ruler has a resolution of one inch.
Of course, we might have other rulers with more markings. But - for the sake of this discussion - we'll always follow the rule that we report measurements to the nearest marking on the ruler.
Accuracy relates to the length of the ruler.
A twelve-inch ruler is ideally twelve inches long. That seems simple enough. But what's an inch? Well, it turns out that there are very precise physical standards against which an inch may be judged accurate or not. The degree to which the ruler in your hand deviates from those standards is a measure of the accuracy of the ruler.
For example, let's say that your ruler is 20% shorter than the standard. (Not likely, of course. Bear with me for the sake of argument. Perhaps you photocopied an actual ruler with the copier set to reduce the image size to fit on a sheet of paper; I dunno...) Then every measurement you make with this ruler will only have an accuracy within 20% of the true dimension.
To be clear, let's assume that this reduced twelve-inch ruler is still marked only in whole inches. You'll still make measurements to the nearest inch; that's what the rule can resolve. But the accuracy of those measurements may be off by up to 20%.
OK, but how does this relate to tuners?
Simply put, a tuner is just another measuring device. Rather than measuring a linear dimension (as a ruler does), the electronic tuner measures a musical pitch. The tuner tells you to which note you've tuned a string and also how closely your string corresponds to the intended pitch. This is a measurement. The measurement necessarily has both a resolution and an accuracy.
But tuners only specify accuracy...
I know. And that's where it gets interesting...
Tuners haven't somehow sidestepped the entire question of resolution. Instead, I think, manufacturers (and players) have simply conflated the meaning of accuracy and resolution.
The conventional wisdom is that, given a tuner which is "accurate" to one cent, each in-tune (according to this tuner) string on your guitar will be at most one cent away from the "ideal" pitch for that string. (Let's dispense with all the usual imperfections of guitars and assume for the sake of argument that we have a perfect guitar that stays tuned precisely where you last adjusted its tuning.)
Let's pretend that's true and see where it leads us...
We intuitively believe that a one-cent "accuracy" means that each string should be in-tune to within one cent its ideal pitch. Some strings may be tuned high by a cent; some may be low by a cent; some may be spot-on; some may be off by less than a cent. Let's not worry about whether that's "good enough" or not; that's a different discussion. All we're concerned about is whether the tuner is telling us that the string is in-tune to the limits of the tuner's ability.
Well, if that's "accuracy"... then what has happened to the "resolution"?
Good question! The answer is: we don't know. Let me say that in a different way: none of the tuner manufacturers make a distinction between resolution and accuracy.
When you use an electronic tuner, the pitch of your "in-tune" strings may deviate from the ideal for (again, assuming a flawless guitar) two reasons:
- The tuner is not accurately calibrated. Going back to the ruler analogy, this corresponds to the deviation between the actual and ideal length of the ruler.
- The tuner can't distinguish a difference between two closely-related pitches. This is like the ruler not having enough markings to resolve, say, anything finer than an inch.
Does a tuner really have both resolution and accuracy?
Yes, a tuner does have both resolution and accuracy.
An electronic tuner has to get its pitch reference from somewhere. The tuner uses the vibration of a quartz crystal or ceramic resonator as its reference. The crystal or resonator vibrates at a very specific pitch (much higher than any pitch on the guitar); the circuitry in the tuner does math to convert that pitch to musical notes. The actual accuracy of the tuner (defined as the deviation of the tuner's pitch reference from the "true" value) is determined by how closely the vibration frequency of the crystal or resonator corresponds to its ideal value.
The resolution of a tuner is closer to what we conventionally think of as a measure of the tuner's "accuracy". We'd like to believe that a tuner that's "accurate" to one cent will tune our strings to within one cent of their true value. If the tuner is absolutely "accurate" - i.e. if its internal reference pitch is spot-on - then the ability of the tuner to distinguish between two very slightly different pitches is the tuner's resolution.
So how does the tuner decide whether a note's in-tune?
If you've ever tuned your guitar to an audible reference pitch, you know to listen for the beat frequency that tells you the string being tuned differs from the reference pitch. You adjust the tuning of the string until the beat frequency slows and "disappears". At that point, your string is "in tune."
A tuner works the same way. It detects the beating between the string's pitch and the reference pitch and displays an indication of the beat frequency. A needle tuner moves the indicator closer to "zero" as the beat frequency slows and stops; a strobe tuner shows decreasing motion - eventually stopping - as the string comes into correspondence with its ideal pitch.
(Never mind that there's also an indication of high versus low relative to the reference pitch. Just take it for given that the indicator somehow does the right thing. The details aren't that complicated, but I'd like to not make this exploration any longer than it already is...)
And this is important because...?
Beat frequencies are important because they tell us something about the theoretical limits of a tuner's resolution.
Let's take the case of tuning the low E of a guitar in standard tuning. The ideal frequency of the low E string is - in round numbers - 82 Hz. (We're concerned about resolution, not accuracy; 82 Hz is close enough for this discussion.)
Let's say that the string is tuned high by one cent (relative to the "ideal" of 82 Hz).
A cent is a ratio, expressed mathematically as the 1,200th root of 2. This is approximately 1.00057779. To save typing, I'm going to round that to 1.0006.
In other words, a one cent difference is approximately a 0.06% change in pitch.
Given our low E as being mistuned one cent high of 82 Hz, we see (after doing some math) that it's actually vibrating at approximately 82.05 Hz.
The difference between the ideal frequency of 82.00 Hz and the actual frequency of 82.05 Hz is 0.05 Hz. That's the beat frequency. What does 0.05 Hz mean? It means that you get one beat every twenty seconds.
Hmm, that's a long time to wait for a beat... A lot of guitars won't sustain that long. How the heck can a tuner so quickly measure something that takes twenty seconds to happen?
Well, think of it this way: a strobe tuner would show one complete rotation of its display in twenty seconds. But you, watching the display, don't need to wait the full twenty seconds to know that the display is moving. The question then becomes one of how small a movement can you detect?
This is where it starts to get interesting...
For the moment, let's concentrate on virtual strobe tuners. I looked at the manual for one such tuner and counted twenty-three segments in the representational display of the strobe disc. Let's round that off to twenty to simplify the math. (Also, I don't think this number is far off-base for all of the variations on virtual-strobe displays in common use.)
Let's go back to our mistuned low E string from the previous section. We know that the beat frequency has a period of twenty seconds. We also know that our hypothetical tuner's strobe wheel is represented by twenty segments. Well, that's convenient, isn't it...?
Whatever pattern this tuner puts in its display will cycle all the way around in twenty seconds, which means that any part of the pattern will take one second to move from one display segment to the next.
Ponder that for a moment...
In order to resolve a one-cent mistuning on the low E string, you have to watch the virtual strobe display for a full second to see whether the display moves.
What if you wanted to resolve a mistuning of one-tenth of one cent? Do the math: The beat frequency would have a period of two hundred seconds. The twenty-segment display would take ten seconds to "rotate" from one point to the next.
That seems like a long time to me... I know that I don't stare at my tuner for a significant fraction of a minute to determine whether a string is in tune.
To be fair, the resolution of the virtual strobe display might be increased using the same techniques used to give the appearance of higher screen resolution for typefaces on computer displays: rather than a spot on the screen being "on" or "off", there may be shades of gray. Even given such techniques, the player must still make a judgement call: is the display changing or not? As you've seen, a higher resolving capacity requires either finer distinctions or a longer evaluation period on the part of the player. In other words, you can't arbitrarily increase resolution without arbitrarily increasing detection time.
Enough already! What does this mean?
I think that - unless I've made a horrible mistake - it's unlikely that any electronic tuner will have a resolution significantly better than one cent when tuning the open low E string. (Higher pitches will have proportionally higher beat frequencies for the same amount of mistuning, but if we're sticking to tuning open strings the high E is only four times the frequency of the low E for a guitar in standard tuning.)
Accuracy is a different quality than resolution. It's entirely conceivable that one manufacturer may calibrate its tuners more accurately than another. To the extent that you care about being in-tune with other tuners, this can be important. However, the tuning of an ensemble will only be as accurate as the least accurate tuner.