From the Peterson website:
Play two pure sine wave tones together, one at 440 Hz (440 cycles per second), one at 441 Hz. The clash you will hear is the one sound beating against (alternately reinforcing and canceling) the other. The repetition of this beating occurs at the difference in frequency of the original tones (441 - 440), which is 1.0 Hz or once every second. This level of out-of-tuneness is already plenty noticeable, but we don't want to be accused of stacking the deck in our favor, so let's use these numbers.
How does this relate to cents scaling? Actually it is a little complicated because the cents difference of the tones depends not only on the difference in frequency but on the size of the frequencies themselves. For this example, at 440 Hz, the cents difference is about 3.9 cents.
So, any tuner that can get you 4 cent (actually ±2 cent) accuracy should be fine, right? Well there's more to it. First, even though almost any tuner these days can have an internal accuracy at this level or greater, what is important to you is the accuracy you can reliably and quickly see on the display. Other tuning display systems generally do not give good results-or usually any results-within about ±3 cents.
But there is still more! Remember, cents scaling changes with the size of the frequencies themselves. If you are so picky that you need your High A (at 880 Hz) to be in tune with everything else (good musicians are funny that way!), any beat frequencies produced must still be below the 1 Hz-once a second-level. This already requires twice the cent accuracy as before or ±1 cents!!
Will it ever end? Not just yet!! We don't generally spend our time listening to laboratory-perfect sine waves (well, we have to sometimes, but we don't recommend it). Real musical tones include a unique and often extended series of overtones (additional sine waves at multiples of the pitch frequency) that gives each sound its timbre or character. Even a flute, which is considered to be relatively pure, has five or more overtones which are significant enough that, if any were to be artificially removed, would leave the tone noticeably wanting. In instruments ranging from guitar-especially with even a touch of over-drive distortion-to woodwinds and brass, overtones at 10 or 15 times the pitch frequency can be significant. This has a huge impact on the human ability to detect tuning.
To take an easy example, an electric guitar string usually has plenty of power in its fourth overtone (and beyond). In our previous example, tuning two strings-one at 440 Hz, one at 441 Hz would sound much worse than the sine wave case. This is because the audible overtones also beat and the frequency of these beats increase along with the frequencies of the overtones. To take a conservative example, the 4th overtone of the 440 Hz tone would be at approximately 2200 Hz and that of the 441 Hz tone would be at 2205 Hz. This makes a beat frequency of 5 Hz or 5 times a second: HORRID! To make these tones sound even reasonably good together, we should make sure that these overtones beat at less than the original "once every second". This will require the fundamental pitches to be tuned to 440 and 440.2 Hz. What's the cents accuracy required now? It's 0.78 cents or ±0.39 cents!
And what about the fact that the 1 Hz beat level that we used in the calculations is really much worse than what anyone with fleshy ears (as opposed to the tin type) would consider to be "in tune"?
And what about accounting for even higher overtones?
And playing higher notes?
And the fact that your tuning will drift with the pressure changes of your thumb in different guitar chord positions or the warmth of your breath into your horn as you play?
At least if you start at "real" 0.1 cent accuracy, you will be able to maintain satisfactory tuning through a few songs!
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When you account for the more accurate starting pitch; to which you then adjust intonation--, it can become VERY apparent when playing with other instruments which are actually playing in tune, like strings, horns, or keyboards, that although you may have intonated the guitar to itself-- it's still out, because the starting pitch wasn't accurate to begin with...
