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  I learned this method of temper-tuning a guitar while I was a professional piano tuner in Ohio
about 24 years ago. This concept is used by the best piano tuners. If you don’t care a farthing
about the theory and want to begin using the method, dispensing with these arcane facts,
skip to "The Method".
  A vibrating guitar string produces a series of tones called the fundamental and it's harmonics (also called partials or overtones). The fundamental is the lowest note--the pitch produced by the entire string length in motion.
  A vibrating string moves in a complex but predictable fashion. In addition to the entire string moving in its parabola (arc), the string can be divided conceptually into halves. Each half of the string also produces its own (almost independent) parabola and, thus, a pitch called a partial. Since the length of this part of the string is exactly half, it vibrates at twice the speed of the whole string and thus produces a pitch that is an octave higher. (E.g., 440=A; 880=A an octave higher). Divide the string into 3 equal lengths (the 7th fret is 1/3 the length of the string!) and you have another section of string with its own vibrating pattern and pitch (i.e., another partial). Divide the string into four equal lengths (at--you guessed it--the 5th fret) and you have yet another partial. These partials are always there, but when you just touch the string at these frets without actually fretting the string, you dampen the fundamental and isolate the partial at that fret, giving the illusion that you have "produced" the harmonic. Pluck an open string and listen for the partials along with the fundamental.
  Inherent Inharmonicity
  This is the phenomenon that necessitates tempered tuning in a stringed instrument. As we saw above, the fundamental of a string produces a tone along with, say, the 1st partial of that string (harmonic at the 12th fret). Let us assume the note is A (oscillating at 440 cycles). In theory, the 1st partial will be oscillating at 880 cycles per second. But it is not! It oscillates slightly faster-- i.e., sharper. Why? The answer lies in the physical properties of a vibrating string. The proportion of stiffness to diameter in a half-section of a string is greater than the proportion in the whole string. And the shorter the section, the greater the difference in the proportion. In any string length (whether fretted or open), the partials are all progressively sharper than the fundamental. This is Inherent Inharmonicity. Gosh, this all happens when you play that note? You bet it does!
  Mathematical Intervals
  Adding to the problem of properly tuning a guitar is this quirk concerning intervals of mathematical purity. You see, each given chord wants intervals of mathematical purity for that particular chord! The mathematically pure intervals of a "C" chord, for example, are different that for a "D" chord. The differences are slight but they are real. That is why when you tune your guitar to "D" while fretting a "D" chord, the "D" will sound great but the other chords will sound slightly wrong. (This has probably driven you to distraction--now you know why!)
   When a string is fretted, the string is slightly stretched as it is pushed down onto the fret. This, of course, minimally sharps the string. To compensate for this slight sharping during fretting and to compensate for some of the Inherent Inharmonicity, the bridge and saddle are traditionally moved away from the nut by a carefully calculated distance (on the Ryan Mission, this amount is .140"). But this cannot fully compensate for all of the above hurdles to excellent intonation. For that, we need Tempered Tuning.
  Tempered Tuning
  This is a method of tuning that addresses all of the above factors. In essence, this method takes the inharmonicity of all six strings and the slight mathematical discrepancy between the whole scales and divides the variation equally among each string. This means that while no one chord or interval is perfect (and it is physically impossible for them all to be perfect), they are all only slightly off. But off by such a small, consistent amount that no ordinary ear can detect any dissonance. What follow are the steps to achieve this tempered tuning. You can learn it quickly. Master it and you will tune your guitar quicker and slicker than the other kids on the block! Fail to master it, and studies show you will spend 7.52 years of your life tuning your guitar.
  Tuning Notes
  A) When you tune the following fretted notes to the harmonics, tune them "beatless"-- i.e., without any hint of "rolling" or pulsating as the two notes synchronize. When two notes get closer, their "beating" slows down until it disappears altogether when they are perfectly in tune. This is very important! This is the skill to be gained!
  B) In each step below, pluck the harmonic first. Then fret and pluck the designated string.
This allows you to hear both notes simultaneously. Then tune the appropriate string.
  1. Tune the D string to a known source
  2. Pluck the 12th fret harmonic of the D then tune the G (fretted at the 7th fret) to this harmonic.
  3. Pluck the same 12th fret harmonic of the D then tune the B (fretted at the 3rd fret) to this harmonic
  4. Pluck the 12th fret harmonic of the G and tune the High E fretted at the 3rd fret to this harmonic
  5. Tune the 12th fret harmonic of the A to the G fretted at the 2nd fret (pluck the harmonic first!)
  6. Tune the 5th fret harmonic of the Low E to the High E open (pluck the harmonic first!)
  NOTE | To apply the tuning method to alternate tunings, all you have to do is find the proper fretted note on the string you are tuning and tune it beatless to a 12th fret harmonic on a string below it. Easy as pie.
  Final advice | Take note that old strings are more difficult to tune than new strings. This is because of uneven stretching of the string and the subsequent erratic vibration patterns. In some instances, old strings are impossible to tune correctly. If you have difficulty achieving good intonation, change strings.
  For easier restringing and faster tuning, the Planet Waves Peg Winder comes with a built in string stretcher...click here