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Virtually all the sources I can find claim that the intervals between adjacent guitar strings in standard tuning (EADGBE) should be perfect fourths, except for one major third. However, since a perfect fourth is 4/3 and a major third is 5/4, this means the interval between the two E strings would be (4/3)**4 * (5/4) = 3.950... which is about 2% flat from the perfect interval, 4 (two perfect octaves).

Wikipedia has a table of "String frequencies of standard tuning" in Hz. If you do the math, you'll find that none of the intervals are actually exact. All the "perfect" fourths are sharp and the major third is also sharp. The really funny thing is they are all sharp by differing percentages. ~~It doesn't seem to be perfectly 12-TET either (semitone = 1.05946..).~~ Actually, maybe it is 12-TET within the given precision.

Is the "perfect" interval tuning just a simplification? Also, I realize this applies to all stringed instruments, not just the guitar.

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    'Perfect thirds' don't exist. On guitar, between G and B is M3. Maybe our ears compensate a little for the 'out of tune'. Also bear in mind that each string is a slightly different length to comensate for intonation caused by the fretwires being perpendicular to all the strings. – Tim Sep 19 at 11:35
  • Are these sources referring to a 4 string guitar? – ggcg Sep 19 at 11:38
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    You are correct, we do not tune the guitar like that. We usually start with relative tuning but adjust by comparing harmonics and open string octaves. In other words the high e is tuned rel to the low E harmonic and intermediate strings adjusted until (at least for acoustics) resonances with open strings are observed. – ggcg Sep 19 at 11:42
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    @Tim Sure a perfect third exists - just climb up the harmonic ladder a bit. It just doesn't "exist" in tempered scales. – Carl Witthoft Sep 19 at 12:43
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    you're confusing the term "perfect" with the term "just". In a sense, a "perfectly in tune interval" would be the just interval, but that's not what the word "perfect" means in a musical context. – Some_Guy Sep 20 at 21:27
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In just intonation, you'd be correct. However, in order for the interval between the top and bottom strings to be exactly two octaves, some compromise needs to be made. (That is, given that the intervals between the open strings are as you say, one major third and all the rest fourths.)

As you say, in just intonation, a perfect fourth is 4:3 and a major third is 5:4. Thus four perfect fourths and a major third are (4/3)^4 * (5/4) = (4^3*5) / (3^4) = 4 * 80/81. Two octaves is a straight 4, so we are too narrow by 81/80, which is a syntonic comma. In terms of cents (1200 cents = 1 octave), a syntonic comma is 21.506 cents. Thus, we must widen our fourths and our major third.

A just perfect fourth (4:3) is 498.045 cents. A just major third (5:4) is 386.314 cents. If we adopt 12-equal temperament, we widen each perfect fourth by 1.955 cents to 500 cents, and widen the major third by 13.686 cents to 400 cents. This widens by a total of 4*1.955 + 13.686 = 21.506 cents as required.

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  • You are correct. I'm beginning to think that the intervals in WP table I referred to are actually 500 cents (2**(500/1200)) within precision. So I should tune my guitar in 12-TET then? Bye perfect just fourths, it was nice knowing you! :) – Emanuel Landeholm Sep 19 at 12:14
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    Isn't this a "12 TET vs Pythagorean scale" issue, and not specific to the guitar? – mkorman Sep 19 at 15:30
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    @mkorman It is indeed not specific to the guitar. I'd say it's an issue of the Pythagorean scale vs Just Intonation. The Pythagorean scale gives us the 81 (it can only produce ratios of powers of 3 to powers of 2). JI gives us the 5. The combo of the two gives us the syntonic comma 81:80. Trying to equate the two entails tempering the s.c. out, hence mean-tone temperaments. These are a family of which 12-equal is one member. You choose how much to narrow a fifth by, then every tone is 2 fifths minus an octave, and a major third is 2 tones. – Rosie F Sep 19 at 16:09
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    It gets worse when considering that on guitar, the frets are straight, so in fact even if you just tune 2 separate strings to each other (perfectly) - playing a cord with the two further up the fretboard - you are back to out of (perfect) tune again. – Stian Yttervik Sep 20 at 13:28
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This applies to all stringed instruments, not just fretted, or even strummed ones. The tuning compromises are messier with a guitar because of the mix of fourths and thirds between strings.

For us bowed instrument players, we tune the open strings as close to perfect fifths (or fourths, for the double bass) so that open strings will resonate "cleanly" against other open strings. We then do some compromising when playing, say, a double-stop with one fingered and one open string to get clean overtones.

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  • Yes, of course... I was being careless with terminology. I actually meant stringed instruments. But I think the piano is a special case since you don't really walk up and down the neck, so to speak... The piano is like a guitar with 88 open strings. – Emanuel Landeholm Sep 19 at 12:52
  • I note that string players playing with unequally tuned keyboards often tune each string to the keyboard, since the fifths on the keyboard will generally be more than 2 cents out of tune. – phoog Sep 23 at 16:38
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Historically, instruments with a chromatic scale of fixed frets have always been tuned in the best approximation to Equal Temperament that the makers could achieve.

That includes guitars and their relations, but not lutes, where the frets were simply loops of gut tied round the neck of the instrument and therefore adjustable by the performer to play in any desired tuning system.

The earliest written records say that each fret was placed 1/18 of the distance between the preceding fret and the bridge. A simple-minded calculation says that is about 1 cent smaller than an exact ET semitone, but that ignores the effect of the height of the action on the intonation of a real instrument which tends to correct the error.

So whatever some modern "guys on the internet" think, real luthiers have known better for several hundred years already, and if you tune an open string and a fretted string in unison, you will automatically get the correctly tempered fourths.

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The issue here is in interpreting “perfect fourth” as meaning “just intonated (perfect) fourth”. The interval from C to F (or E-A, A-D...) is a perfect fourth no matter what intonation is. Here, perfect distinguishes the interval from augmented and diminished fourths, and says nothing about the intonation.

Nowadays at least, guitars are tuned equal temperament, so the intervals across the 6 strings do add up to two octaves.

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  • It seems to be generally accepted that fretted instruments were the vanguard of equal temperament, since it's not easy to temper a fretted instrument otherwise. – phoog Sep 23 at 16:39
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"Perfect fourth" is not a mathematical calculation, but is used in context of music theory, where a perfect fourth would be 5 semitones. A fourth that is not perfect, would be augmented or diminished. Thirds are not described as perfect, instead they are major or minor.

Also, it should be noted that our (Western) scale is mathematically not as simple as dividing an octave into equal parts. There are many different models for calculating the scale. Pythagoras and Vallotti for instance had their ideas. The well tempered tuning is indeed a more equal tuning, but not all instruments use this tuning. String players often tune their instruments to a more 'natural' theme, having the perfect fifth intervals of their strings according to the natural fifth overtone.

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    However, if you tune a guitar naturally (all fourth are exactly 4/3), you'll get problems with many chords sounding quite a bit off. Especially A and D based chords because they have an octave formed by the open A/D string and the 3rd fret on the H/E string. This octave will be significantly too large, and listeners with a good musical ear will notice. Thus, in order to get a sensible guitar tuning, the fourths must be stretched a bit. – cmaster Sep 20 at 9:24
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The perfect fourth in equal temperament is 2 to the power of (5/12), or 1.334839..., not 4/3 = 1.333333...

That said, stringed instruments are not tuned exactly to equal temperament anyway. Firstly, consider the stretch tuning phenomenon exhibited by pianos. Higher notes on the keyboard are sharp relative to the equal temperament math, lower notes slightly flat. This is because the fundamentals of notes in higher octaves are tuned not to clash with the harmonics of notes in lower octaves. The harmonics of a non-ideal, real-world string are sharp compared to what the math says for an idealized string.

Guitarists follow various methods of tuning, some of them even personalized. What comes into play also is the imperfect intonation of the guitar. There are tuning methods that involve matching octaves on non-adjacent strings. For instance, the D string might be tuned by fretting an E, and tuning that against the open E string one octave below. Also, if you naively tune the open strings, the notes above the twelfth fret will likely not be in tune due to intonation imperfections. There are tuning methods involving tuning notes in the middle of the neck to create the best compromise over the fretboard. I usually begin tuning a guitar, or check its tuning, using the 440 Hz A on the B string at the 10th fret.

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