Guitar tuning "perfect" fourths?
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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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.
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.
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.
"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.
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.
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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