Music was the first experimental science. Every idea on this page can be heard, and every number can be checked with a calculator. This is where the Guitar half and the Academy half turn out to be the same subject.
One unit appears throughout: the cent. An octave is 1200 cents, an equal-tempered semitone is 100, and the interval between two frequencies is 1200 × log₂(f₂/f₁).
A pure (just) interval is a simple whole-number frequency ratio. Your guitar's frets are equal-tempered: every semitone is exactly 21/12, so every interval is a power of an irrational number and none of them except the octave is pure. Compare them yourself.
| Interval | Pure ratio | Pure (¢) | Tempered (¢) | Tempered is… | Hear it |
|---|---|---|---|---|---|
| Minor 2nd | 16:15 | 111.73 | 100 | 11.73 ¢ narrow | |
| Major 2nd | 9:8 | 203.91 | 200 | 3.91 ¢ narrow | |
| Minor 3rd | 6:5 | 315.64 | 300 | 15.64 ¢ narrow | |
| Major 3rd | 5:4 | 386.31 | 400 | 13.69 ¢ wide | |
| Perfect 4th | 4:3 | 498.04 | 500 | 1.96 ¢ wide | |
| Tritone | 45:32 | 590.22 | 600 | 9.78 ¢ wide | |
| Perfect 5th | 3:2 | 701.96 | 700 | 1.96 ¢ narrow | |
| Minor 6th | 8:5 | 813.69 | 800 | 13.69 ¢ narrow | |
| Major 6th | 5:3 | 884.36 | 900 | 15.64 ¢ wide | |
| Minor 7th | 9:5 | 1017.60 | 1000 | 17.60 ¢ narrow | |
| Major 7th | 15:8 | 1088.27 | 1100 | 11.73 ¢ wide | |
| Octave | 2:1 | 1200.00 | 1200 | exact |
Pure ratios shown are the standard 5-limit choices; some intervals have other candidates (a minor 7th can also be 16:9, for instance). Both notes start on A3 = 220 Hz and are played as plain sine tones.
What to listen for. On sustained tones the pure major 3rd is smooth and the tempered one shimmers. That shimmer is beating, explained next. It is part of why heavy distortion favours power chords: distortion adds strong upper partials and combination tones, which line up for a nearly pure fifth but clash for a third that is 14 ¢ off.
Two tones at slightly different frequencies drift in and out of phase. You hear the loudness pulse, and the pulse rate equals the difference in frequency: 440 Hz against 443 Hz beats 3 times a second. Tune until the beating slows to nothing and the two strings match.
The same physics shows the tempered fifth is impure. A pure E5 (660 Hz) against a tempered E5 (659.26 Hz) beats about 0.74 times a second. Lower on the neck it is slower still: about 0.19 Hz for the fifth above A2. That is real, but slow enough that most players never notice.
440.0 Hz + 443.0 Hz → 3.0 beats per second
Stack pure fifths (3:2) from C: G, D, A, E… After twelve you should be back at C, seven octaves up. You aren't. (3/2)12 = 129.746…, but 27 = 128. You overshoot by 23.46 cents, the Pythagorean comma.
The proof that it can never close is one line: 312 is odd and 219 is even, so (3/2)12 can never equal 27. Every tuning system is a decision about where to hide this gap. Equal temperament spreads it evenly, shaving 1.955 cents off each of the twelve fifths, which is why your guitar's fifths are very slightly narrow.
A plucked string vibrates at its full length and at halves, thirds, quarters and so on, all at once. Those partials, at 1×, 2×, 3×… the fundamental, give a note its tone. Touch the string lightly over a node and you silence the fundamental, leaving a natural harmonic.
| Harmonic | On the A string | Sounds as | vs. tempered | Touch at fret | Hear it |
|---|---|---|---|---|---|
| 1 | 110 Hz | A2 (open string) | 0 ¢ | – | |
| 2 | 220 Hz | A3: octave | 0 ¢ | 12 | |
| 3 | 330 Hz | E4: octave + 5th | +1.96 ¢ | 7 (or 19) | |
| 4 | 440 Hz | A4: two octaves | 0 ¢ | 5 (or 24) | |
| 5 | 550 Hz | C♯5: major 3rd | −13.69 ¢ | ≈ 3.9 (just behind the 4th) | |
| 6 | 660 Hz | E5: 5th | +1.96 ¢ | ≈ 3.2 | |
| 7 | 770 Hz | ≈ G5: flat minor 7th | −31.17 ¢ | ≈ 2.7 | |
| 8 | 880 Hz | A5: three octaves | 0 ¢ | ≈ 2.3 |
Try it on your guitar: the 4th-fret harmonic on the A string is a C♯ about 14 cents flat of an equal-tempered C♯. That's the pure major third the table above talks about, built into the string itself.
Mersenne's law (1636): frequency f is inversely proportional to the vibrating length L, and proportional to the square root of the tension T divided by the mass per unit length μ. Each part of the formula is something you do with your hands.
Halving L doubles f, so the octave is at the middle of the string. Each fret shortens the string by the same fraction (1 − 2−1/12, about 5.6%), so frets get closer together up the neck.
| Fret | From nut |
|---|---|
| 1 | 36.4 mm |
| 5 | 162.5 mm |
| 7 | 215.5 mm |
| 12 | 324.0 mm (half) |
| 24 | 486.0 mm (¾) |
Because f ∝ √T, tension has to rise with the square of the frequency. Each extra semitone costs more effort than the last.
+26.0% tension
Ideal string, ignoring the slight stretch a bend causes. The old version of this page said "Tension² = Frequency". That's backwards: it is frequency² ∝ tension.
At the same length and tension, a string with 4× the mass per length sounds an octave lower (√4 = 2). That's why low strings are thick and wound: to get low notes without a floppy, slack string.
The same law explains why a heavier set of strings at the same tuning is under more tension and feels stiffer to bend.
Idealisation. Real strings are slightly stiff, so their upper partials run sharp of perfect whole-number multiples. That's why guitars need saddle compensation and why piano tuners stretch octaves. The tuner on this site filters those partials out so they don't skew the reading.