From a legend about blacksmiths' hammers to the reason your guitar is tuned E A D G B E. Every idea here is something you can hear: tap the buttons as you read.
The story goes that Pythagoras, about 2,500 years ago, walked past a smithy and noticed that some hammer blows rang together sweetly while others clashed. He went in, weighed the hammers, and found weights of 12, 9, 8 and 6. The sweet-sounding pairs were in simple ratios:
12 : 6 = 2 : 1 is the octave, 12 : 8 = 3 : 2 is the perfect fifth, 12 : 9 = 4 : 3 is the perfect fourth, and the gap between the middle hammers, 9 : 8, is the whole tone. Those few small numbers are the foundation of almost every scale on Earth.
Sound is vibration. The A string on your guitar vibrates 110 times per second (110 Hz). Double that, 220 Hz, and you get another A, one octave up. Double again: 440 Hz, the A that orchestras tune to.
Why do two different frequencies sound like "the same note"? Look at the waves: the higher one fits exactly twice into the lower one. Every second peak lines up, so the two blend so completely that our ears group them into one note name. That's why the alphabet of notes repeats: A B C D E F G, then A again.
The fifth (3 : 2) also lines up, just a little less often: every 3 cycles of the upper note match 2 of the lower. The simpler the ratio, the smoother the blend. That's the whole secret of consonance.
A plucked string doesn't vibrate in just one way. It swings as a whole, but at the same time in two halves, in three thirds, in four quarters and so on. Each of these partials (or harmonics) is a whole-number multiple of the lowest frequency. You don't hear them as separate notes; together they make the tone of the string.
| Partial | Frequency | Note | Interval from the one below | vs. piano tuning |
|---|
Read the right-hand column: the ratios 2 : 1, 3 : 2, 4 : 3, 5 : 4, 6 : 5 hand us the octave, fifth, fourth, major third and minor third, already sorted from most to less consonant. Nature's own chord, partials 4 : 5 : 6, is a major triad. The 7th partial is a "blue" flat seventh that doesn't match any piano key.
Touch a string lightly (don't press) right above a fret wire and pick it: you get a glassy, bell-like natural harmonic. Your finger stops the string from vibrating as a whole, so only the partials that have a still point (a node) there survive. The magic frets are simply where the string divides into equal parts:
| Touch at | Divides the string | Partial | Sounds | Low E gives |
|---|---|---|---|---|
| 12th fret | in half | 2 | 1 octave up | E3 |
| 7th (or 19th) fret | in thirds | 3 | octave + fifth | B3 |
| 5th (or 24th) fret | in quarters | 4 | 2 octaves | E4 |
| 4th, 9th or 16th fret | in fifths | 5 | 2 octaves + major third | G♯4 (a bit flat) |
The 5th-fret harmonic of one string (4 × its frequency) and the 7th-fret harmonic of the next string (3 × its frequency) are the same note when the strings are a perfect fourth apart. Play them together and tune until the wobble (the "beating") stops.
After the octave, the fifth is the strongest bond between two notes. So here's a recipe almost every musical culture discovered: start on a note and keep going up by fifths (folding back into one octave), collecting the notes you land on. Try it, starting from F:
Five fifths give the pentatonic scale, the scale of folk music, the blues and countless guitar solos. Seven give the diatonic scale: the white keys, the major and minor scales and all their modes. The next five fifths are exactly the black keys, which form a pentatonic scale of their own. Twelve gives the chromatic scale, every fret on your guitar.
Keep stacking fifths and after 12 steps you're (almost) back where you started. Arrange those 12 notes in a circle and you get the most useful map in music. Each key is a slice of 7 neighbouring notes: C major is F C G D A E B. Step one notch clockwise to G major and you swap just one note (F becomes F♯). That's why neighbouring keys sound related and why songs so often move between them.
The circle doesn't quite close. Twelve pure fifths, (3/2)12 ≈ 129.75, overshoot seven octaves, 27 = 128, by a ratio of about 1.0136: 23.46 cents (a cent is 1/100 of a semitone). This tiny gap, the Pythagorean comma, means you cannot tune all twelve notes with pure fifths and still play in every key.
The modern solution is 12-tone equal temperament: divide the octave into 12 exactly equal semitones, each a frequency ratio of 21/12 ≈ 1.0595. Every key sounds the same, which is why your guitar's frets are where they are. The price: every interval except the octave is slightly impure.
| Interval | Pure ratio | Pure (cents) | Equal tempered | Difference | Hear it |
|---|---|---|---|---|---|
| Octave | 2 : 1 | 1200.00 | 1200 | 0 | |
| Perfect fifth | 3 : 2 | 701.96 | 700 | −1.96 (barely) | |
| Perfect fourth | 4 : 3 | 498.04 | 500 | +1.96 | |
| Major third | 5 : 4 | 386.31 | 400 | +13.69 (audible) | |
| Minor third | 6 : 5 | 315.64 | 300 | −15.64 |
Listen to the major thirds: the equal-tempered one has a fast shimmer ("beating") that the pure one doesn't. That's the compromise we all live with. It's also why heavy distortion makes full major chords sound muddy, while power chords (root and fifth, almost pure) stay clean.
Equal temperament was worked out mathematically around 1600 (by Zhu Zaiyu in China and Simon Stevin in the Netherlands), but only became the standard for keyboards in the 1800s. Fretted instruments, with frets straight across all strings, pushed in that direction much earlier.
7 notes from 7 stacked fifths. Sorted in pitch order, the gaps form the famous pattern W W H W W W H (whole and half steps). C D E F G A B C.
Start the same 7 notes on a different note and you get the modes: on D it's Dorian, on E Phrygian, on A natural minor (Aeolian).
5 notes from 5 stacked fifths: C D E G A. It has no half steps, so no two notes in it clash. That's why you can noodle over almost any chord in the key and never hit a "wrong" note.
The same 5 notes starting on A give A minor pentatonic (A C D E G), the backbone of rock and blues soloing.
See them on the fretboard: C major, A minor pentatonic, or any of the 19 scales in the scale explorer.
Look at the gaps between the open strings: they're all perfect fourths (5 frets), except one major third (4 frets) between G and B.
A fourth is 5 frets. With one finger per fret you can cover 4–5 frets without moving your hand, which is exactly enough to reach every note up to the next string's open note. Every position on the neck is a complete little keyboard. Fifths (7 frets, like a violin) would need impossible stretches on a long guitar neck; thirds would shrink the range and need more strings.