Equal Temperament
Why guitars have straight frets, and how the 12-tone equal temperament system allows us to play in any key at the cost of perfect consonance
Just Intonation and Consonant Ratios
When we pluck a string, it vibrates at both its fundamental frequency and at integer multiples: 2×, 3×, 4×, etc. These are the harmonics or overtones.
Just intonation systems build the intervals of a scale from these natural ratios. When two notes have frequencies in simple ratios, their harmonics align and they sound consonant to the human ear. The simpler the ratio, the more consonant the interval: 2:1 (octave) sounds more resolved than 3:2 (fifth), which sounds more "resolved" than 5:4 (major third).
In a just intonation system, the intervals are perfectly consonant, but only in one key.
The Problem with Just Intonation
If you tune a piano (or set guitar frets) using just intonation in, e.g., the key of C, the intervals in other keys will be wrong.
For example, in C-based just intonation, D is tuned as 9/8 of C, and A as 5/3 of C. The D-to-A interval is (5/3)÷(9/8) = 40/27 ≈ 1.481, but a pure fifth is 3/2 = 1.5. That D-A "power chord" interval "fifth" is 22 cents flat. This is enough to sound noticeably sour!
The Circle of Fifths Doesn't "Close"
Stack 12 perfect fifths (3:2 ratio each):
Stack 7 octaves (2:1 ratio each):
They don't match! This ~1.4% discrepancy is called the Pythagorean comma.
This means if you tune by pure fifths starting from C, by the time you get back to C (via B♯), you'll be noticeably sharp. How do we solve this problem?
The Solution: Equally Distributed Errors
12-tone equal temperament (12-TET) solves this by making every semitone exactly the same size: the 12th root of 2.
Each fret multiplies the frequency by this ratio
Now 12 semitones = exactly one octave (1.05946¹² = 2), and every key is equally "in tune", or more accurately, equally out of tune by tiny amounts.
Benefits of Equal Temperament
- Modulate freely between any keys
- Transpose without retuning
- Straight frets on guitars
- Standardized instrument manufacturing
Drawbacks of Equal Temperament
- Major 3rds are 14 cents sharp
- Perfect 5ths are 2 cents flat
- No interval except octave is pure
- Subtle "beating" in sustained chords
Guitar Fret Placement
For a vibrating string, frequency is inversely proportional to length:
To raise the pitch by one semitone, we need to shorten the string by a factor of ¹²√2. Each fret is placed at:
Because this formula only depends on n (the fret number) and L (the scale length), and not on which string you're playing, all six strings can share the same fret positions.
Measuring the Difference: Cents
Musicians measure small pitch differences in cents. One semitone = 100 cents, so one octave = 1200 cents.
The human ear can typically detect differences of about 5-10 cents. The major 3rd in equal temperament is about 14 cents sharp compared to just intonation. This is noticeable if you listen carefully, but acceptable for most music.
Summary
- Just intonation uses simple frequency ratios (3:2, 5:4, etc.) that sound perfectly consonant
- The problem: these ratios don't divide the octave evenly, making key changes impossible
- Equal temperament divides the octave into 12 equal semitones (ratio = ¹²√2)
- This distributes small errors across all intervals, making all keys equally usable
- Guitar frets are placed according to the equal temperament formula, allowing straight frets to work for all strings and keys