Tuning and Temperament

Stack twelve pure perfect fifths at the ratio 3:2. Compare that result with seven pure octaves at 2:1. They nearly meet, but the stack of fifths is higher by about 23.46 cents. That small gap is the Pythagorean comma.

Because the cycle does not close, a fixed keyboard must alter something. It can preserve selected fifths, selected thirds, particular keys or equal transposition, but not all of them at once.

Well temperament is a family of systems, not one tuning. Equal temperament is one specific solution.

An octave doubles frequency. Divide that multiplication into twelve equal steps and each semitone has the ratio 2^(1/12), about 1.05946. Twelve steps multiply back to exactly 2.

The equal-tempered fifth is about two cents narrower than 3:2. Its major third is about fourteen cents wider than 5:4. Those differences introduce slow beating into sustained, simple tones, but the same fingering and interval pattern can move through every key.

Just intervals can sound exceptionally smooth when the harmony and sustained tone make their ratios clear. The problem is that the best frequency for one note depends on its harmonic role.

In C major, an E tuned as a pure 5:4 third above C will not necessarily be the best E for another root. Singers and string players can adjust continuously; a fixed-pitch instrument needs predetermined notes or active retuning.

Ideal equal temperament describes pitch classes. Real piano strings are slightly stiff, which pushes upper partials sharp. Tuners often widen octaves so the partials of low and high notes align more convincingly.

This stretch is an instrument-specific response to inharmonicity. It is separate from the choice to divide the theoretical octave into twelve equal steps.

A4 = 440 Hz is a widespread modern reference, not a law of acoustics. Ensembles may use 442 Hz or another agreed value. Changing the reference shifts every note together; it does not change the interval structure of the temperament.