Keep the octave at exactly 2:1. Split it into N equal steps. Then see how close those steps land on the intervals that sound consonant as pure whole-number ratios. Slide N up from 1 and watch where everything first fits.
| Interval | Ratio | Just | Nearest step | Tempered | Error | Listen |
|---|
Each bar is the largest error among the checked intervals, in cents (1/100 of a 12-EDO semitone). Bars that clear the dashed tolerance line are filled. Hatched bars put two intervals on the same step, or an interval on the unison or octave. Click a bar to jump to it.
Why the fifth dominates. 3:2 is the simplest ratio after the octave, and the fourth (4:3) is its inversion, so their errors always mirror each other. Most small N miss the fifth badly. 5 and 7 get within about 18 and 16 cents of it, but their thirds are far off.
What 12 buys. Twelve is the first N where the fifth is nearly exact (2 cents flat) and every target gets its own step. The thirds land 14–16 cents off, the minor seventh about 18 flat, and the major seventh about 12 sharp. That residue is the price you hear when you A/B the major chord.
Past 12. 19 gets the thirds within about 7 cents. 31 is better still, and 53 is close to pure on everything here. Each costs many more keys, frets, or holes per octave.