TuneHaven Temperaments, and why pure fifths disagree with a piano
Equal temperament makes every key equally usable by making every interval slightly wrong. Why that compromise exists, and when you would want a different one.
Here is the problem every tuning system exists to solve.
Stack twelve pure fifths on top of each other, each one an exact 3:2 frequency ratio, and you should land back on the note you started from, seven octaves up. You do not. You overshoot, by about a quarter of a semitone. This gap is called the Pythagorean comma, and it is not a measurement error. It is arithmetic: powers of three never exactly equal powers of two.
So you cannot have pure fifths and pure octaves and twelve notes that close the circle. Something has to give. A temperament is a decision about what.
Equal temperament
Take that excess and spread it evenly across all twelve fifths. Every semitone becomes exactly the same ratio, the twelfth root of two.
What you get is that every key works and sounds identical in character, transposing is free, and a fretted instrument becomes possible at all, since frets are fixed and cannot know what key you are in.
What you pay is that no interval except the octave is pure. Equal-tempered fifths are about two cents narrow, which nobody notices. Equal-tempered major thirds are about fourteen cents sharp, which you absolutely can hear as a slight restlessness in a sustained major chord.
This is the default in TuneHaven and it is what almost everyone means by “in tune”.
Why your violin disagrees with the piano
String players tuning by ear listen for the beating between two adjacent strings to disappear, which means they are tuning pure fifths, not equal-tempered ones.
Each pure fifth is about two cents wider than the tempered one. That is inaudible on its own. But a violin stacks three of them, from E down to G, and a cello or viola does the same from A down to C. Three times two cents is six, and the outer string ends up measurably off from where a piano puts it.
This is why the advice for playing with a fixed-pitch instrument is to tune each string to the tuner individually rather than chaining fifths by ear. Alone or in a string section, pure fifths sound better and you should keep doing what you are doing; the section will bend to match itself.
Just intonation
Tune the intervals as small whole-number ratios and they lock. A just major third is a 5:4 ratio and it sounds startlingly clean next to the tempered version, with the beating gone entirely.
The catch is that just intonation is only just relative to one key. Retune a keyboard for pure C major and the same instrument in F# major is unusable. Ensembles that sing or play unfretted instruments drift toward just intervals naturally, adjusting in the moment, which is a large part of why a good choir sounds different from a keyboard playing the same chords.
When to change the setting
Most players never should. Equal temperament is the right default for anything involving frets, keyboards, or other people.
Reasons you might:
- Playing period repertoire on instruments set up for it, where a historical temperament is part of the sound
- Matching a specific keyboard that has been tuned to something other than equal
- Solo work in one key where you want the thirds to lock
- Studying the difference, which is a legitimate reason on its own and easier to hear than to read about
Temperament selection is free in TuneHaven, as is A4 calibration. They are independent: A440 decides where the notes sit, temperament decides how they are spaced.