If you start from a given note, let’s say middle “C” on your piano, and follow the natural sequence of harmonics, also called the [overtone] series, you will eventually obtain all the 12 notes you know (in western music) and, consequently, all existing intervals you probably learned with your basic piano lessons (ex.: minor and major third, natural or flat second etc.)
Now, following this natural sequence, also called [harmonic series] in music, you will only get intervals of ‘pure intonation’ or [just intonation]. For instance, if you look to establish the fifth of your starting note “C”, you’ll obtain a “G”. Continuing this logic, you can get the fifth of the “G” to now get a “D”, and so on, doing the fifth of the fifth etc. You will then do that 12 times before you finally end up on the “F” to close the cycle (obviously going through “F#; C#…A#”). Now, if you look again to go up a fifth from the last note you obtained (“F”), (to your amazement), the interval “F – C” will sound completely dissonant or out of tune.
That’s why we need to do some systematic compromising in the whole cycle. Tempering the intervals enable musical possibilities that are impractical using 'just intonation'. The most widely known example (in the western cultures) of this is the use of [equal temperament] to address that problem, allowing for consistent tuning of piano (and other tempered instruments), thus enabling musical composition in, and modulation among, the various keys.
That being said, by preserving only the octave as a perfect interval, the harmonic 2:1 ratio, the so-called ‘perfect fifth’ would have to be slightly closer and respectively a perfect fourth a bit wider. A major third interval on the other hand, would suffer an even more ‘inflicted stretch’ from the pure intonation, to fit in our new model.
You may hear an [example here]. First you'll hear two 'major thirds', followed by two 'major triads'. The first one being played in each case is respectively in 'equal temperament' and the second in 'just intonation'...
Although there is international agreement to establish a standard equal temperament, however, such a ‘temperament’ does remain subjective to a certain extent. Back in the 17th or 18th century, as you have probably noticed, composers (like Bach, Beethoven, Hayden, etc.) were, for example, making a “concerto in A Maj”. For the only reason being to use a temperament as close to the ‘Pure or Just intonation’ as possible. (see Note*(1) below )
Later on, many people came with different theories on how to achieve the ‘perfect' [temperament]. That’s why certain customers, principally with specialized classical training, would prefer a specific ‘temperament’ tuning (ex.: 1799 meantone, Anton Bemetzrieder Pythagorean or equal-beating Jean-Jacques Rousseau to name a few)...
* If asked, we may discuss possibilities. *
Note*(1) : That’s why a “D #” actually did sound different than a “E flat”. Depending on the major key being used, most of ‘non-tempered’ instruments (violins, cellos etc.) were able to deliver that differentiation.