In temperaments, rational intervals are often represented by a shared pitch.

Finally, the quality of a tuning may make tonal relationships more or less explicit or more or less ambiguous. That makes a modulation possible that is impossible in 12.

For example, in 15-tone equal temperament, the sum of three 'tritones' is not an octave and a tritone as in 12, but rather an interval that one recognizes at once as a twelfth. From one tuning to another, sequences of intervals may end up in very different places. The second reason is that alternative tunings may introduce tonal relationships that are simply not available in 12-tone equal temperament. Conversely, when working with instruments which have spectra that are not composed of simple harmonic relationships, tunings which map those more complex relationships may better fit into and represent that timbre. In working with tones with simple harmonic spectra, Just Intonation or a tuning with intervals close to Just will yield intervals which have a quality, to my ears, clarity, quite different from that heard when the tuning has intervals which are distant from Just. An interval can be tuned so that the component parts of the spectra of the individual tones coincide or deviate when combined, and the degree to which they deviate may be controlled. The first is that instead of accepting the quality of an interval as given, a musician has access to a whole range of qualities, from smooth to beating slowly to beating roughly, and from clear to vague in identity. Paying attention to tuning is useful in three regards for me as a composer and musician.

I usually have a nice long answer to the question, but here's more or less what I said today in an attempt at a short answer: In the course of recording a radio interview, I was asked why anyone would want to bother with tunings other than 12-tone equal temperament.

Harmony assistant tunings