Many modern digital tuners offer the ability to tune by various historical temperaments, making the use of historical tuning both easy and fast. However, it is important IMO to understand how the different tuning methods came about, the math behind them and the harmonic implications they entail.

Trying to explain historical tuning using the circle of fifth is actually incorrect, even when it pertains to keyboard instruments. The concept of a cycle of fifths is relatively recent, is not compatible with harmonic theory before the 18th century, and is useless for explaining meantone for example.

It's also important to take into account that the different tuning methods go hand in hand with specific kinds/styles/periods of music. Therefore it's important to present them in a historical context. The website touches on this briefly, but a more comprehensive discussion can be found elsewhere, for example on wikipedia.

In western music one can discern the following periods which correspond with basically universal (disclaimer: the following is grossly generalized):

1. Ancient Greece - the discovery of the mathematical rules of harmony (both musical and celestial) is attributed to Pythagoras, based on the superparticular ratios based on the numbers 1, 2, 3, 4: 2:1 - diapason (octave) 3:2 - diapente (fifth) 4:3 - diatessaron (fourth)

This mathematical framework was superimposed on the Greek system of tetrachords (4-note scales), in their various genera, which always covered a diatessaron (fourth), the most common genus being the diatonic (composed of 2 whole tones and 1 semitone). The so called Pythagorean system divided the diapason into two tetrachords, divided by a whole tone. It is interesting to note that the modern major scale is still basically the same construct as that of the ancient greeks: two diatonic tetrachords with a whole tone in the middle.

2. Middle Ages (up until early 15th century) - the musical and mathematical thought of the middle ages came of course from the classical era, and in the music of the middle ages the prevalent tuning method for all intents and purposes was based on Pythagorean ideas. We now call this method the Pythagorean tuning.

The basic idea is that all ratios are derived from the fifth (3:2). In the key of C, D would be tuned at a ratio of 9:8 (fifth minus fourth), E would be tuned at (9:8)^2 (the product of two whole tones), etc. The advantage of this system, compared to the Greek system, is the ability to wander into other keys. With the advent of Guido d'Arezzo hexachord system with its moveable hexachords, knowing how to tune Bb or F#, or go even further to Ab or G# became a matter of tuning a sequence of fifth.

It is important to note that this does not imply a closed cycle of fifth. In fact, tuning a sequence of 12 fifths (and then canceling the 7 octaves we climbed), starting from C and ending on B#, would give us a note significantly higher than the one we started with. This difference is called the pythagorean comma (about 1/8 tone).

The implications of this system are manifold: - no enharmonic identity - C is not B#, Eb is not D# - # is (relatively) high, b is low - Pythagorean thirds are dissonant: 9:8 * 9:8 = 81:64. Just thirds (based on the natural harmonic series) are 5:4, or 80:64. The resulting difference, 81:80, is called the syntonic comma (about 1/9 tone). In medieval music, cadences include only the stable, consonant intervals of fifths and octaves (with resulting fourths). Thirds and sixths are considered mid-way between consonant and dissonant, and are considered unstable.

3. Renaissance and early baroque - the age of meantone (15th century to late 17th century) - while various experiments have been made during the renaissance with just intonation and with equal tuning, they never took hold. Just intonation means that we lose the ability to move between keys. Equal temperament means that we have lousy thirds (almost as lousy as pythagorean thirds). The musical taste of the renaissance demanded on the one hand consonant,...