Meantone
Meantone, or rarely Syntonic or Didymus, is a widespread historical temperament that forms the basis of Western music theory, where the fifths are flattened to about 696c to produce a diatonic major third tuned to roughly 5/4, enabling the use of 5-limit harmony in the diatonic scale. When all the fifths are tuned the same, Meantone is a regular temperament, where the period is the octave, the generator is 3/2, and four generators stack to reach the 5th harmonic, meaning that the syntonic comma, which is the difference between Pythagorean intervals and nearby 5-limit intervals and has a ratio of 81/80, is tempered out.
As a monocot temperament, Meantone can be notated with standard diatonic notation, and in fact diatonic notation works the best for Meantone as the 5-limit 4:5:6 harmonic triad becomes simply C-E-G on C, and the chromatic semitone is usually smaller than the diatonic semitone. In fact, Leriendil believes the latter point was likely a major factor in favor of its widespread adoption, as it results in a more intuitive 7-form unlike Pythagorean tuning which works best in the 12-form. Unsurprisingly, 7edo supports Meantone, and so does 12edo (which is the simplest ET to do so without exotempering the 5-limit), and so the best tunings of Meantone lie in between those two extremes.
Other primes
If an unmapped (not equated to a stack of anything else, as 5/4 is in blackwood) prime 7 is introduced as a second generator, then the result can be called Didymus.7.
Extensions of Meantone's diatonic structure to include other primes follow.
7/4 as the augmented sixth (12 & 19)
This is the primary extension of Meantone to the 7-limit, where 7/4 is the augmented sixth. It is best tuned with the generator around 696 cents. 5/4 is 384 cents, and 7/4 is 960 cents. It is notable for being the most accurate extension, as well as containing the golden tuning of the diatonic scale, and thus a melodically convenient chromatic and enharmonic scale. This means that the augmented diesis, 128/125, is equated with the septimal quartertone, and the 5-limit supermajor and subminor intervals are equated with their septimal counterparts.
However, one drawback of this temperament is the large degree of complexity required to get to the 11th and 13th harmonics. In fact, there are two main options. In both cases, the tridecimal neutral third 16/13 is conflated with the undecimal neutral third 11/9, representing a characteristic tendency to make 11/9 the sharper of the two 11-limit neutral thirds. (As a result, one might find it useful to irregularly map 11/9.)
11/8 as the double-augmented third (12 & 31)
This is best tuned around 697 cents, and places 11/9 as the double-augmented second and conflates 14/11 with 9/7. 13/8 is mapped to the double-diminished seventh.
11/8 as the double-diminished fifth (19 & 31)
This is best tuned around 696 cents, and places 11/9 as the double-diminished fourth. 13/8 is mapped to the double-augmented fifth.
7/4 as the diminished seventh (19 & 26)
This temperament sets 7/4 equal to the diminished seventh, and is best tuned with the generator around 693 cents. 5/4 is 372 cents, and 7/4 is 963 cents. It is a melodically intuitive extension, as it creates an equiheptatonic scale with a quartertone-sized chroma, and interval sizes tend to match with their corresponding interval categories. For example, it can be easily extended to map prime 11 to the augmented fourth (558 cents) and 13 to the minor sixth (828 cents). 26edo is the usual tuning, though it can be tuned more accurately with 45edo.
Chords
Meantone's main feature is its conflation of the standard harmonic triad 4:5:6 with the diatonic major chord P1-M3-P5. This allows 5-limit harmony to be used in Meantone, and is the foundation of modern Western music theory, which treats P1-M3-P5 as stable and P1-m3-P5 as unstable (10:12:15). Meantone also contains an essentially tempered chord, where 1-9/8-3/2-5/3-2 contains steps of 9/8, 4/3, 9/8, and 6/5. Note that in just intonation, the top interval would be 27/16, not 5/3, or the two whole tones would be different sizes (resulting in a 40/27 wolf fifth).
