Archy

Archy, also called Superpyth (after its tuning range) or Archytas, is the temperament that tempers out the archytas comma, 64/63, equating septal intervals with nearby diatonic ones. In Archy, the generator is a fourth, the period is an octave, and 2 flattened fourths of about 490 cents stack to a sharply tuned 7th harmonic of about 980 cents. Equivalently, two sharpened fifths (octave-reduced) reach a major second that is both the septal supermajor second (8/7) and the diatonic major second, leading to the name "Superpyth".

Archy is usually tuned such that the septal supermajor third and septal subminor third (9/7 and 7/6) are the most accurately tuned; flatter tunings of the fourth lead to more accurate tunings of the 7th harmonic, at the cost of the usability of the diatonic scale, with the tuning leaving the 7th harmonic just being a fourth of around 484 cents.

As a monocot temperament (a temperament generated by a perfect fourth or fifth), Archy can be notated with standard diatonic notation. However, this is somewhat awkward, as Archy is more cleanly analyzed as a 5-form temperament, producing an equipentatonic scale, so perhaps diamond-MOS or KISS notation with pentic would be better suited for it. It is unsurprisingly supported by 5edo, as well as by 17edo, which can be seen as bounds on the tuning range for archy. Consequently, diatonic edos approaching 5edo (22edo, 27edo, etc) also tend to support Archy.

Other primes

If an unmapped (not equated to a stack of anything else, as 5/4 is in blackwood) prime 5 is introduced as a second generator, then the result can be called Archytas.5.

Extensions to prime 5 (interpretations of Archy that incorporate 5/4 into the diatonic structure) follow:

5/4 as limma-flat major third (22 & 27)

The canonical extension, equates 5/4 with the diatonic augmented second, or an octave-reduced stack of 9 fifths, which can be seen in the 5-form as a major third flattened by a diatonic semitone representing the septimal quartertone (36/35, the interval between 5/4 and 9/7) and the syntonic comma. It can be seen as the 22 & 27 temperament, and 22edo is the usual tuning.

5/4 as doubly limma-flat major third (5 & 37)

This is an alternative extension, best tuned sharp of 27edo. Instead of flattening the major third by a diatonic semitone to reach the 5th harmonic, you flatten by two diatonic semitones. In diatonic notation, this means that 5/4 is the double-augmented unison.

Chords

In Archy, the diatonic major and minor chords essentially have their roles swapped from in meantone, as they now represent the supermajor triad and subminor triad respectively, and the minor chord is the more stable of the two. This can be seen by how the supermajor third is, in the 5-form, a flat fourth, serving a somewhat similar role to the diminished fifth in diatonic. The suspended triad is an essentially tempered chord here, stacking 4/3 twice to reach 7/4 in its non-fifth-bounded inversion. Due to the widening of the major second, suspended triads may also be seen as consonant, if somewhat colorless.

Due to existing in 2.3.7, Archy also supports the latal triads (bounded by a fourth, made from intervals near 250c, like 6:7:8), with 1/1-8/7-4/3 in particular appearing as part of the suspended tetrad.