Archy

Archy 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, the pythagorean (9/8) major second is mapped to the same pitch as the septimal (8/7) major second.

Archy is usually tuned such that the subminor (7/6) third is close to 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. The tuning that justly tunes the harmonic seventh places the perfect fourth at 484.4 cents, which leads to a diatonic scale with a small step at the uncomfortable size of 22 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.

Structural theory

Extensions

The following are extensions to prime 5 (i. e. ways to map intervals involving prime 5 onto the existing structure of 2.3.7 archy)

5/4 as limma-flat major third (22 & 27), often called "Superpyth"

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. The preferred tuning range for the fifth in this extension tends to be somewhat flatter than that of archy; tunings where both the supermajor (9/7) and subminor (7/6) thirds are somewhat accurate are preferred.

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.

Compositional theory

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 triad [0 4/3 7/4~14/9] is an important essentially tempered chord, although HKM finds that its other closed-voice inversions do not sound as if they contain septimal intervals unless the fifth is tuned as sharp as that of 37edo.

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.