Gentle tuning

Gentle tuning is the regular tuning of the perfect fifth somewhere between the tunings given by 29edo and 17edo (~703.4 to 705.9 cents), producing MOS diatonic scales with step size ratios between 5:2 and 3:1 (and consequently chromatic scales between 3:2 and 2:1, including the golden tuning of p-chromatic). As a result, the major third generated is between 413.8 and 423.5 cents, and approximates 14/11. It is so named because melodically it functions as a "gentle" version of Superpyth; conversely, it can instead be seen as exaggerating the characteristics of Pythagorean tuning as it compares with 12-tone equal temperament and is usable as a replacement for Pythagorean tuning. One notable characteristic of gentle tuning is that it narrows the diatonic semitone towards a size of 70 cents, described by some theorists as an ideal leading tone; Aura, however, considers it to be between a leading tone and a diesis or quarter tone in function, being able to either lead to the tonic or "skip over" it.

Regular temperament theory

Gentle tuning is associated with the 2.3.(11/7).(13/7) temperament, 29 & 46, that finds 14/11 at the major third and 13/11 at the minor third, which has a number of names depending on the subgroup (see #Names). Therefore, the chromatic semitone is equal to 14/13. This implies the use of these thirds as elements of 13-limit harmony over the 11th and 13th harmonics themselves, similar to, but more intense and approach-specific than, the prioritization of 9/7 and 7/6 in Archy (and in septal harmony as a whole) as opposed to 7/4; both approaches are patterned off of Meantone, which is often described outside of a regular temperament context as tempering the fifth to bring the major and minor third more in-tune with 5/4 and 6/5, where 5/4 is simply treated as "the major third" rather than a generator of just intonation in its own right.

To extend to the 2.3.7.11.13 subgroup, 7/6 is equated to two chromatic semitones (so that 7/4 is C-G##). As a result, 11/8 is placed at the augmented third and 13/8 at the augmented fifth (this is the same interval that becomes 8/5 in Schismic).

For flatter tunings (flat of 46edo), 5/4 is equated to three chromatic semitones (so that a major triad is C-C###-G). There is no stable mapping of 5 for sharper tunings due to 17edo's inaccuracy in approximating prime 5. This is similar to Archy tunings approaching 5edo.

Chords

While it is not particularly simple in just intonation, the primary chords implied by the temperament's structure are the major 22:28:33 and minor 22:26:33 which are complements of one another and also the diatonic major and minor chords (so that their union, P1-m3-M3-P5, is essentially tempered). Standard 5- and 7-limit chords are somewhat unviable when using standard diatonic logic, as a 4:5:6:7 chord is C-C###-G-G##, with all four notes overlapping on two nominals, and the supermajor and subminor triads are C-C##-G and C-Gbb-G, again overlapping.

Names

The most common name for this temperament, applying in the strict sense to the full 2.3.5.7.11.13 temperament, is Leapday. As a 2.3.(11/7).(13/7) temperament, however, the technical name is Pepperoni and as a no-5s temperament the technical name is Leapfrog. Other names that may be used for this temperament or for gentle tuning as a whole are neogothic (in reference to the quality of the generated thirds), Parapyth (technically referring to a rank-3 temperament whose 3/2 generates Pepperoni), and simply gentle.