Father

Father (3 & 5) is a very inaccurate exotemperament that makes 3:4:5 equidistant, in other words equating 5/4 and 4/3 to a single "fourth-third" interval (which the name 'Father' originates from). As a result, it serves as a simplification of 3:4:5-based (naiadic) harmony, in much the same way that Dicot simplifies tertian harmony or Semaphore simplifies chthonic harmony.

Due to tempering out such a large and simple interval as 16/15, there is no accurate tuning for Father. One structurally justifiable tuning, somewhat equivalent to tuning Dicot's 5/4 to a perfect neutral third, involves splitting a just 5/3 in half, resulting in a fourth-third of 442 cents (or a fifth-sixth of 758 cents). However, as with Dicot, it is somewhat preferable to lean the tuning of the generator towards one of the two simple intervals it represents - as flat as about 400 cents to favor 5/4 (as in 3edo), or as sharp as about 480 cents to favor 4/3 (as in 5edo). Another notable tuning is the golden tuning, about 458 cents, which sets the logarithmic ratio of 4/3 and 3/2 to the golden ratio.

In the 5-limit, due to equating two reduced prime (sub)harmonics, it is found in a number of small edos; the simplest edo join, 1 & 2, is an extension of Father, meaning Father can be arguably seen as the simplest 'real' 5-limit temperament. The edo join that gives the best impression of its tuning range is 3 & 5.

Another point of interest in father is its moment-of-symmetry scales. It is likely that Father was originally defined in order to give a simple JI interpretation to the oneirotonic scale, although there are also Father tunings that generate checkertonic.

Melodic interpretation

Father can be thought of as replacing pairs of 5-limit JI intervals with "interordinal" intervals between them, having an interval (around 450 cents) that's 5/4~4/3, a 6/5~9/8 about 300 cents, an 8/5~3/2 of about 750 cents, a 5/3~16/9 of about 900 cents, and 15/8 being equated to the octave (meaning it might be beneficial to flatten the octave about 20-30 cents or so). In this example above, it posseses a secret 10/9 tempered down to 150 cents which defines the difference between its intervals. Due to the new equivalences, a sharper "5/4" condemns a flatter "3/2" (the same is true for for 6/5 ~ 9/8 and 5/3 ~ 16/9.)

Extensions

3 & 5, in the 7-limit, produces Mother, which further equates the generator to 7/5.

However, the perhaps more 'reasonable' extension structurally is to observe that 9/7 is the mediant of 5/4 and 4/3, and therefore equate the fourth-third to 9/7 as well, producing a Trienstonian and Sensamagic temperament. However, due to the tuning instability of 9/7, this is not supported by any patent vals besides 5.

Comparison to other temperaments

Father is distinct from temperaments such as Blackwood (5 & 15), Trienstonian (5 & 18), and Fendo (5 & 7, 2.3.13/5) that equate other major thirds to 4/3 and that are generally more accurate. It is also distinct from more accurate oneirotonic temperaments such as A-Team that are not generated by 4/3, and from the temperament-agnostic golden tuning.