Diminished temperament

Diminished temperament is a linear temperament with a unison vector of [3 4 -4] (648/625). This tuning has a period of 1/4 octave and a generator of around 90 cents. The basic scale has eight notes. It can be extended to a 7-limit temperament in a number of ways; probably the simplest is by setting 7/4 equal to 9/5 (3 periods + 1 generator), which implies a 36/35 unison vector.
A#
  C#
A   E
  C   G
    Eb
      Gb
 P  G  approx. ratio
 0  0  1/1
 0  1  25/24 27/25
 1  0  6/5
 1  1  5/4
 2  0  25/18 36/25
 2  1  3/2
 3  0  5/3
 3  1  9/5
 4  0  2/1

Diminished temperament ET scales

12-ET should need no introduction. The eight-note scale in 12-ET is the familiar octatonic scale.
P=3, G=1. Basic scale: 0 1 3 4 6 7 9 10 12
Example

16-ET is perhaps more familiar for its "faux pelog" scale of 5 small steps and 2 large steps (0 2 4 7 9 11 13 16). But the octatonic scale works just as well.
P=4, G=1. Basic scale: 0 1 4 5 8 9 12 13 16
Example

28-ET
P=7, G=2. Basic scale: 0 2 7 9 14 16 21 23 28
Example

40-ET (inconsistent)
P=10, G=3. Basic scale: 0 3 10 13 20 23 30 33 40
Example

52-ET (inconsistent)
P=13, G=4. Basic scale: 0 4 13 17 26 30 39 43 52
Example

64-ET (inconsistent) is a versatile tuning, which in addition to the diminished temperament can be used as either a meantone or kleismic temperament.
P=16, G=5. Basic scale: 0 5 16 21 32 37 48 53 64
Example