# 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