Half-diminished seventh chord

English Wikipedia has an article on:

Half-diminished seventh chord

A half-diminished seventh chord is a tetrad comprising a root, a minor third, a diminished fifth, and a minor seventh.

In temperaments

In meantone (including 12edo), on which traditional tonal harmony is built, the interval of a diminished fifth tempers together 36/25~64/45~1024/729, and the minor seventh tempers together 9/5~16/9, giving a chord with an intervallic odd limit of 25:

• (Meantone) 1/1 ‒ 6/5 ‒ 36/25 ‒ 9/5, with steps 6/5, 6/5, 5/4

In starling temperament, which tempers out 126/125, the ~36/25 diminished fifth is equated with ~10/7, reducing the chord to the 9-odd-limit as an essentially tempered chord:

• (Starling) 1/1 ‒ 6/5 ‒ 10/7 ‒ 9/5, with steps 6/5, 6/5, 5/4

Septimal meantone, which is well-represented by the historically prevalent quarter-comma meantone, tempers together all three of the tritones (10/7~36/25~64/45) and sevenths (9/5~16/9~25/14), so either of the above interpretations may be relevant for half-diminished seventh chords found in common-practice music. (→ Didymic chords #Dominant seventh chord)

In archytas temperament, which tempers out 64/63, a ~16/9 minor seventh is equated with ~7/4, a ~64/45 diminished fifth is equated with ~7/5, and the ~35/24 ratio between ~7/4 and ~6/5 is equated with a ~3/2 perfect fifth, yielding a 7-odd-limit essentially tempered chord:

• (Archytas) 1/1 ‒ 6/5 ‒ 7/5 ‒ 7/4, with steps 6/5, 7/6, 5/4

Alternately, if we take the ~9/5 minor seventh instead, archytas gives a 9-odd-limit essentially just chord, tempering together 5:6:7:9 with 45:54:64:81:

• (Archytas) 1/1 ‒ 6/5 ‒ 7/5 ‒ 9/5, with steps 6/5, 7/6, 9/7

In just intonation

In the 7-limit:

• 5:6:7:9 is a highly concordant 7-limit interpretation of the half-diminished seventh, found in the Ringer 5 scale and genus 3² ⋅ 5 ⋅ 7, and tempered together with 45:54:64:81 in archytas temperament.
• 60:70:84:105 (1/1 - 7/6 - 7/5 - 7/4) is a subharmonic chord.
• 20:24:28:35 is a condissonant chord found in genus 3 ⋅ 5 ⋅ 7, and tempered together with 45:54:64:80 in archytas temperament.
• 35:42:50:63 is a condissonant chord found in genus 3² ⋅ 5² ⋅ 7, and tempered together with 25:30:36:45 in starling temperament.

In the 5-limit:

• 25:30:36:45 is found on iii^ø7 (5⁄4) and vi^ø7 (5⁄3) in the duodene.
• 45:54:64:80 found on vii^ø7 (15⁄8) in Ptolemy's intense diatonic scale (Zarlino), perhaps the most common 5-limit diatonic. It is also found on V^ø7 (3⁄2) in the duodene.
• 45:54:64:81, another preimage of the meantone half-diminished seventh chord, is found in genus 3⁴ ⋅ 5, and tempered together with 5:6:7:9 in archytas temperament.
• 135:160:192:240 is found on ii^ø7 (9⁄8) and ♯iv^ø7 (45⁄32) in the duodene.