Minor Chord - Just Intonation

Just Intonation

In just intonation, a minor chord is often (but not exclusively) tuned in the frequency ratio 10:12:15 ( play). This is the first occurrence of a minor triad in the harmonic series (if on C: E-G-B). This may be found on iii, vi, ♭vi, ♭iii, and vii. In 12-TET, or twelve-tone equal temperament (now the most common tuning system in the West), a minor chord has 3 semitones between the root and third, 4 between the third and fifth, and 7 between the root and fifth. It is represented by the integer notation 0,3,7. The 12-TET fifth (700 cents) is only two cents narrower than the just perfect fifth (3:2, 701.9 cents), but the 12-TET minor third (300 cents) is noticeably (about 16 cents) narrower than the just minor third (6:5, 315.6 cents). The 12-TET minor third (300 cents) more closely approximates the 19-limit (Limit (music)) minor third 16:19 Play (297.5 cents, the nineteenth harmonic) with only 2 cents error. Ellis proposes that the conflict between mathematicians and physicists on one hand and practicing musicians on the other regarding the supposed inferiority of the minor chord and scale to the major may be explained due to physicists' comparison of just minor and major triads, in which case minor comes out the loser, versus the musicians' comparison of the equal tempered triads, in which case minor comes out the winner since the ET major third is 14 cents sharp from the just major third while the ET minor third closely approximates the consonant 19:16 minor third, which many find pleasing. Other just minor chord tunings include the supertonic triad in just intonation (27:32:40) the false minor triad, Play, 16:19:24 Play, 12:14:18 (6:7:9) Play (Septimal minor third), and the Pythagorean minor triad (54:64:243) Play. More tunings of the minor chord are also available in various equal temperaments other than 12-TET.

Rather than directly from the harmonic series, Sorge derived the minor chord from joining two major triads; for example the A minor triad being the confluence of the F and C major triads. A-C-E= f-A-C-E-g. Given justly tuned major triads this produces a justly tuned minor triad: 10:12:15 on 8/5.