Major and minor triads
Major and minor triads refer to triads containing a fifth alongside a major and minor third respectively.
In tempered scales
In diatonic (5L 2s) scales, "major third" and "minor third" are precisely defined intervals corresponding to 81/64 and 32/27 in Pythagorean tuning, but generated by a tempered fifth.
- In 12edo, major is 400 ¢ and minor is 300 ¢.
- In 19edo, major is 379 ¢ and minor is 316 ¢.
- In 22edo, major is 436 ¢ and minor is 273 ¢.
- If we pretend that 16edo's fifth generates a diatonic scale, this places major at 300 ¢ and minor at 375 ¢, leading to the controversial "harmonic notation" of 16edo.
- A more conventional interpretation labels the 300 ¢ third as "minor" and 375 ¢ as major, which is more sensible in terms of interval size and sound, despite technically not following Pythagorean notation.
The triads closer to the middle of the respective ranges correspond to simple classical or septimal intervals. More extreme than major and minor are tendo and arto, corresponding to interseptimal and tridecimal intervals, and ultimately suspended, corresponding to simple 3-limit intervals; less extreme than major and minor are neutral triads.
As interval regions, minor thirds and major thirds range from, at the widest, 240-340c and 360-460c, although people tend to restrict this further, such as to roughly 260-330c and 370-440c. And as an interval region, Perfect fifths range from around 660 to 740 cents at the widest, often being restricted to 680-720 cents.
In just intonation
In just intonation, 4:5:6 and 10:12:15 are the canonical tunings for the major and minor triads. Major and minor triads may also be tuned to simple septimal intervals, for example, to 14:18:21 and 6:7:9. Further details lie below.
Simple major triads
A major triad is a triad comprising a root, major third, and perfect fifth.
In the 3-limit:
- 64:81:96 is found on the I, IV, and V of the Pythagorean diatonic scale.
In the 5-limit:
In the 7-limit:
- 14:18:21, a supermajor triad, is a 9-odd-limit chord that tunes the third sharper than the 5-limit major.
Simple minor triads
A minor triad is a triad comprising a root, minor third, and fifth.
In the 3-limit:
- 54:64:81 is found on the ii (9⁄8), iii (81⁄64), and vi (27⁄16) of the Pythagorean diatonic scale.
In the 5-limit:
- 10:12:15 is found on the iii (5⁄4) and vi (5⁄3) of Ptolemy's intense diatonic scale (Zarlino), perhaps the most common 5-limit diatonic.
- 27:32:40 is found on the ii (9⁄8) of Ptolemy's intense diatonic scale.
In the 7-limit:
- 6:7:9, a subminor triad, is a 9-odd-limit chord that tunes the third flatter than the 5-limit minor.
SCL files
.SCL files for the classical major and minor triads are provided below:
! majortriad.scl ! The major triad as a wakalix ! Fokblock([25/24, 16/15], [1, 0]) = Fokblock([25/24, 10/9], [2, 0]) = Fokblock([16/15, 10/9], [0, 1]) 3 ! 5/4 3/2 2/1
! minortriad.scl ! The minor triad as a wakalix ! Fokblock([25/24, 16/15], [0, 0]) = Fokblock([25/24, 10/9], [1, 0]) = Fokblock([16/15, 10/9], [1, 0]) 3 ! 6/5 3/2 2/1