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.

In terms of latitude, these triads tend to range from ±5-25°, corresponding 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

English Wikipedia has an article on:

Major chord

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:

• 4:5:6 is found on the I (1⁄1), IV (4⁄3), and V (3⁄2) of Ptolemy's intense diatonic scale (Zarlino).

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

English Wikipedia has an article on:

Minor chord

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