Major and minor triads: Difference between revisions

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Major and minor triads refer to [[triad]]s containing a [[perfect fifth|fifth]] alongside a [[major third|major]] and [[minor third]] respectively.

== ~~Approaches~~ ==

== In tempered scales ==

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 [[7-limit|septimal]] intervals, for example, to [[14:18:21]] and [[6:7:9]].

In diatonic 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 [[5L 2s|diatonic 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 400c and minor is 300c.

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In terms of mediants, minor triads tend to range between a [[mediant (tonality)|mediant]] of 37% and 47%, and major triads tend to range between 53% and 63%, corresponding to simple [[5-limit]] or septimal intervals. More extreme than major and minor are [[Extraclassical tonality|tendo and arto]], corresponding to [[interseptimal]] and [[13-limit|tridecimal]] intervals, and ultimately suspended, corresponding to simple [[3-limit]] intervals; less extreme than major and minor are neutral triads.

== 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 [[7-limit|septimal]] intervals, for example, to [[14:18:21]] and [[6:7:9]]. Further details lie below.

=== Simple major triads ===

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.

In the 5-limit:

* [[4:5:6]] is found on the I ({{Frac|1|1}}), IV ({{Frac|4|3}}), and V ({{Frac|3|2}}) of Ptolemy's intense diatonic scale ([[Zarlino]]).

In the 3-limit:

* [[64:81:96]] is found on the I, IV, and V of the Pythagorean [[5L 2s|diatonic scale]].

== Simple minor triads ==

A '''minor triad''' is a [[triad]] comprising a root, [[minor]] third, and fifth.

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.

In the [[5-limit]]:

* [[10:12:15]] is found on the iii ({{Frac|5|4}}) and vi ({{Frac|5|3}}) of Ptolemy's intense diatonic scale ([[Zarlino]]), perhaps the most common 5-limit diatonic.

* [[27:32:40]] is found on the ii ({{Frac|9|8}}) of Ptolemy's intense diatonic scale.

In the [[3-limit]]:

* [[54:64:81]] is found on the ii ({{Frac|9|8}}), iii ({{Frac|81|64}}), and vi ({{Frac|27|16}}) of the Pythagorean diatonic scale.

== SCL files ==

@@ Line 1: / Line 1: @@
 Major and minor triads refer to [[triad]]s containing a [[perfect fifth|fifth]] alongside a [[major third|major]] and [[minor third]] respectively.
-== Approaches ==
+== In tempered scales ==
-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 [[7-limit|septimal]] intervals, for example, to [[14:18:21]] and [[6:7:9]].
-In diatonic 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 [[5L 2s|diatonic 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 400c and minor is 300c.
@@ Line 12: / Line 11: @@
 In terms of mediants, minor triads tend to range between a [[mediant (tonality)|mediant]] of 37% and 47%, and major triads tend to range between 53% and 63%, corresponding to simple [[5-limit]] or septimal intervals. More extreme than major and minor are [[Extraclassical tonality|tendo and arto]], corresponding to [[interseptimal]] and [[13-limit|tridecimal]] intervals, and ultimately suspended, corresponding to simple [[3-limit]] intervals; less extreme than major and minor are neutral triads.
+== 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 [[7-limit|septimal]] intervals, for example, to [[14:18:21]] and [[6:7:9]]. Further details lie below.
+=== Simple major triads ===
+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.
+In the 5-limit:
+* [[4:5:6]] is found on the I ({{Frac|1|1}}), IV ({{Frac|4|3}}), and V ({{Frac|3|2}}) of Ptolemy's intense diatonic scale ([[Zarlino]]).
+In the 3-limit:
+* [[64:81:96]] is found on the I, IV, and V of the Pythagorean [[5L 2s|diatonic scale]].
+== Simple minor triads ==
+A '''minor triad''' is a [[triad]] comprising a root, [[minor]] third, and fifth.
+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.
+In the [[5-limit]]:
+* [[10:12:15]] is found on the iii ({{Frac|5|4}}) and vi ({{Frac|5|3}}) of Ptolemy's intense diatonic scale ([[Zarlino]]), perhaps the most common 5-limit diatonic.
+* [[27:32:40]] is found on the ii ({{Frac|9|8}}) of Ptolemy's intense diatonic scale.
+In the [[3-limit]]:
+* [[54:64:81]] is found on the ii ({{Frac|9|8}}), iii ({{Frac|81|64}}), and vi ({{Frac|27|16}}) of the Pythagorean diatonic scale.
 == SCL files ==