Silver third

This page presents a topic of primarily mathematical interest. While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown.

The silver third is the octave-reduced second metallic mean, and is either a wide minor third or a narrow supraminor one. It differs from the first metallic mean (acoustic phi) by an interval that can act as a flattone fifth. This is not to be confused with argent tuning, which uses the logarithmic silver ratio.

Interval information

Expression	[math]\displaystyle{ \sigma = \frac{ 1 + \sqrt{2} }{2} }[/math]
Size in cents	325.864¢
Name	silver third
Special properties	reduced

An interesting property of this interval is that a tetrad can be formed with the root, the silver third, the perfect fifth, and a supermajor sixth 600 ¢ above the silver third (925.864 ¢), such that the frequency difference between the sixth and the fifth is the same as that between the root. This means this tetrad has a DR signature of +1 +? +1, a property shared with tetrads like 4:5:6:7 (sometimes called the major tetrad) and 6:7:9:10 (sometimes called the subminor tetrad). This tetrad has this DR property, while also allowing tritone substitution due to the third and sixth being separated by 600 ¢.

Temperaments

It can be used as a generator for many temperaments using a sharpened 6/5, such as keemic, orgone or doublewide, and is closely approximated by 3\11.