Schisma

In

• the difference (in cents) between 8 justly tuned perfect fifths plus a justly tuned major third and 5 octaves;
• the ratio of
  Pythagorean limma
  ;
• the ratio of the syntonic comma and the diaschisma.

Schisma is a Greek word meaning a split or crack (see schism) whose musical sense was introduced by Boethius at the beginning of the 6th century in the 3rd book of his De institutione musica. Boethius was also the first to define the diaschisma.

Andreas Werckmeister defined the grad as the twelfth root of the Pythagorean comma, or equivalently the difference between the justly tuned fifth (3:2) and the equally tempered fifth of 700 cents (2^7/12).^[3] This value, 1.955 cents, may be well approximated by converting the ratio 886:885 to cents.^[4] This interval is also sometimes called a schisma.

Curiously, ${\displaystyle \ 2^{1/12}\ 5^{1/7}\ }$ is very close to 4:3, the just

${\displaystyle \ {\frac {\ 2^{161}\ }{\ 3^{84}\ 5^{12}\ }}\ ,}$

Tempering out the schisma leads to a schismatic temperament.

• Twelfth root of two

References