Trapezoidal rule that Monte-Carlo integration is most likely a better choice, but for 2 and 3 dimensions, equispaced sampling is efficient. This is exploited in computational solid state physics where equispaced sampling over primitive cells in the reciprocal lattice is known as "Monkhorst-Pack integration". For various classes of functions that are not twice-differentiable, the trapezoidal rule has sharper bounds than Simpson's rule. The trapezoidal rule is one of a family of formulas for numerical integration called Newton–Cotes formulas, of which the midpoint rule is similar to the trapezoid rule. Simpson's rule is another member of the same family, and in general has
Rule of the octave Rule of the octave The rule of the octave is a way of harmonizing each note of the diatonic scale, reflecting common practice, and has its origin in the practice of thorough bass, or basso continuo. The rule of the octave also formed the cornerstone of the "regole" (rules) of partimento collections. The rule of the octave was first mentioned in the 17th century in treatises for thorough bass written for lute-like instruments. There is normally a different harmonization for ascending and descending bass lines, and, although called a rule, there are several variants with different chords. Different versions for