Tonality diamond

In music theory and tuning, a tonality diamond is a two-dimensional diagram of ratios in which one dimension is the Otonality and one the Utonality. Thus the n-limit tonality diamond ("limit" here is in the sense of odd limit, not prime limit) is an arrangement in diamond-shape of the set of rational numbers r, ${\displaystyle 1\leq r<2}$, such that the odd part of both the numerator and the denominator of r, when reduced to lowest terms, is less than or equal to the fixed odd number n. Equivalently, the diamond may be considered as a set of pitch classes, where a pitch class is an equivalence class of pitches under octave equivalence. The tonality diamond is often regarded as comprising the set of consonances of the n-limit. Although originally invented by Max Friedrich Meyer, the tonality diamond is now most associated with Harry Partch.

Partch arranged the elements of the tonality diamond in the shape of a rhombus, and subdivided into (n+1)²/4 smaller rhombuses. Along the upper left side of the rhombus are placed the odd numbers from 1 to n, each reduced to the octave (divided by the minimum power of 2 such that ${\displaystyle 1\leq r<2}$). These intervals are then arranged in ascending order. Along the lower left side are placed the corresponding reciprocals, 1 to 1/n, also reduced to the octave (here, multiplied by the minimum power of 2 such that ${\displaystyle 1\leq r<2}$). These are placed in descending order. At all other locations are placed the product of the diagonally upper- and lower-left intervals, reduced to the octave. This gives all the elements of the tonality diamond, with some repetition. Diagonals sloping in one direction form Otonalities and the diagonals in the other direction form Utonalities. One of Partch's instruments, the diamond marimba, is arranged according to the tonality diamond.

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