Tonality diamond

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In music theory, the n-limit tonality diamond is the set of rational numbers r, 1 \le 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 Meyer, the tonality diamond is now most associated with Harry Partch.

[edit] The diamond arrangement

Partch arranged the elements of the tonality diamond in the shape of a rhombus, which we can take to be a square, skew-oriented so that the sides are at 45 degrees to the horizontal, and subdivided into (n+1)2/4 smaller squares, in chessboard style, and hence the name. Along upper left side of the rhombus we place the odd numbers from 1 to n, reduced to the octave, inside the board squares. Along the lower left side we place the corresponding reciprocals, 1 to 1/n, reduced to the octave. At all the other board squares we place the product, reduced to the octave. This gives all the elements of the tonality diamond, with some repetition. All the diagonals sloping in one direction form otonalities and the diagonals in the other direction form utonalities. Harry Partch created an instrument, the diamond marimba, which is laid out as a tonality diamond and which has proven rather popular.

7-limit tonality diamond
         7/4
      3/2    7/5
   5/4   6/5     7/6
1/1   1/1    1/1    1/1
   8/5   5/3    12/7
      4/3   10/7
         8/7

15-limit tonality diamond
                              15/8
                          7/4       5/3
                    13/8      14/9       3/2
                3/2      13/9       7/5      15/11
          11/8       4/3      13/10     14/11      5/4
       5/4     11/9       6/5      13/11      7/6      15/13
    9/8   10/9      11/10     12/11     13/12     14/13     15/14
1/1    1/1      1/1       1/1       1/1       1/1       1/1      1/1
   16/9    9/5      20/11     11/6      24/13     13/7      28/15
       8/5     18/11      5/3      22/13     12/7      26/15
          16/11      3/2      20/13     11/7       8/5
                4/3      18/13     10/7      22/15
                    16/13      9/7       4/3
                          8/7       6/5
                              16/15

[edit] Geometry of the tonality diamond

The five and seven limit tonality diamonds can be viewed as having a highly regular geometry within the modulatory space which makes all the non-unison elements of the diamond at a distance of one unit from the unison. The five-limit diamond then becomes a regular hexagon surrounding the unison, and the seven-limit diamond a cuboctahedron surrounding the unison.

[edit] Size of the tonality diamond

If φ(n) is Euler's totient function, which gives the number of positive integers less than n and relatively prime to n, that is, it counts the integers less than n which share no common factor with n, and if d(n) denotes the size of the n-limit tonality diamond, we have the formula

d(n) = \sum_{m<n \ odd} \phi(m).

From this we can conclude that the rate of growth of the tonality diamond is asymptotically equal to \frac{2}{\pi^2} n^2 . The first few values are the important ones, and the fact that the size of the diamond grows as the square of the size of the odd limit tells us that it becomes large fairly quickly. There are seven members to the 5-limit diamond, 13 to the 7-limit diamond, 19 to the 9-limit diamond, 29 to the 11-limit diamond, 41 to the 13-limit diamond, and 49 to the 15-limit diamond; these suffice for most purposes.