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.[1] 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,
1\ler<2
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
1\ler<2
1\ler<2
A numerary nexus is an identity shared by two or more interval ratios in their numerator or denominator, with different identities in the other.[1] For example, in the Otonality the denominator is always 1, thus 1 is the numerary nexus:
| \begin{array}{cccccc} | 1 | & |
| 1 |
| 2 | & | |
| 1 |
| 3 | & | |
| 1 |
| 4 | & | |
| 1 |
| 5 | |
| 1 |
&etc.\\ &&(
| 3 | |
| 2 |
)&&(
| 5 | |
| 4 |
) \end{array}
In the Utonality the numerator is always 1 and the numerary nexus is thus also 1:
| \begin{array}{cccccc} | 1 | & |
| 1 |
| 1 | & | |
| 2 |
| 1 | & | |
| 3 |
| 1 | & | |
| 4 |
| 1 | |
| 5 |
&etc.\\ &&(
| 4 | |
| 3 |
)&&(
| 8 | |
| 5 |
) \end{array}
For example, in a tonality diamond, such as Harry Partch's 11-limit diamond, each ratio of a right slanting row shares a numerator and each ratio of a left slanting row shares an denominator. Each ratio of the upper left row has 7 as a denominator, while each ratio of the upper right row has 7 (or 14) as a numerator.
This diamond contains three identities (1, 3, 5).
This diamond contains four identities (1, 3, 5, 7).
This diamond contains six identities (1, 3, 5, 7, 9, 11). Harry Partch used the 11-limit tonality diamond, but flipped it 90 degrees.
The five- and seven-limit tonality diamonds exhibit a highly regular geometry within the modulatory space, meaning all non-unison elements of the diamond are only 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.. Further examples of lattices of diamonds ranging from the triadic to the ogdoadic diamond have been realised by Erv Wilson where each interval is given its own unique direction.[4]
See also: Farey sequence.
Three properties of the tonality diamond and the ratios contained:
For example:
| 5-limit tonality diamond, ordered least to greatest | |||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ratio | |||||||||||||||||
| Cents | 0 | 315.64 | 386.31 | 498.04 | 701.96 | 813.69 | 884.36 | 1200 | |||||||||
| Width | 315.64 | 70.67 | 111.73 | 203.91 | 111.73 | 70.67 | 315.64 | ||||||||||
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)=\summ<n\phi(m).
| 2 | |
| \pi2 |
n2
Yuri Landman published an otonality and utonality diagram that clarifies the relationship of Partch's tonality diamonds to the harmonic series and string lengths (as Partch also used in his Kitharas) and Landmans Moodswinger instrument.[6]
In Partch's ratios, the over number corresponds to the amount of equal divisions of a vibrating string and the under number corresponds to the which division the string length is shortened to. for example is derived from dividing the string to 5 equal parts and shortening the length to the 4th part from the bottom. In Landmans diagram these numbers is inverted, changing the frequency ratios into string length ratios.
