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.

, 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,[2] the tonality diamond is now most associated with Harry Partch ("Many theorists of just intonation consider the tonality diamond Partch's greatest contribution to microtonal theory."[3]).

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

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

At all other locations are placed the product of the diagonally upper- and lower-left intervals, reduced to the octave.

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.

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.

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] Three properties of the tonality diamond and the ratios contained: For example: 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 From this we can conclude that the rate of growth of the tonality diamond is asymptotically equal to

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.

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.