Partch 43-tone scale
inverted Monzo lattice

Below is an inverted Monzo lattice of Harry Partch's 43-tone JI scale. Pitches are notated in three formats:

• the 72edo version of HEWM (the letters and accidental symbols);

• the [3 5 7 11] prime-factor vector (the prime factors and their exponents); and

• the ratio (written as a fraction), which is what Partch himself used.

  
  First, the plain lattice:




Next, a shaded version which portrays more readily the 4-dimensionality of the lattice by coloring the planes of the lattice -- each of which incorporates two of the four prime-factors 3, 5, 7, and 11 -- as tho they were panes of transparent tinted glass:

The graph below shows the standard deviation of all EDOs from 43 to 1200. With the desideratum to have the best combination of lowest standard deviation and lowest cardinality, the three clear winners are 72-, 120-, 152-, 183-, 224-, and 270-edo. 270edo even gives a better approximation of Partch's scale than 1200edo (the measurement of cents)!

updated:

2002.10.12 -- added standard deviation of EDOs graph
2002.9.15 -- added shaded lattice
2002.3.5
2000.8.15 -- created

For many more diagrams and explanations of historical tunings, see my book. If you don't understand my theory or the terms I've used, start here or try some definitions.

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