WILSON'S BASIC TEMPLATE FOR LATTICING 13-LIMIT HARMONICALLY BASED STRUCTURES

The Most Common Generalised Lattice used by Wilson.

It is hard to think of a microtonal theorist who has made more headway with developing lattices than Erv Wilson. Here the concentration will be on what he nicknamed  an “Euler“ lattice after where he drew his inspiration.

By working with a template where each prime harmonic has a unique spacing , he managed to avoid clashes when generating lattice of multidimensional harmonically-based structure

There are some other features worth pointing out.  For instance harmonically generated intervals will always appear above the fundamental and the subharmonics below. With a  9-limit system the direction will be both above and to the right , leaving the other quadrant for more complex ratios with the opposite with the subharmonic. This makes it quite easy to understand what is being represented.

Commonly he would use a 10 square to the inch graph paper which explains why he didn’t use a template where all the numbers where divided by 2. This way he had room to notate both ratios and often the scale degree. The latter always followed by a period or dot to separate it from the ratios.



Lately i have been using the following lattice which seems to overcome a few trouble spots. This was plotted using 20 squares per inch.

Alternative lattice by Kraig Grady in cases of slashes with Wilson's Lattice.

Diatonic, Chromatic, Tri-Chromatic, and Enharmonic Forms of Mixolydian

Here is a method of deriving some tetrachordal like shapes starting with the sub 7-14 series borrowing Schlesinger designation for Mixolydian. It is interesting to note that to derive each form, there is a shift up the subhamonic series by multipling by 2, 3, or 4 times shifting the series to 14-28, 21-42 and 28-56 respectfully to derive a class of scale. The term Tri-Chromatic (formed by multiplying by 3 the 7-14 series) is invented here to designate a new scale type in between the chromatic and enharmonic. In each set of four a different interval acts as a disjunction like interval and greatly expands the range of what a disjunction might be. These being in order 8/7, 11/10, 14/13 then these are repeated in the next three to construct tetrachordal shapes where the large interval is at the bottom as opposed to the top. It was an idea Wilson left unexplored past this page it appears. The other tetrachordal permutations are left unexplored.

The Harmonic and Subharmonic Ogdoads with 12 Tones

This quick note written between other papers dated from 1996 shows the possibility of placing a the 13 limit with the harmonic and subharmonic series within a single 12 tone series. Placed at the distance of the 15 means that both series also have the 15th  of their series. The idea of placing these two series apart like this can be seen on page 15 of  http://anaphoria.com/earlylattices12-.pdf
That scale of Gary David illustrates 2 harmonic and 2 subharmonic 9 limit pentads with 12 tones.  He was the first to point out the use of the 15 as a mirror point with 12 tone scales which here we can see is applied in a slightly different way.

Hexany as a 12 tone meeting place

Here Wilson worked out the possibility of using a Hexany  as a meeting place between a harmonic and subharmonic series having no tones in common. the Second example here shows the two complementary series this time up the the 13 (or 15) (sub)harmonics still limited to 12 tones.