Generator gronut

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This page is written from the perspective of User:Ground and may feature proposed concepts.

21\37 and 22\37 applied to a 7\12 gronut, flattened version.

21\37 and 22\37 applied to a 7\12 gronut, 3D render.

Lemniscate-like curve around a 7\12 gronut.

21\37 and 22\37 applied to a 7\12 gronut, simple diagram.

21\37 and 22\37 applied to an 11\19 gronut, simple diagram. This doesn't work as well as the 7\12 gronut in 37edo due to having discontinuities over both types of tritone.

11\19 gronut manifold. Center-crossing lines end up on the bottom, 7/19 of a turn clockwise in longitude. Center-crossing faces approach a width of zero near the center, then twist upside-down so the outside of the shape remains outside.

The circle of fifths is one of the most important concepts in Western music theory, identifying the harmonic position of an interval relative to stacking 3/1 and reducing by 2/1. Because the Pythagorean comma is small, the 12-note circle is the most sensible. There are a few ways of applying this concept to larger microtonal edos.

The simplest is to keep the circle shape, but continue stacking many fifths until the circle closes again, such as a 31-note meantone circle. I don't believe this captures the sense of harmonic distance that the 12-note circle does, especially due to the introduction of pitch drift. Aberrismic theory is based on the concept of a modified unison, with the same harmonic position as unison but wide enough to sound like a step. On a 31-note circle, 1\31 is 12 fifths down, but a key with this note as the tonic doesn't sound extremely distant like its position in the circle implies.

The second and arguably most common is to create a spiral of fifths. 12 fifths still make a full rotation as in the 12-note circle, but they don't close back to unison. Instead, they spiral outward infinitely. This represents harmonic distance accurately, but it doesn't close and is not compatible with a straddled generator. The solution to this lies in the spiral's similarity to a torus. The axis it spirals around is the longitude of the torus, and the 12 spots where the notes line back up with previous ones are the meridians.

The third is to map a harmonic table onto a torus. One of these is documented in the 15edo article, involving a circle of fifths along the longitudes and a circle of major thirds along the meridians. Some form of this can work for any edo. I find it particularly ingenious, but it still doesn't satisfy my intuition.

My version of this concept uses a torus (actually a surface of revolution about the intersection point of a lemniscate) traced with a curve resembling a lemniscate in 3D, connecting notes of adjacent pitches across the center of the torus. I call this shape a gronut (ground donut) for short. It allows any sufficiently large edo to be mapped to any generator circle, but most importantly, allows pitch drift to be visualized geometrically. It is projected from a more idealized 4D shape.

When approximating the 12edo circle of fifths, use two straddled approximations of 7\12, and stack them in the right combination to close the circle. Then stack the two approximations in different orders to fill out the rest of the notes, lining up notes of adjacent pitches on the meridians. Eventually, a note on the top of one meridian will be the same as a note on the bottom of another meridian, so draw a line between them. This represents the feeling of pitch drift, analogous to moving from red to green on a color wheel without moving through any other hues.

See Generator gronut/Mathematics for a mathematical treatment.