Gnome

A gnome is a detemperament created through the following algorithm. It detempers an edo to a temperament it supports, and is intended as a simpler alternative to hobbits, where the calculations can be done by hand rather than requiring advanced matrix math. It is called a "gnome" because it has properties of both a hobbit and an elf.

Generation

Take a mapping matrix V, including a temperament and a set of generators, alongside a target EDO.

The octave is to be ignored as a generator if the temperament is haploid, or if it is not, the maximum number of ploids to be considered is floor(n/2) where the temperament is n-ploid. So, for our blackwood example, we will consider a maximum of 2 periods up or down.

For each step of the EDO, consider all the intervals in this reduced form of the temperament that map to the desired EDOstep when octave-reduced, and sum up the absolute values of their tmonzo entries - this is equivalent to taking the Manhattan distance to each one in tempered interval space. The tempered interval with the smallest Manhattan distance to the unison out of the candidates is chosen for each edostep.

To resolve ties, the Benedetti heights (or equivalently Tenney heights) of the simplest octave-reduced JI intervals (also by Benedetti height) representing each candidate interval in the tie are compared, and the simplest is chosen.

Example

Let us consider the gnome for 5edo in the temperament sensamagic, with generators [~2, ~9/7, ~3/2].

First, the octave is removed, leaving the generator set as [~9/7] and [~3].

For the first step of 5edo, 240c, there is no interval within 1 generator step that matches. However, there are three candidates within two gensteps: ~6/5, ~7/6, and ~9/8. These have been notated with the simplest JI interpretations of those intervals, and the Benedetti heights of those are 30, 42, and 72 respectively. So, ~6/5 is chosen.

For the next step of 5edo, 480c, there are two candidates: ~4/3 and ~9/7. Of these, ~4/3 (simplest representation 4/3) is the simplest.

We proceed with 720c and ~3/2, and 960c and ~5/3.

With that, the gnome is completed: [~6/5, ~4/3, ~3/2, ~5/3, ~2/1]. In CTE tuning, that's [312.2c, 496.3c, 703.7c, 887.8c, 1200c].