Hobbit

A hobbit, or hobbit scale, is a generalization of mos scale for arbitrary regular temperaments which is a sort of cousin to dwarf scales; examples may be found on the Scalesmith page. The idea is that mos scales give us a means of contructing scales for a rank-2 temperament which gives priority to the intervals of least complexity in that temperament, and so makes efficient use of it; a hobbit does the same in higher ranks, and so using them is one way to make higher ranks, including especially the interesting rank-3 case, accessible for musical purposes.

Given a regular temperament and an equal temperament which supports the temperament, there is a unique scale for the temperament, which can be tuned to any tuning of the temperament, containing the same number of notes as the equal temperament.

Definition

A hobbit scale consists of intervals of least complexity in the regular temperament corresponding to each step of the equal temperament. The measure of complexity we use is the octave equivalent Tenney–Euclidean seminorm, or OETES, here denoted T(x) where x is an interval. The OETES complexity of any comma of the temperament, and also of the octave, is 0, encoding the vanishing of the commas and octave equivalence. Note that this means that any given pitch class relative to the unison has a corresponding OETES complexity shared between all of its representative intervals, and additionally T(x) = T(2/x), where x and 2/x are octave complements.

For an edo with an odd number of notes, the selection of notes is unambiguous. However, when an edo is even and thus contains the perfect semioctave, there is an ambiguity, and there are multiple options for the hobbit, differing by the central interval. This is similar to how 12-note Pythagorean tuning has no perfectly symmetrical mode; either the narrow or sharp tritone must be chosen.

The intervals selected by this process are a transversal of the scale, and we may now apply the chosen tuning to the monzos in the transversal, obtaining values (in cents or fractional monzos) defining a scale. The monzos in the transversal are defined only modulo the commas of the temperament, but since these are tempered out this does not affect the definition of the scale.

Example

For an example, consider the 22-note hobbit for minerva temperament, the 11-limit temperament tempering out 99/98 and 176/175. Here the val is ⟨22 35 51 62 76], and an interval of minimal nonzero size for the temperament is 16/15, with monzo [4 -1 -1 0 0⟩. From this we may find a transversal minimizing T(2m − [4 -1 -1 0 0⟩) for each scale step, namely 36/35, 15/14, 11/10, 8/7, 7/6, 40/33, 5/4, 9/7, 4/3, 48/35, 10/7, 22/15, 3/2, 11/7, 8/5, 5/3, 12/7, 7/4, 64/35, 15/8, 64/33, 2/1. A tuning can be defined in various ways, for instance by approximating the above in 53edo, or by using the minimax tuning, which has eigenmonzos 2, 3, and 11.

After applying such a tuning, we discover than there seems to be a certain irregularity or inconsistency in action, in that some of the 11-limit intervals do not stem from the mapping for minerva, but represent additional temperings by 243/242 or 4000/3993. By adding one of these, we can flatten out the irregularity to a corresponding rank-2 temperament; by adding both, we obtain the rank-1 temperament with val ⟨65 103 151 183 225], giving a scale with steps 2, 4, 3, 3, 3, 3, 3, 2, 4, 2, 4, 3, 2, 4, 2, 4, 2, 3, 3, 3, 3, 3. Examples of this sort of inconsistency seem to increase with increasing rank.

Notation

Hobbits are often assumed when a rank-3 temperament is appended with a number (e.g. marvel[9]), similar to how a rank-2 temperament appended with a number (e.g. meantone[7]) denotes a MOS.