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 two regular 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 three case, accessible for musical purposes.

Given a regular temperament and an equal temperament val v which supports (or belongs to) the temperament, there is a unique scale for the temperament, which can be tuned to any tuning of the temperament, containing v₁ notes to the octave.

Definition

To define the hobbit scale we first define a particular seminorm on interval space derived from a regular temperament, the octave equivalent Tenney-Euclidean seminorm or OETES. This seminorm applies to monzos and has the property that the seminorm of any comma of the temperament, and also of the octave, is 0. This seminorm, for any monzo, is a measure of complexity within the temperament of the octave-equivalent pitch class to which the monzo belongs. Roughly speaking, the hobbit is the scale consisting of the interval of lowest OETES complexity for each scale step mapped to the integer i by the val v.

Denoting the OETES for any element x of interval space by T (x), we first define the hobbit of an odd-numbered scale; that is, a scale for which v₁ is an odd number. If v₁ is odd then for each integer j, 0 < j ≤ v₁, we choose a corresponding monzo m such that <v|m> = j, 0 < <J|m> ≤ 1 where J is the JI mapping ⟨log₂2 log₂3 … log₂p], and T (m) is minimal.

If v₁ is even, one approach is to proceed as before, but to break the tie at the midpoint of the scale by choosing the interval of least Benedetti height. Another approach adopted here is to choose a monzo u such that T (u) is minimal under the condition that T (u) > 0; in other words, u is a shortest positive length interval. Then for each integer j, where 0 < j ≤ to v₁, we choose a corresponding monzo m such that <v|m> = j, 0 < <J|m> ≤ 1, and where T (2m - u) is minimal. It should be noted that while this gives a canonical choice, the inverse hobbit is fully equal as a scale to the canonical hobbit.

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.

An alternative and equivalent approach is to work directly with the notes of the temperament, using the temperamental norm defined on the note classes of the temperament modulo period (an octave or fraction of an octave) of the temperament.

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 two temperament; by adding both, we obtain the rank one temperament with val ⟨65 103 151 183 225], giving a scale with steps 2433333242432424233333. Examples of this sort of inconsistency seem to increase with increasing rank.