Hobbit

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A //hobbit scale// is a generalization of [[MOSScales|MOS]] for arbitrary regular temperaments which is a sort of cousin to [[Dwarves|dwarf scales]]; examples may be found on the [[Scalesmith]] page. 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 v[1] notes to the octave.

==Definition==
To define the hobbit scale we first define a particular [[http://mathworld.wolfram.com/Seminorm.html|seminorm]] on interval space derived from a regular temperament, the OE or [[Tenney-Euclidean metrics|octave equivalent seminorm]]. This seminorm applies to [[Monzos and Interval Space|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 of the octave-equivalent pitch class to which the monzo belongs. Roughly speaking, the hobbit is the scale consisting of the interval of lowest OE complexity for each scale step mapped to the integer i by the val v.

Denoting the OE seminorm 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 v[1] is an odd number. If v[1] is odd then for each integer j, 0 less than j less than or equal to v[1], we choose a corresponding monzo m such that <v|m> = j, 0 less than <J|m> less than or equal to 1 where J is the JI mapping <log2(2) log2(3) ... log2(p)|, and T(m) is minimal.

If v[1] is even, we 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 less than j less than or equal to v[1], we choose a corresponding monzo m such that <v|m> = j, 0 less than <J|m> less than or equal to 1, and where T(2m - u) is minimal.

The intervals selected by this process are a [[transversal]] of te 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 [[Tenney-Euclidean metrics|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 inconsistency seem to increase with increasing rank.

Original HTML content:

<html><head><title>Hobbits</title></head><body>A <em>hobbit scale</em> is a generalization of <a class="wiki_link" href="/MOSScales">MOS</a> for arbitrary regular temperaments which is a sort of cousin to <a class="wiki_link" href="/Dwarves">dwarf scales</a>; examples may be found on the <a class="wiki_link" href="/Scalesmith">Scalesmith</a> page. 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 v[1] notes to the octave.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h2>
To define the hobbit scale we first define a particular <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Seminorm.html" rel="nofollow">seminorm</a> on interval space derived from a regular temperament, the OE or <a class="wiki_link" href="/Tenney-Euclidean%20metrics">octave equivalent seminorm</a>. This seminorm applies to <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzos</a> 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 of the octave-equivalent pitch class to which the monzo belongs. Roughly speaking, the hobbit is the scale consisting of the interval of lowest OE complexity for each scale step mapped to the integer i by the val v.<br />
<br />
Denoting the OE seminorm 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 v[1] is an odd number. If v[1] is odd then for each integer j, 0 less than j less than or equal to v[1], we choose a corresponding monzo m such that &lt;v|m&gt; = j, 0 less than &lt;J|m&gt; less than or equal to 1 where J is the JI mapping &lt;log2(2) log2(3) ... log2(p)|, and T(m) is minimal.<br />
<br />
If v[1] is even, we choose a monzo u such that T(u) is minimal under the condition that T(u) &gt; 0; in other words, u is a shortest positive length interval. Then for each integer j, where 0 less than j less than or equal to v[1], we choose a corresponding monzo m such that &lt;v|m&gt; = j, 0 less than &lt;J|m&gt; less than or equal to 1, and where T(2m - u) is minimal.<br />
<br />
The intervals selected by this process are a <a class="wiki_link" href="/transversal">transversal</a> of te 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.<br />
<br />
An alternative and equivalent approach is to work directly with the notes of the temperament, using the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">temperamental norm</a> defined on the note classes of the temperament modulo period (an octave or fraction of an octave) of the temperament.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Example"></a><!-- ws:end:WikiTextHeadingRule:2 -->Example</h2>
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 &lt;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&gt;. From this we may find a transversal minimizing T(2m - |4 -1 -1 0 0&gt;) 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 <a class="wiki_link" href="/53edo">53edo</a>, or by using the minimax tuning, which has eigenmonzos 2, 3, and 11.<br />
<br />
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 &lt;65 103 151 183 225|, giving a scale with steps 2433333242432424233333. Examples of this sort inconsistency seem to increase with increasing rank.</body></html>