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]]. 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. This seminorm applies to [[Monzos and Interval Space|monzos]] and has the property that the seminorm of a comma of the temperament, or of the unison, the octave and any power of two is 0. It may be defined as follows: (1) If o = |1 0 0 ... 0> is the monzo for 2 in the [[Harmonic Limit|p-limit]] group. (2) c1, c2, ..., ci are monzos for a basis for the commas of the temperament. (3) Form the (i+1)**x**n matrix N = [o, c1, c2, ..., ci] whose rows consist of o and the commas ck. (4) Monzo weight N by multiplying on the right by a n**x**n diagonal matrix D consisting of log2(qk) along the diagonal, where qk are the primes from 2 to p, obtaining M = ND. (5) Now find Q = M`M, where M` is the [[RMS tuning|Moore-Penrose pseudoinverse]] of M. On the assumption that the ck form a basis for the commas, then M has linearly independent rows, and by a property of the pseudoinverse, M` = M*(MM*)^(-1), where M* is the transpose of M, so that Q = M*(MM*)^(1)M. (6) Let P = I - Q, where I is the identity matrix. (7) For a p-limit monzo or [[Fractional monzos|fractional monzo]] m we now define the seminorm ||m||_s = ||mDP|| where the norm on the right is the ordinary Euclidean norm. (8) If v[1] is odd then for each integer j, 0 <= j < v[1], we choose a corresponding monzo mj such that <v|m> = j, 0 <= <J|m> < 1 where J is the JI mapping <log2(2) log2(3) ... log2(p)|, and ||m||_s is minimal. (9) If v[1] is even, we choose a monzo u such that ||u||_s > 0 and ||u||_s is minimal. Then for each integer j, 0 <= j < v[1], we choose a corresponding monzo mj such that <v|m> = j, 0 <= <J|m> < 1, and ||m - u/2||_s is minimal. (10) We now apply the chosen tuning to the monzos mj, obtaining values (in cents or fractional monzos) defining a scale. The monzos mj are defined only modulo the commas and the octave o, but since the commas are tempered out and mj is in the octave range from 0 to 1200 cents, this does not affect the definition of the scale. Note that
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>. 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:<h2> --><h2 id="toc0"><a name="x-Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h2>
<br />
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. This seminorm applies to <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzos</a> and has the property that the seminorm of a comma of the temperament, or of the unison, the octave and any power of two is 0. It may be defined as follows:<br />
<br />
(1) If o = |1 0 0 ... 0> is the monzo for 2 in the <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> group.<br />
<br />
(2) c1, c2, ..., ci are monzos for a basis for the commas of the temperament.<br />
<br />
(3) Form the (i+1)<strong>x</strong>n matrix N = [o, c1, c2, ..., ci] whose rows consist of o and the commas ck.<br />
<br />
(4) Monzo weight N by multiplying on the right by a n<strong>x</strong>n diagonal matrix D consisting of log2(qk) along the diagonal, where qk are the primes from 2 to p, obtaining M = ND.<br />
<br />
(5) Now find Q = M`M, where M` is the <a class="wiki_link" href="/RMS%20tuning">Moore-Penrose pseudoinverse</a> of M. On the assumption that the ck form a basis for the commas, then M has linearly independent rows, and by a property of the pseudoinverse, M` = M*(MM*)^(-1), where M* is the transpose of M, so that Q = M*(MM*)^(1)M.<br />
<br />
(6) Let P = I - Q, where I is the identity matrix.<br />
<br />
(7) For a p-limit monzo or <a class="wiki_link" href="/Fractional%20monzos">fractional monzo</a> m we now define the seminorm<br />
<br />
<table class="wiki_table">
<tr>
<td>m<br />
</td>
<td>_s =<br />
</td>
<td>mDP<br />
</td>
</tr>
</table>
<br />
where the norm on the right is the ordinary Euclidean norm.<br />
<br />
(8) If v[1] is odd then for each integer j, 0 <= j < v[1], we choose a corresponding monzo mj such that <v|m> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="j, 0"></a><!-- ws:end:WikiTextHeadingRule:2 --> j, 0 <</h1>
<J|m> < 1 where J is the JI mapping <log2(2) log2(3) ... log2(p)|, and ||m||_s is minimal.<br />
<br />
(9) If v[1] is even, we choose a monzo u such that ||u||_s > 0 and ||u||_s is minimal. Then for each integer j, 0 <= j < v[1], we choose a corresponding monzo mj such that <v|m> <!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="j, 0"></a><!-- ws:end:WikiTextHeadingRule:4 --> j, 0 <</h1>
<J|m> < 1, and ||m - u/2||_s is minimal.<br />
<br />
(10) We now apply the chosen tuning to the monzos mj, obtaining values (in cents or fractional monzos) defining a scale. The monzos mj are defined only modulo the commas and the octave o, but since the commas are tempered out and mj is in the octave range from 0 to 1200 cents, this does not affect the definition of the scale.<br />
<br />
Note that</body></html>