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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | A '''hobbit''', or '''hobbit scale''', is a generalization of [[mos scale]] for arbitrary [[regular temperament]]s which is a sort of cousin to [[dwarf|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. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-04 03:20:45 UTC</tt>.<br>
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| : The original revision id was <tt>167499979</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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.
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| ==Definition==
| | Given a regular temperament and an [[equal temperament]] which [[support]]s 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. |
| 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:
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| (1) If o = |1 0 0 ... 0> is the monzo for 2 in the [[Harmonic Limit|p-limit]] group. | | == 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 [[Tenney-Euclidean metrics #Octave-equivalent TE seminorm|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 {{nowrap| ''T''(''x'') {{=}} ''T''(''2''/''x'') }}, where ''x'' and 2/''x'' are [[octave complement]]s. |
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| (2) c1, c2, ..., ci are monzos for a basis for the commas of the temperament.
| | 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. |
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| (3) Form the (i+1)**x**n matrix N = [o, c1, c2, ..., ci] whose rows consist of o and the commas ck.
| | 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. |
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| (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. | | == 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 {{val| 22 35 51 62 76 }}, and an interval of minimal nonzero size for the temperament is 16/15, with monzo {{monzo| 4 -1 -1 0 0 }}. From this we may find a transversal minimizing {{nowrap| ''T''(2'''m''' − {{monzo| 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. |
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| (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.
| | 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 {{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. |
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| (6) Let P = I - Q, where I is the identity matrix.
| | == Notation == |
| | | Hobbits are often assumed when a rank-3 temperament is appended with a number (e.g. [[Marvel9|marvel[9]]]), similar to how a rank-2 temperament appended with a number (e.g. [[Meantone7|meantone[7]]]) denotes a MOS. |
| (7) For a p-limit monzo or [[Fractional monzos|fractional monzo]] m we now define the seminorm
| | [[Category:Hobbit| ]] <!-- main page --> |
| | | [[Category:Math]] |
| || m ||_s = || mDP ||
| | [[Category:Scale]] |
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| where the norm on the right is the ordinary Euclidean norm.
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| (8) 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 mj 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 ||m||_s is minimal.
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| (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, where 0 less than j less than or equal to v[1], we choose a corresponding monzo mj such that <v|m> = j, 0 less than <J|m> less than or equal to 1, and where ||m - u/2||_s is minimal.
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| (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 less than mj less than or equal to 1200 cents, this does not affect the definition of the scale.
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| ==Example==
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| 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>. The fractional monzo for half of this, corresponding to the square root, is |4 -1/2 -1/2 0 0>, and intervals representing scale steps are 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 eigenmonzsos 2, 3, and 11.
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| 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. This sort of thing seems to happen fairly often with hobbit scales.
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| </pre></div>
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| <h4>Original HTML content:</h4>
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| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><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 />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h2>
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| 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 />
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| <br />
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| (1) If o = |1 0 0 ... 0&gt; is the monzo for 2 in the <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> group.<br />
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| <br />
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| (2) c1, c2, ..., ci are monzos for a basis for the commas of the temperament.<br />
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| <br />
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| (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 />
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| <br />
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| (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 />
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| <br />
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| (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 />
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| <br />
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| (6) Let P = I - Q, where I is the identity matrix.<br />
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| <br />
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| (7) For a p-limit monzo or <a class="wiki_link" href="/Fractional%20monzos">fractional monzo</a> m we now define the seminorm<br />
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| <table class="wiki_table">
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| <tr>
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| <td>m<br />
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| </td>
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| <td>_s =<br />
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| </td>
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| <td>mDP<br />
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| </td>
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| </tr>
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| </table>
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| <br />
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| where the norm on the right is the ordinary Euclidean norm.<br />
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| <br />
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| (8) 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 mj 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 ||m||_s is minimal.<br />
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| <br />
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| (9) If v[1] is even, we choose a monzo u such that ||u||_s &gt; 0 and ||u||_s is minimal. Then for each integer j, where 0 less than j less than or equal to v[1], we choose a corresponding monzo mj such that &lt;v|m&gt; = j, 0 less than &lt;J|m&gt; less than or equal to 1, and where ||m - u/2||_s is minimal.<br />
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| <br />
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| (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 less than mj less than or equal to 1200 cents, this does not affect the definition of the scale.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Example"></a><!-- ws:end:WikiTextHeadingRule:2 -->Example</h2>
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| 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;. The fractional monzo for half of this, corresponding to the square root, is |4 -1/2 -1/2 0 0&gt;, and intervals representing scale steps are 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 eigenmonzsos 2, 3, and 11.<br />
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| <br />
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| 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. This sort of thing seems to happen fairly often with hobbit scales.</body></html></pre></div>
| |
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.