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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>2010-10-24 21:55:51 UTC</tt>.<br>
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| : The original revision id was <tt>173237177</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]] [[val]] ''V'' which [[support]]s (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''<sub>1</sub> notes to the octave. |
| 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) Let o = |1 0 0 ... 0> be the monzo for 2 in the [[Harmonic Limit|p-limit]] group.
| | == 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 [[Tenney-Euclidean metrics #Octave equivalent TE seminorm|octave equivalent Tenney-Euclidean seminorm]] or OETES. 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 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''. |
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| (2) Let c1, c2, ..., ci be monzos for a basis for the commas of the temperament. | | 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 ''v''<sub>1</sub> is an odd number. If ''v''<sub>1</sub> is odd then for each integer ''j'', {{nowrap|0 < ''j'' ≤ ''v''<sub>1</sub>}}, we choose a corresponding monzo '''m''' such that {{nowrap|{{vmp|''V''|'''m'''}} {{=}} ''j''|0 < {{vmp| ''J''|'''m'''}} ≤ 1}} where ''J'' is the [[just tuning map]] {{val| log<sub>2</sub>2 log<sub>2</sub>3 … log<sub>2</sub>''p'' }}, and ''T''('''m''') is minimal. |
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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. | | If ''v''<sub>1</sub> 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 {{nowrap|''T''('''u''') > 0}}; in other words, '''u''' is a shortest positive length interval. Then for each integer ''j'', where {{nowrap|0 < ''j'' ≤ ''v''<sub>1</sub>}}, we choose a corresponding monzo '''m''' such that {{nowrap|{{vmp|''V''|'''m'''}} {{=}} ''j''|0 < {{vmp| ''J''|'''m'''}} ≤ 1}}, and where {{nowrap|''T''(2'''m''' − '''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. |
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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.
| | 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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| (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.
| | An alternative and equivalent approach is to work directly with the notes of the temperament, using the [[Tenney-Euclidean metrics #Temperamental complexity|temperamental norm]] defined on the note classes of the temperament modulo period (an octave or fraction of an octave) of the temperament. |
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| (6) Let P = I - Q, where I is the identity matrix.
| | == 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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| (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.
| | 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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| (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.
| | [[Category:Hobbit| ]] <!-- main page --> |
| | | [[Category:Math]] |
| (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.
| | [[Category:Scale]] |
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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 eigenmonzos 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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| ==Hobbit blocks==
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| Hobbit scales can be generalized and related to [[Fokker blocks]]. As before, we suppose we have a val v supporting the regular temperament, and a seminorm on interval space defined from the temperament.
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| We can find a subgroup of just intonation in which every member of the notes of the temperament, for a particular just tuning, has a unique representative. In that case, the seminorm becomes a norm. The commas of the val v belonging to the subgroup have a unique [[http://www.farcaster.com/papers/sm-thesis/node6.html|Minkowski basis]] in terms of this norm, and we may use these commas, and the reduction of v to the subgroup, to define Fokker blocks in the usual way. The tempering of these blocks by the temperament are the hobbit blocks. This includes the hobbit itself, which is defined so as to be uniquely identified with the temperament and the val but is not otherwise more of interest than the other hobbit blocks. All of the blocks for a given size of scale taken together can be defined in terms of the Minkowski basis of commas, which we may call a hobbit basis.</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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| <!-- 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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| (1) Let o = |1 0 0 ... 0&gt; be the monzo for 2 in the <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> group.<br />
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| (2) Let c1, c2, ..., ci be monzos for a basis for the commas of the temperament.<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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| (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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| (7) For a p-limit monzo or <a class="wiki_link" href="/Fractional%20monzos">fractional monzo</a> m we now define the seminorm ||m||_s = ||mDP|| where the norm on the right is the ordinary Euclidean norm.<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 eigenmonzos 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.<br />
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| <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Hobbit blocks"></a><!-- ws:end:WikiTextHeadingRule:4 -->Hobbit blocks</h2>
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| Hobbit scales can be generalized and related to <a class="wiki_link" href="/Fokker%20blocks">Fokker blocks</a>. As before, we suppose we have a val v supporting the regular temperament, and a seminorm on interval space defined from the temperament.<br />
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| <br />
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| We can find a subgroup of just intonation in which every member of the notes of the temperament, for a particular just tuning, has a unique representative. In that case, the seminorm becomes a norm. The commas of the val v belonging to the subgroup have a unique <a class="wiki_link_ext" href="http://www.farcaster.com/papers/sm-thesis/node6.html" rel="nofollow">Minkowski basis</a> in terms of this norm, and we may use these commas, and the reduction of v to the subgroup, to define Fokker blocks in the usual way. The tempering of these blocks by the temperament are the hobbit blocks. This includes the hobbit itself, which is defined so as to be uniquely identified with the temperament and the val but is not otherwise more of interest than the other hobbit blocks. All of the blocks for a given size of scale taken together can be defined in terms of the Minkowski basis of commas, which we may call a hobbit basis.</body></html></pre></div>
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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 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 just tuning map ⟨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 ≤ 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-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.