Minkowski block - Revision history

Sintel at 18:09, 29 March 2025

2025-03-29T18:09:32Z

← Older revision		Revision as of 18:09, 29 March 2025
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	~~{{Legacy}}~~
	A '''Minkowski block''' is a particular kind of [[Fokker block]] which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the [[Tenney-Euclidean metrics\|OE seminorm]] on [[Monzos and Interval Space\|interval space]] defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.		A '''Minkowski block''' is a particular kind of [[Fokker block]] which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the [[Tenney-Euclidean metrics\|OE seminorm]] on [[Monzos and Interval Space\|interval space]] defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.

Lériendil at 16:46, 17 February 2025

2025-02-17T16:46:38Z

← Older revision		Revision as of 16:46, 17 February 2025
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	{{Legacy~~\|Minkowski_block~~}}		{{Legacy}}
	A '''Minkowski block''' is a particular kind of [[Fokker block]] which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the [[Tenney-Euclidean metrics\|OE seminorm]] on [[Monzos and Interval Space\|interval space]] defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.		A '''Minkowski block''' is a particular kind of [[Fokker block]] which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the [[Tenney-Euclidean metrics\|OE seminorm]] on [[Monzos and Interval Space\|interval space]] defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.

Lériendil: deploying new cat

2025-02-17T16:17:25Z

deploying new cat

← Older revision		Revision as of 16:17, 17 February 2025
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			{{Legacy\|Minkowski_block}}
	A '''Minkowski block''' is a particular kind of [[Fokker block]] which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the [[Tenney-Euclidean metrics\|OE seminorm]] on [[Monzos and Interval Space\|interval space]] defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.		A '''Minkowski block''' is a particular kind of [[Fokker block]] which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the [[Tenney-Euclidean metrics\|OE seminorm]] on [[Monzos and Interval Space\|interval space]] defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.

Lériendil: Changed protection level for "Minkowski block" ([Edit=Allow only administrators] (expires 17:03, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:03, 17 February 2025 (UTC)))

2025-02-17T16:03:29Z

Changed protection level for "Minkowski block" ([Edit=Allow only administrators] (expires 17:03, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:03, 17 February 2025 (UTC)))

← Older revision	Revision as of 16:03, 17 February 2025
(No difference)

Mike Battaglia: Protected "Minkowski block" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

2021-11-15T03:21:58Z

Protected "Minkowski block" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

← Older revision	Revision as of 03:21, 15 November 2021
(No difference)

Xenwolf: lemma bold, simplify links

2021-01-07T06:54:02Z

lemma bold, simplify links

← Older revision		Revision as of 06:54, 7 January 2021
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	A Minkowski block is a particular kind of [[~~Fokker_blocks\|~~Fokker block]] which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the [[Tenney-~~Euclidean_metrics~~\|OE seminorm]] on [[~~Monzos_and_Interval_Space~~\|interval space]] defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.		A '''Minkowski block''' is a particular kind of [[Fokker block]] which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the [[Tenney-Euclidean metrics\|OE seminorm]] on [[Monzos and Interval Space\|interval space]] defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.

	We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to the transversal sublattice 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 to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore [[~~Transversal\|transversals~~]]. This very often but not always includes the [[~~Hobbits\|~~hobbit]] associated with T and v, in which case we may call them hobbit blocks.		We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to the transversal sublattice 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 to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore [[transversal]]s. This very often but not always includes the [[hobbit]] associated with T and v, in which case we may call them hobbit blocks.

	=Example=		== Example ==
	Consider marvel, which is supported by the 11-limit 19et patent val, since it tempers out both 225/224 and 385/384. The transversal sublattice can be taken to be 5-limit JI: every note of marvel has a unique 5-limit JI representative. The 5-limit commas of 19et, in order of the OE seminorm, are 81/80, 4428675/4194304, 273375/262144, 16875/16384, 3125/3072, 15625/15552, 78732/78125... . In terms of the associated 5-limit temperaments, that's meantone, hogzilla, stump, negri, magic, hanson, sensi... . The meantone-hogzilla arena is therefore the arena of the Minkowski blocks for 19-note marvel.		Consider marvel, which is supported by the 11-limit 19et patent val, since it tempers out both 225/224 and 385/384. The transversal sublattice can be taken to be 5-limit JI: every note of marvel has a unique 5-limit JI representative. The 5-limit commas of 19et, in order of the OE seminorm, are 81/80, 4428675/4194304, 273375/262144, 16875/16384, 3125/3072, 15625/15552, 78732/78125... . In terms of the associated 5-limit temperaments, that's meantone, hogzilla, stump, negri, magic, hanson, sensi... . The meantone-hogzilla arena is therefore the arena of the Minkowski blocks for 19-note marvel.
	[[Category:~~fokker_block~~]]
	[[Category:~~math~~]]		[[Category:Math]]
			[[Category:Fokker block]]

Xenwolf: Xenwolf moved page Minkowski blocks to Minkowski block: singular rule for better linking

2021-01-07T06:48:51Z

Xenwolf moved page Minkowski blocks to Minkowski block: singular rule for better linking

← Older revision	Revision as of 06:48, 7 January 2021
(No difference)

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

← Older revision		Revision as of 00:00, 17 July 2018
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	~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~		A Minkowski block is a particular kind of [[Fokker_blocks\|Fokker block]] which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the [[Tenney-Euclidean_metrics\|OE seminorm]] on [[Monzos_and_Interval_Space\|interval space]] defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.
	~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~
	~~: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2014-05-19 23:29:04 UTC</tt>.<br>~~
	~~: The original revision id was <tt>509980780</tt>.<br>~~
	~~: The revision comment was: <tt></tt><br>~~
	~~The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>~~
	~~<h4>Original Wikitext content:</h4>~~
	<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 Minkowski block is a particular kind of [[~~Fokker blocks~~\|Fokker block]] which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the [[Tenney-~~Euclidean metrics~~\|OE seminorm]] on [[~~Monzos and Interval Space~~\|interval space]] defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.

	We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to the transversal sublattice 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 to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore [[~~transversal~~\|transversals]]. This very often but not always includes the [[Hobbits\|hobbit]] associated with T and v, in which case we may call them hobbit blocks.		We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to the transversal sublattice 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 to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore [[Transversal\|transversals]]. This very often but not always includes the [[Hobbits\|hobbit]] associated with T and v, in which case we may call them hobbit blocks.

	=Example=		=Example=
	Consider marvel, which is supported by the 11-limit 19et patent val, since it tempers out both 225/224 and 385/384. The transversal sublattice can be taken to be 5-limit JI: every note of marvel has a unique 5-limit JI representative. The 5-limit commas of 19et, in order of the OE seminorm, are 81/80, 4428675/4194304, 273375/262144, 16875/16384, 3125/3072, 15625/15552, 78732/78125... . In terms of the associated 5-limit temperaments, that's meantone, hogzilla, stump, negri, magic, hanson, sensi... . The meantone-hogzilla arena is therefore the arena of the Minkowski blocks for 19-note marvel.~~</pre></div>~~		Consider marvel, which is supported by the 11-limit 19et patent val, since it tempers out both 225/224 and 385/384. The transversal sublattice can be taken to be 5-limit JI: every note of marvel has a unique 5-limit JI representative. The 5-limit commas of 19et, in order of the OE seminorm, are 81/80, 4428675/4194304, 273375/262144, 16875/16384, 3125/3072, 15625/15552, 78732/78125... . In terms of the associated 5-limit temperaments, that's meantone, hogzilla, stump, negri, magic, hanson, sensi... . The meantone-hogzilla arena is therefore the arena of the Minkowski blocks for 19-note marvel.
	~~<h4>Original HTML content~~:~~</h4>~~		[[Category:fokker_block]]
	~~<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>Minkowski blocks</title></head><body><br />		[[Category:math]]
	A Minkowski block is a particular kind of <a class="wiki_link" href="/Fokker%20blocks">Fokker block</a> which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">OE seminorm</a> on <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a> defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.<br />
	~~<br />~~
	We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to the transversal sublattice 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 to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore <a class="wiki_link" href="/transversal">transversals</a>. This very often but not always includes the <a class="wiki_link" href="/Hobbits">hobbit</a> associated with T and v, in which case we may call them hobbit blocks.<br />
	~~<br />~~
	~~<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule:0 -->Example</h1>~~
	Consider marvel, which is supported by the 11-limit 19et patent val, since it tempers out both 225/224 and 385/384. The transversal sublattice can be taken to be 5-limit JI: every note of marvel has a unique 5-limit JI representative. The 5-limit commas of 19et, in order of the OE seminorm, are 81/80, 4428675/4194304, 273375/262144, 16875/16384, 3125/3072, 15625/15552, 78732/78125... . In terms of the associated 5-limit temperaments, that's meantone, hogzilla, stump, negri, magic, hanson, sensi... . The meantone-hogzilla arena is therefore the arena of the Minkowski blocks for 19-note marvel.</body></html></pre></div>

Wikispaces>genewardsmith: Imported revision 509980780 - Original comment:

2014-05-19T23:29:04Z

**Imported revision 509980780 - Original comment: **

← Older revision		Revision as of 23:29, 19 May 2014
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2014-05-19 14:10:57 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2014-05-19 23:29:04 UTC</tt>.<br>
	: The original revision id was <tt>~~509878950~~</tt>.<br>		: The original revision id was <tt>509980780</tt>.<br>
	: The revision comment was: <tt></tt><br>		: The revision comment was: <tt></tt><br>
	The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>		The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
Line 9:		Line 9:
	A Minkowski block is a particular kind of [[Fokker blocks\|Fokker block]] which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the [[Tenney-Euclidean metrics\|OE seminorm]] on [[Monzos and Interval Space\|interval space]] defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.		A Minkowski block is a particular kind of [[Fokker blocks\|Fokker block]] which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the [[Tenney-Euclidean metrics\|OE seminorm]] on [[Monzos and Interval Space\|interval space]] defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.

	We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to T 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 v~~ to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore [[transversal\|transversals]]. This very often but not always includes the [[Hobbits\|hobbit]] associated with T and v, in which case we may call them hobbit blocks.</pre></div>		We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to the transversal sublattice 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 to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore [[transversal\|transversals]]. This very often but not always includes the [[Hobbits\|hobbit]] associated with T and v, in which case we may call them hobbit blocks.

			=Example=
			Consider marvel, which is supported by the 11-limit 19et patent val, since it tempers out both 225/224 and 385/384. The transversal sublattice can be taken to be 5-limit JI: every note of marvel has a unique 5-limit JI representative. The 5-limit commas of 19et, in order of the OE seminorm, are 81/80, 4428675/4194304, 273375/262144, 16875/16384, 3125/3072, 15625/15552, 78732/78125... . In terms of the associated 5-limit temperaments, that's meantone, hogzilla, stump, negri, magic, hanson, sensi... . The meantone-hogzilla arena is therefore the arena of the Minkowski blocks for 19-note marvel.</pre></div>
	<h4>Original HTML content:</h4>		<h4>Original HTML content:</h4>
	<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>Minkowski blocks</title></head><body><br />		<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>Minkowski blocks</title></head><body><br />
	A Minkowski block is a particular kind of <a class="wiki_link" href="/Fokker%20blocks">Fokker block</a> which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">OE seminorm</a> on <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a> defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.<br />		A Minkowski block is a particular kind of <a class="wiki_link" href="/Fokker%20blocks">Fokker block</a> which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">OE seminorm</a> on <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a> defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.<br />
	<br />		<br />
	We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to T 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 v~~ to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore <a class="wiki_link" href="/transversal">transversals</a>. This very often but not always includes the <a class="wiki_link" href="/Hobbits">hobbit</a> associated with T and v, in which case we may call them hobbit blocks.</body></html></pre></div>		We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to the transversal sublattice 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 to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore <a class="wiki_link" href="/transversal">transversals</a>. This very often but not always includes the <a class="wiki_link" href="/Hobbits">hobbit</a> associated with T and v, in which case we may call them hobbit blocks.<br />
			<br />
			<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Example"></a><!-- ws:end:WikiTextHeadingRule:0 -->Example</h1>
			Consider marvel, which is supported by the 11-limit 19et patent val, since it tempers out both 225/224 and 385/384. The transversal sublattice can be taken to be 5-limit JI: every note of marvel has a unique 5-limit JI representative. The 5-limit commas of 19et, in order of the OE seminorm, are 81/80, 4428675/4194304, 273375/262144, 16875/16384, 3125/3072, 15625/15552, 78732/78125... . In terms of the associated 5-limit temperaments, that's meantone, hogzilla, stump, negri, magic, hanson, sensi... . The meantone-hogzilla arena is therefore the arena of the Minkowski blocks for 19-note marvel.</body></html></pre></div>

Wikispaces>genewardsmith: Imported revision 509878950 - Original comment:

2014-05-19T14:10:57Z

**Imported revision 509878950 - Original comment: **

← Older revision		Revision as of 14:10, 19 May 2014
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:~~xenwolf~~\|~~xenwolf~~]] and made on <tt>~~2010~~-10-~~30 12~~:48:02 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2014-05-19 14:10:57 UTC</tt>.<br>
	: The original revision id was <tt>~~174970613~~</tt>.<br>		: The original revision id was <tt>509878950</tt>.<br>
	: The revision comment was: <tt></tt><br>		: The revision comment was: <tt></tt><br>
	The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>		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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	A Minkowski block is a particular kind of [[Fokker blocks\|Fokker block]] which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the [[Tenney-Euclidean metrics\|OE seminorm]] on [[Monzos and Interval Space\|interval space]] defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.		A Minkowski block is a particular kind of [[Fokker blocks\|Fokker block]] which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the [[Tenney-Euclidean metrics\|OE seminorm]] on [[Monzos and Interval Space\|interval space]] defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.

	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 Minkowski blocks, for which the correspondong Fokker blocks are therefore [[transversal\|transversals]]. This very often but not always includes the [[Hobbits\|hobbit]] associated with T and v, in which case we may call them hobbit blocks.</pre></div>		We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to T 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 v to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore [[transversal\|transversals]]. This very often but not always includes the [[Hobbits\|hobbit]] associated with T and v, in which case we may call them hobbit blocks.</pre></div>
	<h4>Original HTML content:</h4>		<h4>Original HTML content:</h4>
	<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>Minkowski blocks</title></head><body><br />		<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>Minkowski blocks</title></head><body><br />
	A Minkowski block is a particular kind of <a class="wiki_link" href="/Fokker%20blocks">Fokker block</a> which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">OE seminorm</a> on <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a> defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.<br />		A Minkowski block is a particular kind of <a class="wiki_link" href="/Fokker%20blocks">Fokker block</a> which tends to be a good candidate for tempering by a particular regular temperament T. Suppose we have a val v supporting T, and the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">OE seminorm</a> on <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a> defined from the temperament; that is, the seminorm defined by orthogonal projection in interval space orthogonal to the commas of T and 2, the octave.<br />
	<br />		<br />
	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 Minkowski blocks, for which the correspondong Fokker blocks are therefore <a class="wiki_link" href="/transversal">transversals</a>. This very often but not always includes the <a class="wiki_link" href="/Hobbits">hobbit</a> associated with T and v, in which case we may call them hobbit blocks.</body></html></pre></div>		We can find a subspace of interval space in which every note of T has a unique representative, giving a transversal for the temperament in the form of a sublattice of the lattice of intervals T tempers. In that subspace, the seminorm becomes a norm. The commas of v belonging to T 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 v to define Fokker blocks in the usual way. The tempering of these blocks by T are the Minkowski blocks, for which the correspondong Fokker blocks are therefore <a class="wiki_link" href="/transversal">transversals</a>. This very often but not always includes the <a class="wiki_link" href="/Hobbits">hobbit</a> associated with T and v, in which case we may call them hobbit blocks.</body></html></pre></div>