K*N subgroups - Revision history

FloraC: Recategorize

2025-03-31T07:58:54Z

Recategorize

← Older revision		Revision as of 07:58, 31 March 2025
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	For example, to find the 7-limit 2*6 subgroup we first find {{nowrap\|''m'' {{=}} 3 × 5 × 7 {{=}} 105}}. The subgroup of Eu(105) consisting of those intervals mapped to an even number by {{vector\|12 19 28 34}} is 2.5.7. The subgroup of Eu(105<sup>2</sup>) is 2.9.5.7, not the same. However, the subgroup of Eu(105<sup>3</sup>) is again 2.9.5.7. If to this we add 81/80, we still obtain 2.9.5.7. It is clear this is maximal, as 3 is mapped by 12et to an odd number (19), and the rest of the values are prime.		For example, to find the 7-limit 2*6 subgroup we first find {{nowrap\|''m'' {{=}} 3 × 5 × 7 {{=}} 105}}. The subgroup of Eu(105) consisting of those intervals mapped to an even number by {{vector\|12 19 28 34}} is 2.5.7. The subgroup of Eu(105<sup>2</sup>) is 2.9.5.7, not the same. However, the subgroup of Eu(105<sup>3</sup>) is again 2.9.5.7. If to this we add 81/80, we still obtain 2.9.5.7. It is clear this is maximal, as 3 is mapped by 12et to an odd number (19), and the rest of the values are prime.

	[[Category:~~Algorithm~~]]		[[Category:Algorithms]]
	[[Category:Math]]		[[Category:Math]]
	[[Category:~~Theory~~]]		[[Category:Regular temperament theory]]

ArrowHead294 at 18:31, 29 March 2025

2025-03-29T18:31:59Z

← Older revision		Revision as of 18:31, 29 March 2025
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	{{todo\|explain its xenharmonic value}}		{{todo\|explain its xenharmonic value}}
	For any [[~~Harmonic_Limit~~\|prime limit]] p, EDO N and positive integer k, the p-limit kN subgroup is the largest [[~~Just_intonation_subgroups\|~~just intonation subgroup]] of the p-limit on which N-edo and kN-edo approximate intervals to the same values using the mapping supplied by the [[~~Patent_val\|~~patent val]] for k*N-edo. This also means they temper out the same commas.		For any [[Harmonic_limit\|prime limit]] ''p'', EDO ''N'', and positive integer ''k'', the ''p''-limit ''k''''N'' subgroup is the largest [[just intonation subgroup]] of the ''p''-limit on which ''N''-edo and ''k''''N''-edo approximate intervals to the same values using the mapping supplied by the [[patent val]] for ''k''*''N''-edo. This also means they temper out the same commas.

	A procedure for finding the kN subgroup which seems to suffice is to take the product m of the odd primes less than or equal to p, and then find the [[~~Euler_genera\|~~Euler genus]] ~~Euler~~(m^i) for integers 1, 2, and so forth. Here ~~Euler~~(d) for an odd integer d is the set of all divisors of d reduced to an octave, and including 2. For each such genus, select those intervals such that the kN patent val maps the interval to a number divisible by k, and then find the corresponding [[~~Normal_lists~~\|normal interval list]]. When two successive values i and i+1 lead to the same normal list, add to that a basis for the commas of the k*N patent val, and return the normal interval list for that.		A procedure for finding the ''k''''N'' subgroup which seems to suffice is to take the product ''m'' of the odd primes less than or equal to ''p'', and then find the [[Euler genus]] Eu(''m''<sup>''i''</sup>) for integers 1, 2, and so forth. Here Eu(''d'') for an odd integer ''d'' is the set of all divisors of ''d'' reduced to an octave, and including 2. For each such genus, select those intervals such that the ''k''''N'' patent val maps the interval to a number divisible by ''k'', and then find the corresponding [[normal lists\|normal interval list]]. When two successive values ''i'' and {{nowrap\|''i'' + 1}} lead to the same normal list, add to that a basis for the commas of the ''k''*''N'' patent val, and return the normal interval list for that.

	For example, to find the 7-limit 26 subgroup we first find m = 35*7 = 105. The subgroup of ~~Euler~~(105) consisting of those intervals mapped to an even number by ~~<~~12 19 28 34\| is 2.5.7. The subgroup of ~~Euler~~(105^2) is 2.9.5.7, not the same. However, the subgroup of ~~Euler~~(105^3) is again 2.9.5.7. If to this we add 81/80, we still obtain 2.9.5.7. It is clear this is maximal, as 3 is mapped by 12et to an odd number (19), and the rest of the values are prime.		For example, to find the 7-limit 2*6 subgroup we first find {{nowrap\|''m'' {{=}} 3 × 5 × 7 {{=}} 105}}. The subgroup of Eu(105) consisting of those intervals mapped to an even number by {{vector\|12 19 28 34}} is 2.5.7. The subgroup of Eu(105<sup>2</sup>) is 2.9.5.7, not the same. However, the subgroup of Eu(105<sup>3</sup>) is again 2.9.5.7. If to this we add 81/80, we still obtain 2.9.5.7. It is clear this is maximal, as 3 is mapped by 12et to an odd number (19), and the rest of the values are prime.
	[[Category:~~algorithm~~]]
	[[Category:~~math~~]]		[[Category:Algorithm]]
	[[Category:~~theory~~]]		[[Category:Math]]
			[[Category:Theory]]

Sintel: -legacy, todo explain relevance

2025-03-29T18:04:57Z

-legacy, todo explain relevance

← Older revision		Revision as of 18:04, 29 March 2025
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	{{~~Legacy~~}}		{{todo\|explain its xenharmonic value}}
	For any [[Harmonic_Limit\|prime limit]] p, EDO N and positive integer k, the p-limit kN subgroup is the largest [[Just_intonation_subgroups\|just intonation subgroup]] of the p-limit on which N-edo and kN-edo approximate intervals to the same values using the mapping supplied by the [[Patent_val\|patent val]] for k*N-edo. This also means they temper out the same commas.		For any [[Harmonic_Limit\|prime limit]] p, EDO N and positive integer k, the p-limit kN subgroup is the largest [[Just_intonation_subgroups\|just intonation subgroup]] of the p-limit on which N-edo and kN-edo approximate intervals to the same values using the mapping supplied by the [[Patent_val\|patent val]] for k*N-edo. This also means they temper out the same commas.

Lériendil at 16:56, 17 February 2025

2025-02-17T16:56:11Z

← Older revision		Revision as of 16:56, 17 February 2025
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	{{Legacy~~\|K*N_subgroups~~}}		{{Legacy}}
	For any [[Harmonic_Limit\|prime limit]] p, EDO N and positive integer k, the p-limit kN subgroup is the largest [[Just_intonation_subgroups\|just intonation subgroup]] of the p-limit on which N-edo and kN-edo approximate intervals to the same values using the mapping supplied by the [[Patent_val\|patent val]] for k*N-edo. This also means they temper out the same commas.		For any [[Harmonic_Limit\|prime limit]] p, EDO N and positive integer k, the p-limit kN subgroup is the largest [[Just_intonation_subgroups\|just intonation subgroup]] of the p-limit on which N-edo and kN-edo approximate intervals to the same values using the mapping supplied by the [[Patent_val\|patent val]] for k*N-edo. This also means they temper out the same commas.

Lériendil: deploying new cat

2025-02-17T16:11:28Z

deploying new cat

← Older revision		Revision as of 16:11, 17 February 2025
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			{{Legacy\|K*N_subgroups}}
	For any [[Harmonic_Limit\|prime limit]] p, EDO N and positive integer k, the p-limit kN subgroup is the largest [[Just_intonation_subgroups\|just intonation subgroup]] of the p-limit on which N-edo and kN-edo approximate intervals to the same values using the mapping supplied by the [[Patent_val\|patent val]] for k*N-edo. This also means they temper out the same commas.		For any [[Harmonic_Limit\|prime limit]] p, EDO N and positive integer k, the p-limit kN subgroup is the largest [[Just_intonation_subgroups\|just intonation subgroup]] of the p-limit on which N-edo and kN-edo approximate intervals to the same values using the mapping supplied by the [[Patent_val\|patent val]] for k*N-edo. This also means they temper out the same commas.

Lériendil: Changed protection level for "K*N subgroups" ([Edit=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)))

2025-02-17T16:01:26Z

Changed protection level for "K*N subgroups" ([Edit=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)))

← Older revision	Revision as of 16:01, 17 February 2025
(No difference)

Mike Battaglia: Protected "K*N subgroups" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

2021-11-15T03:07:27Z

Protected "K*N subgroups" ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))

← Older revision	Revision as of 03:07, 15 November 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>~~		For any [[Harmonic_Limit\|prime limit]] p, EDO N and positive integer k, the p-limit kN subgroup is the largest [[Just_intonation_subgroups\|just intonation subgroup]] of the p-limit on which N-edo and kN-edo approximate intervals to the same values using the mapping supplied by the [[Patent_val\|patent val]] for k*N-edo. This also means they temper out the same commas.
	~~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>2011-05-30 07:46:56 UTC</tt>.<br>~~
	~~: The original revision id was <tt>232859204</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">For any [[~~Harmonic limit~~\|prime limit]] p, EDO N and positive integer k, the p-limit kN subgroup is the largest [[~~Just intonation subgroups~~\|just intonation subgroup]] of the p-limit on which N-edo and kN-edo approximate intervals to the same values using the mapping supplied by the [[patent val]] for k*N-edo. This also means they temper out the same commas.

	A procedure for finding the kN subgroup which seems to suffice is to take the product m of the odd primes less than or equal to p, and then find the [[~~Euler genera~~\|Euler genus]] Euler(m^i) for integers 1, 2, and so forth. Here Euler(d) for an odd integer d is the set of all divisors of d reduced to an octave, and including 2. For each such genus, select those intervals such that the kN patent val maps the interval to a number divisible by k, and then find the corresponding [[~~Normal lists~~\|normal interval list]]. When two successive values i and i+1 lead to the same normal list, add to that a basis for the commas of the k*N patent val, and return the normal interval list for that.		A procedure for finding the kN subgroup which seems to suffice is to take the product m of the odd primes less than or equal to p, and then find the [[Euler_genera\|Euler genus]] Euler(m^i) for integers 1, 2, and so forth. Here Euler(d) for an odd integer d is the set of all divisors of d reduced to an octave, and including 2. For each such genus, select those intervals such that the kN patent val maps the interval to a number divisible by k, and then find the corresponding [[Normal_lists\|normal interval list]]. When two successive values i and i+1 lead to the same normal list, add to that a basis for the commas of the k*N patent val, and return the normal interval list for that.

	For example, to find the 7-limit 26 subgroup we first find m = 35*7 = 105. The subgroup of Euler(105) consisting of those intervals mapped to an even number by <12 19 28 34\| is 2.5.7. The subgroup of Euler(105^2) is 2.9.5.7, not the same. However, the subgroup of Euler(105^3) is again 2.9.5.7. If to this we add 81/80, we still obtain 2.9.5.7. It is clear this is maximal, as 3 is mapped by 12et to an odd number (19), and the rest of the values are prime.~~</pre></div>~~		For example, to find the 7-limit 26 subgroup we first find m = 35*7 = 105. The subgroup of Euler(105) consisting of those intervals mapped to an even number by <12 19 28 34\| is 2.5.7. The subgroup of Euler(105^2) is 2.9.5.7, not the same. However, the subgroup of Euler(105^3) is again 2.9.5.7. If to this we add 81/80, we still obtain 2.9.5.7. It is clear this is maximal, as 3 is mapped by 12et to an odd number (19), and the rest of the values are prime.
	~~<h4>Original HTML content~~:~~</h4>~~		[[Category:algorithm]]
	~~<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>kN subgroups</title></head><body>For any <a class="wiki_link" href="/Harmonic%20limit">prime limit</a> p, EDO N and positive integer k, the p-limit kN subgroup is the largest <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a> of the p-limit on which N-edo and kN-edo approximate intervals to the same values using the mapping supplied by the <a class="wiki_link" href="/patent%20val">patent val</a> for kN-edo. This also means they temper out the same commas. <br />		[[Category:math]]
	~~<br />~~		[[Category:theory]]
	A procedure for finding the kN subgroup which seems to suffice is to take the product m of the odd primes less than or equal to p, and then find the <a class="wiki_link" href="/Euler%20genera">Euler genus</a> Euler(m^i) for integers 1, 2, and so forth. Here Euler(d) for an odd integer d is the set of all divisors of d reduced to an octave, and including 2. For each such genus, select those intervals such that the kN patent val maps the interval to a number divisible by k, and then find the corresponding <a class="wiki_link" href="/Normal%20lists">normal interval list</a>. When two successive values i and i+1 lead to the same normal list, add to that a basis for the commas of the k*N patent val, and return the normal interval list for that.<br />
	~~<br />~~
	For example, to find the 7-limit 26 subgroup we first find m = 35*7 = 105. The subgroup of Euler(105) consisting of those intervals mapped to an even number by &lt;12 19 28 34\| is 2.5.7. The subgroup of Euler(105^2) is 2.9.5.7, not the same. However, the subgroup of Euler(105^3) is again 2.9.5.7. If to this we add 81/80, we still obtain 2.9.5.7. It is clear this is maximal, as 3 is mapped by 12et to an odd number (19), and the rest of the values are prime.</body></html></pre></div>

Wikispaces>genewardsmith: Imported revision 232859204 - Original comment:

2011-05-30T07:46:56Z

**Imported revision 232859204 - Original comment: **

← Older revision		Revision as of 07:46, 30 May 2011
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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>2011-05-~~28 23~~:35:02 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-05-30 07:46:56 UTC</tt>.<br>
	: The original revision id was <tt>~~232650890~~</tt>.<br>		: The original revision id was <tt>232859204</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 procedure for finding the kN subgroup which seems to suffice is to take the product m of the odd primes less than or equal to p, and then find the [[Euler genera\|Euler genus]] Euler(m^i) for integers 1, 2, and so forth. Here Euler(d) for an odd integer d is the set of all divisors of d reduced to an octave, and including 2. For each such genus, select those intervals such that the kN patent val maps the interval to a number divisible by k, and then find the corresponding [[Normal lists\|normal interval list]]. When two successive values i and i+1 lead to the same normal list, add to that a basis for the commas of the k*N patent val, and return the normal interval list for that.		A procedure for finding the kN subgroup which seems to suffice is to take the product m of the odd primes less than or equal to p, and then find the [[Euler genera\|Euler genus]] Euler(m^i) for integers 1, 2, and so forth. Here Euler(d) for an odd integer d is the set of all divisors of d reduced to an octave, and including 2. For each such genus, select those intervals such that the kN patent val maps the interval to a number divisible by k, and then find the corresponding [[Normal lists\|normal interval list]]. When two successive values i and i+1 lead to the same normal list, add to that a basis for the commas of the k*N patent val, and return the normal interval list for that.

	</pre></div>		For example, to find the 7-limit 26 subgroup we first find m = 35*7 = 105. The subgroup of Euler(105) consisting of those intervals mapped to an even number by <12 19 28 34\| is 2.5.7. The subgroup of Euler(105^2) is 2.9.5.7, not the same. However, the subgroup of Euler(105^3) is again 2.9.5.7. If to this we add 81/80, we still obtain 2.9.5.7. It is clear this is maximal, as 3 is mapped by 12et to an odd number (19), and the rest of the values are prime.</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>kN subgroups</title></head><body>For any <a class="wiki_link" href="/Harmonic%20limit">prime limit</a> p, EDO N and positive integer k, the p-limit kN subgroup is the largest <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a> of the p-limit on which N-edo and kN-edo approximate intervals to the same values using the mapping supplied by the <a class="wiki_link" href="/patent%20val">patent val</a> for kN-edo. This also means they temper out the same commas. <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>kN subgroups</title></head><body>For any <a class="wiki_link" href="/Harmonic%20limit">prime limit</a> p, EDO N and positive integer k, the p-limit kN subgroup is the largest <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a> of the p-limit on which N-edo and kN-edo approximate intervals to the same values using the mapping supplied by the <a class="wiki_link" href="/patent%20val">patent val</a> for kN-edo. This also means they temper out the same commas. <br />
	<br />		<br />
	A procedure for finding the kN subgroup which seems to suffice is to take the product m of the odd primes less than or equal to p, and then find the <a class="wiki_link" href="/Euler%20genera">Euler genus</a> Euler(m^i) for integers 1, 2, and so forth. Here Euler(d) for an odd integer d is the set of all divisors of d reduced to an octave, and including 2. For each such genus, select those intervals such that the kN patent val maps the interval to a number divisible by k, and then find the corresponding <a class="wiki_link" href="/Normal%20lists">normal interval list</a>. When two successive values i and i+1 lead to the same normal list, add to that a basis for the commas of the k*N patent val, and return the normal interval list for that.</body></html></pre></div>		A procedure for finding the kN subgroup which seems to suffice is to take the product m of the odd primes less than or equal to p, and then find the <a class="wiki_link" href="/Euler%20genera">Euler genus</a> Euler(m^i) for integers 1, 2, and so forth. Here Euler(d) for an odd integer d is the set of all divisors of d reduced to an octave, and including 2. For each such genus, select those intervals such that the kN patent val maps the interval to a number divisible by k, and then find the corresponding <a class="wiki_link" href="/Normal%20lists">normal interval list</a>. When two successive values i and i+1 lead to the same normal list, add to that a basis for the commas of the k*N patent val, and return the normal interval list for that.<br />
			<br />
			For example, to find the 7-limit 26 subgroup we first find m = 35*7 = 105. The subgroup of Euler(105) consisting of those intervals mapped to an even number by &lt;12 19 28 34\| is 2.5.7. The subgroup of Euler(105^2) is 2.9.5.7, not the same. However, the subgroup of Euler(105^3) is again 2.9.5.7. If to this we add 81/80, we still obtain 2.9.5.7. It is clear this is maximal, as 3 is mapped by 12et to an odd number (19), and the rest of the values are prime.</body></html></pre></div>

Wikispaces>genewardsmith: Imported revision 232650890 - Original comment:

2011-05-28T23:35:02Z

**Imported revision 232650890 - Original comment: **

← Older revision		Revision as of 23:35, 28 May 2011
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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>2011-05-28 22:10:33 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-05-28 23:35:02 UTC</tt>.<br>
	: The original revision id was <tt>~~232643896~~</tt>.<br>		: The original revision id was <tt>232650890</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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	<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">For any [[Harmonic limit\|prime limit]] p, EDO N and positive integer k, the p-limit kN subgroup is the largest [[Just intonation subgroups\|just intonation subgroup]] of the p-limit on which N-edo and kN-edo approximate intervals to the same values using the mapping supplied by the [[patent val]] for k*N-edo. This also means they temper out the same commas.		<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">For any [[Harmonic limit\|prime limit]] p, EDO N and positive integer k, the p-limit kN subgroup is the largest [[Just intonation subgroups\|just intonation subgroup]] of the p-limit on which N-edo and kN-edo approximate intervals to the same values using the mapping supplied by the [[patent val]] for k*N-edo. This also means they temper out the same commas.

	A procedure for finding the kN subgroup is to take the product m of the odd primes less than or equal to p, and then find the [[Euler genera\|Euler genus]] Euler(m^i) for integers 1, 2, and so forth. Here Euler(d) for an odd integer d is the set of all divisors of d reduced to an octave, and including 2. For each such genus, select those intervals such that the kN patent val maps the interval to a number divisible by k, and then find the corresponding [[Normal lists\|normal interval list]]. When two successive values i and i+1 lead to the same normal list, ~~return~~ that as the ~~canonical list~~ of ~~generators for~~ the k*N ~~subgroup~~.		A procedure for finding the kN subgroup which seems to suffice is to take the product m of the odd primes less than or equal to p, and then find the [[Euler genera\|Euler genus]] Euler(m^i) for integers 1, 2, and so forth. Here Euler(d) for an odd integer d is the set of all divisors of d reduced to an octave, and including 2. For each such genus, select those intervals such that the kN patent val maps the interval to a number divisible by k, and then find the corresponding [[Normal lists\|normal interval list]]. When two successive values i and i+1 lead to the same normal list, add to that a basis for the commas of the k*N patent val, and return the normal interval list for that.

	</pre></div>		</pre></div>
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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>kN subgroups</title></head><body>For any <a class="wiki_link" href="/Harmonic%20limit">prime limit</a> p, EDO N and positive integer k, the p-limit kN subgroup is the largest <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a> of the p-limit on which N-edo and kN-edo approximate intervals to the same values using the mapping supplied by the <a class="wiki_link" href="/patent%20val">patent val</a> for kN-edo. This also means they temper out the same commas. <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>kN subgroups</title></head><body>For any <a class="wiki_link" href="/Harmonic%20limit">prime limit</a> p, EDO N and positive integer k, the p-limit kN subgroup is the largest <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a> of the p-limit on which N-edo and kN-edo approximate intervals to the same values using the mapping supplied by the <a class="wiki_link" href="/patent%20val">patent val</a> for kN-edo. This also means they temper out the same commas. <br />
	<br />		<br />
	A procedure for finding the kN subgroup is to take the product m of the odd primes less than or equal to p, and then find the <a class="wiki_link" href="/Euler%20genera">Euler genus</a> Euler(m^i) for integers 1, 2, and so forth. Here Euler(d) for an odd integer d is the set of all divisors of d reduced to an octave, and including 2. For each such genus, select those intervals such that the kN patent val maps the interval to a number divisible by k, and then find the corresponding <a class="wiki_link" href="/Normal%20lists">normal interval list</a>. When two successive values i and i+1 lead to the same normal list, ~~return~~ that as the ~~canonical list~~ of ~~generators for~~ the k*N ~~subgroup~~.</body></html></pre></div>		A procedure for finding the kN subgroup which seems to suffice is to take the product m of the odd primes less than or equal to p, and then find the <a class="wiki_link" href="/Euler%20genera">Euler genus</a> Euler(m^i) for integers 1, 2, and so forth. Here Euler(d) for an odd integer d is the set of all divisors of d reduced to an octave, and including 2. For each such genus, select those intervals such that the kN patent val maps the interval to a number divisible by k, and then find the corresponding <a class="wiki_link" href="/Normal%20lists">normal interval list</a>. When two successive values i and i+1 lead to the same normal list, add to that a basis for the commas of the k*N patent val, and return the normal interval list for that.</body></html></pre></div>