User:2^67-1/Derivation of some temperaments - Revision history

2^67-1 at 07:18, 30 September 2026

2026-09-30T07:18:36Z

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	This is because the mapping [https://sintel.pythonanywhere.com/result?subgroup=30.2.3.11.275653%2F100000.540281%2F100000&reduce=on&weights=weil&target=&edos=34+5746&submit_edo=submit&commas= Sintel's temperament finder] gives is very complex, with generator ranges reaching 297.		This is because the mapping [https://sintel.pythonanywhere.com/result?subgroup=30.2.3.11.275653%2F100000.540281%2F100000&reduce=on&weights=weil&target=&edos=34+5746&submit_edo=submit&commas= Sintel's temperament finder] gives is very complex, with generator ranges reaching 297.

			Note that [[34ed30]] or [[3ed27/20]] are possible "spines" for this temperament.

2^67-1: /* Polyraider (rank-2, 7&1171, 2.3.5.11.f2.f3) */

2026-09-30T07:14:55Z

Polyraider (rank-2, 7&1171, 2.3.5.11.f2.f3)

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	''Note: f2, f3 refer to the first 2 modes of a bar, or approximately 2.75653 and 5.40281. ''		''Note: f2, f3 refer to the first 2 modes of a bar, or approximately 2.75653 and 5.40281. ''

	Premise: What is the next tuning that ~~approximates the first few harmonics of a string and a bar~~ better ~~than 11EDT up to relative error~~? Can it be possible to join it with 11EDT?		Premise: What is the next tuning that does what 11EDT does but better? Can it be possible to join it with 11EDT?

	== ~~Derivation~~ ==		== First part, solved ==

			The next tuning that approximates the first few harmonics of a string and a bar better than 11EDT up to relative error is [[1171edo]]. It is notable for having a very accurate 2.3.5.11 subgroup.

	== Mapping ==		== Mapping ==

	The mapping which I prefer to use (not Hermite normal form) is:		The mapping which I prefer to use (not Hermite normal form) is:

			<pre>
			[ 7 11 16 24 10 17]
			[-12 -3 15 -5 23 -23]
			</pre>

			This is because the mapping [https://sintel.pythonanywhere.com/result?subgroup=30.2.3.11.275653%2F100000.540281%2F100000&reduce=on&weights=weil&target=&edos=34+5746&submit_edo=submit&commas= Sintel's temperament finder] gives is very complex, with generator ranges reaching 297.

2^67-1: /* Polyraider (rank-2, 7&1171, 2.3.5.11.f2.f3) */

2026-09-30T07:09:36Z

Polyraider (rank-2, 7&1171, 2.3.5.11.f2.f3)

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	''Note: f2, f3 refer to the first 2 modes of a bar, or approximately 2.75653 and 5.40281. ''		''Note: f2, f3 refer to the first 2 modes of a bar, or approximately 2.75653 and 5.40281. ''

			Premise: What is the next tuning that approximates the first few harmonics of a string and a bar better than 11EDT up to relative error? Can it be possible to join it with 11EDT?

			== Derivation ==

			== Mapping ==

			The mapping which I prefer to use (not Hermite normal form) is:

2^67-1 at 07:08, 30 September 2026

2026-09-30T07:08:17Z

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	This is also supported by [[229edo]].		This is also supported by [[229edo]].

			= Polyraider (rank-2, 7&1171, 2.3.5.11.f2.f3) =

			''Note: f2, f3 refer to the first 2 modes of a bar, or approximately 2.75653 and 5.40281. ''

2^67-1 at 11:25, 18 September 2026

2026-09-18T11:25:45Z

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	This page details the derivation of some temperaments I have created.		This page details the derivation of some temperaments I have created.

	= ~~Colian Ultimate~~ (rank-3, 7&b22&b176, 2.3.5.7.11.13.17.19.23.31.47.127) =		= Grand Undetrita (rank-3, 7&b22&b176, 2.3.5.7.11.13.17.19.23.31.47.127) =

	Premise: [[11edt]] is a superb exotemperament in the 2.3.5.11 subgroup. What if we expanded that?		Premise: [[11edt]] is a superb exotemperament in the 2.3.5.11 subgroup. What if we expanded that?

2^67-1 at 05:42, 4 September 2026

2026-09-04T05:42:05Z

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	== Restrictions ==		== Restrictions ==

			=== No-23s ===

	If one removes 23, we get the three-patent-val-EDO-defined temperament 7&111&236:		If one removes 23, we get the three-patent-val-EDO-defined temperament 7&111&236:

	<pre>		<pre>
	[ 7 11 16 20 24 26 29 30 32 35 39 49]		[ 7 11 16 20 24 26 29 30 35 39 49]
	[ 0 0 0 -1 0 -1 -1 -1 -1 -1 -1 -1]		[ 0 0 0 -1 0 -1 -1 -1 -1 -1 -1]
	[-1 0 2 0 0 3 -2 0 -2 -2 1 0]		[-1 0 2 0 0 3 -2 0 -2 1 0]
	</pre>		</pre>

	In this tuning, [[236edo]] makes the semitone 17 steps and the fine tuning increment 2 steps.		In this tuning, [[236edo]] makes the semitone 17 steps and the fine tuning increment 2 steps. Note that the semitone is ''still'' 1/2 the [[11edt]] step.

			=== 2.3.5.7.11.13.19 ===

			Since 17, 23, 31, 47, and 127 are close to 16 (or 18), 24, 32, 48, and 128, it is more difficult to use them in harmony. Restricting it further to 2.3.5.7.11.13.19 gives:

			<pre>
			[ 7 11 16 20 24 26 30]
			[ 0 0 0 -1 0 -1 -1]
			[-1 0 2 0 0 3 0]
			</pre>

	~~Note that the semitone~~ is ~~''still'' 1/2 the~~ [[~~11edt~~]] ~~step~~.		This is also supported by [[229edo]].

2^67-1 at 05:31, 4 September 2026

2026-09-04T05:31:30Z

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	== Utonal approximations of [[11edt]] ==		== Utonal approximations of [[11edt]] ==

	Based off Werckmeister's [[Septenarius]] tuning, I have derived something similar to it in principle, but for ~~11EDT~~.		Based off Werckmeister's [[Septenarius]] tuning, I have derived something similar to it in principle, but for [[11edt]].

	This [https://www.desmos.com/calculator/eoix0qucng Desmos graph] shows the maximum error of the utonal divisions. For every X number of divisions of the string, the "relative error" of the nth step of [[11edt]] is:		This [https://www.desmos.com/calculator/eoix0qucng Desmos graph] shows the maximum error of the utonal divisions. For every X number of divisions of the string, the "relative error" of the nth step of [[11edt]] is:
Line 15:		Line 15:
	</math>		</math>

	Therefore the error metric used in the graph is taken to be the maximum of this function ranging from n = 0 to n = 11. As it turns out, if one divides the string into 231 equal lengths, one has the smallest integer approximation of ~~11EDT~~'s 11 steps with less than 25 percent relative error on all of them.		Therefore the error metric used in the graph is taken to be the maximum of this function ranging from n = 0 to n = 11. As it turns out, if one divides the string into 231 equal lengths, one has the smallest integer approximation of [[11edt]]'s 11 steps with less than 25 percent relative error on all of them.

	== Mashing the steps ==		== Mashing the steps ==
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	[ 0 0 0 -1 0 -1 -1 -1 -1 -1 -1 -1]		[ 0 0 0 -1 0 -1 -1 -1 -1 -1 -1 -1]
	</pre>		</pre>

			== Restrictions ==

			If one removes 23, we get the three-patent-val-EDO-defined temperament 7&111&236:

			<pre>
			[ 7 11 16 20 24 26 29 30 32 35 39 49]
			[ 0 0 0 -1 0 -1 -1 -1 -1 -1 -1 -1]
			[-1 0 2 0 0 3 -2 0 -2 -2 1 0]
			</pre>

			In this tuning, [[236edo]] makes the semitone 17 steps and the fine tuning increment 2 steps.

			Note that the semitone is ''still'' 1/2 the [[11edt]] step.

2^67-1 at 04:40, 4 September 2026

2026-09-04T04:40:25Z

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	Note that there are less than three EDOs or EDTs which support this temperament ([[7edo\|7]] and [[111edo\|111]], and [[22edt\|22]] and [[176edt\|176]] repectively).		Note that there are less than three EDOs or EDTs which support this temperament ([[7edo\|7]] and [[111edo\|111]], and [[22edt\|22]] and [[176edt\|176]] repectively).

			Of these three tunings:

			* [[7edo]] makes the semitone and fine tuning increment 0 steps.
			* [[22edt]] makes the semitone 1 step and fine tuning increment 0 steps.
			* [[111edo]]/[[176edt]] makes the semitone 8 steps and fine tuning increment 1 step.

	== Subsets ==		== Subsets ==
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	=== 7&b22 ===		=== 7&b22 ===

	Also note that if one removes the fine tunings ~~below~~, one obtains [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.7.11.13.17.19.23.31.47.127&reduce=on&weights=weil&target=&edos=7+14%5B-5%2C-13%2C-47%2C-127%5D&submit_edo=submit&commas= this temperament], which in comparison is an exotemperament: however, if one is fine with [[11edt]]'s approximations of 2, 3, 5, and 11 this temperament should be usable. The [[7edo]]-based mapping is:		Also note that if one removes the fine tunings, one obtains [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.7.11.13.17.19.23.31.47.127&reduce=on&weights=weil&target=&edos=7+14%5B-5%2C-13%2C-47%2C-127%5D&submit_edo=submit&commas= this temperament], which in comparison is an exotemperament: however, if one is fine with [[11edt]]'s approximations of 2, 3, 5, and 11 this temperament should be usable. The [[7edo]]-based mapping is:

	<pre>		<pre>

2^67-1 at 04:27, 4 September 2026

2026-09-04T04:27:36Z

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	=== 7&b22 ===		=== 7&b22 ===

	Also note that if one removes the fine tunings below, one obtains [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.7.11.13.17.19.23.31.47.127&reduce=on&weights=weil&target=&edos=7+14%5B-5%2C-13%2C-47%2C-127%5D&submit_edo=submit&commas= this temperament], which in comparison is an exotemperament: however, if one is fine with [[11edt]]'s approximations of 2, 3, 5, and 11 this temperament should be usable.		Also note that if one removes the fine tunings below, one obtains [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.7.11.13.17.19.23.31.47.127&reduce=on&weights=weil&target=&edos=7+14%5B-5%2C-13%2C-47%2C-127%5D&submit_edo=submit&commas= this temperament], which in comparison is an exotemperament: however, if one is fine with [[11edt]]'s approximations of 2, 3, 5, and 11 this temperament should be usable. The [[7edo]]-based mapping is:

			<pre>
			[ 7 11 16 20 24 26 29 30 32 35 39 49]
			[ 0 0 0 -1 0 -1 -1 -1 -1 -1 -1 -1]
			</pre>

2^67-1 at 04:27, 4 September 2026

2026-09-04T04:27:10Z

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	== Subsets ==		== Subsets ==

	== 7&111 ==		=== 7&111 ===

	7&111 maps the semitone to eight of the fine tuning increments. The [[7edo]]-based mapping is:		7&111 maps the semitone to eight of the fine tuning increments. The [[7edo]]-based mapping is:
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	=== b22&b176 ===		=== b22&b176 ===

	This maps the semitone to exactly half of an [[11edt]] step, or one [[22edt]] step. This allows the 3.7.11.19.127 subgroup to be approximated solely using [[22edt]].		This maps the semitone to exactly half of an [[11edt]] step, or one [[22edt]] step. This allows the 3.7.11.19.127 subgroup to be approximated solely using [[22edt]]. The [[22edt]]-based mapping is:

			<pre>
			[14 22 32 39 48 51 57 59 63 69 77 97]
			[-1 0 2 0 0 3 -2 0 -2 -2 1 0]
			</pre>

			=== 7&b22 ===

	Also note that if one removes the fine tunings below, one obtains [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.7.11.13.17.19.23.31.47.127&reduce=on&weights=weil&target=&edos=7+14%5B-5%2C-13%2C-47%2C-127%5D&submit_edo=submit&commas= this temperament], which in comparison is an exotemperament: however, if one is fine with [[11edt]]'s approximations of 2, 3, 5, and 11 this temperament should be usable.		Also note that if one removes the fine tunings below, one obtains [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.7.11.13.17.19.23.31.47.127&reduce=on&weights=weil&target=&edos=7+14%5B-5%2C-13%2C-47%2C-127%5D&submit_edo=submit&commas= this temperament], which in comparison is an exotemperament: however, if one is fine with [[11edt]]'s approximations of 2, 3, 5, and 11 this temperament should be usable.