101ed7 - Revision history

ArrowHead294: /* See also */

2025-06-25T19:13:26Z

FloraC: /* Theory */ note near identity to 93ed6

2025-05-28T13:37:53Z

Theory: note near identity to 93ed6

← Older revision		Revision as of 13:37, 28 May 2025
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	== Theory ==		== Theory ==
	101ed7 is closely related to [[36edo]] (sixth-tone tuning), but with the 7th harmonic rather than the [[2/1\|octave]] being just. The octave is stretched by about 0.770 [[cent]]s. Like 36edo, 101ed7 is [[consistent]] to the [[integer limit\|8-integer-limit]].		101ed7 is closely related to [[36edo]] (sixth-tone tuning), but with the 7th harmonic rather than the [[2/1\|octave]] being just. The octave is stretched by about 0.770 [[cent]]s (almost identical to [[93ed6]], where the octave is stretched by about 0.757 cents). Like 36edo, 101ed7 is [[consistent]] to the [[integer limit\|8-integer-limit]].

	Compared to 36edo, 101ed7 is pretty well optimized for the 2.3.7.13.17 [[subgroup]], with slightly better [[3/1\|3]], [[7/1\|7]], [[13/1\|13]] and [[17/1\|17]], and a slightly worse 2 versus 36edo. Using the [[patent val]], the [[5/1\|5]] is also less accurate. Overall this means 36edo is still better in the [[5-limit]], but 101ed7 is better in the [[13-limit\|13-]] and [[17-limit]], especially when treating it as a dual-5 dual-11 tuning.		Compared to 36edo, 101ed7 is pretty well optimized for the 2.3.7.13.17 [[subgroup]], with slightly better [[3/1\|3]], [[7/1\|7]], [[13/1\|13]] and [[17/1\|17]], and a slightly worse 2 versus 36edo. Using the [[patent val]], the [[5/1\|5]] is also less accurate. Overall this means 36edo is still better in the [[5-limit]], but 101ed7 is better in the [[13-limit\|13-]] and [[17-limit]], especially when treating it as a dual-5 dual-11 tuning.

FloraC: /* Theory */ some rework

2025-05-28T13:35:05Z

Theory: some rework

← Older revision		Revision as of 13:35, 28 May 2025
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	101ed7 is closely related to [[36edo]] (sixth-tone tuning), but with the 7th harmonic rather than the [[2/1\|octave]] being just. The octave is stretched by about 0.770 [[cent]]s. Like 36edo, 101ed7 is [[consistent]] to the [[integer limit\|8-integer-limit]].		101ed7 is closely related to [[36edo]] (sixth-tone tuning), but with the 7th harmonic rather than the [[2/1\|octave]] being just. The octave is stretched by about 0.770 [[cent]]s. Like 36edo, 101ed7 is [[consistent]] to the [[integer limit\|8-integer-limit]].

	Compared to 36edo, 101ed7~~'s harmonics are almost exactly~~ the ~~same~~, ~~but it has a~~ slightly better [[3/1\|3]], [[7/1\|7]], ~~and~~ [[13/1\|13]], and a slightly worse 2 ~~and~~ [[5/1\|5]] ~~versus 36edo~~. Overall this means 36edo is still better in the [[5-limit]], but 101ed7 is better in the [[13-limit~~]]. (The [[7~~-~~limit~~]] and [[11-limit]] ~~could go either way.)~~		Compared to 36edo, 101ed7 is pretty well optimized for the 2.3.7.13.17 [[subgroup]], with slightly better [[3/1\|3]], [[7/1\|7]], [[13/1\|13]] and [[17/1\|17]], and a slightly worse 2 versus 36edo. Using the [[patent val]], the [[5/1\|5]] is also less accurate. Overall this means 36edo is still better in the [[5-limit]], but 101ed7 is better in the [[13-limit\|13-]] and [[17-limit]], especially when treating it as a dual-5 dual-11 tuning.

	~~36edo's 5-limit dominance flips on its head, though, if one approaches it as a [[dual-n\|dual-5]] tuning. In that case~~, ~~the fact that 101ed7's 5/1 is closer to 50% relative error is actually a ''good'' thing, because~~ it ~~means the error on the worse of the two 5/1's is less. So as a single-5 5-limit tuning, 36edo is better. But as a dual-5 5-limit tuning, 101ed7 is better, and~~ as a dual-5, dual-11 ~~[[31-limit]]~~ tuning~~, 101ed7 is exceptional for its size. It is very accurate~~.

	=== Harmonics ===		=== Harmonics ===

FloraC: /* Theory */ correction and misc. cleanup

2025-05-28T13:26:06Z

Theory: correction and misc. cleanup

← Older revision		Revision as of 13:26, 28 May 2025
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	== Theory ==		== Theory ==
	101ed7 is closely related to [[36edo]] (sixth-tone tuning), but with the 7th harmonic rather than the [[2/1\|octave]] being just. The octave is stretched by about 1.23 [[cent]]s. Like 36edo, 101ed7 is [[consistent]] to the [[integer limit\|8-integer-limit]].		101ed7 is closely related to [[36edo]] (sixth-tone tuning), but with the 7th harmonic rather than the [[2/1\|octave]] being just. The octave is stretched by about 0.770 [[cent]]s. Like 36edo, 101ed7 is [[consistent]] to the [[integer limit\|8-integer-limit]].

	Compared to 36edo, 101ed7's harmonics are almost exactly the same, but it has a slightly better [[3/1]], [[7/1]], and [[13/1]], and a slightly worse 2/1 and [[5/1]] versus 36edo. Overall this means 36edo is still better in the [[5-limit]], but 101ed7 is better in the [[13-limit]]. (The [[7-limit]] and [[11-limit]] could go either way.)		Compared to 36edo, 101ed7's harmonics are almost exactly the same, but it has a slightly better [[3/1\|3]], [[7/1\|7]], and [[13/1\|13]], and a slightly worse 2 and [[5/1\|5]] versus 36edo. Overall this means 36edo is still better in the [[5-limit]], but 101ed7 is better in the [[13-limit]]. (The [[7-limit]] and [[11-limit]] could go either way.)

	36edo's 5-limit dominance flips on its head, though, if one approaches it as a [[dual-n\|dual-5]] tuning. In that case, the fact that 101ed7's 5/1 is closer to 50% relative error is actually a ''good'' thing, because it means the error on the worse of the two 5/1's is less. So as a single-5 5-limit tuning, 36edo is better. But as a dual-5 5-limit tuning, 101ed7 is better, and as a dual-5, dual-11 [[31-limit]] tuning, 101ed7 is exceptional for its size. It is very accurate.		36edo's 5-limit dominance flips on its head, though, if one approaches it as a [[dual-n\|dual-5]] tuning. In that case, the fact that 101ed7's 5/1 is closer to 50% relative error is actually a ''good'' thing, because it means the error on the worse of the two 5/1's is less. So as a single-5 5-limit tuning, 36edo is better. But as a dual-5 5-limit tuning, 101ed7 is better, and as a dual-5, dual-11 [[31-limit]] tuning, 101ed7 is exceptional for its size. It is very accurate.

FloraC: /* Theory */ +subsets and supersets

2025-05-27T10:49:36Z

Theory: +subsets and supersets

← Older revision		Revision as of 10:49, 27 May 2025
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	{{Harmonics in equal\|101\|7\|1\|intervals=integer\|columns=11}}		{{Harmonics in equal\|101\|7\|1\|intervals=integer\|columns=11}}
	{{Harmonics in equal\|101\|7\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 101ed7 (continued)}}		{{Harmonics in equal\|101\|7\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 101ed7 (continued)}}

			=== Subsets and supersets ===
			101ed7 is the 26th [[prime equal division\|prime ed7]], so it does not contain any nontrivial subset ed7's.

	== Intervals ==		== Intervals ==

FloraC: Cleanup

2025-05-27T10:48:51Z

Cleanup

← Older revision		Revision as of 10:48, 27 May 2025
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	{{ED intro}}		{{ED intro}}

	101ed7 is related to [[36edo]] (sixth-tone tuning), but with the ~~7/1~~ rather than the 2/1 being just. The octave is stretched by about 1.~~2347~~ [[cent]]s. ~~It tempers out the~~ [[~~syntonic comma~~]] ~~and is consistent~~ to the 8-[[integer-limit]].		== Theory ==
			101ed7 is closely related to [[36edo]] (sixth-tone tuning), but with the 7th harmonic rather than the [[2/1\|octave]] being just. The octave is stretched by about 1.23 [[cent]]s. Like 36edo, 101ed7 is [[consistent]] to the [[integer limit\|8-integer-limit]].

	~~Lookalikes:~~ [[~~21edf~~]], [[36edo]], [[~~57edt~~]], [[~~93ed6~~]], [[~~129ed12~~]]		Compared to 36edo, 101ed7's harmonics are almost exactly the same, but it has a slightly better [[3/1]], [[7/1]], and [[13/1]], and a slightly worse 2/1 and [[5/1]] versus 36edo. Overall this means 36edo is still better in the [[5-limit]], but 101ed7 is better in the [[13-limit]]. (The [[7-limit]] and [[11-limit]] could go either way.)

			36edo's 5-limit dominance flips on its head, though, if one approaches it as a [[dual-n\|dual-5]] tuning. In that case, the fact that 101ed7's 5/1 is closer to 50% relative error is actually a ''good'' thing, because it means the error on the worse of the two 5/1's is less. So as a single-5 5-limit tuning, 36edo is better. But as a dual-5 5-limit tuning, 101ed7 is better, and as a dual-5, dual-11 [[31-limit]] tuning, 101ed7 is exceptional for its size. It is very accurate.

			=== Harmonics ===
			{{Harmonics in equal\|101\|7\|1\|intervals=integer\|columns=11}}
			{{Harmonics in equal\|101\|7\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 101ed7 (continued)}}

	== Intervals ==		== Intervals ==
	{{Interval table}}		{{Interval table}}

	== ~~Harmonics~~ ==		== See also ==
	~~Compared to 36edo, 101ed7’s harmonics are almost exactly the same, but it has a slightly better 3/1, 7/1, and 13/1, and a slightly worse 2/1 and 5/1 versus 36edo.~~		* [[21edf]] – relative edf
			* [[36edo]] – relative edo
	~~Overall this means 36edo is still better in the~~ [[~~5-limit~~]]~~, but 101ed7 is better in the~~ [[~~13-limit~~]]~~. (The~~ [[~~7-limit~~]] ~~and [[11-limit]] could go either way.)~~		* [[57edt]] – relative edt
			* [[93ed6]] – relative ed6
	~~36edo’s 5-limit dominance flips on its head, though, if one approaches it as a~~ [[~~dual-n\|dual-5~~]] ~~tuning. In that case, the fact that 101ed7’s 5/1 is closer to 50%~~ relative ~~error is actually a ''good'' thing, because it means the error on the worse of the two 5/1s is less.~~		* [[129ed12]] – relative ed12

	~~So as a single-5 5-limit tuning, 36edo is better. But as a dual-5 5-limit tuning, 101ed7 is better.~~

	~~And as a dual-5, dual-11~~ [[~~31-limit~~]] ~~tuning, 101ed7 is exceptional for its size. It is very accurate.~~
	~~{{Harmonics in equal\|101\|7\|1\|intervals=prime\|columns=12}}~~


	~~36edo for comparsion:~~
	~~{{Harmonics in equal\|36\|2\|1\|intervals=prime\|columns=12}}~~

	~~{{todo\|expand}}~~
	~~[[Category:Edonoi]]~~
	[[Category:36edo]]		[[Category:36edo]]

ArrowHead294 at 17:20, 21 January 2025

2025-01-21T17:20:08Z

← Older revision		Revision as of 17:20, 21 January 2025
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	{{Infobox ET}}		{{Infobox ET}}
	'''101 divisions of the seventh harmonic''' ('''101ed7''') is related to [[36edo]] (sixth-tone tuning), but with the 7/1 rather than the 2/1 being just. The octave is about 1.2347 [[cent]]s stretched and the step size is about 33.3547 cents. It is consistent to the 8-[[integer-limit]]. This tuning tempers out the [[syntonic comma]].		{{ED intro}}

	Lookalikes: [[21edf]], [[36edo]], [[57edt]], [[93ed6]]		101ed7 is related to [[36edo]] (sixth-tone tuning), but with the 7/1 rather than the 2/1 being just. The octave is stretched by about 1.2347 [[cent]]s. It tempers out the [[syntonic comma]] and is consistent to the 8-[[integer-limit]].

			Lookalikes: [[21edf]], [[36edo]], [[57edt]], [[93ed6]], [[129ed12]]

	== Intervals ==		== Intervals ==
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	== Harmonics ==		== Harmonics ==
	Compared to 36edo, 101ed7’s harmonics are almost exactly the same, but it has a slightly better 3/1, 7/1 and 13/1, and a slightly worse 2/1 and 5/1 versus 36edo.		Compared to 36edo, 101ed7’s harmonics are almost exactly the same, but it has a slightly better 3/1, 7/1, and 13/1, and a slightly worse 2/1 and 5/1 versus 36edo.

	Overall this means 36edo is still better in the [[5-limit]], but 101ed7 is better in the [[13-limit]]. (The [[7-limit]] and [[11-limit]] could go either way.)		Overall this means 36edo is still better in the [[5-limit]], but 101ed7 is better in the [[13-limit]]. (The [[7-limit]] and [[11-limit]] could go either way.)

BudjarnLambeth at 06:38, 7 January 2025

2025-01-07T06:38:59Z

← Older revision		Revision as of 06:38, 7 January 2025
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	{{todo\|expand}}		{{todo\|expand}}
	[[Category:Edonoi]]		[[Category:Edonoi]]
			[[Category:36edo]]

BudjarnLambeth: /* Harmonics */

2025-01-07T06:10:37Z

Harmonics

← Older revision		Revision as of 06:10, 7 January 2025
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	Compared to 36edo, 101ed7’s harmonics are almost exactly the same, but it has a slightly better 3/1, 7/1 and 13/1, and a slightly worse 2/1 and 5/1 versus 36edo.		Compared to 36edo, 101ed7’s harmonics are almost exactly the same, but it has a slightly better 3/1, 7/1 and 13/1, and a slightly worse 2/1 and 5/1 versus 36edo.

	Overall this means 36edo is still better in the [[5-limit]], but 101ed7 better in the [[13-limit]]. ~~And the~~ [[7-limit]] and [[11-limit]] could go either way.		Overall this means 36edo is still better in the [[5-limit]], but 101ed7 is better in the [[13-limit]]. (The [[7-limit]] and [[11-limit]] could go either way.)

	36edo’s 5-limit dominance flips on its head, though, if one approaches it as a [[dual-n\|dual-5]] tuning. In that case, the fact that 101ed7’s 5/1 is closer to 50% relative error is actually a ''good'' thing, because it means the error on the worse of the two 5/1s is less.		36edo’s 5-limit dominance flips on its head, though, if one approaches it as a [[dual-n\|dual-5]] tuning. In that case, the fact that 101ed7’s 5/1 is closer to 50% relative error is actually a ''good'' thing, because it means the error on the worse of the two 5/1s is less.

BudjarnLambeth at 06:10, 7 January 2025

2025-01-07T06:10:04Z

← Older revision		Revision as of 06:10, 7 January 2025
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	{{Harmonics in equal\|36\|2\|1\|intervals=prime\|columns=12}}		{{Harmonics in equal\|36\|2\|1\|intervals=prime\|columns=12}}

			{{todo\|expand}}
	{{~~stub~~}}
	[[Category:Edonoi]]		[[Category:Edonoi]]

← Older revision		Revision as of 19:13, 25 June 2025
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	* [[57edt]] – relative edt		* [[57edt]] – relative edt
	* [[93ed6]] – relative ed6		* [[93ed6]] – relative ed6
	* [[129ed12]] – relative ed12		* [[129ed12]] – relative ed12, close to the zeta-optimized tuning for 36edo

	[[Category:36edo]]		[[Category:36edo]]