113edo - Revision history

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:33:01Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:33, 13 March 2026
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	\| [[Gaster temperament\|Gaster]]		\| [[Gaster temperament\|Gaster]]
	\|}		\|}
	<nowiki />* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|minimal form]] in parentheses if distinct		<nowiki />* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

Lériendil at 03:37, 5 February 2026

2026-02-05T03:37:30Z

← Older revision		Revision as of 03:37, 5 February 2026
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	== Theory ==		== Theory ==
	113edo is [[consistency\|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out\|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.		113edo is [[consistency\|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency (and in the [[15-odd-limit]], only [[15/11]] and its complement are inconsistent). As an equal temperament, it [[tempering out\|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.

	113edo is also notable as a [[31-limit]] system, especially if error on prime 5 is tolerable. In fact, it is consistent in the no-15 no-25 [[29-odd-limit]], which nearly extends to the [[33-odd-limit]], with only two inconsistent interval pairs that both involve 31, being [[31/21]] (50.8% off) and [[31/20]] (55.4% off) and their complements – and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]]. Notably as a slendric system, it is the largest EDO that maps [[64/49]] and [[21/16]] to the same interval consistently.		113edo is also notable as a [[31-limit]] system, especially if error on prime 5 is tolerable. In fact, it is consistent in the no-15 no-25 [[29-odd-limit]], which nearly extends to the [[33-odd-limit]], with only two inconsistent interval pairs that both involve 31, being [[31/21]] (50.8% off) and [[31/20]] (55.4% off) and their complements – and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]]. Notably as a slendric system, it is the largest EDO that maps [[64/49]] and [[21/16]] to the same interval consistently.

Lériendil: /* Theory */

2025-09-14T20:51:52Z

Theory

← Older revision		Revision as of 20:51, 14 September 2025
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	113edo is [[consistency\|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out\|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.		113edo is [[consistency\|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out\|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.

	113edo is also notable as a [[31-limit]] system, especially if error on prime 5 is tolerable. In fact, it is consistent in the no-15 no-25 [[29-odd-limit]] ~~and~~ nearly the [[33-odd-limit]], with only two inconsistent interval pairs ~~involving~~ 31, being [[31/21]] (50.8% off) and [[31/20]] (55.4% off) and their complements – and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]]. Notably as a slendric system, it is the largest EDO that maps [[64/49]] and [[21/16]] to the same interval consistently.		113edo is also notable as a [[31-limit]] system, especially if error on prime 5 is tolerable. In fact, it is consistent in the no-15 no-25 [[29-odd-limit]], which nearly extends to the [[33-odd-limit]], with only two inconsistent interval pairs that both involve 31, being [[31/21]] (50.8% off) and [[31/20]] (55.4% off) and their complements – and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]]. Notably as a slendric system, it is the largest EDO that maps [[64/49]] and [[21/16]] to the same interval consistently.

	=== Prime harmonics ===		=== Prime harmonics ===

Lériendil: /* Theory */

2025-09-14T20:36:00Z

Theory

← Older revision		Revision as of 20:36, 14 September 2025
Line 5:		Line 5:
	113edo is [[consistency\|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out\|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.		113edo is [[consistency\|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out\|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.

	113edo is also notable as a [[31-limit]] system, especially if error on prime 5 is tolerable. In fact, it is consistent in the no-15 no-25 [[29-odd-limit]] and nearly the [[33-odd-limit]], with only two inconsistent interval pairs involving 31, being [[31/21]] (50.8% off) and [[31/20]] (55.4% off) and their complements – and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]].		113edo is also notable as a [[31-limit]] system, especially if error on prime 5 is tolerable. In fact, it is consistent in the no-15 no-25 [[29-odd-limit]] and nearly the [[33-odd-limit]], with only two inconsistent interval pairs involving 31, being [[31/21]] (50.8% off) and [[31/20]] (55.4% off) and their complements – and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]]. Notably as a slendric system, it is the largest EDO that maps [[64/49]] and [[21/16]] to the same interval consistently.

	=== Prime harmonics ===		=== Prime harmonics ===

Lériendil: /* Theory */

2025-09-14T06:37:57Z

Theory

← Older revision		Revision as of 06:37, 14 September 2025
Line 5:		Line 5:
	113edo is [[consistency\|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out\|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.		113edo is [[consistency\|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out\|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.

	113edo is also notable as a [[31-limit]] system, especially if error on prime 5 is tolerable. In fact, it is consistent in the no-15 no-25 [[29-odd-limit]] and nearly the [[33-odd-limit]], as with only two inconsistent interval pairs involving 31, being [[31/21]] (50.8% off) and [[31/20]] (55.4% off) and their complements – and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]].		113edo is also notable as a [[31-limit]] system, especially if error on prime 5 is tolerable. In fact, it is consistent in the no-15 no-25 [[29-odd-limit]] and nearly the [[33-odd-limit]], with only two inconsistent interval pairs involving 31, being [[31/21]] (50.8% off) and [[31/20]] (55.4% off) and their complements – and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]].

	=== Prime harmonics ===		=== Prime harmonics ===

Lériendil at 06:36, 14 September 2025

2025-09-14T06:36:20Z

← Older revision		Revision as of 06:36, 14 September 2025
Line 5:		Line 5:
	113edo is [[consistency\|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out\|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.		113edo is [[consistency\|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out\|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.

	113edo is also notable as a ~~no-fives~~ [[31-limit]] system. In fact, it is consistent in the no-15 no-25 [[29-odd-limit]] and nearly the [[33-odd-limit]], as only [[31/21]] ~~and its complement are inconsistent, being~~ 50.8% off – and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]].		113edo is also notable as a [[31-limit]] system, especially if error on prime 5 is tolerable. In fact, it is consistent in the no-15 no-25 [[29-odd-limit]] and nearly the [[33-odd-limit]], as with only two inconsistent interval pairs involving 31, being [[31/21]] (50.8% off) and [[31/20]] (55.4% off) and their complements – and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]].

	=== Prime harmonics ===		=== Prime harmonics ===

FloraC: Concision

2025-09-12T09:16:32Z

Concision

← Older revision		Revision as of 09:16, 12 September 2025
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	113edo is [[consistency\|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out\|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.		113edo is [[consistency\|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out\|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.

	113edo is also notable as a no-fives [[31-limit]] system, ~~where~~ it is consistent in the [[29-odd-limit]] ~~(additionally, 5 itself, but not the patent 15 or 25, can also be added to this diamond with no inconsistent intervals),~~ and nearly the [[33-odd-limit]], as only [[31/21]] and its complement are inconsistent, being 50.8% off - and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]].		113edo is also notable as a no-fives [[31-limit]] system. In fact, it is consistent in the no-15 no-25 [[29-odd-limit]] and nearly the [[33-odd-limit]], as only [[31/21]] and its complement are inconsistent, being 50.8% off – and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]].

	=== Prime harmonics ===		=== Prime harmonics ===

Lériendil: /* Theory */

2025-09-12T06:22:36Z

Theory

← Older revision		Revision as of 06:22, 12 September 2025
Line 5:		Line 5:
	113edo is [[consistency\|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out\|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.		113edo is [[consistency\|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out\|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.

	113edo is also notable as a no-fives [[31-limit]] system, where it is consistent in the [[29-odd-limit]], and nearly the [[33-odd-limit]], as only [[31/21]] and its complement are inconsistent, being 50.8% off - and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]].		113edo is also notable as a no-fives [[31-limit]] system, where it is consistent in the [[29-odd-limit]] (additionally, 5 itself, but not the patent 15 or 25, can also be added to this diamond with no inconsistent intervals), and nearly the [[33-odd-limit]], as only [[31/21]] and its complement are inconsistent, being 50.8% off - and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]].

	=== Prime harmonics ===		=== Prime harmonics ===

Lériendil at 06:18, 30 August 2025

2025-08-30T06:18:33Z

← Older revision		Revision as of 06:18, 30 August 2025
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	\| 1		\| 1
	\| 34\113		\| 34\113
	\| ~~360~~.06		\| 361.06
	\| 16/13		\| 16/13
	\| [[Phicordial]]		\| [[Phicordial]]

Lériendil: /* Rank-2 temperaments */

2025-08-30T06:18:08Z

Rank-2 temperaments

← Older revision		Revision as of 06:18, 30 August 2025
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	\| 27/25		\| 27/25
	\| [[Quartemka]]		\| [[Quartemka]]
			\|-
			\| 1
			\| 20\113
			\| 212.39
			\| 26/23
			\| [[Shoal]]
	\|-		\|-
	\| 1		\| 1