21st-octave temperaments - Revision history

Overthink: /* Blackmagic */ note primes 17 and 19, etymology

2026-01-15T06:21:36Z

Blackmagic: note primes 17 and 19, etymology

← Older revision		Revision as of 06:21, 15 January 2026
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	== Blackmagic ==		== Blackmagic ==
	Blackmagic is the 63 & 84 temperament, merging two systems which cover many large primes.{{Todo\|review}}		Blackmagic is the 63 & 84 temperament, merging two systems which cover many large primes. It was named by [[User:Overthink\|Overthink]] in 2026 as a twist on "blackjack" (which itself already refers to the 21-note [[MOS scale\|mos]] of [[miracle]]), as well as because of its higher-limit properties. {{Todo\|review}}

	Subgroup: 2.3.5.7		Subgroup: 2.3.5.7
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	=== 2.3.5.7.11.13.23.29.31.43 subgroup ===		=== 2.3.5.7.11.13.23.29.31.43 subgroup ===
			Primes 17 and 19 could be included by mapping them to -1 and 1 generators respectively, though in practice this mapping only works in [[84edo]].

	Subgroup: 2.3.5.7.11.13.23.29.31.43		Subgroup: 2.3.5.7.11.13.23.29.31.43

Overthink: /* Blackmagic */ + badness

2026-01-15T05:47:25Z

Blackmagic: + badness

← Older revision		Revision as of 05:47, 15 January 2026
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	{{Optimal ET sequence\|legend=1\|21, 63, 84, 147}}		{{Optimal ET sequence\|legend=1\|21, 63, 84, 147}}

			[[Badness]] (Sintel): 5.605

	=== 2.3.5.7.11.13.23.29.31.43 subgroup ===		=== 2.3.5.7.11.13.23.29.31.43 subgroup ===
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	Subgroup: 2.3.5.7.11.13.23.29.31.43		Subgroup: 2.3.5.7.11.13.23.29.31.43

	Comma list: 155/154, [[225/224]], [[232/231]], [[300/299]], [[364/363]], 560/559, [[640/637]], [[1716/1715]]		Comma list: 155/154, 225/224, [[232/231]], [[300/299]], [[364/363]], 560/559, [[640/637]], [[1716/1715]]

	{{Mapping\|legend=1\| 21 0 82 59 106 111 95 102 104 114 \| 0 1 -1 0 -1 -1 0 0 0 0 }}		{{Mapping\|legend=1\| 21 0 82 59 106 111 95 102 104 114 \| 0 1 -1 0 -1 -1 0 0 0 0 }}
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	: Mapping generators: ~16807/16384 = 1\21, ~3		: Mapping generators: ~16807/16384 = 1\21, ~3

	[[Optimal tuning]] ([[CWE]]): ~3/2 = 701.742{{C}}		Optimal tuning ([[CWE]]): ~3/2 = 701.742{{C}}

			{{Optimal ET sequence\|legend=0\|21, 63, 84, 147}}

	~~{{Optimal ET sequence\|legend=~~1~~\|21, 63, 84, 147}}~~		Badness (Sintel): 1.317

	{{Navbox fractional-octave}}		{{Navbox fractional-octave}}

Overthink: Add a strong temperament

2026-01-15T05:45:20Z

Add a strong temperament

← Older revision		Revision as of 05:45, 15 January 2026
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	{{Mapping\|legend=2\|21 95}}		{{Mapping\|legend=2\|21 95}}

	: ~~mapping~~ generator: ~529/512 = 1\21		: Mapping generator: ~529/512 = 1\21

	[[Support]]ing [[ET]]s: 21N, N = 1 to 96, largest: [[2016edo]]		[[Support]]ing [[ET]]s: 21N, N = 1 to 96, largest: [[2016edo]]
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	{{Mapping\|legend=1\| 21 0 59 82 \| 0 13 -4 -9 }}		{{Mapping\|legend=1\| 21 0 59 82 \| 0 13 -4 -9 }}

	: ~~mapping~~ generators: ~403368/390625 = 1\21, ~160/147 ~~= 146.305~~		: Mapping generators: ~403368/390625 = 1\21, ~160/147

	[[Optimal tuning]] ([[CTE]]): ~160/147 = 146.305		[[Optimal tuning]] ([[CTE]]): ~160/147 = 146.305
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	{{Mapping\|legend=1\| 21 0 59 82 24 111 114 38 95 \| 0 13 -4 -9 19 -13 -11 20 0}}		{{Mapping\|legend=1\| 21 0 59 82 24 111 114 38 95 \| 0 13 -4 -9 19 -13 -11 20 0}}

	: ~~mapping~~ generators: ~216/209 = 1\21, ~160/147 ~~= 146.308~~		: Mapping generators: ~216/209 = 1\21, ~160/147

	[[Optimal tuning]] ([[CTE]]): ~160/147 = 146.~~305~~		[[Optimal tuning]] ([[CTE]]): ~160/147 = 146.308{{C}}

	[[Support]]ing [[ET]]s: {{EDOs\|525, 861h, 1050f, 1911}}		[[Support]]ing [[ET]]s: {{EDOs\|525, 861h, 1050f, 1911}}

			== Blackmagic ==
			Blackmagic is the 63 & 84 temperament, merging two systems which cover many large primes.{{Todo\|review}}

			Subgroup: 2.3.5.7

			Comma list: [[225/224]], {{Monzo\|27 1 1 -11}}

			{{Mapping\|legend=1\| 21 0 82 59 \| 0 1 -1 0 }}

			: Mapping generators: ~16807/16384 = 1\21, ~3

			[[Optimal tuning]] ([[CWE]]): ~3/2 = 701.120{{C}}

			{{Optimal ET sequence\|legend=1\|21, 63, 84, 147}}

			=== 2.3.5.7.11.13.23.29.31.43 subgroup ===

			Subgroup: 2.3.5.7.11.13.23.29.31.43

			Comma list: 155/154, [[225/224]], [[232/231]], [[300/299]], [[364/363]], 560/559, [[640/637]], [[1716/1715]]

			{{Mapping\|legend=1\| 21 0 82 59 106 111 95 102 104 114 \| 0 1 -1 0 -1 -1 0 0 0 0 }}

			: Mapping generators: ~16807/16384 = 1\21, ~3

			[[Optimal tuning]] ([[CWE]]): ~3/2 = 701.742{{C}}

			{{Optimal ET sequence\|legend=1\|21, 63, 84, 147}}

	{{Navbox fractional-octave}}		{{Navbox fractional-octave}}

Overthink: link, fixes

2026-01-15T05:30:04Z

link, fixes

← Older revision		Revision as of 05:30, 15 January 2026
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	{{Infobox fractional-octave\|21}}		{{Infobox fractional-octave\|21}}
	This page ~~collectes~~ temperaments with a period of 1/21 of an octave.		This page collects temperaments with a period of 1/21 of an [[octave]].

	Although 21edo itself is not remarkably accurate for low-complexity harmonics, some temperaments which are multiples of 21, such as {{EDOs\|441, 1407, and 1848}} are. 441 and 1848 are also members of [[zeta]] edo list.		Although [[21edo]] itself is not remarkably accurate for low-complexity harmonics, some temperaments which are multiples of 21, such as {{EDOs\|441, 1407, and 1848}} are. 441 and 1848 are also members of the [[zeta]] edo list.

	Temperaments discussed elsewhere include		Temperaments discussed elsewhere include

ArrowHead294 at 03:39, 14 February 2025

2025-02-14T03:39:52Z

← Older revision		Revision as of 03:39, 14 February 2025
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	{{~~Fractional~~-octave ~~navigation~~\|21}}		{{Infobox fractional-octave\|21}}
	This page collectes temperaments with a period of 1/21 of an octave.		This page collectes temperaments with a period of 1/21 of an octave.

Lériendil: akjayland was duplicated; landscape page had slightly better information

2024-12-14T03:55:33Z

akjayland was duplicated; landscape page had slightly better information

@@ Line 3: / Line 3: @@
 Although 21edo itself is not remarkably accurate for low-complexity harmonics, some temperaments which are multiples of 21, such as {{EDOs|441, 1407, and 1848}} are. 441 and 1848 are also members of [[zeta]] edo list.
 == 21-23-commatic ==
 Subgroup: 2.23
@@ Line 13: / Line 17: @@
 [[Support]]ing [[ET]]s: 21N, N = 1 to 96, largest: [[2016edo]]
 == Scandium ==

BudjarnLambeth: navbox

2024-12-05T07:20:00Z

navbox

← Older revision		Revision as of 07:20, 5 December 2024
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	[[Support]]ing [[ET]]s: {{EDOs\|525, 861h, 1050f, 1911}}		[[Support]]ing [[ET]]s: {{EDOs\|525, 861h, 1050f, 1911}}

			{{Navbox fractional-octave}}

Eliora: /* Scandium */ documenting scandium, will fill in the rest later

2023-08-15T12:16:02Z

Scandium: documenting scandium, will fill in the rest later

@@ Line 101: / Line 101: @@
 == Scandium ==
-Described as the 525 & 1911 temperament, and named after the 21st element for splitting the octave into 21 parts. Coincidentally, ''Encyclopaedia Britannica'' entry for scandium was written in the year 1911 which was used as the reason for the naming. Remarkably, unlike akjayland, it is ''not'' a landscape system.
+Described as the 525 & 1911 temperament, and named after the 21st element for splitting the octave into 21 parts. Coincidentally, ''Encyclopaedia Britannica'' entry for scandium was written in the year 1911 which was used as the reason for the naming. Remarkably, unlike akjayland or many temperaments in the thousands which contain 3edo as a subset, it is ''not'' a landscape system. [[39/32]] is mapped into 6\21 and [[23/16]] is, as usual, mapped into 11\21.
 Subgroup: 2.3.5.7
-Comma list: 47 -7 -7 -7, -29 0 27 -12
+Comma list: {{monzo|47 -7 -7 -7}}, {{monzo|-29 0 27 -12}}

Eliora at 12:07, 15 August 2023

2023-08-15T12:07:17Z

← Older revision		Revision as of 12:07, 15 August 2023
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	Badness: 0.0199		Badness: 0.0199

	~~[[Category~~:~~21edo]]~~		== Scandium ==
	~~[[Category:Temperament collections]]~~		Described as the 525 & 1911 temperament, and named after the 21st element for splitting the octave into 21 parts. Coincidentally, ''Encyclopaedia Britannica'' entry for scandium was written in the year 1911 which was used as the reason for the naming. Remarkably, unlike akjayland, it is ''not'' a landscape system.
	~~[[Category~~:~~Rank 2]]~~
			Subgroup: 2.3.5.7

			Comma list: 47 -7 -7 -7, -29 0 27 -12

Eliora: /* 21-23-commatic */

2023-08-11T13:48:49Z

21-23-commatic

← Older revision		Revision as of 13:48, 11 August 2023
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	Comma list: {{monzo\|95 0 0 0 0 0 0 0 -21}}		Comma list: {{monzo\|95 0 0 0 0 0 0 0 -21}}

	{{Mapping\|legend=2\|21 95}		{{Mapping\|legend=2\|21 95}}

	: mapping generator: ~529/512 = 1\21		: mapping generator: ~529/512 = 1\21