17th-octave temperaments - Revision history

FloraC: Text replacement - "Mapping: {{mapping| " to "{{Mapping|legend=0| "

2026-09-24T12:32:31Z

Text replacement - "Mapping: {{mapping| " to "{{Mapping|legend=0| "

← Older revision		Revision as of 12:32, 24 September 2026
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	Comma list: 64/63, 99/98, 243/242		Comma list: 64/63, 99/98, 243/242

	~~Mapping:~~ {{~~mapping~~\| 17 27 0 48 59 \| 0 0 1 0 0 }}		{{Mapping\|legend=0\| 17 27 0 48 59 \| 0 0 1 0 0 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 64/63, 78/77, 99/98, 144/143		Comma list: 64/63, 78/77, 99/98, 144/143

	~~Mapping:~~ {{~~mapping~~\| 17 27 0 48 59 63 \| 0 0 1 0 0 0 }}		{{Mapping\|legend=0\| 17 27 0 48 59 63 \| 0 0 1 0 0 0 }}

	Optimal tunings:		Optimal tunings:

FloraC: /* Chlorine */

2026-08-20T09:34:43Z

Chlorine

← Older revision		Revision as of 09:34, 20 August 2026
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	: ''For extensions, see [[Ragismic microtemperaments #Chlorine]].''		: ''For extensions, see [[Ragismic microtemperaments #Chlorine]].''

	Chlorine has a period of 1/17 octave. It tempers out {{monzo\| -52 -17 34 }}, the [[septendecima]], by which 17 classical chromatic semitones ((25/24)<sup>17</sup>) exceed an octave~~. This temperament can be described as 289 & 323 temperament, which tempers out {{monzo\| -49 4 22 -3 }} as well as the ragisma. Not only the semitwelfth, but also the ~5/4 can be used as a generator~~.		Chlorine has a period of 1/17 octave. It tempers out {{monzo\| -52 -17 34 }}, the [[septendecima]], by which 17 classical chromatic semitones ((25/24)<sup>17</sup>) exceed an octave.

	The name of this temperament comes from chlorine, the 17th element, and has no relation to the [[chlorisma]].		The name of this temperament comes from chlorine, the 17th element, and has no relation to the [[chlorisma]].

FloraC: - septimal chlorine (properly addressed in Ragismic)

2026-08-20T09:31:12Z

- septimal chlorine (properly addressed in Ragismic)

@@ Line 73: / Line 73: @@
 == Chlorine ==
 Chlorine has a period of 1/17 octave. It tempers out {{monzo| -52 -17 34 }}, the [[septendecima]], by which 17 classical chromatic semitones ((25/24)<sup>17</sup>) exceed an octave. This temperament can be described as 289 & 323 temperament, which tempers out {{monzo| -49 4 22 -3 }} as well as the ragisma. Not only the semitwelfth, but also the ~5/4 can be used as a generator.
@@ Line 93: / Line 95: @@
 [[Badness]] (Sintel): 1.81
 {{Navbox fractional-octave}}
 [[Category:17edo]]

FloraC: Delete leaves

2026-08-20T09:26:01Z

Delete leaves

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 Badness (Sintel): 2.11
 {{Navbox fractional-octave}}
 [[Category:17edo]]

FloraC: Switch to Sintel's badness, WE & CWE tunings (2/)

2026-08-17T13:49:01Z

Switch to Sintel's badness, WE & CWE tunings (2/)

← Older revision		Revision as of 13:49, 17 August 2026
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	== Chlorine ==		== Chlorine ==
	~~The name~~ of ~~chlorine temperament comes from chlorine~~, ~~the 17th element, and has no relation to~~ the [[~~chlorisma~~]].		Chlorine has a period of 1/17 octave. It tempers out {{monzo\| -52 -17 34 }}, the [[septendecima]], by which 17 classical chromatic semitones ((25/24)<sup>17</sup>) exceed an octave. This temperament can be described as 289 & 323 temperament, which tempers out {{monzo\| -49 4 22 -3 }} as well as the ragisma. Not only the semitwelfth, but also the ~5/4 can be used as a generator.

	~~Chlorine~~ temperament has ~~a period of 1/17 octave. It tempers out~~ the [[~~septendecima~~]], {{monzo\| -52 -17 34 }}, by which 17 chromatic semitones (25/24) exceed an octave. This temperament can be described as 289 & 323 temperament, which tempers out {{monzo\| -49 4 22 -3 }} as well as the ragisma. Not only the semitwelfth, but also the ~5/4 can be used as a generator.		The name of this temperament comes from chlorine, the 17th element, and has no relation to the [[chlorisma]].

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5
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	{{Mapping\|legend=1\| 17 0 26 \| 0 2 1 }}		{{Mapping\|legend=1\| 17 0 26 \| 0 2 1 }}

	: mapping generators: ~25/24, ~{{monzo\| 26 9 -17 }}		: mapping generators: ~25/24, ~{{monzo\| 26 9 -17 }}

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[~~CTE~~]]: ~25/24 = 70.~~588~~{{c}} ~~(1\17)~~, ~{{monzo\| 26 9 -17 }} = 950.~~982~~{{c}}		* [[WE]]: ~25/24 = 70.5887{{c}}, ~{{monzo\| 26 9 -17 }} = 950.9807{{c}}
	* [[CWE]]: ~25/24 = 70.~~588~~{{c}} ~~(1\17)~~, ~{{monzo\| 26 9 -17 }} = 950.~~978~~{{c}}		: [[error map]]: {{val\| +0.008 +0.006 -0.027 }}
			* [[CWE]]: ~25/24 = 70.5882{{c}}, ~{{monzo\| 26 9 -17 }} = 950.9776{{c}}
			: error map: {{val\| 0.000 -0.000 -0.042 }}

	{{Optimal ET sequence\|legend=1\| 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797 }}		{{Optimal ET sequence\|legend=1\| 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797 }}
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[~~CTE~~]]: ~25/24 = 70.~~588~~{{c}} ~~(1\17)~~, ~{{monzo\| 24 -5 -9 2 }} = 950.~~998~~{{c}}		* [[WE]]: ~25/24 = 70.5880{{c}}, ~{{monzo\| 24 -5 -9 2 }} = 950.9962{{c}}
	* [[CWE]]: ~25/24 = 70.~~588~~{{c}} ~~(1\17)~~, ~{{monzo\| 24 -5 -9 2 }} = 950.~~999~~{{c}}		: [[error map]]: {{val\| -0.004 +0.037 -0.030 -0.019 }}
			* [[CWE]]: ~25/24 = 70.5882{{c}}, ~{{monzo\| 24 -5 -9 2 }} = 950.9990{{c}}
			: error map: {{val\| 0.000 +0.043 -0.021 -0.013 }}

	{{Optimal ET sequence\|legend=1\| 289, 323, 612, 935, 1547 }}		{{Optimal ET sequence\|legend=1\| 289, 323, 612, 935, 1547, 3706, 5253 }}

	[[Badness]] (Sintel): 1.05		[[Badness]] (Sintel): 1.05
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	Optimal tunings:		Optimal tunings:
	* ~~CTE~~: ~25/24 = 70.~~588~~{{c}} ~~(1\17)~~, ~693/400 = ~~950~~.~~978~~{{c}}		* WE: ~25/24 = 70.5905{{c}}, ~693/400 = 951.0054{{c}}
	* CWE: ~25/24 = 70.~~588~~{{c}} ~~(1\17)~~, ~693/400 = 950.~~975~~{{c}}		* CWE: ~25/24 = 70.5882{{c}}, ~693/400 = 950.9754{{c}}

	{{Optimal ET sequence\|legend=0\| 289, 323, 612 }}		{{Optimal ET sequence\|legend=0\| 289, 323, 612, 3349de, 3961de, …, 5797ddee }}

	Badness (Sintel): 2.11		Badness (Sintel): 2.11

FloraC: Switch to Sintel's badness, WE & CWE tunings (1/)

2026-08-17T13:22:11Z

Switch to Sintel's badness, WE & CWE tunings (1/)

← Older revision		Revision as of 13:22, 17 August 2026
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	== Gothic ==		== Gothic ==
	The gothic temperament is associated with the [[17-comma]]. It used to be known as '''septendecic'''.		The gothic temperament is associated with the [[17-comma]]. It used to be known as ''septendecic''.

			It can be extended to the higher limits using the identical mapping to 17et in the no-5 subgroups. While this comes at the cost of greater error for primes [[7/1\|7]] and [[11/1\|11]], it is the only [[strong extension]] for these primes that makes practical sense.

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5
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	{{mapping\|legend=1\| 17 27 0 \| 0 0 1 }}		{{mapping\|legend=1\| 17 27 0 \| 0 0 1 }}

	: mapping generators: ~256/243, ~5		: mapping generators: ~256/243, ~5

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[~~CTE~~]]: ~256/243 = 70.~~5882~~{{c}} ~~(1\17)~~, ~5/4 = ~~386~~.~~3137~~{{c}} ~~(~20480/19683 = 33.3725~~{{c}})		* [[WE]]: ~256/243 = 70.5152{{c}}, ~5/4 = 388.7908{{c}}
	* [[CWE]]: ~256/243 = 70.5882{{c}} ~~(1\17)~~, ~5/4 = 388.~~2316~~{{c}} ~~(~20480/19683 = 35.2904~~{{c}})		: [[error map]]: {{val\| -1.241 +1.956 -0.006 }}
			* [[CWE]]: ~256/243 = 70.5882{{c}}, ~5/4 = 388.2315{{c}}
			: error map: {{val\| 0.000 +3.927 +1.918 }}

	{{Optimal ET sequence\|legend=1\| 17c, 34, 323bbcc, 357bbcc, 391bbcc }}		{{Optimal ET sequence\|legend=1\| 17c, 34, 323bbcc, 357bbcc, 391bbcc }}
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	=== 7-limit ===		=== 7-limit ===
	~~Gothic is identical to 17et in the no-5 subgroups, but has an independent generator for prime 5.~~

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[~~CTE~~]]: ~28/27 = 70.~~588~~{{c}} ~~(1\17)~~, ~5/4 = ~~386~~.~~314~~{{c}}		* [[WE]]: ~28/27 = 70.4030{{c}}, ~5/4 = 392.5654{{c}}
	* [[CWE]]: ~28/27 = 70.~~588~~{{c}} ~~(1\17)~~, ~5/4 = 391.~~765~~{{c}}		: [[error map]]: {{val\| -3.149 -1.075 -0.047 +10.517 }}
			* [[CWE]]: ~28/27 = 70.5882{{c}}, ~5/4 = 391.7654{{c}}
			: error map: {{val\| 0.000 +3.927 +5.452 +19.409 }}

	{{Optimal ET sequence\|legend=1\| 17c, 34d }}		{{Optimal ET sequence\|legend=1\| 17c, 34d, 85cdd }}

	[[Badness]] (Sintel): 3.44		[[Badness]] (Sintel): 3.44
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	Comma list: 64/63, 99/98, 243/242		Comma list: 64/63, 99/98, 243/242

	Mapping: {{~~Mapping~~\| 17 27 0 48 59 \| 0 0 1 0 0 }}		Mapping: {{mapping\| 17 27 0 48 59 \| 0 0 1 0 0 }}

	Optimal tunings:		Optimal tunings:
	* ~~CTE~~: ~28/27 = 70.~~588~~{{c}} ~~(1\17)~~, ~5/4 = ~~386~~.~~314~~{{c}}		* WE: ~28/27 = 70.3922{{c}}, ~5/4 = 392.9298{{c}}
	* CWE: ~28/27 = 70.~~588~~{{c}} ~~(1\17)~~, ~5/4 = 392.~~472~~{{c}}		* CWE: ~28/27 = 70.5882{{c}}, ~5/4 = 392.4722{{c}}

	{{Optimal ET sequence\|legend=0\| 17c, 34d }}		{{Optimal ET sequence\|legend=0\| 17c, 34d, 85cdde }}

	Badness (Sintel): 1.77		Badness (Sintel): 1.77
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	Comma list: 64/63, 78/77, 99/98, 144/143		Comma list: 64/63, 78/77, 99/98, 144/143

	Mapping: {{~~Mapping~~\| 17 27 0 48 59 63 \| 0 0 1 0 0 0 }}		Mapping: {{mapping\| 17 27 0 48 59 63 \| 0 0 1 0 0 0 }}

	Optimal tunings:		Optimal tunings:
	* ~~CTE~~: ~27/26 = 70.~~588~~{{c}} ~~(1\17)~~, ~5/4 = ~~386~~.~~314~~{{c}}		* WE: ~27/26 = 70.4103{{c}}, ~5/4 = 392.3109{{c}}
	* CWE: ~27/26 = 70.~~588~~{{c}} ~~(1\17)~~, ~5/4 = 392.~~129~~{{c}}		* CWE: ~27/26 = 70.5882{{c}}, ~5/4 = 392.1288{{c}}

	{{Optimal ET sequence\|legend=0\| 17c, 34d }}		{{Optimal ET sequence\|legend=0\| 17c, 34d }}

Xenllium at 02:14, 23 February 2026

2026-02-23T02:14:50Z

← Older revision		Revision as of 02:14, 23 February 2026
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	=== 7-limit ===		=== 7-limit ===
			Gothic is identical to 17et in the no-5 subgroups, but has an independent generator for prime 5.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

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	Mapping: {{mapping\| 17 0 26 -87 207 \| 0 2 1 10 -11 }}		Mapping: {{mapping\| 17 0 26 -87 207 \| 0 2 1 10 -11 }}

	[[Optimal ~~tuning]]s~~:		Optimal tunings:
	* [[CTE]]: ~25/24 = 70.588{{c}} (1\17), ~693/400 = 950.978{{c}}		* CTE: ~25/24 = 70.588{{c}} (1\17), ~693/400 = 950.978{{c}}
	* [[CWE]]: ~25/24 = 70.588{{c}} (1\17), ~693/400 = 950.975{{c}}		* CWE: ~25/24 = 70.588{{c}} (1\17), ~693/400 = 950.975{{c}}

	{{Optimal ET sequence\|legend=0\| 289, 323, 612 }}		{{Optimal ET sequence\|legend=0\| 289, 323, 612 }}
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	== Leaves ==		== Leaves ==
	Defined as the 323 & 2023 temperament. 2 generators reach [[17/13]], 7 generators reach [[5/4]], 10 generators produce [[13/11]].		Defined as the {{nowrap\|323 & 2023}} temperament. 2 generators reach [[17/13]], 7 generators reach [[5/4]], 10 generators produce [[13/11]].

			[[Subgroup]]: 2.3.5.7

			[[Comma list]]: 1729951171875/1727094849536, {{monzo\|10 39 -14 -14}}

			[[Mapping]]: [{{val\|17 10 31 9}}, {{val\|0 14 7 32}}]

			: mapping generators: ~25/24, ~6125/5832

			[[Optimal tuning]]s:
			* [[CTE]]: ~25/24 = 70.588{{c}} (1\17), ~6125/5832 = 85.427{{c}}
			* [[CWE]]: ~25/24 = 70.588{{c}} (1\17), ~6125/5832 = 85.426{{c}}

			{{Optimal ET sequence\|legend=1\| 323, 1700d, 2023, 2346}}

			[[Badness]] (Sintel): 32.032

			=== 11-limit ===
			Subgroup: 2.3.5.7.11

			Comma list: 160083/160000, 198359290368/198165034375, 1729951171875/1727094849536

			Mapping: [{{val\|17 10 31 9 106}}, {{val\|0 14 7 32 -39}}]

			Optimal tunings:
			* CTE: ~25/24 = 70.588{{c}} (1\17), ~6125/5832 = 85.421{{c}}
			* CWE: ~25/24 = 70.588{{c}} (1\17), ~6125/5832 = 85.420{{c}}

			{{Optimal ET sequence\|legend=0\| 323, 1700d, 2023, 2346e}}

			Badness (Sintel): 20.456

			=== 13-limit ===
	Subgroup: 2.3.5.7.11.13		Subgroup: 2.3.5.7.11.13

	Comma list: 160083/160000, 928125/927472, 1990656/1990625, ~~20726199~~/~~20706224~~		Comma list: 160083/160000, 928125/927472, 1990656/1990625, 4831530/4826809

	Mapping: [{{val\|17 10 31 9 106 98}}, {{val\|0 14 7 32 -39 -29}}]		Mapping: [{{val\|17 10 31 9 106 98}}, {{val\|0 14 7 32 -39 -29}}]

	~~Mapping generators~~: ~25/24, ~1024/975		Optimal tunings:
			* CTE: ~25/24 = 70.588{{c}} (1\17), ~1024/975 = 85.421{{c}}
			* CWE: ~25/24 = 70.588{{c}} (1\17), ~1024/975 = 85.420{{c}}

	Optimal ~~tuning (CTE): ~1024/975~~ = ~~85.421~~		{{Optimal ET sequence\|legend=0\| 323, 1700d, 2023, 2346e}}

	~~{{Optimal ET sequence\|legend=1\| 323, 1700, 2023}}~~		Badness (Sintel): 10.096

	=== 17-limit ===		=== 17-limit ===
	Subgroup: 2.3.5.7.11.13.17		Subgroup: 2.3.5.7.11.13.17

	Comma list: 57375/57344, 111537/111475, 140800/140777~~, 111537/111475, 1026675/1026256~~		Comma list: 39325/39304, 57375/57344, 71874/71825, 111537/111475, 140800/140777

	Mapping: [{{val\|17 10 31 9 106 98 107}}, {{val\|0 14 7 32 -39 -29 -31}}]		Mapping: [{{val\|17 10 31 9 106 98 107}}, {{val\|0 14 7 32 -39 -29 -31}}]

	~~Mapping generators~~: ~25/24, ~765/728		Optimal tunings:
			* CTE: ~25/24 = 70.588{{c}} (1\17), ~765/728 = 85.421{{c}}
			* CWE: ~25/24 = 70.588{{c}} (1\17), ~765/728 = 85.421{{c}}

	Optimal ~~tuning (CTE): ~765/728~~ = ~~85.421~~		{{Optimal ET sequence\|legend=0\| 323, 1700d, 2023, 2346e}}

	~~{{Optimal ET sequence\|legend=1\| 323, 1700, 2023}}~~		Badness (Sintel): 6.678

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

	[[Category:17edo]]		[[Category:17edo]]

Xenllium at 00:24, 21 February 2026

2026-02-21T00:24:02Z

@@ Line 21: / Line 21: @@
 [[Badness]] (Sintel): 12.7
 == Chlorine ==
@@ Line 50: / Line 95: @@
 {{Mapping|legend=1| 17 0 26 -87 | 0 2 1 10 }}
-[[Optimal tuning]] ([[POTE]]): ~{{monzo| 24 -5 -9 2 }} = 950.9995
+[[Optimal tuning]]s:
 {{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547 }}
@@ Line 61: / Line 108: @@
 Comma list: 4375/4374, 41503/41472, 1879453125/1879048192
-Mapping: {{mapping|  17 0 26 -87 207 | 0 2 1 10 -11  }}
+Mapping: {{mapping| 17 0 26 -87 207 | 0 2 1 10 -11 }}
-Optimal tuning (POTE): ~{{monzo| 24 -5 -9 2 }} = 950.9749
+[[Optimal tuning]]s:
 {{Optimal ET sequence|legend=0| 289, 323, 612 }}

Xenllium at 23:57, 20 February 2026

2026-02-20T23:57:32Z

@@ Line 15: / Line 15: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~256/243 = 1\17, ~5/4 = 386.3137 (~20480/19683 = 33.3725)
+* [[CTE]]: ~256/243 = 70.5882{{c}} (1\17), ~5/4 = 386.3137{{c}} (~20480/19683 = 33.3725{{c}})
-* [[CWE]]: ~256/243 = 1\17, ~5/4 = 388.2316 (~20480/19683 = 35.2904)
+* [[CWE]]: ~256/243 = 70.5882{{c}} (1\17), ~5/4 = 388.2316{{c}} (~20480/19683 = 35.2904{{c}})
 {{Optimal ET sequence|legend=1| 17c, 34, 323bbcc, 357bbcc, 391bbcc }}
-[[Badness]]: 0.541
+[[Badness]] (Sintel): 12.7
 == Chlorine ==
@@ Line 61: / Line 35: @@
 : mapping generators: ~25/24, ~{{monzo| 26 9 -17 }}
-[[Optimal tuning]] ([[POTE]]): ~{{monzo| 26 9 -17 }} = 950.9746
+[[Optimal tuning]]s:
 {{Optimal ET sequence|legend=1| 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797 }}
-[[Badness]]: 0.077072
+[[Badness]] (Sintel): 1.81
 === 7-limit ===
@@ Line 78: / Line 54: @@
 {{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547 }}
-[[Badness]]: 0.041658
+[[Badness]] (Sintel): 1.05
 === 11-limit ===
@@ Line 89: / Line 65: @@
 Optimal tuning (POTE): ~{{monzo| 24 -5 -9 2 }} = 950.9749
-{{Optimal ET sequence|legend=1| 289, 323, 612 }}
+{{Optimal ET sequence|legend=0| 289, 323, 612 }}
-Badness: 0.063706
+{{Optimal ET sequence|legend=1| 323, 1700, 2023}}
 {{Navbox fractional-octave}}
 [[Category:17edo]]

Jerdle: /* Chlorine */ Mentioned the lack of connection with the chlorisma.

2025-06-15T01:50:31Z

Chlorine: Mentioned the lack of connection with the chlorisma.

← Older revision		Revision as of 01:50, 15 June 2025
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	== Chlorine ==		== Chlorine ==
	The name of chlorine temperament comes from ~~Chlorine~~, the 17th element.		The name of chlorine temperament comes from chlorine, the 17th element, and has no relation to the [[chlorisma]].

	Chlorine temperament has a period of 1/17 octave. It tempers out the [[septendecima]], {{monzo\| -52 -17 34 }}, by which 17 chromatic semitones (25/24) exceed an octave. This temperament can be described as 289 & 323 temperament, which tempers out {{monzo\| -49 4 22 -3 }} as well as the ragisma. Not only the semitwelfth, but also the ~5/4 can be used as a generator.		Chlorine temperament has a period of 1/17 octave. It tempers out the [[septendecima]], {{monzo\| -52 -17 34 }}, by which 17 chromatic semitones (25/24) exceed an octave. This temperament can be described as 289 & 323 temperament, which tempers out {{monzo\| -49 4 22 -3 }} as well as the ragisma. Not only the semitwelfth, but also the ~5/4 can be used as a generator.