Smate family - Revision history

FloraC: Normalize mappings. Review optimal ET sequences.

2026-03-11T10:07:50Z

Normalize mappings. Review optimal ET sequences.

Xenllium at 12:05, 29 January 2026

2026-01-29T12:05:29Z

← Older revision		Revision as of 12:05, 29 January 2026
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	{{Mapping\|legend=1\| 1 3 2 \| 0 -4 1 }}		{{Mapping\|legend=1\| 1 3 2 \| 0 -4 1 }}

	: ~~mapping~~ generators: ~2, ~5/4		: Mapping generators: ~2, ~5/4

	[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.000, ~5/4 = ~~420~~.~~855~~		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1191.896{{c}}, ~5/4 = 418.013{{c}}
			* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 421.637{{c}}

	{{Optimal ET sequence\|legend=1\| 3, 11, 14, 17c, 20c, 37c }}		{{Optimal ET sequence\|legend=1\| 3, 11, 14, 17c, 20c, 37c }}

	[[Badness]] (~~Smith~~): 0.~~178624~~		[[Badness]] (Sintel): 4.190

	== Septimal smate ==		== Septimal smate ==
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	{{Mapping\|legend=1\| 1 3 2 6 \| 0 -4 1 -9 }}		{{Mapping\|legend=1\| 1 3 2 6 \| 0 -4 1 -9 }}

	[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.000, ~5/4 = ~~422~~.~~275~~		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1191.900{{c}}, ~5/4 = 419.425{{c}}
			* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 423.046{{c}}

	{{Optimal ET sequence\|legend=1\| 3d, 11d, 14, 17c, 37ccdd }}		{{Optimal ET sequence\|legend=1\| 3d, 11d, 14, 17c, 37ccdd }}

	[[Badness]] (~~Smith~~): 0.~~077871~~		[[Badness]] (Sintel): 1.971

	=== 11-limit ===		=== 11-limit ===
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	Mapping: {{mapping\| 1 3 2 6 7 \| 0 -4 1 -9 -10 }}		Mapping: {{mapping\| 1 3 2 6 7 \| 0 -4 1 -9 -10 }}

	Optimal ~~tuning (POTE)~~: ~2 = 1200.000, ~5/4 = ~~422~~.~~217~~		Optimal tunings:
			* WE: ~2 = 1191.914{{c}}, ~5/4 = 419.372{{c}}
			* CWE: ~2 = 1200.000{{c}}, ~5/4 = 423.010{{c}}

	{{Optimal ET sequence\|legend=0\| 3de, 14, 17c, 37ccddee }}		{{Optimal ET sequence\|legend=0\| 3de, 14, 17c, 37ccddee }}

	Badness (~~Smith~~): 0.~~042518~~		Badness (Sintel): 1.406

	=== 13-limit ===		=== 13-limit ===
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	Mapping: {{mapping\| 1 3 2 6 7 3 \| 0 -4 1 -9 -10 2 }}		Mapping: {{mapping\| 1 3 2 6 7 3 \| 0 -4 1 -9 -10 2 }}

	Optimal ~~tuning (POTE)~~: ~2 = 1200.000, ~5/4 = 423.~~020~~		Optimal tunings:
			* WE: ~2 = 1194.045{{c}}, ~5/4 = 420.921{{c}}
			* CWE: ~2 = 1200.000{{c}}, ~5/4 = 423.402{{c}}

	{{Optimal ET sequence\|legend=0\| 3de, 14, 17c }}		{{Optimal ET sequence\|legend=0\| 3de, 14, 17c }}

	Badness (~~Smith~~): 0.~~036836~~		Badness (Sintel): 1.522

	== Hemismate ==		== Hemismate ==
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	{{Mapping\|legend=1\| 1 3 2 3 \| 0 -8 2 -1 }}		{{Mapping\|legend=1\| 1 3 2 3 \| 0 -8 2 -1 }}
			: Mapping generators: ~2, ~8/7

	[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.000, ~8/7 = 210.~~452~~		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1192.090{{c}}, ~8/7 = 209.065{{c}}
			* [[CWE]]: ~2 = 1200.000{{c}}, ~8/7 = 210.778{{c}}

	{{Optimal ET sequence\|legend=1\| 6, 11, 17c, 40bcd }}		{{Optimal ET sequence\|legend=1\| 6, 11, 17c, 40bcd }}

	[[Badness]] (~~Smith~~): 0.~~154301~~		[[Badness]] (Sintel): 3.905

	=== 11-limit ===		=== 11-limit ===
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	Mapping: {{mapping\| 1 3 2 3 4 \| 0 -8 2 -1 -3 }}		Mapping: {{mapping\| 1 3 2 3 4 \| 0 -8 2 -1 -3 }}

	Optimal ~~tuning (POTE)~~: ~2 = 1200.000, ~8/7 = 210.~~481~~		Optimal tunings:
			* WE: ~2 = 1192.679{{c}}, ~8/7 = 209.197{{c}}
			* CWE: ~2 = 1200.000{{c}}, ~8/7 = 210.780{{c}}

	{{Optimal ET sequence\|legend=0\| 6, 11, 17c, 40bcde }}		{{Optimal ET sequence\|legend=0\| 6, 11, 17c, 40bcde }}

	Badness (~~Smith~~): 0.~~065528~~		Badness (Sintel): 2.166

	=== 13-limit ===		=== 13-limit ===
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	Mapping: {{mapping\| 1 3 2 3 4 3 \| 0 -8 2 -1 -3 4 }}		Mapping: {{mapping\| 1 3 2 3 4 3 \| 0 -8 2 -1 -3 4 }}

	Optimal ~~tuning (POTE)~~: ~2 = 1200.000, ~8/7 = ~~210~~.~~974~~		Optimal tunings:
			* WE: ~2 = 1194.135{{c}}, ~8/7 = 209.943{{c}}
			* CWE: ~2 = 1200.000{{c}}, ~8/7 = 211.116{{c}}

	{{Optimal ET sequence\|legend=0\| 6, 11, 17c }}		{{Optimal ET sequence\|legend=0\| 6, 11, 17c }}

	Badness (~~Smith~~): 0.~~050472~~		Badness (Sintel): 2.086

	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:Smate family\| ]] <!-- main article -->		[[Category:Smate family\| ]] <!-- main article -->
	[[Category:Rank 2]]		[[Category:Rank 2]]

FloraC: Text replacement - "Category:Temperament families↵Category:Pages with mostly numerical content" to "Category:Temperament families"

2025-11-27T12:26:56Z

Text replacement - "Category:Temperament families↵Category:Pages with mostly numerical content" to "Category:Temperament families"

← Older revision		Revision as of 12:26, 27 November 2025
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	[[Category:Temperament families]]		[[Category:Temperament families]]
	~~[[Category:Pages with mostly numerical content]]~~
	[[Category:Smate family\| ]] <!-- main article -->		[[Category:Smate family\| ]] <!-- main article -->
	[[Category:Rank 2]]		[[Category:Rank 2]]

Lériendil: Text replacement - "Category:Temperament families" to "Category:Temperament families Category:Pages with mostly numerical content"

2025-06-24T00:37:01Z

Text replacement - "Category:Temperament families" to "Category:Temperament families Category:Pages with mostly numerical content"

← Older revision		Revision as of 00:37, 24 June 2025
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	[[Category:Temperament families]]		[[Category:Temperament families]]
			[[Category:Pages with mostly numerical content]]
	[[Category:Smate family\| ]] <!-- main article -->		[[Category:Smate family\| ]] <!-- main article -->
	[[Category:Rank 2]]		[[Category:Rank 2]]

VectorGraphics at 04:39, 12 June 2025

2025-06-12T04:39:55Z

← Older revision		Revision as of 04:39, 12 June 2025
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	{{Mapping\|legend=1\| 1 3 2 6 \| 0 -4 1 -9 }}		{{Mapping\|legend=1\| 1 3 2 6 \| 0 -4 1 -9 }}

	~~{{Multival\|legend=1\| 4 -1 9 -11 3 24 }}~~

	[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~5/4 = 422.275		[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~5/4 = 422.275
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	{{Mapping\|legend=1\| 1 3 2 3 \| 0 -8 2 -1 }}		{{Mapping\|legend=1\| 1 3 2 3 \| 0 -8 2 -1 }}

	~~{{Multival\|legend=1\| 8 -2 1 -22 -21 8 }}~~

	[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 210.452		[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 210.452

FloraC: Text replacement - "{{Technical data page}}

" to "{{Technical data page}}"

2025-04-13T13:43:31Z

Text replacement - "{{Technical data page}}<br><br>" to "{{Technical data page}}"

← Older revision		Revision as of 13:43, 13 April 2025
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	{{Technical data page}}~~<br><br>~~		{{Technical data page}}
	The '''smate family''' of temperaments tempers out [[2048/1875]], the smate comma, resulting in equation of four [[5/4\|just major thirds (5/4)]] with the [[8/3\|just perfect eleventh (8/3)]]. It therefore requires an extremely sharp tuning of the just major third. [[17edo]] and [[20edo]] provide it and make for good tunings.		The '''smate family''' of temperaments tempers out [[2048/1875]], the smate comma, resulting in equation of four [[5/4\|just major thirds (5/4)]] with the [[8/3\|just perfect eleventh (8/3)]]. It therefore requires an extremely sharp tuning of the just major third. [[17edo]] and [[20edo]] provide it and make for good tunings.

BudjarnLambeth: {{Technical data page}}

2025-04-13T11:52:38Z

{{Technical data page}}<br><br>

← Older revision		Revision as of 11:52, 13 April 2025
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			{{Technical data page}}<br><br>
	The '''smate family''' of temperaments tempers out [[2048/1875]], the smate comma, resulting in equation of four [[5/4\|just major thirds (5/4)]] with the [[8/3\|just perfect eleventh (8/3)]]. It therefore requires an extremely sharp tuning of the just major third. [[17edo]] and [[20edo]] provide it and make for good tunings.		The '''smate family''' of temperaments tempers out [[2048/1875]], the smate comma, resulting in equation of four [[5/4\|just major thirds (5/4)]] with the [[8/3\|just perfect eleventh (8/3)]]. It therefore requires an extremely sharp tuning of the just major third. [[17edo]] and [[20edo]] provide it and make for good tunings.

Jerdle: Discussed the link to skwares.

2025-02-27T10:43:48Z

Discussed the link to skwares.

← Older revision		Revision as of 10:43, 27 February 2025
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	The '''smate family''' of temperaments tempers out [[2048/1875]], the smate comma, resulting in equation of four [[5/4\|just major thirds (5/4)]] with the [[8/3\|just perfect eleventh (8/3)]]. It therefore requires an extremely sharp tuning of the just major third. [[17edo]] and [[20edo]] provide it and make for good tunings.		The '''smate family''' of temperaments tempers out [[2048/1875]], the smate comma, resulting in equation of four [[5/4\|just major thirds (5/4)]] with the [[8/3\|just perfect eleventh (8/3)]]. It therefore requires an extremely sharp tuning of the just major third. [[17edo]] and [[20edo]] provide it and make for good tunings.

			These temperaments resemble [[skwares]], which also splits 8/3 into four major thirds, but those major thirds are flattish [[9/7]]s rather than the sharp 5/4s used here.

	== Smate ==		== Smate ==

FloraC: Spell 1\1 in decimal; note the badness metric being used; misc. cleanup

2025-01-01T11:33:16Z

Spell 1\1 in decimal; note the badness metric being used; misc. cleanup

← Older revision		Revision as of 11:33, 1 January 2025
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	{{Mapping\|legend=1\| 1 3 2 \| 0 -4 1 }}		{{Mapping\|legend=1\| 1 3 2 \| 0 -4 1 }}

	[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~5/4 = 420.855		: mapping generators: ~2, ~5/4

			[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~5/4 = 420.855

	{{Optimal ET sequence\|legend=1\| 3, 11, 14, 17c, 20c, 37c }}		{{Optimal ET sequence\|legend=1\| 3, 11, 14, 17c, 20c, 37c }}

	[[Badness]]: 0.178624		[[Badness]] (Smith): 0.178624

	== Septimal smate ==		== Septimal smate ==
	~~{{See also\| Mint temperaments #Smate }}~~

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

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	{{Multival\|legend=1\| 4 -1 9 -11 3 24 }}		{{Multival\|legend=1\| 4 -1 9 -11 3 24 }}

	[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~5/4 = 422.275		[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~5/4 = 422.275

	{{Optimal ET sequence\|legend=1\| 3d, 11d, 14, 17c, 37ccdd }}		{{Optimal ET sequence\|legend=1\| 3d, 11d, 14, 17c, 37ccdd }}

	[[Badness]]: 0.077871		[[Badness]] (Smith): 0.077871

	=== 11-limit ===		=== 11-limit ===
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	Mapping: {{mapping\| 1 3 2 6 7 \| 0 -4 1 -9 -10 }}		Mapping: {{mapping\| 1 3 2 6 7 \| 0 -4 1 -9 -10 }}

	Optimal tuning (POTE): ~2 = ~~1\1~~, ~5/4 = 422.217		Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 422.217

	{{Optimal ET sequence\|legend=1\| 3de, 14, 17c, 37ccddee }}		{{Optimal ET sequence\|legend=0\| 3de, 14, 17c, 37ccddee }}

	Badness: 0.042518		Badness (Smith): 0.042518

	=== 13-limit ===		=== 13-limit ===
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	Mapping: {{mapping\| 1 3 2 6 7 3 \| 0 -4 1 -9 -10 2 }}		Mapping: {{mapping\| 1 3 2 6 7 3 \| 0 -4 1 -9 -10 2 }}

	Optimal tuning (POTE): ~2 = ~~1\1~~, ~5/4 = 423.020		Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 423.020

	{{Optimal ET sequence\|legend=1\| 3de, 14, 17c }}		{{Optimal ET sequence\|legend=0\| 3de, 14, 17c }}

	Badness: 0.036836		Badness (Smith): 0.036836

	== Hemismate ==		== Hemismate ==
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	{{Multival\|legend=1\| 8 -2 1 -22 -21 8 }}		{{Multival\|legend=1\| 8 -2 1 -22 -21 8 }}

	[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~8/7 = 210.452		[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 210.452

	{{Optimal ET sequence\|legend=1\| 6, 11, 17c, 40bcd }}		{{Optimal ET sequence\|legend=1\| 6, 11, 17c, 40bcd }}

	[[Badness]]: 0.154301		[[Badness]] (Smith): 0.154301

	=== 11-limit ===		=== 11-limit ===
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	Mapping: {{mapping\| 1 3 2 3 4 \| 0 -8 2 -1 -3 }}		Mapping: {{mapping\| 1 3 2 3 4 \| 0 -8 2 -1 -3 }}

	Optimal tuning (POTE): ~2 = ~~1\1~~, ~8/7 = 210.481		Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 210.481

	{{Optimal ET sequence\|legend=1\| 6, 11, 17c, 40bcde }}		{{Optimal ET sequence\|legend=0\| 6, 11, 17c, 40bcde }}

	Badness: 0.065528		Badness (Smith): 0.065528

	=== 13-limit ===		=== 13-limit ===
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	Mapping: {{mapping\| 1 3 2 3 4 3 \| 0 -8 2 -1 -3 4 }}		Mapping: {{mapping\| 1 3 2 3 4 3 \| 0 -8 2 -1 -3 4 }}

	Optimal tuning (POTE): ~2 = ~~1\1~~, ~8/7 = 210.974		Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 210.974

	{{Optimal ET sequence\|legend=1\| 6, 11, 17c }}		{{Optimal ET sequence\|legend=0\| 6, 11, 17c }}

	Badness: 0.050472		Badness (Smith): 0.050472

	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:Smate family\| ]] <!-- main article -->		[[Category:Smate family\| ]] <!-- main article -->
	[[Category:Rank 2]]		[[Category:Rank 2]]

FloraC: Update keys

2023-08-06T10:48:44Z

Update keys

← Older revision		Revision as of 10:48, 6 August 2023
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	== Smate ==		== Smate ==
	Subgroup: 2.3.5		[[Subgroup]]: 2.3.5

	[[Comma list]]: 2048/1875		[[Comma list]]: 2048/1875

	~~[[Mapping]]: [~~{{~~val~~\| 1 3 2 ~~}}, {{val~~\| 0 -4 1 }}]		{{Mapping\|legend=1\| 1 3 2 \| 0 -4 1 }}

	[[POTE ~~generator~~]]: ~5/4 = 420.855		[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 420.855

	{{Optimal ET sequence\|legend=1\| 3, 11, 14, 17c, 20c, 37c }}		{{Optimal ET sequence\|legend=1\| 3, 11, 14, 17c, 20c, 37c }}
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	== Septimal smate ==		== Septimal smate ==
	{{~~see~~ also\| Mint temperaments #Smate }}		{{See also\| Mint temperaments #Smate }}

	Subgroup: 2.3.5.7		[[Subgroup]]: 2.3.5.7

	[[Comma list]]: 36/35, 2048/1875		[[Comma list]]: 36/35, 2048/1875

	~~[[Mapping]]: [~~{{~~val~~\| 1 3 2 6 ~~}}, {{val~~\| 0 -4 1 -9 }}]		{{Mapping\|legend=1\| 1 3 2 6 \| 0 -4 1 -9 }}

	{{Multival\|legend=1\| 4 -1 9 -11 3 24 }}		{{Multival\|legend=1\| 4 -1 9 -11 3 24 }}

	[[POTE ~~generator~~]]: ~5/4 = 422.275		[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 422.275

	{{Optimal ET sequence\|legend=1\| 3d, 11d, 14, 17c, ~~37cd~~ }}		{{Optimal ET sequence\|legend=1\| 3d, 11d, 14, 17c, 37ccdd }}

	[[Badness]]: 0.077871		[[Badness]]: 0.077871
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	Comma list: 36/35, 56/55, 243/242		Comma list: 36/35, 56/55, 243/242

	Mapping: [{{~~val~~\| 1 3 2 6 7 ~~}}, {{val~~\| 0 -4 1 -9 -10 }}]		Mapping: {{mapping\| 1 3 2 6 7 \| 0 -4 1 -9 -10 }}

	POTE ~~generator~~: ~5/4 = 422.217		Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 422.217

	{{Optimal ET sequence\|legend=1\| 14, 17c, ~~37cde~~ }}		{{Optimal ET sequence\|legend=1\| 3de, 14, 17c, 37ccddee }}

	Badness: 0.042518		Badness: 0.042518
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	Comma list: 26/25, 36/35, 56/55, 243/242		Comma list: 26/25, 36/35, 56/55, 243/242

	Mapping: [{{~~val~~\| 1 3 2 6 7 3 ~~}}, {{val~~\| 0 -4 1 -9 -10 2 }}]		Mapping: {{mapping\| 1 3 2 6 7 3 \| 0 -4 1 -9 -10 2 }}

	POTE ~~generator~~: ~5/4 = 423.020		Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 423.020

	{{Optimal ET sequence\|legend=1\| 14, 17c }}		{{Optimal ET sequence\|legend=1\| 3de, 14, 17c }}

	Badness: 0.036836		Badness: 0.036836

	== Hemismate ==		== Hemismate ==
	Subgroup: 2.3.5.7		[[Subgroup]]: 2.3.5.7

	[[Comma list]]: 256/245, 392/375		[[Comma list]]: 256/245, 392/375

	~~[[Mapping]]: [~~{{~~val~~\| 1 3 2 3 ~~}}, {{val~~\| 0 -8 2 -1 }}]		{{Mapping\|legend=1\| 1 3 2 3 \| 0 -8 2 -1 }}

	{{Multival\|legend=1\| 8 -2 1 -22 -21 8 }}		{{Multival\|legend=1\| 8 -2 1 -22 -21 8 }}

	[[POTE ~~generator~~]]: ~8/7 = 210.452		[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 210.452

	{{Optimal ET sequence\|legend=1\| 6, 11, 17c, 40bcd }}		{{Optimal ET sequence\|legend=1\| 6, 11, 17c, 40bcd }}
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	Comma list: 56/55, 77/75, 256/245		Comma list: 56/55, 77/75, 256/245

	Mapping: [{{~~val~~\| 1 3 2 3 4 ~~}}, {{val~~\| 0 -8 2 -1 -3 }}]		Mapping: {{mapping\| 1 3 2 3 4 \| 0 -8 2 -1 -3 }}

	POTE ~~generator~~: ~8/7 = 210.481		Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 210.481

	{{Optimal ET sequence\|legend=1\| 6, 11, 17c, 40bcde }}		{{Optimal ET sequence\|legend=1\| 6, 11, 17c, 40bcde }}
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	Comma list: 26/25, 56/55, 77/75, 256/245		Comma list: 26/25, 56/55, 77/75, 256/245

	Mapping: [{{~~val~~\| 1 3 2 3 4 3 ~~}}, {{val~~\| 0 -8 2 -1 -3 4 }}]		Mapping: {{mapping\| 1 3 2 3 4 3 \| 0 -8 2 -1 -3 4 }}

	POTE ~~generators~~: ~8/7 = 210.974		Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 210.974

	{{Optimal ET sequence\|legend=1\| 6, 11, 17c }}		{{Optimal ET sequence\|legend=1\| 6, 11, 17c }}
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	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:Smate]]		[[Category:Smate family\| ]] <!-- main article -->
	[[Category:Rank 2]]		[[Category:Rank 2]]