168edo - Revision history

Francium: changed EDO intro to ED intro

2025-02-19T18:05:21Z

changed EDO intro to ED intro

← Older revision		Revision as of 18:05, 19 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro}}		{{ED intro}}

	== Theory ==		== Theory ==

FloraC: Sectioning

2025-02-06T10:07:28Z

Sectioning

← Older revision		Revision as of 10:07, 6 February 2025
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	{{EDO intro}}		{{EDO intro}}

			== Theory ==
	168edo is closely related to [[84edo]], but the [[patent val]]s differ on the mapping for [[11/1\|11]] and [[17/1\|17]]. It is [[contorted]] in the 7-limit, [[tempering out]] [[225/224]], [[1728/1715]], and [[78732/78125]]. Using the patent val, it tempers out [[243/242]], 2420/2401, [[3025/3024]], and 43923/43750 in the 11-limit; [[351/350]], [[625/624]], [[640/637]], [[847/845]], and [[1573/1568]] in the 13-limit; [[375/374]], [[561/560]], [[715/714]], 891/884, [[936/935]], and 1331/1326 in the 17-limit. Using the 168d val, it tempers out [[3136/3125]], [[19683/19600]], and 33075/32768 in the 7-limit; 243/242, [[385/384]], 3773/3750, and [[9801/9800]] in the 11-limit.		168edo is closely related to [[84edo]], but the [[patent val]]s differ on the mapping for [[11/1\|11]] and [[17/1\|17]]. It is [[contorted]] in the 7-limit, [[tempering out]] [[225/224]], [[1728/1715]], and [[78732/78125]]. Using the patent val, it tempers out [[243/242]], 2420/2401, [[3025/3024]], and 43923/43750 in the 11-limit; [[351/350]], [[625/624]], [[640/637]], [[847/845]], and [[1573/1568]] in the 13-limit; [[375/374]], [[561/560]], [[715/714]], 891/884, [[936/935]], and 1331/1326 in the 17-limit. Using the 168d val, it tempers out [[3136/3125]], [[19683/19600]], and 33075/32768 in the 7-limit; 243/242, [[385/384]], 3773/3750, and [[9801/9800]] in the 11-limit.

	Stacking alternating steps of 43 and 53 produces an optimal [[~~whitewood~~]][14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds. [[Substitute harmonic#Dotcom\|Dotcom]] is also supported.		Stacking alternating steps of 43 and 53 produces an optimal [[Whitewood]][14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds. [[Substitute harmonic#Dotcom\|Dotcom]] is also supported.

	== Odd harmonics ==		=== Odd harmonics ===
	{{Harmonics in equal\|168}}		{{Harmonics in equal\|168}}

			=== Subsets and supersets ===
			Since 168 factors into 2<sup>3</sup> × 3 × 7, 168edo has subset edos {{EDOs\| 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, and 84 }}.

	== Notation ==		== Notation ==
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	{{sharpness-sharp14-qt1\|168}}		{{sharpness-sharp14-qt1\|168}}

	~~== Subsets and supersets ==~~
	~~Since 168 factors into {{factorization\|168}}, 168edo has subset edos {{EDOs\| 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, and 84 }}.~~

ArrowHead294: /* Notation */

2025-02-05T19:50:26Z

Notation

← Older revision		Revision as of 19:50, 5 February 2025
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	== Notation ==		== Notation ==
	168edo can be notated using [[ups and downs notation]], in which [[Helmholtz–Ellis]] arrow accidentals can be used.		168edo can be notated using [[ups and downs notation]], in which [[Helmholtz–Ellis]] arrow accidentals can be used in combination with Stein–Zimmerman [[24edo#Notation\|quarter tone]] accidentals:

	{{sharpness-sharp14-qt1\|168}}		{{sharpness-sharp14-qt1\|168}}

ArrowHead294 at 19:06, 24 December 2024

2024-12-24T19:06:56Z

← Older revision		Revision as of 19:06, 24 December 2024
Line 6:		Line 6:
	Stacking alternating steps of 43 and 53 produces an optimal [[whitewood]][14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds. [[Substitute harmonic#Dotcom\|Dotcom]] is also supported.		Stacking alternating steps of 43 and 53 produces an optimal [[whitewood]][14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds. [[Substitute harmonic#Dotcom\|Dotcom]] is also supported.

	=== Odd harmonics ===		== Odd harmonics ==
	{{Harmonics in equal\|168}}		{{Harmonics in equal\|168}}

	=== Subsets and supersets ===		== Notation ==
			168edo can be notated using [[ups and downs notation]], in which [[Helmholtz–Ellis]] arrow accidentals can be used.

			{{sharpness-sharp14-qt1\|168}}

			== Subsets and supersets ==
	Since 168 factors into {{factorization\|168}}, 168edo has subset edos {{EDOs\| 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, and 84 }}.		Since 168 factors into {{factorization\|168}}, 168edo has subset edos {{EDOs\| 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, and 84 }}.

FloraC: +subsets and supersets

2024-05-05T08:40:01Z

+subsets and supersets

← Older revision		Revision as of 08:40, 5 May 2024
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	=== Odd harmonics ===		=== Odd harmonics ===
	{{Harmonics in equal\|168}}		{{Harmonics in equal\|168}}

			=== Subsets and supersets ===
			Since 168 factors into {{factorization\|168}}, 168edo has subset edos {{EDOs\| 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, and 84 }}.

FloraC: Cleanup

2024-05-05T08:37:33Z

Cleanup

← Older revision		Revision as of 08:37, 5 May 2024
Line 2:		Line 2:
	{{EDO intro}}		{{EDO intro}}

	It is closely related to [[84edo]], but the patent ~~vals~~ differ on the mapping for 11 and 17. It is [[contorted]] in the 7-limit, tempering out 225/224, 1728/1715, and 78732/78125. Using the patent val, it tempers out 243/242, 2420/2401, 3025/3024, and 43923/43750 in the 11-limit; 351/350, 625/624, 640/637, 847/845, and 1573/1568 in the 13-limit; 375/374, 561/560, 715/714, 891/884, 936/935, and 1331/1326 in the 17-limit. Using the 168d val, it tempers out 3136/3125, 19683/19600, and 33075/32768 in the 7-limit; 243/242, 385/384, 3773/3750, and 9801/9800 in the 11-limit.		168edo is closely related to [[84edo]], but the [[patent val]]s differ on the mapping for [[11/1\|11]] and [[17/1\|17]]. It is [[contorted]] in the 7-limit, [[tempering out]] [[225/224]], [[1728/1715]], and [[78732/78125]]. Using the patent val, it tempers out [[243/242]], 2420/2401, [[3025/3024]], and 43923/43750 in the 11-limit; [[351/350]], [[625/624]], [[640/637]], [[847/845]], and [[1573/1568]] in the 13-limit; [[375/374]], [[561/560]], [[715/714]], 891/884, [[936/935]], and 1331/1326 in the 17-limit. Using the 168d val, it tempers out [[3136/3125]], [[19683/19600]], and 33075/32768 in the 7-limit; 243/242, [[385/384]], 3773/3750, and [[9801/9800]] in the 11-limit.

	Stacking alternating steps of 43 and 53 produces an optimal [[whitewood]] [14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds. [[Substitute harmonic#Dotcom\|Dotcom]] is also supported.		Stacking alternating steps of 43 and 53 produces an optimal [[whitewood]][14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds. [[Substitute harmonic#Dotcom\|Dotcom]] is also supported.

	=== Odd harmonics ===		=== Odd harmonics ===
	{{Harmonics in equal\|168}}		{{Harmonics in equal\|168}}

	~~[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->~~

BudjarnLambeth at 22:57, 3 February 2024

2024-02-03T22:57:15Z

← Older revision		Revision as of 22:57, 3 February 2024
Line 4:		Line 4:
	It is closely related to [[84edo]], but the patent vals differ on the mapping for 11 and 17. It is [[contorted]] in the 7-limit, tempering out 225/224, 1728/1715, and 78732/78125. Using the patent val, it tempers out 243/242, 2420/2401, 3025/3024, and 43923/43750 in the 11-limit; 351/350, 625/624, 640/637, 847/845, and 1573/1568 in the 13-limit; 375/374, 561/560, 715/714, 891/884, 936/935, and 1331/1326 in the 17-limit. Using the 168d val, it tempers out 3136/3125, 19683/19600, and 33075/32768 in the 7-limit; 243/242, 385/384, 3773/3750, and 9801/9800 in the 11-limit.		It is closely related to [[84edo]], but the patent vals differ on the mapping for 11 and 17. It is [[contorted]] in the 7-limit, tempering out 225/224, 1728/1715, and 78732/78125. Using the patent val, it tempers out 243/242, 2420/2401, 3025/3024, and 43923/43750 in the 11-limit; 351/350, 625/624, 640/637, 847/845, and 1573/1568 in the 13-limit; 375/374, 561/560, 715/714, 891/884, 936/935, and 1331/1326 in the 17-limit. Using the 168d val, it tempers out 3136/3125, 19683/19600, and 33075/32768 in the 7-limit; 243/242, 385/384, 3773/3750, and 9801/9800 in the 11-limit.

	Stacking alternating steps of 43 and 53 produces an optimal [[whitewood]] [14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds. [[Substitute harmonic#Dotcom\|~~dotcom~~]] is also supported.		Stacking alternating steps of 43 and 53 produces an optimal [[whitewood]] [14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds. [[Substitute harmonic#Dotcom\|Dotcom]] is also supported.

	=== Odd harmonics ===		=== Odd harmonics ===

BudjarnLambeth: Added edo intro

2024-02-03T22:56:54Z

Added edo intro

← Older revision		Revision as of 22:56, 3 February 2024
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	~~'''168edo''' is the [[~~EDO~~\|equal division of the octave]] into 168 parts of 7.1429 cents each.~~ It is closely related to [[84edo]], but the patent vals differ on the mapping for 11 and 17. It is [[contorted]] in the 7-limit, tempering out 225/224, 1728/1715, and 78732/78125. Using the patent val, it tempers out 243/242, 2420/2401, 3025/3024, and 43923/43750 in the 11-limit; 351/350, 625/624, 640/637, 847/845, and 1573/1568 in the 13-limit; 375/374, 561/560, 715/714, 891/884, 936/935, and 1331/1326 in the 17-limit. Using the 168d val, it tempers out 3136/3125, 19683/19600, and 33075/32768 in the 7-limit; 243/242, 385/384, 3773/3750, and 9801/9800 in the 11-limit.		{{EDO intro}}

			It is closely related to [[84edo]], but the patent vals differ on the mapping for 11 and 17. It is [[contorted]] in the 7-limit, tempering out 225/224, 1728/1715, and 78732/78125. Using the patent val, it tempers out 243/242, 2420/2401, 3025/3024, and 43923/43750 in the 11-limit; 351/350, 625/624, 640/637, 847/845, and 1573/1568 in the 13-limit; 375/374, 561/560, 715/714, 891/884, 936/935, and 1331/1326 in the 17-limit. Using the 168d val, it tempers out 3136/3125, 19683/19600, and 33075/32768 in the 7-limit; 243/242, 385/384, 3773/3750, and 9801/9800 in the 11-limit.

	Stacking alternating steps of 43 and 53 produces an optimal [[whitewood]] [14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds. [[Substitute harmonic#Dotcom\|dotcom]] is also supported.		Stacking alternating steps of 43 and 53 produces an optimal [[whitewood]] [14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds. [[Substitute harmonic#Dotcom\|dotcom]] is also supported.

BudjarnLambeth: Added link to whitewood, added mention of dotcom

2024-02-03T22:55:15Z

Added link to whitewood, added mention of dotcom

← Older revision		Revision as of 22:55, 3 February 2024
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	'''168edo''' is the [[EDO\|equal division of the octave]] into 168 parts of 7.1429 cents each. It is closely related to [[84edo]], but the patent vals differ on the mapping for 11 and 17. It is [[contorted]] in the 7-limit, tempering out 225/224, 1728/1715, and 78732/78125. Using the patent val, it tempers out 243/242, 2420/2401, 3025/3024, and 43923/43750 in the 11-limit; 351/350, 625/624, 640/637, 847/845, and 1573/1568 in the 13-limit; 375/374, 561/560, 715/714, 891/884, 936/935, and 1331/1326 in the 17-limit. Using the 168d val, it tempers out 3136/3125, 19683/19600, and 33075/32768 in the 7-limit; 243/242, 385/384, 3773/3750, and 9801/9800 in the 11-limit. Stacking alternating steps of 43 and 53 produces an optimal ~~Whitewood~~ [14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds.		'''168edo''' is the [[EDO\|equal division of the octave]] into 168 parts of 7.1429 cents each. It is closely related to [[84edo]], but the patent vals differ on the mapping for 11 and 17. It is [[contorted]] in the 7-limit, tempering out 225/224, 1728/1715, and 78732/78125. Using the patent val, it tempers out 243/242, 2420/2401, 3025/3024, and 43923/43750 in the 11-limit; 351/350, 625/624, 640/637, 847/845, and 1573/1568 in the 13-limit; 375/374, 561/560, 715/714, 891/884, 936/935, and 1331/1326 in the 17-limit. Using the 168d val, it tempers out 3136/3125, 19683/19600, and 33075/32768 in the 7-limit; 243/242, 385/384, 3773/3750, and 9801/9800 in the 11-limit.

			Stacking alternating steps of 43 and 53 produces an optimal [[whitewood]] [14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds. [[Substitute harmonic#Dotcom\|dotcom]] is also supported.

	=== Odd harmonics ===		=== Odd harmonics ===

FloraC: Fix the error table

2022-12-20T10:28:36Z

Fix the error table

← Older revision		Revision as of 10:28, 20 December 2022
Line 2:		Line 2:
	'''168edo''' is the [[EDO\|equal division of the octave]] into 168 parts of 7.1429 cents each. It is closely related to [[84edo]], but the patent vals differ on the mapping for 11 and 17. It is [[contorted]] in the 7-limit, tempering out 225/224, 1728/1715, and 78732/78125. Using the patent val, it tempers out 243/242, 2420/2401, 3025/3024, and 43923/43750 in the 11-limit; 351/350, 625/624, 640/637, 847/845, and 1573/1568 in the 13-limit; 375/374, 561/560, 715/714, 891/884, 936/935, and 1331/1326 in the 17-limit. Using the 168d val, it tempers out 3136/3125, 19683/19600, and 33075/32768 in the 7-limit; 243/242, 385/384, 3773/3750, and 9801/9800 in the 11-limit. Stacking alternating steps of 43 and 53 produces an optimal Whitewood [14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds.		'''168edo''' is the [[EDO\|equal division of the octave]] into 168 parts of 7.1429 cents each. It is closely related to [[84edo]], but the patent vals differ on the mapping for 11 and 17. It is [[contorted]] in the 7-limit, tempering out 225/224, 1728/1715, and 78732/78125. Using the patent val, it tempers out 243/242, 2420/2401, 3025/3024, and 43923/43750 in the 11-limit; 351/350, 625/624, 640/637, 847/845, and 1573/1568 in the 13-limit; 375/374, 561/560, 715/714, 891/884, 936/935, and 1331/1326 in the 17-limit. Using the 168d val, it tempers out 3136/3125, 19683/19600, and 33075/32768 in the 7-limit; 243/242, 385/384, 3773/3750, and 9801/9800 in the 11-limit. Stacking alternating steps of 43 and 53 produces an optimal Whitewood [14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds.

	{{Harmonics in equal\|~~159\|columns=12~~}}		=== Odd harmonics ===
			{{Harmonics in equal\|168}}

	[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->		[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->

← Older revision		Revision as of 10:28, 20 December 2022
Line 2:		Line 2:
	'''168edo''' is the [[EDO\|equal division of the octave]] into 168 parts of 7.1429 cents each. It is closely related to [[84edo]], but the patent vals differ on the mapping for 11 and 17. It is [[contorted]] in the 7-limit, tempering out 225/224, 1728/1715, and 78732/78125. Using the patent val, it tempers out 243/242, 2420/2401, 3025/3024, and 43923/43750 in the 11-limit; 351/350, 625/624, 640/637, 847/845, and 1573/1568 in the 13-limit; 375/374, 561/560, 715/714, 891/884, 936/935, and 1331/1326 in the 17-limit. Using the 168d val, it tempers out 3136/3125, 19683/19600, and 33075/32768 in the 7-limit; 243/242, 385/384, 3773/3750, and 9801/9800 in the 11-limit. Stacking alternating steps of 43 and 53 produces an optimal Whitewood [14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds.		'''168edo''' is the [[EDO\|equal division of the octave]] into 168 parts of 7.1429 cents each. It is closely related to [[84edo]], but the patent vals differ on the mapping for 11 and 17. It is [[contorted]] in the 7-limit, tempering out 225/224, 1728/1715, and 78732/78125. Using the patent val, it tempers out 243/242, 2420/2401, 3025/3024, and 43923/43750 in the 11-limit; 351/350, 625/624, 640/637, 847/845, and 1573/1568 in the 13-limit; 375/374, 561/560, 715/714, 891/884, 936/935, and 1331/1326 in the 17-limit. Using the 168d val, it tempers out 3136/3125, 19683/19600, and 33075/32768 in the 7-limit; 243/242, 385/384, 3773/3750, and 9801/9800 in the 11-limit. Stacking alternating steps of 43 and 53 produces an optimal Whitewood [14] scale of 19 5 19 5 19 5 19 5 19 5 19 5 19 5 that spreads the overall flatness evenly between the major and minor thirds.

	{{Harmonics in equal\|~~159\|columns=12~~}}		=== Odd harmonics ===
			{{Harmonics in equal\|168}}

	[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->		[[Category:Equal divisions of the octave\|###]] <!-- 3-digit number -->