Ennealimmal - Revision history

Inthar: /* Extensions */

2026-04-25T10:51:39Z

Extensions

← Older revision		Revision as of 10:51, 25 April 2026
Line 159:		Line 159:
	The most natural extension to prime 11 is a weak extension Hemiennealimmal (72 & 198) which has a 1\18 period and tempers out 9801/9800 = S99, equating 99/98 to 100/99 (dividing 50/49 into equal halves and thus splitting 2/1 into two equal parts too). Hemiennealimmal can, of course, be extended to prime 17 via S50.		The most natural extension to prime 11 is a weak extension Hemiennealimmal (72 & 198) which has a 1\18 period and tempers out 9801/9800 = S99, equating 99/98 to 100/99 (dividing 50/49 into equal halves and thus splitting 2/1 into two equal parts too). Hemiennealimmal can, of course, be extended to prime 17 via S50.

	There is no canonical 13 extension (you can temper out 4096/4095 = S64, but you could instead temper out a different 13-limit comma to extend from an 11-limit strong extension); however, [[270edo]] is an excellent edo tuning in the 13-limit.		There is no canonical 13 extension (you can temper out 4096/4095 = S64 to extend from the 7-limit, but you could instead temper out a different 13-limit comma to extend from an 11-limit strong extension); however, [[270edo]] is an excellent edo tuning in the 13-limit.

	== Praxis ==		== Praxis ==

Inthar at 18:40, 18 April 2026

2026-04-18T18:40:59Z

← Older revision		Revision as of 18:40, 18 April 2026
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	'''Ennealimmal''' (from "ennea-" = 9 and "large limma" = 27/25), [[72edo\|72]] & [[99edo\|99]], is a 7-limit rank-2 microtemperament that tempers out the two smallest 7-limit [[superparticular]] ratios:		'''Ennealimmal''' (from "ennea-" = 9 and "large limma" = 27/25), [[72edo\|72]] & [[99edo\|99]], is a 7-limit rank-2 microtemperament that tempers out the two smallest 7-limit [[superparticular]] ratios:
	* 2401/2400 = [[Square-superparticular\|S49]], the difference between the 49/40 neutral third and its 3/2-complement 60/49, or the difference between 49/48 and 50/49		* 2401/2400 = [[Square-superparticular\|S49]], the difference between the 49/40 neutral third and its 3/2-complement 60/49, or the difference between 49/48 and 50/49
	* 4375/4374 = S25/S27, the difference between (27/25)<sup>2</sup> and 7/6		* 4375/4374 = S25/S27, the difference between (27/25)<sup>2</sup> and 7/6 or equivalently the difference between (28/27) * (25/24) and 27/25
	This implies a period of 1\9 (representing 27/25) and a generator of 49/40. The generator can also be taken to be 5/3.		This implies a period of 1\9 (representing 27/25) and a generator of 49/40. The generator can also be taken to be 5/3.
	== Theory ==		== Theory ==

Inthar: /* Derivation of 1\9 period */

2026-04-18T18:37:19Z

Derivation of 1\9 period

← Older revision		Revision as of 18:37, 18 April 2026
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	{{Adv\|This can be derived by showing that (1) (27/25)^3 is equated to 63/50 and (2) (63/50)^3 is equated to 2/1.}}		{{Adv\|This can be derived by showing that (1) (27/25)^3 is equated to 63/50 and (2) (63/50)^3 is equated to 2/1.}}

	{{Adv\|(1) is easy (~= means "is equated to"):}}		{{Adv\|(1) is easy (~{{=}} means "is equated to"):}}
	(27/25)^3		(27/25)^3
	~= 27/25 * 7/6		~= 27/25 * 7/6

Inthar: /* Derivation of 1\9 period */

2026-04-18T18:37:08Z

Derivation of 1\9 period

← Older revision		Revision as of 18:37, 18 April 2026
Line 139:		Line 139:
	{{Adv\|This can be derived by showing that (1) (27/25)^3 is equated to 63/50 and (2) (63/50)^3 is equated to 2/1.}}		{{Adv\|This can be derived by showing that (1) (27/25)^3 is equated to 63/50 and (2) (63/50)^3 is equated to 2/1.}}

	{{Adv\|(1) is easy:}}		{{Adv\|(1) is easy (~= means "is equated to"):}}
	(27/25)^3		(27/25)^3
	~= 27/25 * 7/6		~= 27/25 * 7/6

Inthar: /* Extensions */

2026-04-13T04:05:34Z

Extensions

← Older revision		Revision as of 04:05, 13 April 2026
Line 159:		Line 159:
	The most natural extension to prime 11 is a weak extension Hemiennealimmal (72 & 198) which has a 1\18 period and tempers out 9801/9800 = S99, equating 99/98 to 100/99 (dividing 50/49 into equal halves and thus splitting 2/1 into two equal parts too). Hemiennealimmal can, of course, be extended to prime 17 via S50.		The most natural extension to prime 11 is a weak extension Hemiennealimmal (72 & 198) which has a 1\18 period and tempers out 9801/9800 = S99, equating 99/98 to 100/99 (dividing 50/49 into equal halves and thus splitting 2/1 into two equal parts too). Hemiennealimmal can, of course, be extended to prime 17 via S50.

	There is no canonical 13 extension; however, [[270edo]] is an excellent edo tuning in the 13-limit.		There is no canonical 13 extension (you can temper out 4096/4095 = S64, but you could instead temper out a different 13-limit comma to extend from an 11-limit strong extension); however, [[270edo]] is an excellent edo tuning in the 13-limit.

	== Praxis ==		== Praxis ==

Inthar: /* Extensions */

2026-04-09T14:26:06Z

Extensions

← Older revision		Revision as of 14:26, 9 April 2026
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	== Extensions ==		== Extensions ==
	Ennealimmal readily extends to prime 17 by tempering out S50, equating 49/48~50/49 to 51/50. [[171edo]] is an excellent tuning for this extension.		Ennealimmal readily extends to prime 17 by tempering out 2500/2499 = S50, equating 49/48~50/49 to 51/50. [[171edo]] is an excellent tuning for this extension.

	The most natural extension to prime 11 is a weak extension Hemiennealimmal (72 & 198) which has a 1\18 period and tempers out 9801/9800 = S99, equating 99/98 to 100/99 (dividing 50/49 into equal halves and thus splitting 2/1 into two equal parts too). Hemiennealimmal can, of course, be extended to prime 17 via S50.		The most natural extension to prime 11 is a weak extension Hemiennealimmal (72 & 198) which has a 1\18 period and tempers out 9801/9800 = S99, equating 99/98 to 100/99 (dividing 50/49 into equal halves and thus splitting 2/1 into two equal parts too). Hemiennealimmal can, of course, be extended to prime 17 via S50.

Inthar: /* Extensions */

2026-04-09T12:47:22Z

Extensions

← Older revision		Revision as of 12:47, 9 April 2026
Line 157:		Line 157:
	Ennealimmal readily extends to prime 17 by tempering out S50, equating 49/48~50/49 to 51/50. [[171edo]] is an excellent tuning for this extension.		Ennealimmal readily extends to prime 17 by tempering out S50, equating 49/48~50/49 to 51/50. [[171edo]] is an excellent tuning for this extension.

	The most natural extension to prime 11 is a weak extension Hemiennealimmal which has a 1\18 period and tempers out 9801/9800 = S99, equating 99/98 to 100/99 (dividing 50/49 into equal halves and thus splitting 2/1 into two equal parts too). Hemiennealimmal can be extended to prime 17 via ~~the 2.3.7.11.17 comma 2058/2057 which ends up being a weak extension of the above 2.3.5.7.17 extension~~.		The most natural extension to prime 11 is a weak extension Hemiennealimmal (72 & 198) which has a 1\18 period and tempers out 9801/9800 = S99, equating 99/98 to 100/99 (dividing 50/49 into equal halves and thus splitting 2/1 into two equal parts too). Hemiennealimmal can, of course, be extended to prime 17 via S50.

	There is no canonical 13 extension; however, [[270edo]] is an excellent edo tuning in the 13-limit.		There is no canonical 13 extension; however, [[270edo]] is an excellent edo tuning in the 13-limit.

Inthar: /* Extensions */

2026-04-09T12:38:18Z

Extensions

← Older revision		Revision as of 12:38, 9 April 2026
Line 155:		Line 155:

	== Extensions ==		== Extensions ==
	Ennealimmal readily extends to prime 17 by tempering out S50, equating 49/48~50/49 to 51/50.		Ennealimmal readily extends to prime 17 by tempering out S50, equating 49/48~50/49 to 51/50. [[171edo]] is an excellent tuning for this extension.

	The most natural extension to prime 11 is a weak extension Hemiennealimmal which has a 1\18 period and tempers out 9801/9800 = S99, equating 99/98 to 100/99 (dividing 50/49 into equal halves and thus splitting 2/1 into two equal parts too). Hemiennealimmal can be extended to prime 17 via the 2.3.7.11.17 comma 2058/2057 which ends up being a weak extension of the above 2.3.5.7.17 extension.		The most natural extension to prime 11 is a weak extension Hemiennealimmal which has a 1\18 period and tempers out 9801/9800 = S99, equating 99/98 to 100/99 (dividing 50/49 into equal halves and thus splitting 2/1 into two equal parts too). Hemiennealimmal can be extended to prime 17 via the 2.3.7.11.17 comma 2058/2057 which ends up being a weak extension of the above 2.3.5.7.17 extension.

Inthar: /* Extensions */

2026-04-09T12:37:01Z

Extensions

← Older revision		Revision as of 12:37, 9 April 2026
Line 159:		Line 159:
	The most natural extension to prime 11 is a weak extension Hemiennealimmal which has a 1\18 period and tempers out 9801/9800 = S99, equating 99/98 to 100/99 (dividing 50/49 into equal halves and thus splitting 2/1 into two equal parts too). Hemiennealimmal can be extended to prime 17 via the 2.3.7.11.17 comma 2058/2057 which ends up being a weak extension of the above 2.3.5.7.17 extension.		The most natural extension to prime 11 is a weak extension Hemiennealimmal which has a 1\18 period and tempers out 9801/9800 = S99, equating 99/98 to 100/99 (dividing 50/49 into equal halves and thus splitting 2/1 into two equal parts too). Hemiennealimmal can be extended to prime 17 via the 2.3.7.11.17 comma 2058/2057 which ends up being a weak extension of the above 2.3.5.7.17 extension.

	There is no canonical 13 extension; however, [[270edo]] is an excellent ~~rank-1 Ennealimmal~~ tuning in the 13-limit.		There is no canonical 13 extension; however, [[270edo]] is an excellent edo tuning in the 13-limit.

	== Praxis ==		== Praxis ==

Inthar: /* Extensions */

2026-04-09T12:33:25Z

Extensions

← Older revision		Revision as of 12:33, 9 April 2026
Line 159:		Line 159:
	The most natural extension to prime 11 is a weak extension Hemiennealimmal which has a 1\18 period and tempers out 9801/9800 = S99, equating 99/98 to 100/99 (dividing 50/49 into equal halves and thus splitting 2/1 into two equal parts too). Hemiennealimmal can be extended to prime 17 via the 2.3.7.11.17 comma 2058/2057 which ends up being a weak extension of the above 2.3.5.7.17 extension.		The most natural extension to prime 11 is a weak extension Hemiennealimmal which has a 1\18 period and tempers out 9801/9800 = S99, equating 99/98 to 100/99 (dividing 50/49 into equal halves and thus splitting 2/1 into two equal parts too). Hemiennealimmal can be extended to prime 17 via the 2.3.7.11.17 comma 2058/2057 which ends up being a weak extension of the above 2.3.5.7.17 extension.

	There is no canonical 13 extension; however, [[~~270~~]] is an excellent rank-1 Ennealimmal tuning in the 13-limit.		There is no canonical 13 extension; however, [[270edo]] is an excellent rank-1 Ennealimmal tuning in the 13-limit.

	== Praxis ==		== Praxis ==