99edo - Revision history

Inthar: /* Hemiwürschmidt/Würschmidt/Hemiwur */

2026-05-18T03:15:25Z

Hemiwürschmidt/Würschmidt/Hemiwur

← Older revision		Revision as of 03:15, 18 May 2026
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	The Würschmidt generator, which is the classic major third ~[[5/4]] (near-just), is 32\99 in 99edo, so it is divisible by 2 or 4 but not by 3 (seen with [[65edo]].		The Würschmidt generator, which is the classic major third ~[[5/4]] (near-just), is 32\99 in 99edo, so it is divisible by 2 or 4 but not by 3 (seen with [[65edo]].

	==== Hemiwürschmidt/Würschmidt~~/Hemiwur~~ ====		==== Hemiwürschmidt/Würschmidt ====
	Division by 2 to get 16\99 yields [[Didacus\|Hemiwürschmidt]] with a slightly flat septimal) middle whole tone ~[[28/25]] for the divided generator, with a scale [[6L 1s]] (16:3 step ratio). This mapping only splits the [[Würschmidt]] in half to get greater range (over four octaves) than when splitting it in quarters, but at the cost of missing many notes in each octave. Despite the missing notes, [[Bryan Deister]] has demonstrated this mapping in [https://www.youtube.com/shorts/p9OUaFuTUek ''99edo waltz''] (2025).		Division by 2 to get 16\99 yields [[Didacus\|Hemiwürschmidt]] with a slightly flat septimal) middle whole tone ~[[28/25]] for the divided generator, with a scale [[6L 1s]] (16:3 step ratio). This mapping only splits the [[Würschmidt]] in half to get greater range (over four octaves) than when splitting it in quarters, but at the cost of missing many notes in each octave. Despite the missing notes, [[Bryan Deister]] has demonstrated this mapping in [https://www.youtube.com/shorts/p9OUaFuTUek ''99edo waltz''] (2025).
	{{Lumatone edo mapping\|n=99\|start=40\|xstep=16\|ystep=-13}}		{{Lumatone edo mapping\|n=99\|start=40\|xstep=16\|ystep=-13}}

Inthar: /* Hemiwürschmidt/Würschmidt/Hemiwur */

2026-05-18T03:15:12Z

Hemiwürschmidt/Würschmidt/Hemiwur

← Older revision		Revision as of 03:15, 18 May 2026
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	==== Hemiwürschmidt/Würschmidt/Hemiwur ====		==== Hemiwürschmidt/Würschmidt/Hemiwur ====
	Division by 2 to get 16\99 yields [[~~Hemimean_clan#Hemiwürschmidt~~\|Hemiwürschmidt~~/Würschmidt/Hemiwur~~]] with a slightly flat septimal) middle whole tone ~[[28/25]] for the divided generator, with a scale [[6L 1s]] (16:3 step ratio). This mapping only splits the [[Würschmidt]] in half to get greater range (over four octaves) than when splitting it in quarters, but at the cost of missing many notes in each octave. Despite the missing notes, [[Bryan Deister]] has demonstrated this mapping in [https://www.youtube.com/shorts/p9OUaFuTUek ''99edo waltz''] (2025).		Division by 2 to get 16\99 yields [[Didacus\|Hemiwürschmidt]] with a slightly flat septimal) middle whole tone ~[[28/25]] for the divided generator, with a scale [[6L 1s]] (16:3 step ratio). This mapping only splits the [[Würschmidt]] in half to get greater range (over four octaves) than when splitting it in quarters, but at the cost of missing many notes in each octave. Despite the missing notes, [[Bryan Deister]] has demonstrated this mapping in [https://www.youtube.com/shorts/p9OUaFuTUek ''99edo waltz''] (2025).
	{{Lumatone edo mapping\|n=99\|start=40\|xstep=16\|ystep=-13}}		{{Lumatone edo mapping\|n=99\|start=40\|xstep=16\|ystep=-13}}

Inthar: /* Theory */

2026-04-19T16:54:51Z

Theory

← Older revision		Revision as of 16:54, 19 April 2026
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	* [[Wurschmidt]]		* [[Wurschmidt]]

	~~=== Derivation ===~~
	==== Tempering the 7-prime-limited 45-odd-limit ====		==== Tempering the 7-prime-limited 45-odd-limit ====
	99edo can be derived from any three of the following non-JI identities between melodic steps in the 7-prime-limited 35-odd-limit, namely 9/8, 28/25, 10/9, 35/32, 27/25, 15/14, 16/15, 21/20, 25/24, 28/27, 36/35:		99edo can be derived from any three of the following non-JI identities between melodic steps in the 7-prime-limited 35-odd-limit, namely 9/8, 28/25, 10/9, 35/32, 27/25, 15/14, 16/15, 21/20, 25/24, 28/27, 36/35:
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	* 16/15 = 36/35 * 28/27		* 16/15 = 36/35 * 28/27

	===~~= DKW derivation =~~===		=== Derivation ===
	The following square-superparticulars:		The following square-superparticulars:

Inthar: /* Intervals made equidistant by 99edo */

2026-04-19T02:02:43Z

Intervals made equidistant by 99edo

← Older revision		Revision as of 02:02, 19 April 2026
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	Runs of intervals separated by 3\99:		Runs of intervals separated by 3\99:
	# 36/35 ↔<sub>h</sub> 21/20 ↔<sub>i</sub> 15/14 ↔<sub>h</sub> 35/32 ↔<sub>j</sub> 28/25 ↔<sub>i</sub> 8/7 ↔<sub>h</sub> 7/6 ↔<sub>i</sub> 25/21		# 36/35 ↔<sub>h</sub> 21/20 ↔<sub>i</sub> 15/14 ↔<sub>h</sub> 35/32 ↔<sub>j</sub> 28/25 ↔<sub>i</sub> 8/7 ↔<sub>h</sub> 7/6 ↔<sub>i</sub> 25/21
			# 48/35 ↔<sub>h</sub> 7/5 ↔<sub>i</sub> 10/7 ↔<sub>h</sub> 35/24

	The separating intervals (all equated to 3\99):		The separating intervals (all equated to 3\99):

Inthar: /* Intervals made equidistant by 99edo */

2026-04-19T01:58:14Z

Intervals made equidistant by 99edo

← Older revision		Revision as of 01:58, 19 April 2026
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	# ↔<sub>f</sub> = 64/63		# ↔<sub>f</sub> = 64/63
	# ↔<sub>g</sub> = 875/864		# ↔<sub>g</sub> = 875/864

			Runs of intervals separated by 3\99:
			# 36/35 ↔<sub>h</sub> 21/20 ↔<sub>i</sub> 15/14 ↔<sub>h</sub> 35/32 ↔<sub>j</sub> 28/25 ↔<sub>i</sub> 8/7 ↔<sub>h</sub> 7/6 ↔<sub>i</sub> 25/21

			The separating intervals (all equated to 3\99):
			# ↔<sub>h</sub> = 49/48
			# ↔<sub>i</sub> = 50/49
			# ↔<sub>j</sub> = 128/125

	=== Temperaments ===		=== Temperaments ===

Inthar: /* Intervals made equidistant by 99edo */

2026-04-19T01:10:23Z

Intervals made equidistant by 99edo

← Older revision		Revision as of 01:10, 19 April 2026
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	# 25/18 ↔<sub>c</sub> 7/5 ↔<sub>b</sub> 45/32 ↔<sub>d</sub> 64/45 ↔<sub>b</sub> 10/7 ↔<sub>c</sub> 36/25		# 25/18 ↔<sub>c</sub> 7/5 ↔<sub>b</sub> 45/32 ↔<sub>d</sub> 64/45 ↔<sub>b</sub> 10/7 ↔<sub>c</sub> 36/25

	The separating intervals (all equated):		The separating intervals (all equated to 1\99):
	# ↔<sub>a</sub> = 245/243, the [[Sensamagic]] comma		# ↔<sub>a</sub> = 245/243, the [[Sensamagic]] comma
	# ↔<sub>b</sub> = 225/224, the [[Marvel]] comma		# ↔<sub>b</sub> = 225/224, the [[Marvel]] comma
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	# 32/25 ↔<sub>g</sub> 35/27 ↔<sub>e</sub> 21/16 ↔<sub>f</sub> 4/3 ↔<sub>e</sub> 27/20 ↔<sub>f</sub> 48/35 ↔<sub>g</sub> 25/18 ↔<sub>e</sub> 45/32 ↔<sub>f</sub> 10/7		# 32/25 ↔<sub>g</sub> 35/27 ↔<sub>e</sub> 21/16 ↔<sub>f</sub> 4/3 ↔<sub>e</sub> 27/20 ↔<sub>f</sub> 48/35 ↔<sub>g</sub> 25/18 ↔<sub>e</sub> 45/32 ↔<sub>f</sub> 10/7

	The separating intervals (all equated):		The separating intervals (all equated to 2\99):
	# ↔<sub>e</sub> = 81/80		# ↔<sub>e</sub> = 81/80
	# ↔<sub>f</sub> = 64/63		# ↔<sub>f</sub> = 64/63

Inthar: /* From the 7-prime-limited 45-odd-limit */

2026-04-19T01:02:52Z

From the 7-prime-limited 45-odd-limit

← Older revision		Revision as of 01:02, 19 April 2026
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	=== Derivation ===		=== Derivation ===
	==== ~~From~~ the 7-prime-limited 45-odd-limit ====		==== Tempering the 7-prime-limited 45-odd-limit ====
	99edo can be derived from any three of the following non-JI identities between melodic steps in the 7-prime-limited 35-odd-limit, namely 9/8, 28/25, 10/9, 35/32, 27/25, 15/14, 16/15, 21/20, 25/24, 28/27, 36/35:		99edo can be derived from any three of the following non-JI identities between melodic steps in the 7-prime-limited 35-odd-limit, namely 9/8, 28/25, 10/9, 35/32, 27/25, 15/14, 16/15, 21/20, 25/24, 28/27, 36/35:

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	* 15/14 = 36/35 * 25/24		* 15/14 = 36/35 * 25/24
	* 16/15 = 36/35 * 28/27		* 16/15 = 36/35 * 28/27

	==== DKW derivation ====		==== DKW derivation ====
	The following square-superparticulars:		The following square-superparticulars:

Inthar: /* Derivation */

2026-04-18T22:01:46Z

Derivation

← Older revision		Revision as of 22:01, 18 April 2026
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	=== Derivation ===		=== Derivation ===
			==== From the 7-prime-limited 45-odd-limit ====
	99edo can be derived from any three of the following non-JI identities between melodic steps in the 7-prime-limited 35-odd-limit, namely 9/8, 28/25, 10/9, 35/32, 27/25, 15/14, 16/15, 21/20, 25/24, 28/27, 36/35:		99edo can be derived from any three of the following non-JI identities between melodic steps in the 7-prime-limited 35-odd-limit, namely 9/8, 28/25, 10/9, 35/32, 27/25, 15/14, 16/15, 21/20, 25/24, 28/27, 36/35:

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	* 15/14 = 36/35 * 25/24		* 15/14 = 36/35 * 25/24
	* 16/15 = 36/35 * 28/27		* 16/15 = 36/35 * 28/27
			==== DKW derivation ====
			The following square-superparticulars:

			* S4 ('''16'''/15)
			* S5 (25/'''24''')
			* S6 ('''36'''/35)
			* S7 (49/'''48''')
			can be placed in the ratio of 1/16:1/24:1/36:1/48 - that is, 9:6:4:3, which specifies 99edo, much analogously to how [[34edo]] in the 5-limit is specified by S3, S4, and S5.

	== 99edo on a Lumatone ==		== 99edo on a Lumatone ==

Inthar: /* Intervals made equidistant by 99edo */

2026-04-18T21:46:15Z

Intervals made equidistant by 99edo

← Older revision		Revision as of 21:46, 18 April 2026
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	==== Intervals made equidistant by 99edo ====		==== Intervals made equidistant by 99edo ====
	Runs of 7-PL 45-OL intervals separated by 1\99:		Runs of 7-prime-limited 45-odd-limit intervals separated by 1\99:
	# 36/35 ↔<sub>a</sub> 28/27 ↔<sub>b</sub> 25/24 ↔<sub>c</sub> 21/20		# 36/35 ↔<sub>a</sub> 28/27 ↔<sub>b</sub> 25/24 ↔<sub>c</sub> 21/20
	# 16/15 ↔<sub>b</sub> 15/14 ↔<sub>c</sub> 27/25		# 16/15 ↔<sub>b</sub> 15/14 ↔<sub>c</sub> 27/25

Inthar: /* Edostep interpretations */

2026-04-18T21:41:44Z

Edostep interpretations

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 * 50/49
 * 128/125
 === Temperaments ===