99edo - Revision history

Overthink: /* Theory */ step size

2026-05-20T01:15:24Z

Theory: step size

← Older revision		Revision as of 01:15, 20 May 2026
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	Being a [[zeta peak edo]], 99edo is also a very strong no-11 no-13 system, where it is consistent to the [[29-odd-limit]] with a sharp tendency. This favors the sharp mapping of 11 and 13, and allows these relatively weak approximations to somewhat blend with the rest for a full [[29-limit]] (or [[31-limit]], using the sharp-tending 99efk val) temperament. In fact, the 99efk val is the first to achieve [[diamond monotone]] in the [[31-odd-limit]], though it fails in the [[33-odd-limit]] due to mapping [[33/32]] to 5 steps, while [[32/31]] is mapped to 4 steps.		Being a [[zeta peak edo]], 99edo is also a very strong no-11 no-13 system, where it is consistent to the [[29-odd-limit]] with a sharp tendency. This favors the sharp mapping of 11 and 13, and allows these relatively weak approximations to somewhat blend with the rest for a full [[29-limit]] (or [[31-limit]], using the sharp-tending 99efk val) temperament. In fact, the 99efk val is the first to achieve [[diamond monotone]] in the [[31-odd-limit]], though it fails in the [[33-odd-limit]] due to mapping [[33/32]] to 5 steps, while [[32/31]] is mapped to 4 steps.

			One step of 99edo is close to [[144/143]], the grossma. Unfortunately, neither 99ef nor the patent val map it consistently, though [[198edo]] does.

	=== Prime harmonics ===		=== Prime harmonics ===

2026-05-19T09:55:31Z

Regular temperament properties: update

← Older revision		Revision as of 09:55, 19 May 2026
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	\| 339.394		\| 339.394
	\| 128/105		\| 128/105
	\| [[Amity]] (99ef) / hitchcock (99)		\| [[Amity]] (99ef) / stalagmite (99ef) / hitchcock (99)
	\|-		\|-
	\| 1		\| 1
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	\| 48.485		\| 48.485
	\| 36/35		\| 36/35
	\| [[Ennealimmal]] (~~99e~~) / ~~ennealimmia~~ (99) / <br>ennealimnic (99ef) / ennealim (99e) / ennealiminal (99)		\| [[Ennealimmal]] / enneabiotic (99ef) / ennealympic (99) / <br>ennealimnic (99ef) / ennealim (99e) / ennealiminal (99)
	\|-		\|-
	\| 11		\| 11

2026-05-12T14:11:30Z

Notation: SZG notation

← Older revision		Revision as of 14:11, 12 May 2026
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	== Notation ==		== Notation ==
	=== ~~Ups~~ and downs notation ===		[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
	99edo can be notated with [[Kite's ups and downs notation]]. Note that quip (quintuple-up) is the same as quudsharp (quadruple-down sharp) and that quid (quintuple-down) is the same as quupflat (quadruple-up flat):		{{Sharpness-sharp10-qt1-szg}}

			=== Kite's ups and downs notation ===
			99edo can also be notated with [[Kite's ups and downs notation\|Kite's ups and downs]]. Note that quip (quintuple-up) is the same as quudsharp (quadruple-down sharp) and that quid (quintuple-down) is the same as quupflat (quadruple-up flat):
	{{Ups and downs sharpness\|99\|true}}		{{Ups and downs sharpness\|99\|true}}

	Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5\|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
	~~{{Sharpness-sharp10-qt1}}~~

	== Approximation to JI ==		== Approximation to JI ==

2026-04-19T01:16:18Z

Intervals made equidistant by 99edo

2026-04-18T21:46:35Z

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

2026-04-18T21:41:34Z

7-prime-limited odd-limit analysis

@@ Line 208: / Line 208: @@
 </pre>
 }}
 === Interval mappings ===
 {{Q-odd-limit intervals|99}}

2026-04-17T19:24:58Z

- extra newline

← Older revision		Revision as of 19:24, 17 April 2026
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	\|}		\|}
	<nowiki/>* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct


	== Octave stretch or compression ==		== Octave stretch or compression ==

2026-04-17T14:22:22Z

← Older revision		Revision as of 14:22, 17 April 2026
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	}}		}}
	=== Interval mappings ===		=== Interval mappings ===
	{{Q-odd-limit intervals\|99~~\|31~~}}		{{Q-odd-limit intervals\|99}}
	{{Q-odd-limit intervals\|99.1~~\|31~~\|apx=val\|header=none\|tag=none\|title=31-odd-limit intervals by ~~99efk~~ val mapping}}		{{Q-odd-limit intervals\|99.1\|apx=val\|header=none\|tag=none\|title=15-odd-limit intervals by 99ef val mapping}}

	== Regular temperament properties ==		== Regular temperament properties ==

2026-04-17T11:00:44Z

Moved to correct section

← Older revision		Revision as of 11:00, 17 April 2026
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	</pre>		</pre>
	}}		}}
			=== Interval mappings ===
			{{Q-odd-limit intervals\|99\|31}}
			{{Q-odd-limit intervals\|99.1\|31\|apx=val\|header=none\|tag=none\|title=31-odd-limit intervals by 99efk val mapping}}

	== Regular temperament properties ==		== Regular temperament properties ==
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	\|}		\|}
	<nowiki/>* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct
	~~=== Interval mappings ===~~
	~~{{Q-odd-limit intervals\|99\|31}}~~
	~~{{Q-odd-limit intervals\|99.1\|31\|apx=val\|header=none\|tag=none\|title=31-odd-limit intervals by 99efk val mapping}}~~

	== Octave stretch or compression ==		== Octave stretch or compression ==

2026-04-17T10:57:18Z

Regular temperament properties: Added 31-odd-limit interval mappings and 99efk val interval mapping

← Older revision		Revision as of 10:57, 17 April 2026
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	\|}		\|}
	<nowiki/>* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct
			=== Interval mappings ===
			{{Q-odd-limit intervals\|99\|31}}
			{{Q-odd-limit intervals\|99.1\|31\|apx=val\|header=none\|tag=none\|title=31-odd-limit intervals by 99efk val mapping}}

	== Octave stretch or compression ==		== Octave stretch or compression ==