Schismic rank-3 family - Revision history

Hotcrystal0: .

2026-09-24T20:51:25Z

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	Orthoschismic extends rank-3 schismic to the 11-limit in one of the most efficient ways possible.		Orthoschismic extends rank-3 schismic to the 11-limit in one of the most efficient ways possible.

	Orthoschismic can be notated with [[chain-of-fifths notation]] with two additional set of accidentals, one for the generic comma step (one should choose whether this step represents the syntonic comma, which is more characteristic in schismic, or the septimal comma, which is more characteristic in [[olympic]]), and the other for the generic aberschisma step which stands in for the [[~~garischisma~~]] and the [[aberschisma]].		Orthoschismic can be notated with [[chain-of-fifths notation]] with two additional set of accidentals, one for the generic comma step (one should choose whether this step represents the syntonic comma, which is more characteristic in schismic, or the septimal comma, which is more characteristic in [[olympic]]), and the other for the generic aberschisma step which stands in for the [[septischisma]] and the [[aberschisma]].

	It was named by [[Flora Canou]] in 2023, using the Greek prefix [[wiktionary: ortho- #English\|ortho-]] to signify the additional rank.		It was named by [[Flora Canou]] in 2023, using the Greek prefix [[wiktionary: ortho- #English\|ortho-]] to signify the additional rank.

FloraC: Text replacement - "Mapping: {{mapping| " to "{{Mapping|legend=0| "

2026-09-24T12:32:30Z

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	Comma list: 540/539, 729/728, 4096/4095		Comma list: 540/539, 729/728, 4096/4095

	~~Mapping:~~ {{~~mapping~~\| 1 0 15 0 17 -3 \| 0 1 -8 0 -5 6 \| 0 0 0 1 -2 -1 }}		{{Mapping\|legend=0\| 1 0 15 0 17 -3 \| 0 1 -8 0 -5 6 \| 0 0 0 1 -2 -1 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 10985/10976, 32805/32768, 161280/161051		Comma list: 10985/10976, 32805/32768, 161280/161051

	~~Mapping:~~ {{~~mapping~~\| 1 2 -1 3 3 5 \| 0 -3 24 -3 3 -11 \| 0 0 0 5 1 5 }}		{{Mapping\|legend=0\| 1 2 -1 3 3 5 \| 0 -3 24 -3 3 -11 \| 0 0 0 5 1 5 }}
	: mapping generators: ~2, ~208/189, ~33/32		: mapping generators: ~2, ~208/189, ~33/32

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2026-09-17T07:06:17Z

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 [[Category:Schismic rank-3 family| ]] <!-- main article -->
 [[Category:Temperament families]]
 [[Category:Catalogs of rank-3 temperaments]]

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			[[Category:Schismic rank-3 family\| ]] <!-- main article -->
			[[Category:Temperament families]]
			[[Category:Catalogs of rank-3 temperaments]]

FloraC: Birth

2026-09-17T06:48:43Z

Birth

New page

The '''schismic rank-3 family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the schisma, [[32805/32768]]. If nothing else is tempered out in the 7-limit, we have a [[rank-3 temperament]].

== Rank-3 schismic ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768

{{Mapping|legend=1| 1 0 15 0 | 0 1 -8 0 | 0 0 0 1 }}
: mapping generators: ~2, ~3, ~7

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0749{{c}}, ~3/2 = 701.7797{{c}}, ~7/4 = 968.6761{{c}}
: [[error map]]: {{val| +0.075 -0.100 -0.027 -0.000 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7308{{c}}, ~7/4 = 968.6783{{c}}
: error map: {{val| 0.000 -0.224 -0.160 -0.148 }}

{{Optimal ET sequence|legend=1| 12, 24, 29, 36, 41, 53, 94, 118, 130, 171, 814, 985, 1156, 1209, 1380, 1551, 1722, 2023, 2194, 2365, 2536 }}

[[Badness]] (Sintel): 0.610

== Orthoschismic ==
Orthoschismic extends rank-3 schismic to the 11-limit in one of the most efficient ways possible.

Orthoschismic can be notated with [[chain-of-fifths notation]] with two additional set of accidentals, one for the generic comma step (one should choose whether this step represents the syntonic comma, which is more characteristic in schismic, or the septimal comma, which is more characteristic in [[olympic]]), and the other for the generic aberschisma step which stands in for the [[garischisma]] and the [[aberschisma]].

It was named by [[Flora Canou]] in 2023, using the Greek prefix [[wiktionary: ortho- #English|ortho-]] to signify the additional rank.

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 540/539, 32805/32768

{{Mapping|legend=1| 1 0 15 0 17 | 0 1 -8 0 -5 | 0 0 0 1 -2 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9335{{c}}, ~3/2 = 701.6817{{c}}, ~7/4 = 969.6199{{c}}
: [[error map]]: {{val| -0.067 -0.340 -0.233 +0.661 +0.502 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7248{{c}}, ~7/4 = 969.6728{{c}}
: error map: {{val| 0.000 -0.230 -0.112 +0.847 +0.713 }}

{{Optimal ET sequence|legend=1| 41, 53, 89, 94, 130, 183, 224, 354, 537, 578, 761d, 985d, 1115de }}

[[Badness]] (Sintel): 1.41

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 4096/4095

Mapping: {{mapping| 1 0 15 0 17 -3 | 0 1 -8 0 -5 6 | 0 0 0 1 -2 -1 }}

Optimal tunings:
* WE: ~2 = 1199.9399{{c}}, ~3/2 = 701.6899{{c}}, ~7/4 = 969.6107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7264{{c}}, ~7/4 = 969.6672{{c}}

{{Optimal ET sequence|legend=0| 41, 53, 89, 94, 130, 183, 224, 354, 578, 985d }}

Badness (Sintel): 0.779