Trisected: Difference between revisions

Trisected
Subgroups	2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
Comma basis	128/125, 1029/1000 (7-limit);; 56/55, 128/125, 1029/1000 (11-limit);; 56/55, 91/90, 128/125, 1029/1000 (13-limit)
Reduced mapping	⟨3; 3 0 -1 -1 7]
ET join	15 & 36
Generators (CWE)	~10/7 = 635.0 ¢
MOS scales	6L 9s, 15L 6s, 15L 21s
Ploidacot	triploid tricot
Pergen	(P8/3, P5/3)
Minimax error	9-odd-limit: 16.1 ¢; ; 13-limit 21-odd-limit: 17.5 ¢
Target scale size	9-odd-limit: 36 notes; ; 13-limit 21-odd-limit: 36 notes

Revision as of 19:41, 19 March 2026

Trisected is the rank-2 temperament tempering out 128/125, 1029/1000, and 1029/1024 in the 7-limit, making it a member of the augmented family, keegic temperaments, and gamelismic clan. Since it tempers out 128/125, the octave is split into 3 ~5/4's, each tuned to 400 ¢ if the octave is pure. Since it tempers out 1029/1024, the perfect fifth is split into three intervals of ~8/7.Since it tempers out 1029/1000, the tritave is split into three intervals of 10/7. This means that every Pythagorean interval is split into three equal parts.

In the 11-limit, the perfect fourth is split into three ~11/10's, thus tempering out 4000/3993. Additionally, the 1/3-octave period represents 14/11, tempering out 56/55 and 176/175. The 13-limit extension equates the ~10/7 with 13/9, tempering out 91/90 and 2197/2187.

Todo: complete section

Todo: complete section

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 | Odd limit 2 = 13-limit 21 | Mistuning 2 = 17.5 | Complexity 2 = 36
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-'''Trisected''' is the [[rank-2 temperament]] tempering out [[128/125]], [[1029/1000]], and [[1029/1024]] in the [[7-limit]], making it a member of the [[augmented family]], [[keegic temperaments]], and [[gamelismic clan]].
+'''Trisected''' is the [[rank-2 temperament]] tempering out [[128/125]], [[1029/1000]], and [[1029/1024]] in the [[7-limit]], making it a member of the [[augmented family]], [[keegic temperaments]], and [[gamelismic clan]]. Since it tempers out 128/125, the [[2/1|octave]] is split into 3 ~[[5/4]]'s, each tuned to 400{{C}} if the octave is pure. Since it tempers out 1029/1024, the [[3/2|perfect fifth]] is split into three intervals of ~[[8/7]].Since it tempers out [[1029/1000]], the [[3/1|tritave]] is split into three intervals of [[10/7]]. This means that every [[Pythagorean tuning|Pythagorean]] interval is split into three equal parts.
+In the [[11-limit]], the [[4/3|perfect fourth]] is split into three ~[[11/10]]'s, thus tempering out [[4000/3993]]. Additionally, the 1/3-octave period represents [[14/11]], tempering out [[56/55]] and [[176/175]]. The [[13-limit]] extension equates the ~10/7 with [[13/9]], tempering out [[91/90]] and [[2197/2187]].
 For technical data, see [[Augmented family #Trisected]].