Trisected: Difference between revisions
→Tunings: + tuning spectrum |
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| Odd limit 2 = 13-limit 21 | Mistuning 2 = 17.5 | Complexity 2 = 36 | | Odd limit 2 = 13-limit 21 | Mistuning 2 = 17.5 | Complexity 2 = 36 | ||
}} | }} | ||
'''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 [[ | '''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{{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]]. | 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]]. | ||
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=== Target tunings === | === Target tunings === | ||
{{ | {| class="wikitable center-all mw-collapsible mw-collapsed" | ||
|+ style="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings | |||
|- | |||
! rowspan="2" | Target | |||
! colspan="2" | Minimax | |||
|- | |||
! Generator | |||
! Eigenmonzo* | |||
|- | |||
| 7-odd-limit | |||
| ~10/7 = 633.282{{C}} | |||
| 7/6 | |||
|- | |||
| 9-odd-limit | |||
| ~10/7 = 633.583{{C}} | |||
| 9/7 | |||
|- | |||
| 11-odd-limit | |||
| ~10/7 = 633.760{{C}} | |||
| 77/45 | |||
|- | |||
| 13-odd-limit | |||
| ~10/7 = 633.962{{C}} | |||
| 13/7 | |||
|- | |||
| 15-odd-limit | |||
| ~10/7 = 633.962{{C}} | |||
| 13/7 | |||
|- | |||
| 13-limit 21-odd-limit | |||
| ~10/7 = 634.129{{C}} | |||
| 45/44 | |||
|} | |||
=== Tuning spectrum === | === Tuning spectrum === | ||
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| [[7/6]] | | [[7/6]] | ||
| 633.282 | | 633.282 | ||
| | | 7-odd-limit minimax | ||
|- | |- | ||
| [[36edo|19\36]] | | [[36edo|19\36]] | ||
| Line 212: | Line 244: | ||
| [[9/7]] | | [[9/7]] | ||
| 633.583 | | 633.583 | ||
| | | 9-odd-limit minimax | ||
|- | |- | ||
| | | | ||
| Line 222: | Line 254: | ||
| [[13/7]] | | [[13/7]] | ||
| 633.962 | | 633.962 | ||
| | | 13- and 15-odd-limit minimax | ||
|- | |- | ||
| | | | ||