270edo: Difference between revisions
m →Rank-2 temperaments: (''See regular temperament for more about what all this means and how to use it.'') Tag: Reverted |
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== Theory == | == Theory == | ||
270edo is an extremely strong [[13-limit]] system, [[distinctly consistent]] through the [[15-odd-limit]] with all intervals in the 15-odd-limit being approximated with less than 25% relative error with only the exception of [[15/13]] which barely misses (corresponding to the fact of tempering out [[676/675]]). This results in it being a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limit. It is [[The Riemann zeta function and tuning #Zeta EDO lists|the 11th zeta gap edo, the 13th zeta integral edo, the 23rd zeta peak edo, and the 18th zeta peak integer edo]], making it a strict zeta edo, and is the first [[Trivial temperament|non-trivial]] edo to be consistent in the 16-[[Odd prime sum limit|odd-prime-sum-limit]]. | 270edo is an extremely strong [[13-limit]] system, [[distinctly consistent]] through the [[15-odd-limit]] with all intervals in the 15-odd-limit being approximated with less than 25% relative error with only the exception of [[15/13]] which barely misses (corresponding to the fact of tempering out [[676/675]]). This results in it being a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limit. It is [[The Riemann zeta function and tuning #Zeta EDO lists|the 11th zeta gap edo, the 13th zeta integral edo, the 23rd zeta peak edo, and the 18th zeta peak integer edo]], making it a strict zeta edo, and is the first [[Trivial temperament|non-trivial]] edo to be consistent in the 16-[[Odd prime sum limit|odd-prime-sum-limit]]. | ||
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In the [[11-limit]], it tempers out [[3025/3024]], [[5632/5625]], and [[9801/9800]], meaning it tempers out the four smallest [[superparticular]] commas in the 11-limit (2401/2400, 3025/3024, 4375/4374, and 9801/9800). In addition to these, it also tempers out both the [[nexus comma]] (1771561/1769472) and the [[quartisma]] (117440512/117406179), which, in turn means that the [[symbiotic comma]] (19712/19683) is tempered out as well. | In the [[11-limit]], it tempers out [[3025/3024]], [[5632/5625]], and [[9801/9800]], meaning it tempers out the four smallest [[superparticular]] commas in the 11-limit (2401/2400, 3025/3024, 4375/4374, and 9801/9800). In addition to these, it also tempers out both the [[nexus comma]] (1771561/1769472) and the [[quartisma]] (117440512/117406179), which, in turn means that the [[symbiotic comma]] (19712/19683) is tempered out as well. | ||
Finally, in the [[13-limit]] it is not quite as accurate but still very accurate, as it tempers out [[676/675]], [[1001/1000]], [[1716/1715]], and [[2080/2079]], making it an [[The Archipelago|archipelago]] tuning, and the [[optimal patent val]] for some of the archipelago temperaments such as [[hemiennealimmal]], [[vulture]], [[eagle]], and [[avicenna (temperament)|avicenna]]. | Finally, in the [[13-limit]] it is not quite as accurate but still very accurate, as it tempers out [[676/675]], [[1001/1000]], [[1716/1715]], and [[2080/2079]], making it an [[The Archipelago|archipelago]] tuning, and the [[optimal patent val]] for some of the archipelago temperaments such as [[hemiennealimmal]], [[vulture]], [[eagle]], and [[avicenna (temperament)|avicenna]]. | ||
The excellent tuning accuracy does not bar it from the utility of [[essentially tempered chord]]s, including [[sinbadmic chords]] in the 13-odd-limit, and [[island chords]] in the 15-odd-limit. | The excellent tuning accuracy does not bar it from the utility of [[essentially tempered chord]]s, including [[sinbadmic chords]] in the 13-odd-limit, and [[island chords]] in the 15-odd-limit. | ||
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=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||