1547edo: Difference between revisions
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m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
1547edo is [[consistent]] to the [[15-odd-limit]] and is excellent in the 7-limit. | 1547edo is [[consistent]] to the [[15-odd-limit]] and is excellent in the 7-limit. As an equal temperament, it [[tempering out|tempers out]] [[4375/4374]] and it is a member of the [[optimal ET sequence]] for the rank-3 temperament associated with this comma. | ||
In the 5-limit, it supports [[gross]], which is a very high-accuracy temperament. The 118-tone [[maximal evenness]] scale produced by gross is [[concoctic]], since it uses 118\1547 as the generator. In addition, 1547edo tempers out the [[septendecima]] and thus supports the [[chlorine]] temperament in 5-limit and also in the 7-limit. 1547edo tempers out the 5-limit comma {{monzo| 236 -61 -60 }}, thus associating a stack of sixty [[15/8]]'s with [[4/3]], and sixty-one of them make [[5/4]]. | In the 5-limit, it supports [[gross]], which is a very high-accuracy temperament. The 118-tone [[maximal evenness]] scale produced by gross is [[concoctic]], since it uses 118\1547 as the generator. In addition, 1547edo tempers out the [[septendecima]] and thus supports the [[chlorine]] temperament in 5-limit and also in the 7-limit. 1547edo tempers out the 5-limit comma {{monzo| 236 -61 -60 }}, thus associating a stack of sixty [[15/8]]'s with [[4/3]], and sixty-one of them make [[5/4]]. | ||
In the 7-limit, it provides the [[optimal patent val]] for 7-limit [[brahmagupta]], the 441 & 1106 temperament, and supports an alternative 11-limit extension to it. It also supports [[semidimi]], the 171 & 1376 temperament. | In the 7-limit, it provides the [[optimal patent val]] for 7-limit [[brahmagupta]], the {{nowrap|441 & 1106}} temperament, and supports an alternative 11-limit extension to it. It also supports [[semidimi]], the {{nowrap|171 & 1376}} temperament. | ||
In the 11-limit, 1547edo provides the optimal patent val for the [[aluminium]] temperament, which maps 135/128 to 1/13th of the occtave. It also tempers out 117649/117612, and is a tuning for the rank-3 temperament [[heimdall]]. In higher limits, it supports 91th-octave temperament [[protactinium]]. | In the 11-limit, 1547edo provides the optimal patent val for the [[aluminium]] temperament, which maps 135/128 to 1/13th of the occtave. It also tempers out 117649/117612, and is a tuning for the rank-3 temperament [[heimdall]]. In higher limits, it supports 91th-octave temperament [[protactinium]]. | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 1547 factors into | Since 1547 factors into 7 × 13 × 17, 1547edo has subset edos {{EDOs| 7, 13, 17, 91, 119, and 221 }}. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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| {{monzo| 2452 -1547 }} | | {{monzo| 2452 -1547 }} | ||
| {{mapping| 1547 2542 }} | | {{mapping| 1547 2542 }} | ||
| | | −0.015 | ||
| 0.015 | | 0.015 | ||
| 1.99 | | 1.99 | ||
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| {{monzo| -52 -17 34 }}, {{monzo| 40 -56 21 }} | | {{monzo| -52 -17 34 }}, {{monzo| 40 -56 21 }} | ||
| {{mapping| 1547 2542 3592 }} | | {{mapping| 1547 2542 3592 }} | ||
| | | −0.008 | ||
| 0.017 | | 0.017 | ||
| 2.14 | | 2.14 | ||
| Line 46: | Line 46: | ||
| 4375/4374, {{monzo| -1 4 11 -11 }}, {{monzo| 46 -14 -3 -6 }} | | 4375/4374, {{monzo| -1 4 11 -11 }}, {{monzo| 46 -14 -3 -6 }} | ||
| {{mapping| 1547 2542 3592 4343 }} | | {{mapping| 1547 2542 3592 4343 }} | ||
| | | −0.007 | ||
| 0.014 | | 0.014 | ||
| 1.86 | | 1.86 | ||
| Line 53: | Line 53: | ||
| 4375/4374, 117649/117612, 234375/234256, 2097152/2096325 | | 4375/4374, 117649/117612, 234375/234256, 2097152/2096325 | ||
| {{mapping| 1547 2542 3592 4343 5352 }} | | {{mapping| 1547 2542 3592 4343 5352 }} | ||
| | | −0.017 | ||
| 0.024 | | 0.024 | ||
| 3.10 | | 3.10 | ||
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| 4096/4095, 4375/4374, 6656/6655, 78125/78078, 85750/85683 | | 4096/4095, 4375/4374, 6656/6655, 78125/78078, 85750/85683 | ||
| {{mapping| 1547 2542 3592 4343 5352 5725 }} | | {{mapping| 1547 2542 3592 4343 5352 5725 }} | ||
| | | −0.029 | ||
| 0.034 | | 0.034 | ||
| 4.42 | | 4.42 | ||
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| [[Protactinium]] | | [[Protactinium]] | ||
|} | |} | ||
<nowiki />* [[Normal | <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 | ||
== Music == | == Music == | ||