Xenharmonic Wiki

1547edo: Difference between revisions

← Older edit
VisualWikitext
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,351 edits
→Regular temperament properties: completed rtp table
ArrowHead294 (talk | contribs)
14,512 edits
mNo edit summary
 
(23 intermediate revisions by 5 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|1547}}
{{ED intro}}
==Theory==
{{harmonics in equal|1547}}
1547edo is excellent in the 7-limit. It tempers out [[4375/4374]] and it is a member of the [[optimal GPV 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 60 [[15/8]]<nowiki/>s with [[8/5]], and 61 of them make a [[5/4]].  
== Theory ==
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 7-limit, it supports [[semidimi]]. Another edo which is quite strong in the 7-limit is like 1547edo is 441edo, and 1547edo thus supports the [[brahmagupta]] temperament produced by merging 441 & 1547.  
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 11-limit, it is a tuning for the rank-3 temperament [[heimdall]].  
In the 7-limit, it provides the [[optimal patent val]] for 7-limit [[brahmagupta]], the {{nowrap|441 &amp; 1106}} temperament, and supports an alternative 11-limit extension to it. It also supports [[semidimi]], the {{nowrap|171 &amp; 1376}} temperament.  


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]].


1547's divisors are {{EDOs|1, 7, 13, 17, 91, 119, 221}}.
=== Prime harmonics ===
==Regular temperament properties==
{{Harmonics in equal|1547}}
 
=== Subsets and supersets ===
Since 1547 factors into 7 × 13 × 17, 1547edo has subset edos {{EDOs| 7, 13, 17, 91, 119, and 221 }}.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal 8ve
Stretch (¢)
! colspan="2" |Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! rowspan="2" | [[Subgroup]]
![[TE simple badness|Relative]] (%)
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|2.3
! [[TE error|Absolute]] (¢)
|{{monzo|2452 -1547}}
! [[TE simple badness|Relative]] (%)
|[{{val|1547 2542}}]
|<nowiki>-0.015</nowiki>
|0.015
|1.99
|-
|-
|2.3.5
| 2.3
|{{monzo|-52 -17 34}}, {{monzo|40 -56 21}}
| {{monzo| 2452 -1547 }}
|[{{val|1547 2542 3592}}]
| {{mapping| 1547 2542 }}
|<nowiki>-0.008</nowiki>
| −0.015
|0.017
| 0.015
|2.14
| 1.99
|-
|-
|2.3.5.7
| 2.3.5
|4375/4374, {{monzo|-1 4 11 -11}}, {{monzo|46 -14 -3  -6}}
| {{monzo| -52 -17 34 }}, {{monzo| 40 -56 21 }}
|[{{val|1547 2542 3592 4343}}]
| {{mapping| 1547 2542 3592 }}
| -0.007
| −0.008
|0.014
| 0.017
|1.86
| 2.14
|-
|-
|2.3.5.7.11
| 2.3.5.7
|4375/4374, 151263/151250, 820125/819896, 2097152/2096325
| 4375/4374, {{monzo| -1 4 11 -11 }}, {{monzo| 46 -14 -3  -6 }}
|[{{val|1547 2542 3592 4343 5352}}]
| {{mapping| 1547 2542 3592 4343 }}
|<nowiki>-0.017</nowiki>
| −0.007
|0.024
| 0.014
|3.10
| 1.86
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11
|4375/4374, 4096/4095, 10648/10647, 91125/91091, 105644/105625
| 4375/4374, 117649/117612, 234375/234256, 2097152/2096325
|[{{val|1547 2542 3592 4343 5352 5725}}]
| {{mapping| 1547 2542 3592 4343 5352 }}
|<nowiki>-0.029</nowiki>
| −0.017
|0.034
| 0.024
|4.42
| 3.10
|-
| 2.3.5.7.11.13
| 4096/4095, 4375/4374, 6656/6655, 78125/78078, 85750/85683
| {{mapping| 1547 2542 3592 4343 5352 5725 }}
| −0.029
| 0.034
| 4.42
|}
|}
===Rank-2 temperaments===
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|118\1547
| 118\1547
|91.532
| 91.532
|[9 -32 18>
| {{monzo| 9 -32 18 }}
|[[Gross]]
| [[Gross]]
|-
| 1
| 579\1547
| 449.127
| 35/27
| [[Semidimi]]
|-
|-
|1
| 7
|579\1547
| 670\1547<br />(7\1547)
|449.127
| 519.715<br />(5.429)
|35/27
| 27/20<br />(325/324)
|[[Semidimi]]
| [[Brahmagupta]] (7-limit)
|-
|-
|7
| 7
|670\1547<br>(7\1547)
| 11\1547
|519.715<br>(5.429)
| 8.533
|27/20<br>(325/324)
| 1029/1024
|[[Brahmagupta]]
| [[Nitrogen]]
|-
| 13
| 642\1547<br />(47\1547)
| 497.996<br />(36.458)
| 4/3<br />(?)
| [[Aluminium]]
|-
| 17
| 321\1547<br />(48\1547)
| 248.998<br />(37.233)
| {{monzo| -23 5 9 -2 }}<br />(100352/98415)
| [[Chlorine]]
|-
|-
| 91
| 91
| 905\1547<br>(4\1547)
| 642\1547<br />(13\1547)
| 702.003<br>(3.103)
| 497.996<br />(10.084)
| 3/2<br>(?)
| 4/3<br />(176/175)
| [[Protactinium]]
| [[Protactinium]]
|}
|}
[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=oac5JZ9FtB8 ''Prelude and Fugue in Two Elements'']
 
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/1547edo"