1547edo: Difference between revisions

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{{EDO intro|1547}}
{{Infobox ET}}
==Theory==
{{ED intro}}
{{harmonics in equal|1547}}
1547edo is excellent in the 7-limit.


In the 5-limit, it supports [[gross]].  
== 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 17-limit, it supports 91th-octave temperament protactinium.
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]].
==Regular temperament properties==
 
===Rank-2 temperaments by generator===
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]].
 
=== Prime harmonics ===
{{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"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 2452 -1547 }}
| {{mapping| 1547 2542 }}
| −0.015
| 0.015
| 1.99
|-
| 2.3.5
| {{monzo| -52 -17 34 }}, {{monzo| 40 -56 21 }}
| {{mapping| 1547 2542 3592 }}
| −0.008
| 0.017
| 2.14
|-
| 2.3.5.7
| 4375/4374, {{monzo| -1 4 11 -11 }}, {{monzo| 46 -14 -3  -6 }}
| {{mapping| 1547 2542 3592 4343 }}
| −0.007
| 0.014
| 1.86
|-
| 2.3.5.7.11
| 4375/4374, 117649/117612, 234375/234256, 2097152/2096325
| {{mapping| 1547 2542 3592 4343 5352 }}
| −0.017
| 0.024
| 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 ===
{| 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 octave
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
| 1
| 118\1547
| 91.532
| {{monzo| 9 -32 18 }}
| [[Gross]]
|-
| 1
| 579\1547
| 449.127
| 35/27
| [[Semidimi]]
|-
| 7
| 670\1547<br />(7\1547)
| 519.715<br />(5.429)
| 27/20<br />(325/324)
| [[Brahmagupta]] (7-limit)
|-
| 7
| 11\1547
| 8.533
| 1029/1024
| [[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]]
|}
|}
<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]]

Latest revision as of 23:08, 20 February 2025

← 1546edo 1547edo 1548edo →
Prime factorization 7 × 13 × 17
Step size 0.775695 ¢ 
Fifth 905\1547 (702.004 ¢)
Semitones (A1:m2) 147:116 (114 ¢ : 89.98 ¢)
Consistency limit 15
Distinct consistency limit 15

1547 equal divisions of the octave (abbreviated 1547edo or 1547ed2), also called 1547-tone equal temperament (1547tet) or 1547 equal temperament (1547et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1547 equal parts of about 0.776 ¢ each. Each step represents a frequency ratio of 21/1547, or the 1547th root of 2.

Theory

1547edo is consistent to the 15-odd-limit and is excellent in the 7-limit. As an equal temperament, it 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 [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 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.

Prime harmonics

Approximation of prime harmonics in 1547edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.049 -0.018 +0.017 +0.201 +0.326 -0.237 +0.354 +0.039 -0.230 -0.110
Relative (%) +0.0 +6.3 -2.3 +2.2 +25.9 +42.0 -30.5 +45.6 +5.0 -29.7 -14.2
Steps
(reduced) 		1547
(0) 		2452
(905) 		3592
(498) 		4343
(1249) 		5352
(711) 		5725
(1084) 		6323
(135) 		6572
(384) 		6998
(810) 		7515
(1327) 		7664
(1476)

Subsets and supersets

Since 1547 factors into 7 × 13 × 17, 1547edo has subset edos 7, 13, 17, 91, 119, and 221.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [2452 -1547⟩ [⟨1547 2542]] −0.015 0.015 1.99
2.3.5 [-52 -17 34⟩, [40 -56 21⟩ [⟨1547 2542 3592]] −0.008 0.017 2.14
2.3.5.7 4375/4374, [-1 4 11 -11⟩, [46 -14 -3  -6⟩ [⟨1547 2542 3592 4343]] −0.007 0.014 1.86
2.3.5.7.11 4375/4374, 117649/117612, 234375/234256, 2097152/2096325 [⟨1547 2542 3592 4343 5352]] −0.017 0.024 3.10
2.3.5.7.11.13 4096/4095, 4375/4374, 6656/6655, 78125/78078, 85750/85683 [⟨1547 2542 3592 4343 5352 5725]] −0.029 0.034 4.42

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 118\1547 91.532 [9 -32 18⟩ Gross
1 579\1547 449.127 35/27 Semidimi
7 670\1547
(7\1547) 		519.715
(5.429) 		27/20
(325/324) 		Brahmagupta (7-limit)
7 11\1547 8.533 1029/1024 Nitrogen
13 642\1547
(47\1547) 		497.996
(36.458) 		4/3
(?) 		Aluminium
17 321\1547
(48\1547) 		248.998
(37.233) 		[-23 5 9 -2⟩
(100352/98415) 		Chlorine
91 642\1547
(13\1547) 		497.996
(10.084) 		4/3
(176/175) 		Protactinium

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Eliora
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