1547edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|1547}}
{{EDO intro|1547}}
==Theory==
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|4/3]], and 61 of them make a [[5/4]].
== Theory ==
1547edo is [[consistent]] to the [[15-odd-limit]] and 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 sixty [[15/8]]'s with [[4/3]], and sixty-one of them make [[5/4]].


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 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.  
Line 11: Line 12:


In higher limits, it supports 91th-octave temperament [[protactinium]].
In higher limits, it supports 91th-octave temperament [[protactinium]].
=== Prime harmonics ===
{{Harmonics in equal|1547}}
=== Divisors ===
=== Divisors ===
1547's divisors are {{EDOs|1, 7, 13, 17, 91, 119, 221}}.
1547 has subset edos {{EDOs| 7, 13, 17, 91, 119, and 221 }}.  
=== Prime harmonics ===
{{harmonics in equal|1547}}


==Regular temperament properties==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal 8ve  
! rowspan="2" | Optimal 8ve<br>Stretch (¢)
Stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" |Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3
|{{monzo|2452 -1547}}
| {{monzo| 2452 -1547 }}
|[{{val|1547 2542}}]
| [{{val| 1547 2542 }}]
|<nowiki>-0.015</nowiki>
| -0.015
|0.015
| 0.015
|1.99
| 1.99
|-
|-
|2.3.5
| 2.3.5
|{{monzo|-52 -17 34}}, {{monzo|40 -56 21}}
| {{monzo| -52 -17 34 }}, {{monzo| 40 -56 21 }}
|[{{val|1547 2542 3592}}]
| [{{val| 1547 2542 3592 }}]
|<nowiki>-0.008</nowiki>
| -0.008
|0.017
| 0.017
|2.14
| 2.14
|-
|-
|2.3.5.7
| 2.3.5.7
|4375/4374, {{monzo|-1 4 11 -11}}, {{monzo|46 -14 -3  -6}}
| 4375/4374, {{monzo| -1 4 11 -11 }}, {{monzo| 46 -14 -3  -6 }}
|[{{val|1547 2542 3592 4343}}]
| [{{val| 1547 2542 3592 4343 }}]
| -0.007
| -0.007
|0.014
| 0.014
|1.86
| 1.86
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|4375/4374, 151263/151250, 820125/819896, 2097152/2096325
| 4375/4374, 151263/151250, 820125/819896, 2097152/2096325
|[{{val|1547 2542 3592 4343 5352}}]
| [{{val| 1547 2542 3592 4343 5352 }}]
|<nowiki>-0.017</nowiki>
| -0.017
|0.024
| 0.024
|3.10
| 3.10
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|4375/4374, 4096/4095, 10648/10647, 91125/91091, 105644/105625
| 4375/4374, 4096/4095, 10648/10647, 91125/91091, 105644/105625
|[{{val|1547 2542 3592 4343 5352 5725}}]
| [{{val| 1547 2542 3592 4343 5352 5725 }}]
|<nowiki>-0.029</nowiki>
| -0.029
|0.034
| 0.034
|4.42
| 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
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator<br>(Reduced)
! Cents<br>(reduced)
! Cents<br>(Reduced)
! Associated<br>ratio
! Associated<br>Ratio
! Temperaments
! Temperaments
|-
|-
|1
| 1
|118\1547
| 118\1547
|91.532
| 91.532
|{{monzo|9 -32 18}}
| {{monzo| 9 -32 18 }}
|[[Gross]]
| [[Gross]]
|-
|-
|1
| 1
|579\1547
| 579\1547
|449.127
| 449.127
|35/27
| 35/27
|[[Semidimi]]
| [[Semidimi]]
|-
|-
|7
| 7
|670\1547<br>(7\1547)
| 670\1547<br>(7\1547)
|519.715<br>(5.429)
| 519.715<br>(5.429)
|27/20<br>(325/324)
| 27/20<br>(325/324)
|[[Brahmagupta]]
| [[Brahmagupta]]
|-
|-
|13
| 13
|642\1547<br>(47\1547)
| 642\1547<br>(47\1547)
|497.996<br>(36.458)
| 497.996<br>(36.458)
|4/3<br>(?)
| 4/3<br>(?)
|[[Aluminium]]
| [[Aluminium]]
|-
|-
|17
| 17
|321\1547<br>(48\1547)
| 321\1547<br>(48\1547)
|248.998<br>(37.233)
| 248.998<br>(37.233)
|{{monzo|-23 5 9 -2}}<br>(100352/98415)
| {{monzo| -23 5 9 -2 }}<br>(100352/98415)
|[[Chlorine]]
| [[Chlorine]]
|-
|-
|91
| 91
|642\1547<br>(13\1547)
| 642\1547<br>(13\1547)
|497.996<br>(10.084)
| 497.996<br>(10.084)
|4/3<br>(176/175)
| 4/3<br>(176/175)
|[[Protactinium]]
| [[Protactinium]]
|}
|}
[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->

Revision as of 03:23, 8 January 2023

← 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

Template:EDO intro

Theory

1547edo is consistent to the 15-odd-limit and 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 [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 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 11-limit, it 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)

Divisors

1547 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, 151263/151250, 820125/819896, 2097152/2096325 [⟨1547 2542 3592 4343 5352]] -0.017 0.024 3.10
2.3.5.7.11.13 4375/4374, 4096/4095, 10648/10647, 91125/91091, 105644/105625 [⟨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
(Reduced) 		Cents
(Reduced) 		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
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
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