1547edo: Difference between revisions

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== Theory ==

1547edo is [[consistent]] to the [[15-odd-limit]] and is excellent in the 7-limit. 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.

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 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]].

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=== Subsets and supersets ===

Since 1547 factors into ~~{{factorization|1547}}~~, 1547edo has subset edos {{EDOs| 7, 13, 17, 91, 119, and 221 }}.

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

== Regular temperament properties ==

{~~{comma basis begin}}~~

{| 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

| 0.015

| 1.99

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| {{monzo| -52 -17 34 }}, {{monzo| 40 -56 21 }}

| {{mapping| 1547 2542 3592 }}

| ~~−0~~.008

| −0.008

| 0.017

| 2.14

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| 4375/4374, {{monzo| -1 4 11 -11 }}, {{monzo| 46 -14 -3 -6 }}

| {{mapping| 1547 2542 3592 4343 }}

| ~~−0~~.007

| −0.007

| 0.014

| 1.86

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| 4375/4374, 117649/117612, 234375/234256, 2097152/2096325

| {{mapping| 1547 2542 3592 4343 5352 }}

| ~~−0~~.017

| −0.017

| 0.024

| 3.10

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| 4096/4095, 4375/4374, 6656/6655, 78125/78078, 85750/85683

| {{mapping| 1547 2542 3592 4343 5352 5725 }}

| ~~−0~~.029

| −0.029

| 0.034

| 4.42

~~{{comma basis end}~~}

|}

=== Rank-2 temperaments ===

{{rank-2 ~~begin}}~~

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods<br />per 8ve

! Generator*

! Cents*

! Associated<br />ratio*

! Temperaments

|-

| 1

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| 4/3<br />(176/175)

| [[Protactinium]]

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

<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 ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|1547}}
+{{ED intro}}
 == Theory ==
-edo is [[consistent]] to the [[15-odd-limit]] and is excellent in the 7-limit. 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.
+edo 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 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 &amp; 1106}} temperament, and supports an alternative 11-limit extension to it. It also supports [[semidimi]], the {{nowrap|171 &amp; 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]].
@@ Line 15: / Line 15: @@
 === Subsets and supersets ===
-Since 1547 factors into {{factorization|1547}}, 1547edo has subset edos {{EDOs| 7, 13, 17, 91, 119, and 221 }}.
+Since 1547 factors into 7 × 13 × 17, 1547edo has subset edos {{EDOs| 7, 13, 17, 91, 119, and 221 }}.
 == Regular temperament properties ==
-{{comma basis begin}}
+{| 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 }}
-| &minus;0.015
+| −0.015
 | 0.015
 | 1.99
@@ Line 30: / Line 39: @@
 | {{monzo| -52 -17 34 }}, {{monzo| 40 -56 21 }}
 | {{mapping| 1547 2542 3592 }}
-| &minus;0.008
+| −0.008
 | 0.017
 | 2.14
@@ Line 37: / Line 46: @@
 | 4375/4374, {{monzo| -1 4 11 -11 }}, {{monzo| 46 -14 -3  -6 }}
 | {{mapping| 1547 2542 3592 4343 }}
-| &minus;0.007
+| −0.007
 | 0.014
 | 1.86
@@ Line 44: / Line 53: @@
 | 4375/4374, 117649/117612, 234375/234256, 2097152/2096325
 | {{mapping| 1547 2542 3592 4343 5352 }}
-| &minus;0.017
+| −0.017
 | 0.024
 | 3.10
@@ Line 51: / Line 60: @@
 | 4096/4095, 4375/4374, 6656/6655, 78125/78078, 85750/85683
 | {{mapping| 1547 2542 3592 4343 5352 5725 }}
-| &minus;0.029
+| −0.029
 | 0.034
 | 4.42
-{{comma basis end}}
+|}
 === Rank-2 temperaments ===
-{{rank-2 begin}}
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
 |-
 | 1
@@ Line 100: / Line 116: @@
 | 4/3<br />(176/175)
 | [[Protactinium]]
-{{rank-2 end}}
+|}
-{{orf}}
+<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 ==