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 ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal ~~8ve~~ ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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| {{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

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=== Rank-2 temperaments ===

{| 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 per 8ve

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

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|-

| 7

| 670\1547 (7\1547)

| 670\1547 (7\1547)

| 519.715 (5.429)

| 519.715 (5.429)

| 27/20 (325/324)

| 27/20 (325/324)

| [[Brahmagupta]] (7-limit)

|-

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|-

| 13

| 642\1547 (47\1547)

| 642\1547 (47\1547)

| 497.996 (36.458)

| 497.996 (36.458)

| 4/3 (?)

| 4/3 (?)

| [[Aluminium]]

|-

| 17

| 321\1547 (48\1547)

| 321\1547 (48\1547)

| 248.998 (37.233)

| 248.998 (37.233)

| {{monzo| -23 5 9 -2 }} (100352/98415)

| {{monzo| -23 5 9 -2 }} (100352/98415)

| [[Chlorine]]

|-

| 91

| 642\1547 (13\1547)

| 642\1547 (13\1547)

| 497.996 (10.084)

| 497.996 (10.084)

| 4/3 (176/175)

| 4/3 (176/175)

| [[Protactinium]]

|}

<nowiki>*~~</nowiki>~~ [[Normal lists|~~octave~~-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if ~~it is~~ distinct

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|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 ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal 8ve<br>Stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 31: / Line 32: @@
 | {{monzo| 2452 -1547 }}
 | {{mapping| 1547 2542 }}
-| -0.015
+| −0.015
 | 0.015
 | 1.99
@@ Line 38: / Line 39: @@
 | {{monzo| -52 -17 34 }}, {{monzo| 40 -56 21 }}
 | {{mapping| 1547 2542 3592 }}
-| -0.008
+| −0.008
 | 0.017
 | 2.14
@@ Line 45: / Line 46: @@
 | 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
@@ Line 52: / Line 53: @@
 | 4375/4374, 117649/117612, 234375/234256, 2097152/2096325
 | {{mapping| 1547 2542 3592 4343 5352 }}
-| -0.017
+| −0.017
 | 0.024
 | 3.10
@@ Line 59: / Line 60: @@
 | 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
@@ Line 66: / Line 67: @@
 === Rank-2 temperaments ===
 {| 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
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 87: / Line 88: @@
 |-
 | 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]] (7-limit)
 |-
@@ Line 99: / Line 100: @@
 |-
 | 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]]
 |-
 | 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]]
 |-
 | 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]]
 |}
-<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Music ==