897edo: Difference between revisions

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== Theory ==

897edo ~~has sharp approximations~~ to ~~harmonics~~ [[3/1|3]] ~~and~~ [[5/1|5]] and flat ones starting with [[7/1|7]]. In the 13-limit at least, we want to consider the [[patent val]] {{val| 897 1422 2083 '''2518''' '''3103''' '''3319''' }} as well as the 897def [[val]] {{val| 897 1422 2083 '''2519''' '''3104''' '''3320''' }}.

897edo shares the same approximation to [[3/1|harmonic 3]] with [[299edo]]. It has sharp approximations to [[5/1|5]] and flat ones starting with [[7/1|7]]. In the 13-limit at least, we want to consider the [[patent val]] {{val| 897 1422 2083 '''2518''' '''3103''' '''3319''' }} as well as the 897def [[val]] {{val| 897 1422 2083 '''2519''' '''3104''' '''3320''' }}.

The patent val tempers out [[250047/250000]] in the 7-limit; [[3025/3024]], [[5632/5625]], 102487/102400, [[131072/130977]], 160083/160000, and 172032/171875 in the 11-limit; [[676/675]], [[1001/1000]], [[2080/2079]], [[4096/4095]], and [[4225/4224]] in the 13-limit.

The patent val tempers out [[250047/250000]] in the 7-limit; [[3025/3024]], [[5632/5625]], 42592/42525, 102487/102400, [[131072/130977]], 160083/160000, and 172032/171875 in the 11-limit; [[676/675]], [[1001/1000]], [[2080/2079]], [[4096/4095]], and [[4225/4224]] in the 13-limit.

The 897def val tempers out [[4375/4374]] in the 7-limit; [[4000/3993]], 46656/46585, and 172032/171875 in the 11-limit; [[1716/1715]], and [[10648/10647]] in the 13-limit.

=== ~~Prime~~ harmonics ===

=== Odd harmonics ===

=== Subsets and supersets ===

Since 897 factors into {{factorization|897}}, 897edo has subset edos {{EDOs| 3, 13, 23, 39, 69, and 299 }}.

== 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]] (¢)

! [[TE simple badness|Relative]] (%)

|-

~~| 2.3~~

~~| {{monzo| 474 -299 }}~~

~~| {{mapping| 897 1422 }}~~

~~| -0.1218~~

~~| 0.1218~~

~~| 9.10~~

|-

| 2.3.5

| {{monzo| 33 -34 9 }}, {{monzo| 70 -9 -24 }}

| {{mapping| 897 1422 2083 }}

| -0.1255

| −0.1255

| 0.~~010~~

| 0.0996

| 0.75

| 7.44

|-

| 2.3.5.7

| 250047/250000, 67108864/66976875, 28824005/28697814

| {{mapping| 897 1422 2083 2518 }} (897)

| -0.0706

| −0.0706

| 0.1283

| 9.59

|-

| 2.3.5.7.11

| 3025/3024, 5632/5625, ~~160083~~/~~160000~~, 28824005/28697814

| 3025/3024, 5632/5625, 102487/102400, 28824005/28697814

| {{mapping| 897 1422 2083 2518 3103 }} (897)

| -0.0480

| −0.0480

| 0.1234

| 9.22

|-

| 2.3.5.7.11.13

| ~~2080~~/~~2079~~, 1001/1000, 3025/3024, 4096/4095, ~~2058857500~~/~~2051893701~~

| 676/675, 1001/1000, 3025/3024, 4096/4095, 28824005/28697814

| {{mapping| 897 1422 2083 2518 3103 3319 }} (897)

| -0.0222

| −0.0222

| 0.1265

| 9.46

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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

|-

| 1

| 419\897

| 560.~~535~~

| 560.54

| 864/625

| [[Whoosh]]

|-

| 1

| 440\897

| 588.63

| 128/91

| [[Countritonic]] (897df)

|-

| 3

| 103\897

| 137.79

| 13/12

| [[Avicenna (temperament)|Avicenna]] (897)

|}

<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 ==

; [[User:Francium|Francium]]

* [https://www.youtube.com/watch?v=opR71xbJ-PQ ''13|3|23''] – 3-, 13- and 23edo polymicrotonal

* [https://www.youtube.com/watch?v=opR71xbJ-PQ ''13|3|23''] (2023) – 3-, 13-, and 23edo polymicrotonal

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|897}}
+{{ED intro}}
 == Theory ==
-edo has sharp approximations to harmonics [[3/1|3]] and [[5/1|5]] and flat ones starting with [[7/1|7]]. In the 13-limit at least, we want to consider the [[patent val]] {{val| 897 1422 2083 '''2518''' '''3103''' '''3319''' }} as well as the 897def [[val]] {{val| 897 1422 2083 '''2519''' '''3104''' '''3320''' }}.
+edo shares the same approximation to [[3/1|harmonic 3]] with [[299edo]]. It has sharp approximations to [[5/1|5]] and flat ones starting with [[7/1|7]]. In the 13-limit at least, we want to consider the [[patent val]] {{val| 897 1422 2083 '''2518''' '''3103''' '''3319''' }} as well as the 897def [[val]] {{val| 897 1422 2083 '''2519''' '''3104''' '''3320''' }}.
-The patent val tempers out [[250047/250000]] in the 7-limit; [[3025/3024]], [[5632/5625]], 102487/102400, [[131072/130977]], 160083/160000, and 172032/171875 in the 11-limit; [[676/675]], [[1001/1000]], [[2080/2079]], [[4096/4095]], and [[4225/4224]] in the 13-limit.
+The patent val tempers out [[250047/250000]] in the 7-limit; [[3025/3024]], [[5632/5625]], 42592/42525, 102487/102400, [[131072/130977]], 160083/160000, and 172032/171875 in the 11-limit; [[676/675]], [[1001/1000]], [[2080/2079]], [[4096/4095]], and [[4225/4224]] in the 13-limit.
 The 897def val tempers out [[4375/4374]] in the 7-limit; [[4000/3993]], 46656/46585, and 172032/171875 in the 11-limit; [[1716/1715]], and [[10648/10647]] in the 13-limit.
-=== Prime harmonics ===
+=== Odd harmonics ===
 {{Harmonics in equal|897}}
+=== Subsets and supersets ===
+Since 897 factors into {{factorization|897}}, 897edo has subset edos {{EDOs| 3, 13, 23, 39, 69, and 299 }}.
 == 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<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
 ! [[TE simple badness|Relative]] (%)
-|-
-| 2.3
-| {{monzo| 474 -299 }}
-| {{mapping| 897 1422 }}
-| -0.1218
-| 0.1218
-| 9.10
 |-
 | 2.3.5
 | {{monzo| 33 -34 9 }}, {{monzo| 70 -9 -24 }}
 | {{mapping| 897 1422 2083 }}
-| -0.1255
+| −0.1255
-| 0.010
+| 0.0996
-| 0.75
+| 7.44
 |-
 | 2.3.5.7
 | 250047/250000, 67108864/66976875, 28824005/28697814
 | {{mapping| 897 1422 2083 2518 }} (897)
-| -0.0706
+| −0.0706
 | 0.1283
 | 9.59
 |-
 | 2.3.5.7.11
-| 3025/3024, 5632/5625, 160083/160000, 28824005/28697814
+| 3025/3024, 5632/5625, 102487/102400, 28824005/28697814
 | {{mapping| 897 1422 2083 2518 3103 }} (897)
-| -0.0480
+| −0.0480
 | 0.1234
 | 9.22
 |-
 | 2.3.5.7.11.13
-| 2080/2079, 1001/1000, 3025/3024, 4096/4095, 2058857500/2051893701
+| 676/675, 1001/1000, 3025/3024, 4096/4095, 28824005/28697814
 | {{mapping| 897 1422 2083 2518 3103 3319 }} (897)
-| -0.0222
+| −0.0222
 | 0.1265
 | 9.46
@@ Line 61: / Line 58: @@
 === 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
 |-
 | 1
 | 419\897
-| 560.535
+| 560.54
 | 864/625
 | [[Whoosh]]
+|-
+| 1
+| 440\897
+| 588.63
+| 128/91
+| [[Countritonic]] (897df)
+|-
+| 3
+| 103\897
+| 137.79
+| 13/12
+| [[Avicenna (temperament)|Avicenna]] (897)
 |}
-<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 ==
 ; [[User:Francium|Francium]]
-* [https://www.youtube.com/watch?v=opR71xbJ-PQ ''13|3|23''] – 3-, 13- and 23edo polymicrotonal
+* [https://www.youtube.com/watch?v=opR71xbJ-PQ ''13|3|23''] (2023) – 3-, 13-, and 23edo polymicrotonal
+[[Category:Listen]]