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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|897}}
{{ED intro}}


== Theory ==
== 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.  
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}}
{{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 ==
== 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]]
! rowspan="2" | [[Mapping]]
! 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 error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 474 -299 }}
| {{mapping| 897 1422 }}
| -0.1218
| 0.1218
| 9.10
|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 33 -34 9 }}, {{monzo| 70 -9 -24 }}
| {{monzo| 33 -34 9 }}, {{monzo| 70 -9 -24 }}
| {{mapping| 897 ​1422​ 2083​ }}
| {{mapping| 897 ​1422​ 2083​ }}
| -0.1255
| −0.1255
| 0.010
| 0.0996
| 0.75
| 7.44
|-
|-
| 2.3.5.7
| 2.3.5.7
| 250047/250000, 67108864/66976875, 28824005/28697814
| 250047/250000, 67108864/66976875, 28824005/28697814
| {{mapping| 897​ 1422 ​2083 ​2518 }} (897)
| {{mapping| 897​ 1422 ​2083 ​2518 }} (897)
| -0.0706
| −0.0706
| 0.1283
| 0.1283
| 9.59
| 9.59
|-
|-
| 2.3.5.7.11
| 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)
| {{mapping| 897 ​1422​ 2083 ​2518 ​3103 }} (897)
| -0.0480
| −0.0480
| 0.1234
| 0.1234
| 9.22
| 9.22
|-
|-
| 2.3.5.7.11.13
| 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)
| {{mapping| 897​ 1422 ​2083 ​2518​ 3103​ 3319 }} (897)
| -0.0222
| −0.0222
| 0.1265
| 0.1265
| 9.46
| 9.46
Line 61: Line 58:
=== 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
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
| 1
| 1
| 419\897
| 419\897
| 560.535
| 560.54
| 864/625
| 864/625
| [[Whoosh]]
| [[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 ==
== Music ==
; [[User:Francium|Francium]]
; [[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]]

Latest revision as of 06:30, 21 February 2025

← 896edo 897edo 898edo →
Prime factorization 3 × 13 × 23
Step size 1.33779 ¢ 
Fifth 525\897 (702.341 ¢) (→ 175\299)
Semitones (A1:m2) 87:66 (116.4 ¢ : 88.29 ¢)
Consistency limit 7
Distinct consistency limit 7

897 equal divisions of the octave (abbreviated 897edo or 897ed2), also called 897-tone equal temperament (897tet) or 897 equal temperament (897et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 897 equal parts of about 1.34 ¢ each. Each step represents a frequency ratio of 21/897, or the 897th root of 2.

Theory

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

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.

Odd harmonics

Approximation of odd harmonics in 897edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.386 +0.308 -0.264 -0.566 -0.147 -0.394 -0.643 -0.608 -0.523 +0.122 +0.488
Relative (%) +28.9 +23.0 -19.7 -42.3 -11.0 -29.4 -48.1 -45.4 -39.1 +9.1 +36.5
Steps
(reduced) 		1422
(525) 		2083
(289) 		2518
(724) 		2843
(152) 		3103
(412) 		3319
(628) 		3504
(813) 		3666
(78) 		3810
(222) 		3940
(352) 		4058
(470)

Subsets and supersets

Since 897 factors into 3 × 13 × 23, 897edo has subset edos 3, 13, 23, 39, 69, and 299.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 [33 -34 9⟩, [70 -9 -24⟩ [⟨897 ​1422​ 2083​]] −0.1255 0.0996 7.44
2.3.5.7 250047/250000, 67108864/66976875, 28824005/28697814 [⟨897​ 1422 ​2083 ​2518]] (897) −0.0706 0.1283 9.59
2.3.5.7.11 3025/3024, 5632/5625, 102487/102400, 28824005/28697814 [⟨897 ​1422​ 2083 ​2518 ​3103]] (897) −0.0480 0.1234 9.22
2.3.5.7.11.13 676/675, 1001/1000, 3025/3024, 4096/4095, 28824005/28697814 [⟨897​ 1422 ​2083 ​2518​ 3103​ 3319]] (897) −0.0222 0.1265 9.46

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 419\897 560.54 864/625 Whoosh
1 440\897 588.63 128/91 Countritonic (897df)
3 103\897 137.79 13/12 Avicenna (897)

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

Music

Francium
  • 13|3|23 (2023) – 3-, 13-, and 23edo polymicrotonal
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