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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|897}}
{{EDO intro|897}}
==Theory==
 
897et tempers out [[250047/250000]] in the 7-limit; 100663296/100656875, 2097152/2096325, [[131072/130977]], 172032/171875, [[5632/5625]], 47265625/47258883, [[3025/3024]], 160083/160000, 102487/102400, [[1771561/1769472]] in the 11-limit.
== 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''' }}.
 
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 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 ===
{{Harmonics in equal|897}}
{{Harmonics in equal|897}}
==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<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
| 2.3
|{{monzo| 474 -299}}
| {{monzo| 474 -299 }}
|{{val| 897 1422}}
| {{mapping| 897 1422 }}
| -0.1218
| -0.1218
| 0.1218
| 0.1218
| 9.10
| 9.10
|-
|-
|2.3.5
| 2.3.5
|{{monzo| 33 -34 9}}, {{monzo| 70 -9 -24}}
| {{monzo| 33 -34 9 }}, {{monzo| 70 -9 -24 }}
|{{val| 897 ​1422​ 2083​}}
| {{mapping| 897 ​1422​ 2083​ }}
| -0.1255
| -0.1255
| 0.010
| 0.010
| 0.75
| 0.75
|-
|-
|2.3.5.7
| 2.3.5.7
|250047/250000, 67108864/66976875, 28824005/28697814
| 250047/250000, 67108864/66976875, 28824005/28697814
|{{val| 897​ 1422 ​2083 ​2518}}
| {{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, 160083/160000, 28824005/28697814
|{{val| 897 ​1422​ 2083 ​2518 ​3103}}
| {{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
| 2080/2079, 1001/1000, 3025/3024, 4096/4095, 2058857500/2051893701
|{{val| 897​ 1422 ​2083 ​2518​ 3103​ 3319}}
| {{mapping| 897​ 1422 ​2083 ​2518​ 3103​ 3319 }} (897)
| -0.0222
| -0.0222
| 0.1265
| 0.1265
Line 55: Line 63:
|+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*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>Ratio
! Temperaments
! Temperaments
|-
|-
|1
| 1
|419\897
| 419\897
|560.535
| 560.535
|864/625
| 864/625
|[[Whoosh]]
| [[Whoosh]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


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

Revision as of 09:03, 20 October 2023

← 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

Template:EDO intro

Theory

897edo has sharp approximations to harmonics 3 and 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, 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

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)

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [474 -299⟩ [⟨897 1422]] -0.1218 0.1218 9.10
2.3.5 [33 -34 9⟩, [70 -9 -24⟩ [⟨897 ​1422​ 2083​]] -0.1255 0.010 0.75
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, 160083/160000, 28824005/28697814 [⟨897 ​1422​ 2083 ​2518 ​3103]] (897) -0.0480 0.1234 9.22
2.3.5.7.11.13 2080/2079, 1001/1000, 3025/3024, 4096/4095, 2058857500/2051893701 [⟨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\897 560.535 864/625 Whoosh

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

Music

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