1889edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|1889}}  
{{ED intro}}  
== Theory ==
== Theory ==
1889edo is strong in the [[23-limit]], though [[1578edo|1578]], which among other things has a lower 23-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]], rather puts it in the shade. It is distinctly [[consistent]] through the 27-odd-limit, but not, unlike 1578, to the 29-odd-limit. Even so, it should be noted that it supplies the [[optimal patent val]] for the 7-limit [[monzismic]] temperament.  
1889edo is strong in the [[23-limit]], though [[1578edo|1578]], which among other things has a lower 23-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]], rather puts it in the shade. It is distinctly [[consistent]] through the 27-odd-limit, but not, unlike 1578, to the 29-odd-limit. Even so, it should be noted that it supplies the [[optimal patent val]] for the 7-limit [[monzismic]] temperament.  
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== 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" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|-
|2.3
! rowspan="2" | [[Subgroup]]
|{{monzo|2994 -1889}}
! rowspan="2" | [[Comma list]]
|{{mapping|1889 2994}}
! rowspan="2" | [[Mapping]]
| -0.0012
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo|2994 -1889}}
| {{mapping|1889 2994}}
| −0.0012
| 0.0012
| 0.0012
| 0.19
| 0.19
|-
|-
|2.3.5
| 2.3.5
|{{monzo|54 -37 2}}, {{monzo|-66 -36 53}}
| {{monzo|54 -37 2}}, {{monzo|-66 -36 53}}
|{{mapping|1889 2994 4386}}
| {{mapping|1889 2994 4386}}
| +0.0104
| +0.0104
| 0.0163
| 0.0163
| 2.57
| 2.57
|-
|-
|2.3.5.7
| 2.3.5.7
|4375/4374, 3955078125/3954653486, {{monzo|-57 16 10 3}}
| 4375/4374, 3955078125/3954653486, {{monzo|-57 16 10 3}}
|{{mapping|1889 2994 4386 5303}}
| {{mapping|1889 2994 4386 5303}}
| +0.0131
| +0.0131
| 0.0149
| 0.0149
| 2.35
| 2.35
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|4375/4374, 151263/151250, 820125/819896, 1879453125/1879048192
| 4375/4374, 151263/151250, 820125/819896, 1879453125/1879048192
|{{mapping|1889 2994 4386 5303 6535}}
| {{mapping|1889 2994 4386 5303 6535}}
| +0.0055
| +0.0055
| 0.0201
| 0.0201
| 3.16
| 3.16
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|4375/4374, 4225/4224, 6656/6655, 4100625/4100096, 151263/151250
| 4375/4374, 4225/4224, 6656/6655, 4100625/4100096, 151263/151250
|{{mapping|1889 2994 4386 5303 6535 6990}}
| {{mapping|1889 2994 4386 5303 6535 6990}}
| +0.0084
| +0.0084
| 0.0194
| 0.0194
| 3.05
| 3.05
|-
|-
|2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
|4375/4374, 12376/12375, 4225/4224, 14400/14399, 14875/14872, 194481/194480
| 4375/4374, 12376/12375, 4225/4224, 14400/14399, 14875/14872, 194481/194480
|{{mapping|1889 2994 4386 5303 6535 6990 7721}}
| {{mapping|1889 2994 4386 5303 6535 6990 7721}}
| +0.0120
| +0.0120
| 0.0200
| 0.0200
| 3.15
| 3.15
|-
|-
|2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
|4375/4374, 12376/12375, 5985/5984, 4225/4224, 14400/14399, 6175/6174, 61965/61952
| 4375/4374, 12376/12375, 5985/5984, 4225/4224, 14400/14399, 6175/6174, 61965/61952
|{{mapping|1889 2994 4386 5303 6535 6990 7721 8024}}
| {{mapping|1889 2994 4386 5303 6535 6990 7721 8024}}
| +0.0167
| +0.0167
| 0.0226
| 0.0226
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=== 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
|-
! Generator<br>(reduced)*
! Periods<br />per 8ve
! Cents<br>(reduced)*
! Generator*
! Associated<br>Ratio*
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|392\1889
| 392\1889
|249.021
| 249.021
|{{monzo|-26 18 1}}
| {{monzo|-26 18 1}}
|[[Monzismic]]
| [[Monzismic]]
|-
|-
|1
| 1
|707\1889
| 707\1889
|449.127
| 449.127
|35/27
| 35/27
|[[Semidimi]]
| [[Semidimi]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Music ==
== Music ==

Revision as of 14:10, 20 February 2025

← 1888edo 1889edo 1890edo →
Prime factorization 1889 (prime)
Step size 0.635257 ¢ 
Fifth 1105\1889 (701.959 ¢)
Semitones (A1:m2) 179:142 (113.7 ¢ : 90.21 ¢)
Consistency limit 27
Distinct consistency limit 27

1889 equal divisions of the octave (abbreviated 1889edo or 1889ed2), also called 1889-tone equal temperament (1889tet) or 1889 equal temperament (1889et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1889 equal parts of about 0.635 ¢ each. Each step represents a frequency ratio of 21/1889, or the 1889th root of 2.

Theory

1889edo is strong in the 23-limit, though 1578, which among other things has a lower 23-limit relative error, rather puts it in the shade. It is distinctly consistent through the 27-odd-limit, but not, unlike 1578, to the 29-odd-limit. Even so, it should be noted that it supplies the optimal patent val for the 7-limit monzismic temperament.

Prime harmonics

Approximation of prime harmonics in 1889edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.004 -0.078 -0.059 +0.085 -0.083 -0.138 -0.213 -0.005 +0.174 -0.303
Relative (%) +0.0 +0.6 -12.2 -9.3 +13.4 -13.1 -21.7 -33.5 -0.9 +27.4 -47.7
Steps
(reduced) 		1889
(0) 		2994
(1105) 		4386
(608) 		5303
(1525) 		6535
(868) 		6990
(1323) 		7721
(165) 		8024
(468) 		8545
(989) 		9177
(1621) 		9358
(1802)

Subsets and supersets

1889edo is the 290th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [2994 -1889⟩ [⟨1889 2994]] −0.0012 0.0012 0.19
2.3.5 [54 -37 2⟩, [-66 -36 53⟩ [⟨1889 2994 4386]] +0.0104 0.0163 2.57
2.3.5.7 4375/4374, 3955078125/3954653486, [-57 16 10 3⟩ [⟨1889 2994 4386 5303]] +0.0131 0.0149 2.35
2.3.5.7.11 4375/4374, 151263/151250, 820125/819896, 1879453125/1879048192 [⟨1889 2994 4386 5303 6535]] +0.0055 0.0201 3.16
2.3.5.7.11.13 4375/4374, 4225/4224, 6656/6655, 4100625/4100096, 151263/151250 [⟨1889 2994 4386 5303 6535 6990]] +0.0084 0.0194 3.05
2.3.5.7.11.13.17 4375/4374, 12376/12375, 4225/4224, 14400/14399, 14875/14872, 194481/194480 [⟨1889 2994 4386 5303 6535 6990 7721]] +0.0120 0.0200 3.15
2.3.5.7.11.13.17.19 4375/4374, 12376/12375, 5985/5984, 4225/4224, 14400/14399, 6175/6174, 61965/61952 [⟨1889 2994 4386 5303 6535 6990 7721 8024]] +0.0167 0.0226 3.56

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 392\1889 249.021 [-26 18 1⟩ Monzismic
1 707\1889 449.127 35/27 Semidimi

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

Music

Francium
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