3889edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|3889}} == Theory == 3889edo is consistent to the 27-odd-limit, tempering out 12376/12375, 14400/14399, 6175/6174, 8625/8624..."
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|3889}}
{{ED intro}}


== Theory ==
== Theory ==
3889edo is [[consistent]] to the [[27-odd-limit]], [[tempering out]] [[12376/12375]], [[14400/14399]], [[6175/6174]], 8625/8624, 89376/89375, [[123201/123200]], 1549184/1549125 and [[1990656/1990625]] in the [[23-limit]]. It is strong in the 2.3.5.7.13.17.19.23.31 [[subgroup]], tempering out 6175/6174, 14365/14364, 426496/426465, 52326/52325, 52003/52000, 1549184/1549125, 1990656/1990625 and 22816/22815. The equal temperament also tempers out [[19251/19250]] in the 2.3.5.7.11.23.31 subgroup and [[5440/5439]] in the 2.3.5.7.17.37 subgroup.
3889edo is [[consistent]] to the [[27-odd-limit]], and except for [[29/19]] and its [[octave complement]], it is consistent to the [[31-odd-limit]].
 
As an equal temperament, it [[tempering out|tempers out]] the [[pirate comma]], the [[starscape comma]], and the [[quartisma]]. Some of the simpler commas it tempers out in the higher limits include [[6656/6655]], [[123201/123200]] and [[1990656/1990625]] in the [[13-limit]]; [[12376/12375]], [[14400/14399]] in the [[17-limit]]; [[5776/5775]], [[6175/6174]], 14365/14364, 23409/23408, 28900/28899, 43681/43680 in the [[19-limit]]; [[5083/5082]], 7866/7865, and 8625/8624 in the [[23-limit]].  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|3889}}
{{Harmonics in equal|3889|columns=11}}
{{Harmonics in equal|3889|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 3889edo (continued)}}


=== Subsets and supersets ===
=== Subsets and supersets ===
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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" | [[Subgroup]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Comma list]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | [[Mapping]]
! colspan="2" |Tuning Error
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo|6164 -3889}}
| {{Monzo| 6164 -3889 }}
| {{mapping|3889 6164}}
| {{Mapping| 3889 6164 }}
| -0.0079
| −0.0079
| 0.0079
| 0.0079
| 2.56
| 2.56
|-
|-
| 2.3.5
| 2.3.5
| {{monzo|-90 -15 49}}, {{monzo|56 -91 38}}
| {{Monzo| -90 -15 49 }}, {{monzo| 56 -91 38 }}
| {{mapping|3889 6164 9030}}
| {{Mapping| 3889 6164 9030 }}
| -0.0062
| −0.0062
| 0.0068
| 0.0068
| 2.20
| 2.20
|-
|-
| 2.3.5.7
| 2.3.5.7
| {{monzo|-4 17 1 -9}}, {{monzo|2 -20 14 -1}}, {{monzo|46 -14 -3 -6}}
| 645700815/645657712, {{monzo| 2 -20 14 -1 }}, {{monzo| 46 -14 -3 -6 }}
| {{mapping|3889 6164 9030 10918}}
| {{Mapping| 3889 6164 9030 10918 }}
| -0.0101
| −0.0101
| 0.0089
| 0.0089
| 2.88
| 2.88
|-
|-
| 2.3.5.7.11
| 2.3.5.7.11
| 21437500/21434787, 47265625/47258883, 56953125/56942116, 104857600/104825259
| 1771875/1771561, 3294225/3294172, 14348907/14348180, 104857600/104825259
| {{mapping|3889 6164 9030 10918 13454}}
| {{Mapping| 3889 6164 9030 10918 13454 }}
| -0.0129
| −0.0129
| 0.0098
| 0.0098
| 3.18
| 3.18
|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 123201/123200, 6656/6655, 1990656/1990625, 492128/492075, 29046875/29042496
| 6656/6655, 123201/123200, 492128/492075, 823680/823543, 1990656/1990625
| {{mapping|3889 6164 9030 10918 13454 14391}}
| {{Mapping| 3889 6164 9030 10918 13454 14391 }}
| -0.0106
| −0.0106
| 0.0103
| 0.0103
| 3.34
| 3.34
|-
|-
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 12376/12375, 14400/14399, 123201/123200, 37180/37179, 1990656/1990625, 361250/361179
| 6656/6655, 12376/12375, 28561/28560, 37180/37179, 361250/361179, 937125/937024
| {{mapping|3889 6164 9030 10918 13454 14391 15896}}
| {{Mapping| 3889 6164 9030 10918 13454 14391 15896 }}
| -0.0075
| −0.0075
| 0.0121
| 0.0121
| 3.92
| 3.92
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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
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 602\3889
| 602\3889
| 185.755
| 185.755
| {{monzo|24 4 -13}}
| {{Monzo| 24 4 -13 }}
| [[Pirate]]
| [[Pirate]]
|}
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
== Music ==
; [[Francium]]
* "Don't Stop Interrupting People" from ''Don't'' (2025) – [https://open.spotify.com/track/2W1M2taD9SRU2MIz7YqpxQ Spotify] | [https://francium223.bandcamp.com/track/dont-stop-interrupting-people Bandcamp] | [https://www.youtube.com/watch?v=nOoXjJ6dqhk YouTube]

Latest revision as of 14:46, 17 April 2026

← 3888edo 3889edo 3890edo →
Prime factorization 3889 (prime)
Step size 0.308563 ¢ 
Fifth 2275\3889 (701.98 ¢)
Semitones (A1:m2) 369:292 (113.9 ¢ : 90.1 ¢)
Consistency limit 27
Distinct consistency limit 27

3889 equal divisions of the octave (abbreviated 3889edo or 3889ed2), also called 3889-tone equal temperament (3889tet) or 3889 equal temperament (3889et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3889 equal parts of about 0.309 ¢ each. Each step represents a frequency ratio of 21/3889, or the 3889th root of 2.

Theory

3889edo is consistent to the 27-odd-limit, and except for 29/19 and its octave complement, it is consistent to the 31-odd-limit.

As an equal temperament, it tempers out the pirate comma, the starscape comma, and the quartisma. Some of the simpler commas it tempers out in the higher limits include 6656/6655, 123201/123200 and 1990656/1990625 in the 13-limit; 12376/12375, 14400/14399 in the 17-limit; 5776/5775, 6175/6174, 14365/14364, 23409/23408, 28900/28899, 43681/43680 in the 19-limit; 5083/5082, 7866/7865, and 8625/8624 in the 23-limit.

Prime harmonics

Approximation of prime harmonics in 3889edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.025 +0.007 +0.061 +0.083 -0.003 -0.044 -0.059 -0.041 +0.096 +0.040
Relative (%) +0.0 +8.1 +2.2 +19.7 +27.0 -1.0 -14.3 -19.0 -13.2 +31.2 +13.1
Steps
(reduced) 		3889
(0) 		6164
(2275) 		9030
(1252) 		10918
(3140) 		13454
(1787) 		14391
(2724) 		15896
(340) 		16520
(964) 		17592
(2036) 		18893
(3337) 		19267
(3711)
Approximation of prime harmonics in 3889edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) +0.134 +0.148 +0.079 +0.063 +0.036 +0.124 +0.112 -0.006 -0.113 -0.070 -0.124
Relative (%) +43.6 +48.0 +25.6 +20.4 +11.7 +40.1 +36.2 -2.1 -36.7 -22.8 -40.3
Steps
(reduced) 		20260
(815) 		20836
(1391) 		21103
(1658) 		21602
(2157) 		22276
(2831) 		22878
(3433) 		23065
(3620) 		23591
(257) 		23916
(582) 		24072
(738) 		24515
(1181)

Subsets and supersets

3889edo is the 539th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [6164 -3889⟩ [⟨3889 6164]] −0.0079 0.0079 2.56
2.3.5 [-90 -15 49⟩, [56 -91 38⟩ [⟨3889 6164 9030]] −0.0062 0.0068 2.20
2.3.5.7 645700815/645657712, [2 -20 14 -1⟩, [46 -14 -3 -6⟩ [⟨3889 6164 9030 10918]] −0.0101 0.0089 2.88
2.3.5.7.11 1771875/1771561, 3294225/3294172, 14348907/14348180, 104857600/104825259 [⟨3889 6164 9030 10918 13454]] −0.0129 0.0098 3.18
2.3.5.7.11.13 6656/6655, 123201/123200, 492128/492075, 823680/823543, 1990656/1990625 [⟨3889 6164 9030 10918 13454 14391]] −0.0106 0.0103 3.34
2.3.5.7.11.13.17 6656/6655, 12376/12375, 28561/28560, 37180/37179, 361250/361179, 937125/937024 [⟨3889 6164 9030 10918 13454 14391 15896]] −0.0075 0.0121 3.92

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 602\3889 185.755 [24 4 -13⟩ Pirate

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

Music

Francium
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