1289edo: Difference between revisions

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m →Theory: nullity-1 temps are best given by commas
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|1289}}
{{ED intro}}


== Theory ==
== Theory ==
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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 simple badness|Relative]] (%)
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
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| {{monzo|-16 35 -17}}, {{monzo|91 -12 -31}}
| {{monzo|-16 35 -17}}, {{monzo|91 -12 -31}}
| {{mapping|1289 2043 2993}}
| {{mapping|1289 2043 2993}}
| -0.0014
| −0.0014
| 0.0097
| 0.0097
| 1.04
| 1.04
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| 2460375/2458624, 78125000/78121827, 12884901888/12867859375
| 2460375/2458624, 78125000/78121827, 12884901888/12867859375
| {{mapping|1289 2043 2993 3619}}
| {{mapping|1289 2043 2993 3619}}
| -0.0275
| −0.0275
| 0.0461
| 0.0461
| 4.95
| 4.95
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| 3025/3024, 180224/180075, 2460375/2458624, 50014503/50000000
| 3025/3024, 180224/180075, 2460375/2458624, 50014503/50000000
| {{mapping|1289 2043 2993 3619 4459}}
| {{mapping|1289 2043 2993 3619 4459}}
| -0.0109
| −0.0109
| 0.0530
| 0.0530
| 5.69
| 5.69
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| 3025/3024, 1716/1715, 4096/4095, 91125/91091, 5282739/5281250
| 3025/3024, 1716/1715, 4096/4095, 91125/91091, 5282739/5281250
| {{mapping|1289 2043 2993 3619 4459 4770}}
| {{mapping|1289 2043 2993 3619 4459 4770}}
| -0.0146
| −0.0146
| 0.0491
| 0.0491
| 5.27
| 5.27
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| 3025/3024, 1716/1715, 4096/4095, 2500/2499, 37180/37179, 3536379/3536000
| 3025/3024, 1716/1715, 4096/4095, 2500/2499, 37180/37179, 3536379/3536000
| {{mapping|1289 2043 2993 3619 4459 4770 5269}}
| {{mapping|1289 2043 2993 3619 4459 4770 5269}}
| -0.0210
| −0.0210
| 0.0481
| 0.0481
| 5.17
| 5.17
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| 3025/3024, 1716/1715, 2376/2375, 4096/4095, 2500/2499, 270864/270725, 75735/75712
| 3025/3024, 1716/1715, 2376/2375, 4096/4095, 2500/2499, 270864/270725, 75735/75712
| {{mapping|1289 2043 2993 3619 4459 4770 5269 5476}}
| {{mapping|1289 2043 2993 3619 4459 4770 5269 5476}}
| -0.0299
| −0.0299
| 0.0508
| 0.0508
| 5.46
| 5.46
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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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| [[Fortune]]
| [[Fortune]]
|}
|}
<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>*</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]]
* "Baking With Honey" from ''The Scallop Disco Accident'' (2025) – [https://open.spotify.com/track/4MS4KuEo2jocbD1F968mBE Spotify] | [https://francium223.bandcamp.com/track/baking-with-honey Bandcamp] | [https://www.youtube.com/watch?v=mh0-dZt9NI8 YouTube] – in Honeybrookic, 1289edo tuning
* "Those Peas, With That Hat?" from ''Questions, Vol. 2'' (2025) – [https://open.spotify.com/track/3huEuNHvcAeKM6smXRSALR Spotify] | [https://francium223.bandcamp.com/track/those-peas-with-that-hat Bandcamp] | [https://www.youtube.com/watch?v=6us1wSQjn7w YouTube]

Latest revision as of 13:32, 13 March 2026

← 1288edo 1289edo 1290edo →
Prime factorization 1289 (prime)
Step size 0.930954 ¢ 
Fifth 754\1289 (701.939 ¢)
Semitones (A1:m2) 122:97 (113.6 ¢ : 90.3 ¢)
Consistency limit 9
Distinct consistency limit 9

1289 equal divisions of the octave (abbreviated 1289edo or 1289ed2), also called 1289-tone equal temperament (1289tet) or 1289 equal temperament (1289et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1289 equal parts of about 0.931 ¢ each. Each step represents a frequency ratio of 21/1289, or the 1289th root of 2.

Theory

1289edo is consistent to the 9-odd-limit. As an equal temperament, it tempers out [-16 35 -17⟩ (minortone comma) in the 5-limit. Using the patent val, it tempers out 3025/3024, 180224/180075, 2460375/2458624 and 50014503/50000000 in the 11-limit; 1716/1715, 4096/4095, 91125/91091 and 5282739/5281250 in the 13-limit. In the 2.3.13.23.29.31 subgroup it tempers out 19344/19343, in the 2.3.5.7.11.23.31 subgroup 19251/19250.

Prime harmonics

Approximation of prime harmonics in 1289edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.016 +0.032 +0.297 -0.193 +0.124 +0.242 +0.392 +0.120 +0.058 +0.038
Relative (%) +0.0 -1.7 +3.5 +32.0 -20.7 +13.3 +26.0 +42.1 +12.9 +6.2 +4.1
Steps
(reduced) 		1289
(0) 		2043
(754) 		2993
(415) 		3619
(1041) 		4459
(592) 		4770
(903) 		5269
(113) 		5476
(320) 		5831
(675) 		6262
(1106) 		6386
(1230)

Subsets and supersets

1289edo is the 209th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-2043 1289⟩ [⟨1289 2043]] +0.0049 0.0049 0.53
2.3.5 [-16 35 -17⟩, [91 -12 -31⟩ [⟨1289 2043 2993]] −0.0014 0.0097 1.04
2.3.5.7 2460375/2458624, 78125000/78121827, 12884901888/12867859375 [⟨1289 2043 2993 3619]] −0.0275 0.0461 4.95
2.3.5.7.11 3025/3024, 180224/180075, 2460375/2458624, 50014503/50000000 [⟨1289 2043 2993 3619 4459]] −0.0109 0.0530 5.69
2.3.5.7.11.13 3025/3024, 1716/1715, 4096/4095, 91125/91091, 5282739/5281250 [⟨1289 2043 2993 3619 4459 4770]] −0.0146 0.0491 5.27
2.3.5.7.11.13.17 3025/3024, 1716/1715, 4096/4095, 2500/2499, 37180/37179, 3536379/3536000 [⟨1289 2043 2993 3619 4459 4770 5269]] −0.0210 0.0481 5.17
2.3.5.7.11.13.17.19 3025/3024, 1716/1715, 2376/2375, 4096/4095, 2500/2499, 270864/270725, 75735/75712 [⟨1289 2043 2993 3619 4459 4770 5269 5476]] −0.0299 0.0508 5.46

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 142\1289 132.196 [-38 5 13⟩ Astro
1 196\1289 182.467 10/9 Minortone
1 238\1289 221.567 8388608/7381125 Fortune

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

Music

Francium
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