1559edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|1559}}
{{ED intro}}


== Theory ==
== Theory ==
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== Regular temperament properties ==
== Regular temperament properties ==
{{comma basis begin}}
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo| 2471 -1559 }}
| {{monzo| 2471 -1559 }}
| {{mapping| 1559 2471 }}
| {{mapping| 1559 2471 }}
| -0.0106
| −0.0106
| 0.0106
| 0.0106
| 1.38
| 1.38
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| {{monzo| 38 -2 -15 }}, {{monzo| 71 -99 37 }}
| {{monzo| 38 -2 -15 }}, {{monzo| 71 -99 37 }}
| {{mapping| 1559 2471 3620 }}
| {{mapping| 1559 2471 3620 }}
| &minus;0.0196
| −0.0196
| 0.0155
| 0.0155
| 2.01
| 2.01
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| 43046721/43025920, 78125000/78121827, 12884901888/12867859375
| 43046721/43025920, 78125000/78121827, 12884901888/12867859375
| {{mapping| 1559 2471 3620 4377 }}
| {{mapping| 1559 2471 3620 4377 }}
| &minus;0.0376
| −0.0376
| 0.0339
| 0.0339
| 4.40
| 4.40
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| 3025/3024, 180224/180075, 50014503/50000000, 56953125/56942116
| 3025/3024, 180224/180075, 50014503/50000000, 56953125/56942116
| {{mapping| 1559 2471 3620 4377 5393 }}
| {{mapping| 1559 2471 3620 4377 5393 }}
| &minus;0.0188
| −0.0188
| 0.0483
| 0.0483
| 6.27
| 6.27
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| 1716/1715, 3025/3024, 4096/4095, 492128/492075, 5282739/5281250
| 1716/1715, 3025/3024, 4096/4095, 492128/492075, 5282739/5281250
| {{mapping| 1559 2471 3620 4377 5393 5769 }}
| {{mapping| 1559 2471 3620 4377 5393 5769 }}
| &minus;0.0162
| −0.0162
| 0.0445
| 0.0445
| 5.78
| 5.78
{{comma basis end}}
|}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{{rank-2 begin}}
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
|-
| 1
| 1
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| 8000/6561
| 8000/6561
| [[Raider]]
| [[Raider]]
{{rank-2 end}}
|}
{{orf}}
<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


== Music ==
== Music ==
; [[Francium]]
; [[Francium]]
* "confluenceprettype" from ''albumwithoutspaces'' (2024) &ndash; [https://open.spotify.com/track/0jCzkUrwVEHevs7xAWgsvh Spotify] | [https://francium223.bandcamp.com/track/confluenceprettype Bandcamp] | [https://www.youtube.com/watch?v=4rxv7ec4BEM YouTube] &ndash; luna[25] in 1559edo tuning
* "confluenceprettype" from ''albumwithoutspaces'' (2024) – [https://open.spotify.com/track/0jCzkUrwVEHevs7xAWgsvh Spotify] | [https://francium223.bandcamp.com/track/confluenceprettype Bandcamp] | [https://www.youtube.com/watch?v=4rxv7ec4BEM YouTube] – luna[25] in 1559edo tuning


[[Category:Listen]]
[[Category:Listen]]

Latest revision as of 13:33, 13 March 2026

← 1558edo 1559edo 1560edo →
Prime factorization 1559 (prime)
Step size 0.769724 ¢ 
Fifth 912\1559 (701.988 ¢)
Semitones (A1:m2) 148:117 (113.9 ¢ : 90.06 ¢)
Consistency limit 9
Distinct consistency limit 9

1559 equal divisions of the octave (abbreviated 1559edo or 1559ed2), also called 1559-tone equal temperament (1559tet) or 1559 equal temperament (1559et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1559 equal parts of about 0.77 ¢ each. Each step represents a frequency ratio of 21/1559, or the 1559th root of 2.

Theory

1559edo is consistent to the 9-odd-limit. Using the patent val, it tempers out [38 -2 -15⟩ (luna comma) and [71 -99 37⟩ (raider comma) in the 5-limit; 43046721/43025920, 78125000/78121827 and [32 1 -6 -7⟩ in the 7-limit, 3025/3024, 180224/180075, 50014503/50000000 and 56953125/56942116 in the 11-limit.

Prime harmonics

Approximation of prime harmonics in 1559edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.033 +0.088 +0.257 -0.195 +0.011 -0.273 +0.370 -0.179 +0.314 +0.314
Relative (%) +0.0 +4.3 +11.4 +33.4 -25.4 +1.4 -35.5 +48.1 -23.3 +40.8 +40.8
Steps
(reduced) 		1559
(0) 		2471
(912) 		3620
(502) 		4377
(1259) 		5393
(716) 		5769
(1092) 		6372
(136) 		6623
(387) 		7052
(816) 		7574
(1338) 		7724
(1488)

Subsets and supersets

1559edo is the 246th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [2471 -1559⟩ [⟨1559 2471]] −0.0106 0.0106 1.38
2.3.5 [38 -2 -15⟩, [71 -99 37⟩ [⟨1559 2471 3620]] −0.0196 0.0155 2.01
2.3.5.7 43046721/43025920, 78125000/78121827, 12884901888/12867859375 [⟨1559 2471 3620 4377]] −0.0376 0.0339 4.40
2.3.5.7.11 3025/3024, 180224/180075, 50014503/50000000, 56953125/56942116 [⟨1559 2471 3620 4377 5393]] −0.0188 0.0483 6.27
2.3.5.7.11.13 1716/1715, 3025/3024, 4096/4095, 492128/492075, 5282739/5281250 [⟨1559 2471 3620 4377 5393 5769]] −0.0162 0.0445 5.78

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 251\1559 193.201 262144/234375 Luna
1 446\1559 343.297 8000/6561 Raider

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

Music

Francium
  • "confluenceprettype" from albumwithoutspaces (2024) – Spotify | Bandcamp | YouTube – luna[25] in 1559edo tuning
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