1559edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
VisualWikitext
Francium (talk | contribs)
4,850 edits
Created page with "{{Infobox ET}} {{EDO intro|1559}} == Theory == 1559edo is consistent to the 9-odd-limit. Using the patent val, it tempers out 43046721/43025920, 78125000/781218..."
 
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
(7 intermediate revisions by 3 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|1559}}
{{ED intro}}


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


=== Prime harmonics ===
=== Prime harmonics ===
Line 9: Line 9:


=== Subsets and supersets ===
=== Subsets and supersets ===
1559edo is the 246th [[prime EDO]].
1559edo is the 246th [[prime edo]].


== 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
| {{monzo|2471 -1559}}
| {{monzo| 2471 -1559 }}
| {{mapping|1559 2471}}
| {{mapping| 1559 2471 }}
| -0.0106
| −0.0106
| 0.0106
| 0.0106
| 1.38
| 1.38
|-
|-
| 2.3.5
| 2.3.5
| {{monzo|38 -2 -15}}, {{monzo|71 -99 37}}
| {{monzo| 38 -2 -15 }}, {{monzo| 71 -99 37 }}
| {{mapping|1559 2471 3620}}
| {{mapping| 1559 2471 3620 }}
| -0.0196
| −0.0196
| 0.0155
| 0.0155
| 2.01
| 2.01
Line 38: Line 39:
| 2.3.5.7
| 2.3.5.7
| 43046721/43025920, 78125000/78121827, 12884901888/12867859375
| 43046721/43025920, 78125000/78121827, 12884901888/12867859375
| {{mapping|1559 2471 3620 4377}}
| {{mapping| 1559 2471 3620 4377 }}
| -0.0376
| −0.0376
| 0.0339
| 0.0339
| 4.40
| 4.40
Line 45: Line 46:
| 2.3.5.7.11
| 2.3.5.7.11
| 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 }}
| -0.0188
| −0.0188
| 0.0483
| 0.0483
| 6.27
| 6.27
|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 3025/3024, 1716/1715, 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 }}
| -0.0162
| −0.0162
| 0.0445
| 0.0445
| 5.78
| 5.78
Line 60: Line 61:
=== 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
|-
|-
Line 79: Line 81:
| [[Raider]]
| [[Raider]]
|}
|}
<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 ==
; [[Francium]]
* "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


<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
[[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
Retrieved from "https://en.xen.wiki/index.php?title=1559edo&oldid=225958"