1559edo: Difference between revisions

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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..."
 
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== 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" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[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
Line 30: Line 30:
|-
|-
| 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
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| 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
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| 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
Line 51: Line 51:
|-
|-
| 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
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| [[Raider]]
| [[Raider]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is 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

Revision as of 07:44, 19 May 2024

← 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

Template:EDO intro

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 it is distinct

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