1619edo: Difference between revisions

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{{Infobox ET
{{Infobox ET
| Prime factorization = 1619 (is prime)
| Prime factorization = 1619 (prime)
| Step size = 0.741¢
| Step size = 0.741198¢
| Fifth = 947\1619(701.9148¢)
| Fifth = 947\1619 (701.915¢)
| Major 2nd = 275\1619 (203.8295¢)
| Major 2nd = 275\1619 (203.830¢)
| Semitones = 275:153
| Semitones = 153:122 (113.403¢ : 90.426¢)
}}
}}
1619edo divides the octave into parts of 741 millicents each. It is the 256th [[Prime EDO]].
'''1619edo''' divides the octave into parts of about 0.741 cents each.  


== Theory ==
== Theory ==
{{Harmonics in equal|1619|columns=10}}
1619edo is excellent in the 13-limit. It supports an extension of the [[Ragismic family|ragismic]] temperament with 2 extra dimensions in several ways. First, it supports the 441 & 270 & 1619 rank-3 temperament tempering out [[4225/4224]], [[4375/4374]], [[123201/123200]], 655473/655360, 1664000/1663893, and 6470695/6469632. Second, it supports 72 & 494 & 270 & 1619 temperament tempering out [[6656/6655]], 2912000/2910897, and 29115625/29113344.  
1619edo is excellent in the 13-limit. It supports an extension of the [[Ragismic family|ragismic]] temperament with 2 extra dimensions in several ways. First, it supports the 441 & 270 & 1619 rank 3 temperament tempering out 4225/4224, 4375/4374, 123201/123200, 655473/655360, 1664000/1663893, and 6470695/6469632. Second, it supports 72 & 494 & 270 & 1619 temperament tempering out 6656/6655, 2912000/2910897, and 29115625/29113344.  


In general, 1619edo supports vidar, with the comma set 4225/4224, 4375/4374, and 6656/6655.
In general, 1619edo supports vidar, with the comma set 4225/4224, 4375/4374, and 6656/6655.


1619edo has 7/6 on 360th step, a highly divisible number, 27/25 on 180th, and 33/32 on 72nd as a consequence of tempering out the commas. This means that 72ed33/32 is equivalent to 1619edo. When it comes to using 33/32 as the generator, 1619edo supports the temperament that tempers out 196625/196608, 200000/199927, 2912000/2910897, 3764768/3764475.
1619edo has 7/6 on 360th step, a highly divisible number, 27/25 on 180th, and 33/32 on 72nd as a consequence of tempering out the commas. This means that 72ed33/32 is equivalent to 1619edo. When it comes to using 33/32 as the generator, 1619edo supports the temperament that tempers out 196625/196608, 200000/199927, 2912000/2910897, 3764768/3764475.
1619edo is the 256th [[Prime edo]].
=== Prime harmonics ===
{{Harmonics in equal|1619|columns=10}}


== 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]]
! rowspan="2" | [[Comma list]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal
! rowspan="2" | Optimal<br>8ve stretch (¢)
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|-2566 1619}}
| {{monzo| -2566 1619 }}
|[{{val|1619 2566}}]
| [{{val| 1619 2566 }}]
|0.013
| 0.013
|0.013
| 0.013
|1.7
| 1.7
|-
|-
|2.3.5
| 2.3.5
|{{monzo|-69, 45, -1}}, {{monzo|-82, -1, 36}}
| {{monzo| -69 45 -1 }}, {{monzo| -82 -1 36 }}
|[{{val|1619 2566 3759}}]
| [{{val| 1619 2566 3759 }}]
|0.030
| 0.030
|0.026
| 0.026
|3.5
| 3.5
|-
|-
|2.3.5.7
| 2.3.5.7
|4375/4374, {{monzo|-6 3 9 -7}}, {{monzo|-67 14 6 11}}
| 4375/4374, {{monzo| -6 3 9 -7 }}, {{monzo| -67 14 6 11 }}
|[{{val|1619 2566 3759 4545}}]
| [{{val| 1619 2566 3759 4545 }}]
|0.030
| 0.030
|0.023
| 0.023
|3.1
| 3.1
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|117649/117612, 151263/151250, 759375/758912, [[117440512/117406179]]
| 117649/117612, 151263/151250, 759375/758912, [[117440512/117406179]]
|[{{val|1619 2566 3759 4545 5601}}]
| [{{val| 1619 2566 3759 4545 5601 }}]
|0.016
| 0.016
|0.034
| 0.034
|4.0
| 4.0
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|4225/4224, 43940/43923, 151263/151250, 91125/91091, 123201/123200
| 4225/4224, 43940/43923, 151263/151250, 91125/91091, 123201/123200
|[{{val|1619 2566 3759 4545 5601 5991}}]
| [{{val| 1619 2566 3759 4545 5601 5991 }}]
|0.013
| 0.013
|0.032
| 0.032
|4.2
| 4.2
|-
|-
|}
|}
[[Category:Ragismic family]]
 
[[Category:Equal divisions of the octave]]

Revision as of 18:13, 12 April 2022

← 1618edo 1619edo 1620edo →
Prime factorization 1619 (prime)
Step size 0.741198 ¢ 
Fifth 947\1619 (701.915 ¢)
Semitones (A1:m2) 153:122 (113.4 ¢ : 90.43 ¢)
Consistency limit 15
Distinct consistency limit 15

1619edo divides the octave into parts of about 0.741 cents each.

Theory

1619edo is excellent in the 13-limit. It supports an extension of the ragismic temperament with 2 extra dimensions in several ways. First, it supports the 441 & 270 & 1619 rank-3 temperament tempering out 4225/4224, 4375/4374, 123201/123200, 655473/655360, 1664000/1663893, and 6470695/6469632. Second, it supports 72 & 494 & 270 & 1619 temperament tempering out 6656/6655, 2912000/2910897, and 29115625/29113344.

In general, 1619edo supports vidar, with the comma set 4225/4224, 4375/4374, and 6656/6655.

1619edo has 7/6 on 360th step, a highly divisible number, 27/25 on 180th, and 33/32 on 72nd as a consequence of tempering out the commas. This means that 72ed33/32 is equivalent to 1619edo. When it comes to using 33/32 as the generator, 1619edo supports the temperament that tempers out 196625/196608, 200000/199927, 2912000/2910897, 3764768/3764475.

1619edo is the 256th Prime edo.

Prime harmonics

Approximation of prime harmonics in 1619edo
Harmonic 2 3 5 7 11 13 17 19 23 29
Error Absolute (¢) +0.000 -0.040 -0.149 -0.080 +0.134 -0.009 +0.295 -0.293 +0.262 -0.053
Relative (%) +0.0 -5.4 -20.2 -10.8 +18.0 -1.2 +39.8 -39.5 +35.3 -7.1
Steps
(reduced) 		1619
(0) 		2566
(947) 		3759
(521) 		4545
(1307) 		5601
(744) 		5991
(1134) 		6618
(142) 		6877
(401) 		7324
(848) 		7865
(1389)

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-2566 1619⟩ [⟨1619 2566]] 0.013 0.013 1.7
2.3.5 [-69 45 -1⟩, [-82 -1 36⟩ [⟨1619 2566 3759]] 0.030 0.026 3.5
2.3.5.7 4375/4374, [-6 3 9 -7⟩, [-67 14 6 11⟩ [⟨1619 2566 3759 4545]] 0.030 0.023 3.1
2.3.5.7.11 117649/117612, 151263/151250, 759375/758912, 117440512/117406179 [⟨1619 2566 3759 4545 5601]] 0.016 0.034 4.0
2.3.5.7.11.13 4225/4224, 43940/43923, 151263/151250, 91125/91091, 123201/123200 [⟨1619 2566 3759 4545 5601 5991]] 0.013 0.032 4.2
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