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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 1619 (prime)
{{ED intro}}
| Step size = 0.741198¢
| Fifth = 947\1619 (701.915¢)
| Major 2nd = 275\1619 (203.830¢)
| Semitones = 153:122 (113.403¢ : 90.426¢)
}}
'''1619edo''' divides the octave into parts of about 0.741 cents each.


== Theory ==
== Theory ==
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, where it tempers out [[4225/4224]], [[4375/4374]], [[6656/6655]], 78125/78078, and 117649/117612. It also notably tempers out [[quartisma]] (117440512/117406179) and [[123201/123200]].


In general, 1619edo supports vidar, with the comma set 4225/4224, 4375/4374, and 6656/6655.
1619edo tunes [[keenanisma]] very finely, to 6 steps, and can use it as a microchroma. 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 virtually equivalent to 1619edo. When it comes to using 33/32 as the generator, 1619edo supports the [[ravine]] temperament, which tempers out 196625/196608, 200000/199927, 2912000/2910897, and 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.
Since 33/32 is close to 1\45, 7\6 is close to 1\9, and 385/384 is close to 1\270, 1619edo can be thought of as [[1620edo]] where one step was extracted and all others were moved into a more harmonically just position. It achieves this because 1620edo is contorted 270edo in the 11-limit, and its 13/8 is on the flat side coming from 324edo, and thus when it is octave stretched, steps sharpen enough to arrive at 1619edo's 13-limit excellence.


1619edo is the 256th [[Prime edo]].
1619edo supports a very precise rank two temperament, {{nowrap|19 & 1619}}, which uses [[6/5]] as a generator and has a comma basis 4375/4374, 91125/91091, 196625/196608, and 54925000/54908469. 
 
1619edo supports the keenanose temperament, which has comma basis 4225/4224, 4375/4374, 6656/6655, and 151263/151250. Keenanisma is the generator in the keenanose temperament, {{nowrap|270 & 1619}}, in which it highlights the relationship between 270 keenanismas and the octave. It also achieves this since {{nowrap|270 × 6 {{=}} 1620}}, and 1619 is 1 short of that and also excellent in the 13-limit. 
 
Another temperament which highlights the interval relationships in 1619edo is {{nowrap|45 & 1619}}, called ''decigrave'', since 10 steps make a 7/6, which is referred to as the grave minor third sometimes. It has a comma basis 4225/4224, 4375/4374, 6656/6655, {{monzo|23  5 13 -23  1 0}} in the 13-limit. Its generator is 36 steps, which represents 65/64 and 66/65 tempered together, and 2 of them make 33/32. 5 of them make [[27/25]], and 10 of them make 7/6. 
 
1619edo supports the {{nowrap|494 & 1619}} temperament called moulin, with the comma basis of 4225/4224, 4375/4374, 6656/6655, 91125/91091. The 25-tone scale of moulin is capable of supporting the 8:11:13 triad, as it takes less than 25 notes to map the 11th and 13th harmonics. 
 
=== The Vidarines ===
1619edo supports [[vidar]], which has the comma basis 4225/4224, 4375/4374, and 6656/6655. In addition, it contains a wealth of rank-two 13-limit temperaments that are produced by adding one comma on top of the vidar comma basis;. Temperaments described above such as decigrave, keenanose, moulin, are members of this collection. Eliora proposes the name ''The Vidarines'' for this collection of temperaments.
 
A quick summary is shown below.
 
{| class="wikitable"
|+ style="font-size: 105%;" | The Vidarines in 1619edo (named and unnamed)
|-
! Temperament
! Generator<br />associated ratio
! Completing comma
|-
| Keenanose ({{nowrap|270 &amp; 1619}})
| 385/384
| 151263/151250
|-
| Decigrave ({{nowrap|45 &amp; 1619}})
| 66/65 ~ 65/64
| {{monzo|23  5 13 -23  1 0}}
|-
| Moulin ({{nowrap|494 &amp; 1619}})
| 13/11
| 91125/91091
|-
| {{nowrap|46 &amp; 1619}}
| 3328/3087
| {{monzo| -18  9 -2 8 -3 -1 }}
|-
| {{nowrap|178 &amp; 1619}}
| 4429568/4084101
| {{monzo| -29 10  2 12 -3 -4 }}
|-
| {{nowrap|224 &amp; 1619}}
| 256/175
| 18753525/18743296
|-
| {{nowrap|764 &amp; 1619}}
| 12375/8918
| 52734375/52706752
|-
| {{nowrap|901 &amp; 1619}}
| 104/99
| 34875815625/34843787264
|}
While [[abigail]] is a member of the vidarines, 1619edo does not support it because abigail is a period-2 temperament, and 1619 is an odd number.


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|1619|columns=10}}
{{Harmonics in equal|1619}}
 
=== Subsets and supersets ===
1619edo is the 256th [[prime edo]].
 
== Selected intervals ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 1619edo
|-
! Step
! Cents
! Ratio
! Name*
|-
| 0
| 0.000
| 1/1
| prime, unison
|-
| 6
| 4.447
| 385/384
| keenanisma
|-
| 72
| 53.366
| 33/32
| al-Farabi quarter-tone
|-
| 360
| 266.831
| 7/6
| septimal subminor third, grave minor third
|-
| 744
| 551.451
| 11/8
| 11th harmonic, undecimal superfourth
|-
| 1134
| 840.519
| 13/8
| 13th harmonic, tridecimal neutral sixth
|-
| 1619
| 1200.000
| 2/1
| perfect octave
|}
<nowiki />* Named in accordance to their most just 13-limit counterpart using the names accepted on the wiki.


== 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<br>8ve stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 31: Line 127:
! [[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
| +0.0127
| 0.013
| 0.0127
| 1.7
| 1.71
|-
|-
| 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
| +0.0299
| 0.026
| 0.0265
| 3.5
| 3.57
|-
|-
| 2.3.5.7
| 2.3.5.7
| 4375/4374, {{monzo| -6 3 9 -7 }}, {{monzo| -67 14 6 11 }}
| 4375/4374, 52734375/52706752, {{monzo| -67 14 6 11 }}
| [{{val| 1619 2566 3759 4545 }}]
| [{{val| 1619 2566 3759 4545 }}]
| 0.030
| +0.0295
| 0.023
| 0.0229
| 3.1
| 3.09
|-
|-
| 2.3.5.7.11
| 2.3.5.7.11
| 117649/117612, 151263/151250, 759375/758912, [[117440512/117406179]]
| 4375/4374, 117649/117612, 759375/758912, {{monzo| 24 -6 0 1 -5 }}
| [{{val| 1619 2566 3759 4545 5601 }}]
| [{{val| 1619 2566 3759 4545 5601 }}]
| 0.016
| +0.0159
| 0.034
| 0.0341
| 4.0
| 4.60
|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 4225/4224, 43940/43923, 151263/151250, 91125/91091, 123201/123200
| 4225/4224, 4375/4374, 6656/6655, 78125/78078, 117649/117612
| [{{val| 1619 2566 3759 4545 5601 5991 }}]
| [{{val| 1619 2566 3759 4545 5601 5991 }}]
| 0.013
| +0.0136
| 0.032
| 0.0315
| 4.2
| 4.26
|}
 
=== Rank-2 temperaments ===
{| 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
| 6\1619
| 4.447
| 385/384
| [[Keenanose]]
|-
| 1
| 36\1619
| 26.683
| 65/64
| [[Decigrave]]
|-
| 1
| 72\1619
| 53.366
| 33/32
| [[Ravine]]
|-
| 1
| 112\1619
| 83.014
| 1573/1500
| [[Acrosextilifourths]]
|-
| 1
| 390\1619
| 289.067
| 13/11
| [[Moulin]]
|-
| 1
| 426\1619
| 315.750
| 6/5
| [[Oviminor]]
|-
| 1
| 587\1619
| 435.083
| 9/7
| [[Supermajor]]
|-
| 1
| 672\1619
| 498.085
| 4/3
| [[Counterschismic]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if  distinct
== Music ==
; [[Francium]]
* "Le's Cancel Monday" from ''The Scallop Disco Accident'' (2025) – [https://open.spotify.com/track/5yxExt1gC5KA1grtcefU2m Spotify] | [https://francium223.bandcamp.com/track/les-cancel-monday Bandcamp] | [https://www.youtube.com/watch?v=TWAsePkJvtI YouTube]
* "this you?" from ''Questions, Vol. 2'' (2025) – [https://open.spotify.com/track/3ZdhHP0wAyzg9aQkKwQIar Spotify] | [https://francium223.bandcamp.com/track/this-you Bandcamp] | [https://www.youtube.com/watch?v=28NveBGA3-U YouTube]
* "Derpy Cat" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/1j301ZrWIbkw1b8Ar5Ww5L Spotify] | [https://francium223.bandcamp.com/track/derpy-cat Bandcamp] | [https://www.youtube.com/watch?v=qjNJoR__pT4 YouTube]


[[Category:Equal divisions of the octave]]
[[Category:Quartismic]]
Retrieved from "https://en.xen.wiki/w/1619edo"