1619edo: Difference between revisions

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== Theory ==

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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.

~~=== The Vidarines ===~~

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 [[~~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, ~~such as decigrave~~, ~~keenanose~~, ~~moulin~~, ~~etc~~. ~~Eliora proposes the name ''The Vidarines'' for this collection~~ of ~~temperaments~~.

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.

~~One such temperament is~~ the ~~keenanose~~ temperament, ~~which has~~ comma basis 4225/4224, 4375/4374, 6656/6655, ~~and 151263~~/~~151250~~. ~~Keenanisma~~ is the ~~generator in the keenanose temperament~~, ~~270 & 1619, in which~~ it ~~highlights~~ the ~~relationship between 270 keenanismas~~ and ~~the octave. It also achieves this since 270 × 6 = 1620, and 1619 is 1 short of that and also excellent in the 13-limit~~.

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.

~~Another temperament which highlights the interval relationships in 1619edo (and is also a member of~~ The Vidarines ~~collection) is 45 & 1619, and if it had a name, it would be 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~~.

=== 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.

~~1619edo supports the 494 & 1619 temperament called moulin, which~~ is ~~also a member of The Vidarines collection. In this case, 91125/91091 has to be added to the three commas to produce a rank-2 temperament~~.

A quick summary is shown below.

~~A quick summary is shown below.~~

{| class="wikitable"

|+The Vidarines in 1619edo (named and unnamed)

|+ style="font-size: 105%;" | The Vidarines in 1619edo (named and unnamed)

!Temperament

|-

!Generator

! Temperament

associated ratio

! Generator associated ratio

!Completing comma

! Completing comma

|-

| Keenanose ({{nowrap| 270 & 1619 }})

| 385/384

| 151263/151250

|-

| Decigrave ({{nowrap| 45 & 1619 }})

| 66/65 ~ 65/64

| {{Monzo| 23 5 13 -23 1 0 }}

|-

| Moulin ({{nowrap| 494 & 1619 }})

| 13/11

| 91125/91091

|-

|~~Keenanose~~

| {{Nowrap| 46 & 1619 }}

|~~385~~/~~384~~

| 3328/3087

|~~151263/151250~~

| {{Monzo| -18 9 -2 8 -3 -1 }}

|-

|~~Decigrave~~

| {{Nowrap| 178 & 1619 }}

|~~66/65 ~ 65~~/64

| 4429568/4084101

|{{~~monzo~~|~~23 5 13~~ -~~23 1 0~~}}

| {{Monzo| -29 10 2 12 -3 -4 }}

|-

|~~Moulin~~

| {{nowrap| 224 & 1619 }}

|13/11

| 256/175

|~~91125~~/~~91091~~

| 18753525/18743296

|-

|~~224~~ & 1619

| {{nowrap| 764 & 1619 }}

|~~256~~/~~175~~

| 12375/8918

|~~18753525~~/~~18743296~~

| 52734375/52706752

|-

|901 & 1619

| {{nowrap| 901 & 1619 }}

|104/99

| 104/99

|34875815625/34843787264

| 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.

While [[abigail]] is a member of the vidarines, 1619edo does not support it because abigail has a semi-octave period, and 1619 is an odd number.

=== Prime harmonics ===

=== ~~Miscellaneous properties~~ ===

=== Subsets and supersets ===

1619edo is the 256th [[prime edo]].

== Selected intervals ==

{| class="wikitable mw-collapsible mw-collapsed"

|+ style=white-space:nowrap | Table of intervals in 1619edo

|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 1619edo

|-

! Step

! Cents

! Ratio

! Name~~<nowiki>~~*~~</nowiki>~~

! Name*

|-

| 0

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| 266.831

| 7/6

| septimal subminor third

| 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

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| perfect octave

|}

<nowiki>*~~</nowiki> named~~ in accordance to their most just 13-limit counterpart using the names accepted on the wiki.

<nowiki/>* Named in accordance to their most just 13-limit counterpart using the names accepted on the wiki.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

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|-

| 2.3

| {{~~monzo~~| -2566 1619 }}

| {{Monzo| -2566 1619 }}

| [{{~~val~~| 1619 2566 }}]

| {{Mapping| 1619 2566 }}

| +0.0127

| 0.0127

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|-

| 2.3.5

| {{~~monzo~~| -69 45 -1 }}, {{monzo| -82 -1 36 }}

| {{Monzo| -69 45 -1 }}, {{monzo| -82 -1 36 }}

| [{{~~val~~| 1619 2566 3759 }}]

| {{Mapping| 1619 2566 3759 }}

| +0.0299

| 0.0265

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| 2.3.5.7

| 4375/4374, 52734375/52706752, {{monzo| -67 14 6 11 }}

| [{{~~val~~| 1619 2566 3759 4545 }}]

| {{Mapping| 1619 2566 3759 4545 }}

| +0.0295

| 0.0229

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| 2.3.5.7.11

| 4375/4374, 117649/117612, 759375/758912, {{monzo| 24 -6 0 1 -5 }}

| [{{~~val~~| 1619 2566 3759 4545 5601 }}]

| {{Mapping| 1619 2566 3759 4545 5601 }}

| +0.0159

| 0.0341

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| 2.3.5.7.11.13

| 4225/4224, 4375/4374, 6656/6655, 78125/78078, 117649/117612

| [{{~~val~~| 1619 2566 3759 4545 5601 5991 }}]

| {{Mapping| 1619 2566 3759 4545 5601 5991 }}

| +0.0136

| 0.0315

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=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

! Periods per ~~Octave~~

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Generator~~ (Reduced)~~

|-

! Cents~~ (Reduced)~~

! Periods per 8ve

! Associated ~~Ratio~~

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

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| [[Keenanose]]

|-

|1

| 1

|36\1619

| 36\1619

|26.683

| 26.683

|65/64 ~~~ 66/65~~

| 65/64

|[[Decigrave]]

| [[Decigrave]]

|-

| 1

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| [[Ravine]]

|-

|1

| 1

|390\1619

| 112\1619

|289.067

| 83.014

|13/11

| 1573/1500

|[[Moulin]]

| [[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 (temperament)|Supermajor]]

|-

| 1

| 672\1619

| 498.085

| 4/3

| [[Counterschismic]]

|}

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct

[[~~Category~~:~~Equal divisions of the octave~~|~~####~~]]

== 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:Quartismic]]

~~{{Todo| review }}~~

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|1619}}
+{{ED intro}}
 == Theory ==
@@ Line 9: / Line 9: @@
 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.
-=== The Vidarines ===
+edo supports a very precise rank two temperament, {{nowrap|19 &amp; 1619}}, which uses [[6/5]] as a generator and has a comma basis 4375/4374, 91125/91091, 196625/196608, and 54925000/54908469.
-edo 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, such as decigrave, keenanose, moulin, etc. Eliora proposes the name ''The Vidarines'' for this collection of temperaments.
+edo 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.
-One such temperament is the keenanose temperament, which has comma basis 4225/4224, 4375/4374, 6656/6655, and 151263/151250. Keenanisma is the generator in the keenanose temperament, 270 & 1619, in which it highlights the relationship between 270 keenanismas and the octave. It also achieves this since 270 × 6 = 1620, and 1619 is 1 short of that and also excellent in the 13-limit.
+edo 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.
-Another temperament which highlights the interval relationships in 1619edo (and is also a member of The Vidarines collection) is 45 & 1619, and if it had a name, it would be 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.
+=== The Vidarines ===
+edo 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.
-edo supports the 494 & 1619 temperament called moulin, which is also a member of The Vidarines collection. In this case, 91125/91091 has to be added to the three commas to produce a rank-2 temperament.
+A quick summary is shown below.
-A quick summary is shown below.
 {| class="wikitable"
-|+The Vidarines in 1619edo (named and unnamed)
+|+ style="font-size: 105%;" | The Vidarines in 1619edo (named and unnamed)
-!Temperament
+|-
-!Generator
+! Temperament
-associated ratio
+! Generator<br>associated ratio
-!Completing comma
+! Completing comma
+|-
+| Keenanose ({{nowrap| 270 & 1619 }})
+| 385/384
+| 151263/151250
+|-
+| Decigrave ({{nowrap| 45 & 1619 }})
+| 66/65 ~ 65/64
+| {{Monzo| 23 5 13 -23 1 0 }}
+|-
+| Moulin ({{nowrap| 494 & 1619 }})
+| 13/11
+| 91125/91091
 |-
-|Keenanose
+| {{Nowrap| 46 & 1619 }}
-|385/384
+| 3328/3087
-|151263/151250
+| {{Monzo| -18 9 -2 8 -3 -1 }}
 |-
-|Decigrave
+| {{Nowrap| 178 & 1619 }}
-|66/65 ~ 65/64
+| 4429568/4084101
-|{{monzo|23  5 13 -23  1 0}}
+| {{Monzo| -29 10 2 12 -3 -4 }}
 |-
-|Moulin
+| {{nowrap| 224 & 1619 }}
-|13/11
+| 256/175
-|91125/91091
+| 18753525/18743296
 |-
-|224 & 1619
+| {{nowrap| 764 & 1619 }}
-|256/175
+| 12375/8918
-|18753525/18743296
+| 52734375/52706752
 |-
-|901 & 1619
+| {{nowrap| 901 & 1619 }}
-|104/99
+| 104/99
-|34875815625/34843787264
+| 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.
+While [[abigail]] is a member of the vidarines, 1619edo does not support it because abigail has a semi-octave period, and 1619 is an odd number.
 === Prime harmonics ===
-{{Harmonics in equal|1619|columns=10}}
+{{Harmonics in equal|1619}}
-=== Miscellaneous properties ===
+=== Subsets and supersets ===
 edo is the 256th [[prime edo]].
 == Selected intervals ==
 {| class="wikitable mw-collapsible mw-collapsed"
-|+ style=white-space:nowrap | Table of intervals in 1619edo
+|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 1619edo
+|-
 ! Step
 ! Cents
 ! Ratio
-! Name<nowiki>*</nowiki>
+! Name*
 |-
 | 0
@@ Line 80: / Line 96: @@
 | 266.831
 | 7/6
-| septimal subminor third
+| 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
@@ Line 87: / Line 113: @@
 | perfect octave
 |}
-<nowiki>*</nowiki> named in accordance to their most just 13-limit counterpart using the names accepted on the wiki.
+<nowiki/>* Named in accordance to their most just 13-limit counterpart using the names accepted on the wiki.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
@@ Line 101: / Line 128: @@
 |-
 | 2.3
-| {{monzo| -2566 1619 }}
+| {{Monzo| -2566 1619 }}
-| [{{val| 1619 2566 }}]
+| {{Mapping| 1619 2566 }}
 | +0.0127
 | 0.0127
@@ Line 108: / Line 135: @@
 |-
 | 2.3.5
-| {{monzo| -69 45 -1 }}, {{monzo| -82 -1 36 }}
+| {{Monzo| -69 45 -1 }}, {{monzo| -82 -1 36 }}
-| [{{val| 1619 2566 3759 }}]
+| {{Mapping| 1619 2566 3759 }}
 | +0.0299
 | 0.0265
@@ Line 116: / Line 143: @@
 | 2.3.5.7
 | 4375/4374, 52734375/52706752, {{monzo| -67 14 6 11 }}
-| [{{val| 1619 2566 3759 4545 }}]
+| {{Mapping| 1619 2566 3759 4545 }}
 | +0.0295
 | 0.0229
@@ Line 123: / Line 150: @@
 | 2.3.5.7.11
 | 4375/4374, 117649/117612, 759375/758912, {{monzo| 24 -6 0 1 -5 }}
-| [{{val| 1619 2566 3759 4545 5601 }}]
+| {{Mapping| 1619 2566 3759 4545 5601 }}
 | +0.0159
 | 0.0341
@@ Line 130: / Line 157: @@
 | 2.3.5.7.11.13
 | 4225/4224, 4375/4374, 6656/6655, 78125/78078, 117649/117612
-| [{{val| 1619 2566 3759 4545 5601 5991 }}]
+| {{Mapping| 1619 2566 3759 4545 5601 5991 }}
 | +0.0136
 | 0.0315
@@ Line 138: / Line 165: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-! Periods<br>per Octave
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Generator<br>(Reduced)
+|-
-! Cents<br>(Reduced)
+! Periods<br>per 8ve
-! Associated<br>Ratio
+! Generator*
+! Cents*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 150: / Line 179: @@
 | [[Keenanose]]
 |-
-|1
+| 1
-|36\1619
+| 36\1619
-|26.683
+| 26.683
-|65/64 ~ 66/65
+| 65/64
-|[[Decigrave]]
+| [[Decigrave]]
 |-
 | 1
@@ Line 162: / Line 191: @@
 | [[Ravine]]
 |-
-|1
+| 1
-|390\1619
+| 112\1619
-|289.067
+| 83.014
-|13/11
+| 1573/1500
-|[[Moulin]]
+| [[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 (temperament)|Supermajor]]
+|-
+| 1
+| 672\1619
+| 498.085
+| 4/3
+| [[Counterschismic]]
 |}
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if  distinct
-[[Category:Equal divisions of the octave|####]]
+== 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]
-*
-<!-- 4-digit number -->
 [[Category:Quartismic]]
-{{Todo| review }}