1619edo: Difference between revisions

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== Theory ==

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]].

~~It supports~~ [[~~vidar~~]], ~~which~~ has ~~the comma basis 4225~~/~~4224~~, ~~4375~~/~~4374~~, and ~~6656~~/~~6655~~, ~~and other unnamed expansions of~~ the [[~~ragismic~~]] temperament ~~such as the 270 & 441 & 1619~~, ~~tempering~~ out ~~4225~~/~~4224~~, ~~4375~~/~~4374~~, ~~655473~~/~~655360~~, ~~or the 72 & 270 & 494 & 1619 temperament tempering out 6656/6655~~ and ~~2912000~~/~~2910897~~.

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 ~~supports~~ the ~~rank~~-~~5 temperament tempering out the jacobin comma~~, ~~6656/6655~~ and its ~~fifth-order maximal evenness scale~~ is ~~represented by every 3rd step of~~ the ~~72 & 270 & 494 & 1619 temperament{{clarify}}~~.

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 ~~tunes [[keenanisma]]~~ very ~~finely, to 6 steps, and can use it as a microchroma. In addition, it can use the keenanisma as a 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~~ 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 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.

~~Another~~ temperament ~~which highlights the interval relationships in 1619edo 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~~.

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.

~~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~~.

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 associated ratio

! 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

|-

| {{nowrap|46 & 1619}}

| 3328/3087

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

|-

| {{nowrap|178 & 1619}}

| 4429568/4084101

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

|-

| {{nowrap|224 & 1619}}

| 256/175

| 18753525/18743296

|-

| {{nowrap|764 & 1619}}

| 12375/8918

| 52734375/52706752

|-

| {{nowrap|901 & 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 ===

=== ~~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

Line 47:

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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]]

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

! rowspan="2" | Optimal 8ve stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

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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

Line 129:

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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]]

|-

| 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

[[~~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 ==
 edo 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]].
-It supports [[vidar]], which has the comma basis 4225/4224, 4375/4374, and 6656/6655, and other unnamed expansions of the [[ragismic]] temperament such as the 270 & 441 & 1619, tempering out 4225/4224, 4375/4374, 655473/655360, or the 72 & 270 & 494 & 1619 temperament tempering out 6656/6655 and 2912000/2910897.
+edo 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.
-edo supports the rank-5 temperament tempering out the jacobin comma, 6656/6655 and its fifth-order maximal evenness scale is represented by every 3rd step of the 72 & 270 & 494 & 1619 temperament{{clarify}}.
+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.
-edo tunes [[keenanisma]] very finely, to 6 steps, and can use it as a microchroma. In addition, it can use the keenanisma as a 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 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.
+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.
-Another temperament which highlights the interval relationships in 1619edo 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.
+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 &amp; 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.
-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.
+Another temperament which highlights the interval relationships in 1619edo is {{nowrap|45 &amp; 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.
+edo supports the {{nowrap|494 &amp; 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 ===
+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.
+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 ===
-{{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 47: / 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 54: / 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]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 105: / 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 117: / Line 179: @@
 | [[Keenanose]]
 |-
-|1
+| 1
-|36\1619
+| 36\1619
-|26.683
+| 26.683
-|65/64 ~ 66/65
+| 65/64
-|[[Decigrave]]
+| [[Decigrave]]
 |-
 | 1
@@ Line 129: / 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]]
+|-
+| 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
-[[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 }}