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.

1619edo supports a very precise rank two temperament, 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 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, 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 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 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.

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 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.

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

! Generator associated ratio

! Completing comma

|-

| Keenanose (270 & 1619)

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

| 385/384

| 151263/151250

|-

| Decigrave (45 & 1619)

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

| 66/65 ~ 65/64

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

|-

| Moulin (494 & 1619)

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

| 13/11

| 91125/91091

|-

| 46 & 1619

| {{nowrap|46 & 1619}}

| 3328/3087

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

|-

| 178 & 1619

| {{nowrap|178 & 1619}}

| 4429568/4084101

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

|-

| 224 & 1619

| {{nowrap|224 & 1619}}

| 256/175

| 18753525/18743296

|-

| 764 & 1619

| {{nowrap|764 & 1619}}

| 12375/8918

| 52734375/52706752

|-

| 901 & 1619

| {{nowrap|901 & 1619}}

| 104/99

| 34875815625/34843787264

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== Regular temperament properties ==

{~~{comma basis begin}}~~

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

|-

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

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

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

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

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3

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| 0.0315

| 4.26

~~{{comma basis end}~~}

|}

=== Rank-2 temperaments ===

{{rank-2 ~~begin}}~~

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

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

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

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| 83.014

| 1573/1500

| [[~~Acrosextilififths~~]]

| [[Acrosextilifourths]]

|-

| 1

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| 4/3

| [[Counterschismic]]

~~{{rank~~-2 ~~end}}~~

|}

~~{{orf}}~~

<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: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.
-edo supports a very precise rank two temperament, 19 & 1619, which uses [[6/5]] as a generator and has a comma basis 4375/4374, 91125/91091, 196625/196608, and 54925000/54908469.
+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 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 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.
-Another temperament which highlights the interval relationships in 1619edo is 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.
+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 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.
+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
+! Generator<br />associated ratio
 ! Completing comma
 |-
-| Keenanose (270 & 1619)
+| Keenanose ({{nowrap|270 &amp; 1619}})
 | 385/384
 | 151263/151250
 |-
-| Decigrave (45 & 1619)
+| Decigrave ({{nowrap|45 &amp; 1619}})
 | 66/65 ~ 65/64
 | {{monzo|23  5 13 -23  1 0}}
 |-
-| Moulin (494 & 1619)
+| Moulin ({{nowrap|494 &amp; 1619}})
 | 13/11
 | 91125/91091
 |-
-| 46 & 1619
+| {{nowrap|46 &amp; 1619}}
 | 3328/3087
 | {{monzo| -18  9 -2 8 -3 -1 }}
 |-
-| 178 & 1619
+| {{nowrap|178 &amp; 1619}}
 | 4429568/4084101
 | {{monzo| -29 10  2 12 -3 -4 }}
 |-
-| 224 & 1619
+| {{nowrap|224 &amp; 1619}}
 | 256/175
 | 18753525/18743296
 |-
-| 764 & 1619
+| {{nowrap|764 &amp; 1619}}
 | 12375/8918
 | 52734375/52706752
 |-
-| 901 & 1619
+| {{nowrap|901 &amp; 1619}}
 | 104/99
 | 34875815625/34843787264
@@ Line 115: / Line 116: @@
 == Regular temperament properties ==
-{{comma basis begin}}
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
 |-
 | 2.3
@@ Line 151: / Line 161: @@
 | 0.0315
 | 4.26
-{{comma basis end}}
+|}
 === Rank-2 temperaments ===
-{{rank-2 begin}}
+{| 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
@@ Line 178: / Line 195: @@
 | 83.014
 | 1573/1500
-| [[Acrosextilififths]]
+| [[Acrosextilifourths]]
 |-
 | 1
@@ Line 203: / Line 220: @@
 | 4/3
 | [[Counterschismic]]
-{{rank-2 end}}
+|}
-{{orf}}
+<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:Quartismic]]
-{{Todo| review }}