Orwell: Difference between revisions

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{{interwiki

| en = Orwell

| de = Orwell

~~| en = Orwell~~

| es =

| ja =

}}

{{Infobox regtemp

| Title = Orwell

| Subgroups = 2.3.5.7, 2.3.5.7.11

| Comma basis = [[225/224]], [[1728/1715]] (7-limit); [[99/98]], [[121/120]], [[176/175]] (11-limit)

| Edo join 1 = 22 | Edo join 2 = 31

| Mapping = 1; 7 -3 8 2

| Generators = 7/6 | Generators tuning = 271.5 | Optimization method = CWE

| MOS scales = [[4L 1s]], [[4L 5s]], [[9L 4s]], [[9L 13s]]

| Pergen = (P8, cP5/7)

| Odd limit 1 = 7 | Mistuning 1 = 4.27 | Complexity 1 = 13

| Odd limit 2 = 11-limit 21 | Mistuning 2 = 9.32 | Complexity 2 = 22

}}

[[File:Orwell generator in 31.jpg|thumb|Martin Aurell's diagram showing Orwell[9] generated in 31 tone equal temperament.]]

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'''Orwell''' – so named because 19 steps of [[84edo]], i.e. 19\84, is a possible generator – is an excellent [[7-limit]] [[regular temperament|temperament]] and an amazing [[11-limit]] temperament because of the simplicity of [[harmonic]] [[11/1|11]].

In orwell, ~~the~~ [[3/~~1|just perfect twelfth (3/1)~~]] is divided into 7 equal steps~~. One~~ of ~~these steps represents~~ [[7/6]]~~; three represent~~ [[8/5]]. ~~Alternately,~~ the [[5/1|5th harmonic (5/1)]] divided into 3 equal steps ~~also makes a good orwell generator, being~~ [[~]][[12/7]].

In orwell, [[8/5]] is divided into three equal steps, each of which represent [[7/6]], so that [[1728/1715]] ({{S|6/S7}}) is tempered out. This means that the [[5/1|5th harmonic (5/1)]] is divided into three equal steps that represent [[~]][[12/7]]. After two 8/5's (six generators), [[9/7]] is found by [[tempering out]] the marvel comma, [[225/224]], and thus the [[3/1|just perfect twelfth (3/1)]] is divided into 7 equal steps.

In the 11-limit, two generators are equated to [[11/8]] (meaning [[99/98]] is tempered out). This means that three stacked generators makes the [[orwell tetrad]] 1–7/6–11/8–8/5, a chord in which every interval is a (tempered) 11-odd-limit consonance. Other such chords in orwell are the [[keenanismic chords]] and the [[swetismic chords]].

In the 11-limit, two generators are equated to [[15/11]] and [[11/8]] (meaning [[99/98]] and [[121/120]] are tempered out). This means that three stacked generators makes the [[orwell tetrad]] 1–7/6–11/8–8/5, a chord in which every interval is a (tempered) 11-odd-limit consonance. Other such chords in undecimal orwell are the [[keenanismic chords]] and the [[swetismic chords]]. A far more complicated mapping of 11 at 33 generators, tempering out [[441/440]] instead, is also possible and is known as [[newspeak]] temperament; these two mappings unite on 31edo.

Compatible equal temperaments include [[22edo]], [[31edo]], [[53edo]], and [[84edo]]. Orwell is in better tune in lower limits than higher ones; the [[optimal patent val]] is [[296edo]] in the 5-limit, [[137edo]] in the 7-limit, and [[53edo]] in the 11-limit. It tempers out the semicomma in the 5-limit, and so belongs to the [[semicomma family]]. In the 7-limit it tempers out [[225/224]], [[1728/1715]], [[2430/2401]] and [[6144/6125]], and in the 11-limit, [[99/98]], [[121/120]], [[176/175]], [[385/384]] and [[540/539]]. By adding [[275/273]] to the list of commas it can be extended to the 13-limit as [[Semicomma family #Orwell|tridecimal orwell]], and by adding instead [[66/65]], [[Semicomma family #Winston|winston temperament]]. See [[Orwell/Extensions]] for details about 13-limit extensions.

Compatible [[equal temperaments]] include [[22edo]], [[31edo]], [[53edo]], and [[84edo]] (though in 84edo, 11-limit orwell uses the 84e [[val]]). Orwell is in better tune in lower limits than higher ones; the [[optimal patent val]] is [[296edo]] in the 5-limit, [[137edo]] in the 7-limit, and [[53edo]] in the 11-limit.

See [[Semicomma family #Orwell]] for technical details.

See [[Semicomma family #Orwell]] for technical details. See [[Orwell extensions]] for details about 13-limit extensions.

== Interval chain ==

== Theory ==

=== Interval chain ===

Odd harmonics 1–21 and their inverses are in '''bold'''.

{| class="wikitable center-1 right-2"

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| 5

| 157.28

| 12/11~~, 11/10~~, 35/32

| 11/10, 12/11, 35/32

|-

| 6

| 428.73

| 14/11~~, 9/7~~, 32/25

| 9/7, 14/11, 32/25

|-

| 7

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| 9

| 43.10

| 49/48, 36/35, 33/32

| 33/32, 36/35, 49/48

|-

| 10

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| 63/32

|}

<nowiki/>* In 11-limit CWE tuning

<nowiki/>* In 11-limit CWE tuning, octave reduced

== Chords and harmony ==

=== Chords and harmony ===

{{See also| Chords of orwell | Functional harmony in rank-2 temperaments }}

~~{{See also~~| Functional harmony in rank-2 temperaments }}

The fundamental otonal consonance of orwell, voiced in a roughly {{w|tertian harmony|tertian}} manner, is 4:5:6:7:9:11. In terms of generator steps this is 0–(-3)–7–8–14–2, only available in a 22-tone mos. However, some subsets of this chord are way simpler, such as 8:11:12:14, which is 1–11/8–3/2–7/4 (0–2–7–8).

The fundamental otonal consonance of orwell, voiced in a roughly {{w|tertian harmony|tertian}} manner, is 4:5:6:7:9:11. In terms of generator steps this is 0–(−3)–7–8–14–2, only available in a 22-tone mos. However, some subsets of this chord are way simpler, such as 8:11:12:14, which is 1–11/8–3/2–7/4 (0–2–7–8).

The generator, ~7/6, is a septimal interval, so chords could instead be built around it as 1–7/6–3/2 (0–1–7), 1–7/4–3 (0–8–7), or tetrads such as 1–7/6–3/2–7/4 (0–1–7–8).

To 1–7/6–3/2–7/4 we may add 11/8, or to 1–11/8–3/2–7/4 we may add 7/6, to form an essentially tempered pentad, 1–7/6–11/8–3/2–7/4 (0–1–2–7–8). Its inverse is 1–12/11–9/7–3/2–12/7 (0–5–6–7–(-1)), which can serve as a minor counterpart. This is similar, but also in clear contrast to the 1–5/4–3/2 (0–4–1) and 1–6/5–3/2 (0–(−3)–1) chords of [[meantone]]. Two approaches to functional harmony thus arise.

To 1–7/6–3/2–7/4 we may add 11/8, or to 1–11/8–3/2–7/4 we may add 7/6, to form an essentially tempered pentad, 1–7/6–11/8–3/2–7/4 (0–1–2–7–8). Its inverse is 1–12/11–9/7–3/2–12/7 (0–5–6–7–(−1)), which can serve as a minor counterpart. This is similar, but also in clear contrast to the 1–5/4–3/2 (0–4–1) and 1–6/5–3/2 (0–(−3)–1) chords of [[meantone]]. Two approaches to functional harmony thus arise.

First, we can treat the septimal chords above as the basis of harmony, but swapping the roles of 3 and 7 according to their temperamental complexities (number of generator steps). Thus a "dominant" chord is either 7/6 or 12/7 over tonic; a "subdominant" chord is either 7/6 or 12/7 under tonic. This leads to an approach closely adherent to mos scales. The 9-tone mos contains a tonic and a "dominant" triad. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions, triads to pentads alike.

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== Tunings ==

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

|+ style="font-size: 105%; white-space: nowrap;" | 7-limit ~~prime~~-~~optimized~~ tunings

|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings

|-

! rowspan="2" |

! colspan="2" | Euclidean

! colspan="3" | Euclidean

|-

! Constrained

! Constrained & skewed

! Destretched

|-

! Equilateral

| CEE: ~7/6 = 271.3553{{c}}

| CSEE: ~7/6 = 271.3339{{c}}

| POEE: ~7/6 = 271.3727{{c}}

|-

! Tenney

| CTE: ~7/6 = 271.5130{{c}}

| CWE: ~7/6 = 271.5097{{c}}

| POTE: ~7/6 = 271.5087{{c}}

|-

! Benedetti, Wilson

| CBE: ~7/6 = 271.5725{{c}}

| CSBE: ~7/6 = 271.5741{{c}}

| POBE: ~7/6 = 271.5576{{c}}

|}

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

|+ style="font-size: 105%; white-space: nowrap;" | 11-limit ~~prime~~-~~optimized~~ tunings

|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings

|-

! rowspan="2" |

! colspan="2" | Euclidean

! colspan="3" | Euclidean

|-

! Constrained

! Constrained & skewed

! Destretched

|-

! Equilateral

| CEE: ~7/6 = 271.4920{{c}}

| CSEE: ~7/6 = 271.3038{{c}}

| POEE: ~7/6 = 271.1665{{c}}

|-

! Tenney

| CTE: ~7/6 = 271.5597{{c}}

| CWE: ~7/6 = 271.4552{{c}}

| POTE: ~7/6 = 271.4261{{c}}

|-

! Benedetti, Wilson

| CBE: ~7/6 = 271.5915{{c}}

| CSBE: ~7/6 = 271.5302{{c}}

| POBE: ~7/6 = 271.5174{{c}}

|}

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! 53tet

|-

| [[~~Marvel family|~~Marvel]]

| [[Marvel]]

|

| Negri, septimin, august, amavil, enneaportent

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|

|-

| [[Porwell ~~family|Hewuermity~~]]

| [[Porwell]]

|

| Triforce, armodue, twothirdtonic

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| Amity, hemischis

|-

| [[~~Orwellismic family|~~Orwellismic]]

| [[Orwellismic]]

|

| Beep, secund, infraorwell, niner

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| Quartonic, buzzard

|-

| [[~~Nuwell family|~~Nuwell]]

| [[Nuwell]]

|

| Progression, superpelog

| Quasisuper, hedgehog

| Squares, nusecond

| ~~Tricot~~, hamity

| Alphatrimot, hamity

|-

|

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| Quasisupra, hedgehog

| Squares, nusecond

| ~~Tricot~~, hamity

| Alphatrimot, hamity

|-

| [[~~Horwell family|~~Horwell]]

| [[Horwell]]

|

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; [[Claudi Meneghin]]

* [https://www.youtube.com/watch?v=zWrOiih7raY ''Orwell Canon 3 in 1 upon a Ground for Baroque Oboe, Viola, Clarinet, and Viola da Gamba''] (2024)

* [https://www.youtube.com/shorts/g7C2OrFd-nk ''Orwell Micro Trio, for Organ (Just: 7 Orwells = 1 Twelfth)''] (2025) — in open-ended Orwell tuning, but with the generator adjusted to be extremely close to 12\53, at 271.71{{c}}

; [[Herman Miller]]

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== Keyboards ==

{{See also| Orwell on an isomorphic keyboard | Lumatone mapping for orwell }}

~~{{See also~~| Lumatone mapping for orwell}}

To play interactive versions of these keyboards, check out [https://github.com/vsicurella/SuperVirtualKeyboard Vito Sicurella's plugin], which works with REAPER:

@@ Line 1: / Line 1: @@
 {{interwiki
+| en = Orwell
 | de = Orwell
-| en = Orwell
 | es =
 | ja =
+}}
+{{Infobox regtemp
+| Title = Orwell
+| Subgroups = 2.3.5.7, 2.3.5.7.11
+| Comma basis = [[225/224]], [[1728/1715]] (7-limit); <br> [[99/98]], [[121/120]], [[176/175]] (11-limit)
+| Edo join 1 = 22 | Edo join 2 = 31
+| Mapping = 1; 7 -3 8 2
+| Generators = 7/6 | Generators tuning = 271.5 | Optimization method = CWE
+| MOS scales = [[4L 1s]], [[4L 5s]], [[9L 4s]], [[9L 13s]]
+| Pergen = (P8, cP5/7)
+| Odd limit 1 = 7 | Mistuning 1 = 4.27 | Complexity 1 = 13
+| Odd limit 2 = 11-limit 21 | Mistuning 2 = 9.32 | Complexity 2 = 22
 }}
 [[File:Orwell generator in 31.jpg|thumb|Martin Aurell's diagram showing Orwell[9] generated in 31 tone equal temperament.]]
@@ Line 9: / Line 21: @@
 '''Orwell''' – so named because 19 steps of [[84edo]], i.e. 19\84, is a possible generator – is an excellent [[7-limit]] [[regular temperament|temperament]] and an amazing [[11-limit]] temperament because of the simplicity of [[harmonic]] [[11/1|11]].
-In orwell, the [[3/1|just perfect twelfth (3/1)]] is divided into 7 equal steps. One of these steps represents [[7/6]]; three represent [[8/5]]. Alternately, the [[5/1|5th harmonic (5/1)]] divided into 3 equal steps also makes a good orwell generator, being [[~]][[12/7]].
+In orwell, [[8/5]] is divided into three equal steps, each of which represent [[7/6]], so that [[1728/1715]] ({{S|6/S7}}) is tempered out. This means that the [[5/1|5th harmonic (5/1)]] is divided into three equal steps that represent [[~]][[12/7]]. After two 8/5's (six generators), [[9/7]] is found by [[tempering out]] the marvel comma, [[225/224]], and thus the [[3/1|just perfect twelfth (3/1)]] is divided into 7 equal steps.
-In the 11-limit, two generators are equated to [[11/8]] (meaning [[99/98]] is tempered out). This means that three stacked generators makes the [[orwell tetrad]] 1–7/6–11/8–8/5, a chord in which every interval is a (tempered) 11-odd-limit consonance. Other such chords in orwell are the [[keenanismic chords]] and the [[swetismic chords]].
+In the 11-limit, two generators are equated to [[15/11]] and [[11/8]] (meaning [[99/98]] and [[121/120]] are tempered out). This means that three stacked generators makes the [[orwell tetrad]] 1–7/6–11/8–8/5, a chord in which every interval is a (tempered) 11-odd-limit consonance. Other such chords in undecimal orwell are the [[keenanismic chords]] and the [[swetismic chords]]. A far more complicated mapping of 11 at 33 generators, tempering out [[441/440]] instead, is also possible and is known as [[newspeak]] temperament; these two mappings unite on 31edo.
-Compatible equal temperaments include [[22edo]], [[31edo]], [[53edo]], and [[84edo]]. Orwell is in better tune in lower limits than higher ones; the [[optimal patent val]] is [[296edo]] in the 5-limit, [[137edo]] in the 7-limit, and [[53edo]] in the 11-limit. It tempers out the semicomma in the 5-limit, and so belongs to the [[semicomma family]]. In the 7-limit it tempers out [[225/224]], [[1728/1715]], [[2430/2401]] and [[6144/6125]], and in the 11-limit, [[99/98]], [[121/120]], [[176/175]], [[385/384]] and [[540/539]]. By adding [[275/273]] to the list of commas it can be extended to the 13-limit as [[Semicomma family #Orwell|tridecimal orwell]], and by adding instead [[66/65]], [[Semicomma family #Winston|winston temperament]]. See [[Orwell/Extensions]] for details about 13-limit extensions.
+Compatible [[equal temperaments]] include [[22edo]], [[31edo]], [[53edo]], and [[84edo]] (though in 84edo, 11-limit orwell uses the 84e [[val]]). Orwell is in better tune in lower limits than higher ones; the [[optimal patent val]] is [[296edo]] in the 5-limit, [[137edo]] in the 7-limit, and [[53edo]] in the 11-limit.
-See [[Semicomma family #Orwell]] for technical details.
+See [[Semicomma family #Orwell]] for technical details. See [[Orwell extensions]] for details about 13-limit extensions.
-== Interval chain ==
+== Theory ==
+=== Interval chain ===
 Odd harmonics 1–21 and their inverses are in '''bold'''.
 {| class="wikitable center-1 right-2"
@@ Line 47: / Line 60: @@
 | 5
 | 157.28
-| 12/11, 11/10, 35/32
+| 11/10, 12/11, 35/32
 |-
 | 6
 | 428.73
-| 14/11, 9/7, 32/25
+| 9/7, 14/11, 32/25
 |-
 | 7
@@ Line 63: / Line 76: @@
 | 9
 | 43.10
-| 49/48, 36/35, 33/32
+| 33/32, 36/35, 49/48
 |-
 | 10
@@ Line 117: / Line 130: @@
 | 63/32
 |}
-<nowiki/>* In 11-limit CWE tuning
+<nowiki/>* In 11-limit CWE tuning, octave reduced
-== Chords and harmony ==
+=== Chords and harmony ===
-{{Main| Chords of orwell }}
+{{See also| Chords of orwell | Functional harmony in rank-2 temperaments }}
-{{See also| Functional harmony in rank-2 temperaments }}
-The fundamental otonal consonance of orwell, voiced in a roughly {{w|tertian harmony|tertian}} manner, is 4:5:6:7:9:11. In terms of generator steps this is 0–(-3)–7–8–14–2, only available in a 22-tone mos. However, some subsets of this chord are way simpler, such as 8:11:12:14, which is 1–11/8–3/2–7/4 (0–2–7–8).
+The fundamental otonal consonance of orwell, voiced in a roughly {{w|tertian harmony|tertian}} manner, is 4:5:6:7:9:11. In terms of generator steps this is 0–(−3)–7–8–14–2, only available in a 22-tone mos. However, some subsets of this chord are way simpler, such as 8:11:12:14, which is 1–11/8–3/2–7/4 (0–2–7–8).
 The generator, ~7/6, is a septimal interval, so chords could instead be built around it as 1–7/6–3/2 (0–1–7), 1–7/4–3 (0–8–7), or tetrads such as 1–7/6–3/2–7/4 (0–1–7–8).
-To 1–7/6–3/2–7/4 we may add 11/8, or to 1–11/8–3/2–7/4 we may add 7/6, to form an essentially tempered pentad, 1–7/6–11/8–3/2–7/4 (0–1–2–7–8). Its inverse is 1–12/11–9/7–3/2–12/7 (0–5–6–7–(-1)), which can serve as a minor counterpart. This is similar, but also in clear contrast to the 1–5/4–3/2 (0–4–1) and 1–6/5–3/2 (0–(−3)–1) chords of [[meantone]]. Two approaches to functional harmony thus arise.
+To 1–7/6–3/2–7/4 we may add 11/8, or to 1–11/8–3/2–7/4 we may add 7/6, to form an essentially tempered pentad, 1–7/6–11/8–3/2–7/4 (0–1–2–7–8). Its inverse is 1–12/11–9/7–3/2–12/7 (0–5–6–7–(−1)), which can serve as a minor counterpart. This is similar, but also in clear contrast to the 1–5/4–3/2 (0–4–1) and 1–6/5–3/2 (0–(−3)–1) chords of [[meantone]]. Two approaches to functional harmony thus arise.
 First, we can treat the septimal chords above as the basis of harmony, but swapping the roles of 3 and 7 according to their temperamental complexities (number of generator steps). Thus a "dominant" chord is either 7/6 or 12/7 over tonic; a "subdominant" chord is either 7/6 or 12/7 under tonic. This leads to an approach closely adherent to mos scales. The 9-tone mos contains a tonic and a "dominant" triad. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions, triads to pentads alike.
@@ Line 274: / Line 286: @@
 == Tunings ==
 {| class="wikitable mw-collapsible mw-collapsed"
-|+ style="font-size: 105%; white-space: nowrap;" | 7-limit prime-optimized tunings
+|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
 |-
 ! rowspan="2" |
-! colspan="2" | Euclidean
+! colspan="3" | Euclidean
 |-
 ! Constrained
 ! Constrained & skewed
+! Destretched
 |-
 ! Equilateral
 | CEE: ~7/6 = 271.3553{{c}}
 | CSEE: ~7/6 = 271.3339{{c}}
+| POEE: ~7/6 = 271.3727{{c}}
 |-
 ! Tenney
 | CTE: ~7/6 = 271.5130{{c}}
 | CWE: ~7/6 = 271.5097{{c}}
+| POTE: ~7/6 = 271.5087{{c}}
 |-
 ! Benedetti, <br>Wilson
 | CBE: ~7/6 = 271.5725{{c}}
 | CSBE: ~7/6 = 271.5741{{c}}
+| POBE: ~7/6 = 271.5576{{c}}
 |}
 {| class="wikitable mw-collapsible mw-collapsed"
-|+ style="font-size: 105%; white-space: nowrap;" | 11-limit prime-optimized tunings
+|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
 |-
 ! rowspan="2" |
-! colspan="2" | Euclidean
+! colspan="3" | Euclidean
 |-
 ! Constrained
 ! Constrained & skewed
+! Destretched
 |-
 ! Equilateral
 | CEE: ~7/6 = 271.4920{{c}}
 | CSEE: ~7/6 = 271.3038{{c}}
+| POEE: ~7/6 = 271.1665{{c}}
 |-
 ! Tenney
 | CTE: ~7/6 = 271.5597{{c}}
 | CWE: ~7/6 = 271.4552{{c}}
+| POTE: ~7/6 = 271.4261{{c}}
 |-
 ! Benedetti, <br>Wilson
 | CBE: ~7/6 = 271.5915{{c}}
 | CSBE: ~7/6 = 271.5302{{c}}
+| POBE: ~7/6 = 271.5174{{c}}
 |}
@@ Line 506: / Line 526: @@
 ! 53tet
 |-
-| [[Marvel family|Marvel]]
+| [[Marvel]]
 |
 | Negri, septimin, august,<br>amavil, enneaportent
@@ Line 534: / Line 554: @@
 |
 |-
-| [[Porwell family|Hewuermity]]
+| [[Porwell]]
 |
 | Triforce, armodue,<br>twothirdtonic
@@ Line 555: / Line 575: @@
 | Amity, hemischis
 |-
-| [[Orwellismic family|Orwellismic]]
+| [[Orwellismic]]
 |
 | Beep, secund, infraorwell,<br>niner
@@ Line 576: / Line 596: @@
 | Quartonic, buzzard
 |-
-| [[Nuwell family|Nuwell]]
+| [[Nuwell]]
 |
 | Progression, superpelog
 | Quasisuper, hedgehog
 | Squares, nusecond
-| Tricot, hamity
+| Alphatrimot, hamity
 |-
 |
@@ Line 588: / Line 608: @@
 | Quasisupra, hedgehog
 | Squares, nusecond
-| Tricot, hamity
+| Alphatrimot, hamity
 |-
-| [[Horwell family|Horwell]]
+| [[Horwell]]
 |
 |
@@ Line 627: / Line 647: @@
 ; [[Claudi Meneghin]]
 * [https://www.youtube.com/watch?v=zWrOiih7raY ''Orwell Canon 3 in 1 upon a Ground for Baroque Oboe, Viola, Clarinet, and Viola da Gamba''] (2024)
+* [https://www.youtube.com/shorts/g7C2OrFd-nk ''Orwell Micro Trio, for Organ (Just: 7 Orwells = 1 Twelfth)''] (2025) &mdash; in open-ended Orwell tuning, but with the generator adjusted to be extremely close to 12\53, at 271.71{{c}}
 ; [[Herman Miller]]
@@ Line 643: / Line 664: @@
 == Keyboards ==
-{{See also| Orwell on an isomorphic keyboard }}
+{{See also| Orwell on an isomorphic keyboard | Lumatone mapping for orwell }}
-{{See also| Lumatone mapping for orwell}}
 To play interactive versions of these keyboards, check out [https://github.com/vsicurella/SuperVirtualKeyboard Vito Sicurella's plugin], which works with REAPER: