Orwell: Difference between revisions

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

{{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

| Generator = 7/6 | Generator tuning = 271.5 | Optimization method = CTE

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

| Mapping = 1; 7 -3 8 2

| Pergen = (P8, cP5/7)

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

| Odd limit 2 = (L11) 21 | Mistuning 2 = ??? | Complexity 2 = 22

}}

'''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]] (alternatively, the [[5/1|5th harmonic (5/1)]] is divided into 3 equal steps that represent [[~]][[12/7]]), so that [[1728/1715]] = {{S|6/S7}} is tempered out. After two 8/5s (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]].

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.

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 }}
 [[File:Orwell generator in 31.jpg|thumb|Martin Aurell's diagram showing Orwell[9] generated in 31 tone equal temperament.]]
+{{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
+| Generator = 7/6 | Generator tuning = 271.5 | Optimization method = CTE
+| MOS scales = [[4L 1s]], [[4L 5s]], [[9L 4s]], [[9L 13s]]
+| Mapping = 1; 7 -3 8 2
+| Pergen = (P8, cP5/7)
+| Odd limit 1 = 7 | Mistuning 1 = ??? | Complexity 1 = 13
+| Odd limit 2 = (L11) 21 | Mistuning 2 = ??? | Complexity 2 = 22
+}}
 '''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]] (alternatively, the [[5/1|5th harmonic (5/1)]] is divided into 3 equal steps that represent [[~]][[12/7]]), so that [[1728/1715]] = {{S|6/S7}} is tempered out. After two 8/5s (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]].
 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.