Semicomma family: Difference between revisions

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The 5-limit parent comma for the '''semicomma family''' is the [[semicomma]]~~, 2109375/2097152 =~~ {{monzo| -21 3 7 }}. This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor sixths.

The [[5-limit]] parent [[comma]] for the '''semicomma family''' of [[regular temperament|temperaments]] is the [[semicomma]] ({{monzo|legend=1| -21 3 7 }}, [[ratio]]: 2109375/2097152). This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor sixths.

== Orson ==

'''Orson''', first discovered by [[Erv Wilson]], is the [[5-limit]] temperament tempering out the semicomma. It has a [[generator]] of [[75/64]], which is sharper than [[7/6]] by [[225/224]] when untempered, and less sharp than that in any good orson tempering, for example [[53edo]] or [[84edo]]. These give tunings to the generator which are sharp of 7/6 by less than five [[cent]]s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.

'''Orson''', first discovered by [[Erv Wilson]]{{citation needed}}, is the [[5-limit]] temperament [[tempering out]] the semicomma. It has a [[generator]] of [[~]][[75/64]], seven of which give the [[3/1|perfect twelfth]]; its [[ploidacot]] is alpha-heptacot. The generator is sharper than [[7/6]] by [[225/224]] when untempered, and less sharp than that in any good orson tempering, for example [[53edo]] or [[84edo]]. These give tunings to the generator which are sharp of 7/6 by less than five [[cent]]s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.

[[Subgroup]]: 2.3.5

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* 5-odd-limit [[diamond monotone]]: ~75/64 = [257.143, 276.923] (3\14 to 3\13)

* 5-odd-limit [[diamond tradeoff]]: ~75/64 = [271.229, 271.708] (1/3-comma to 2/7-comma)

* 5-odd-limit diamond monotone and tradeoff: ~75/64 = [271.229, 271.708]

{{Optimal ET sequence|legend=1| 22, 31, 53, 190, 243, 296, 645c }}

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=== Overview to extensions ===

The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. Adding 65625/65536 (or 225/224) leads to orwell, but we could also add

* 1029/1024, leading to the 31&~~amp;~~159 temperament (triwell) with wedgie {{multival| 21 -9 -7 -63 -70 9 }}, or

* 1029/1024, leading to the {{nowrap| 31 & 159 }} temperament (triwell) with wedgie {{multival| 21 -9 -7 -63 -70 9 }}, or

* 2401/2400, giving the 31&~~amp;~~243 temperament (quadrawell) with wedgie {{multival| 28 -12 1 -84 -77 36 }}, or

* 2401/2400, giving the {{nowrap| 31 & 243 }} temperament (quadrawell) with wedgie {{multival| 28 -12 1 -84 -77 36 }}, or

* 4375/4374, giving the 53&~~amp;~~243 temperament (sabric) with wedgie {{multival| 7 -3 61 -21 77 150 }}.

* 4375/4374, giving the {{nowrap| 53 & 243 }} temperament (sabric) with wedgie {{multival| 7 -3 61 -21 77 150 }}.

== Orwell ==

So called because 19\84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It is compatible with [[22edo|22]], [[31edo|31]], [[53edo|53]] and [[84edo|84]] equal, and may be described as the 22 &~~amp;~~ 31 temperament. It is a good system in the [[7-limit]] and naturally extends into the [[11-limit]]. [[84edo]], with the 19\84 generator, provides a good tuning for the 5-, 7- and 11-limit, but it does use its second-closest approximation to 11. However, the 19\84 generator is remarkably close to the 11-limit [[POTE tuning]], as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. [[53edo]] might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of it slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out [[2430/2401]], the nuwell comma, [[1728/1715]], the orwellisma, [[225/224]], the septimal kleisma, and [[6144/6125]], the porwell comma.

So called because 19\84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It is compatible with [[22edo|22]], [[31edo|31]], [[53edo|53]] and [[84edo|84]] equal, and may be described as the {{nowrap| 22 & 31 }} temperament. It is a good system in the [[7-limit]] and naturally extends into the [[11-limit]]. [[84edo]], with the 19\84 generator, provides a good tuning for the 5-, 7- and 11-limit, but it does use its second-closest approximation to 11. However, the 19\84 generator is remarkably close to the 11-limit [[POTE tuning]], as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. [[53edo]] might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of it slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out [[2430/2401]], the nuwell comma, [[1728/1715]], the orwellisma, [[225/224]], the septimal kleisma, and [[6144/6125]], the porwell comma.

The 11-limit version of orwell tempers out [[99/98]], which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered ~~1-7~~/~~6-11~~/~~8-8~~/5 chord is natural to orwell.

The 11-limit version of orwell tempers out [[99/98]], which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1–7/6–11/8–8/5 chord is natural to orwell.

Orwell has [[mos scale]]s of size 9, 13, 22 and 31. The 9-note mos is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has [[Retuning 12edo to Orwell9|considerable harmonic resources]] despite its absence of 5-limit triads. The 13-note mos has those, and of course the 22- and 31-note mos are very well supplied with everything.

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* 7-odd-limit [[diamond tradeoff]]: ~7/6 = [266.871, 271.708]

* 9-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]

* 7-odd-limit diamond monotone and tradeoff: ~7/6 = [266.871, 271.708]

* 9-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 272.514]

[[Algebraic generator]]: Sabra3, the real root of 12''x<sup>3</sup> - 7''x'' - 48.

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* 11-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)

* 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]

* 11-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 272.727]

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* 13- and 15-odd-limit diamond monotone: ~7/6 = [270.968, 271.698] (7\31 to 12\53)

* 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]

* 13- and 15-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 271.698]

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* 13- and 15-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)

* 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 281.691]

* 13- and 15-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 272.727]

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* 13- and 15-odd-limit diamond monotone: ~13/12 = [135.484, 136.364] (7\62 to 5\44)

* 13- and 15-odd-limit diamond tradeoff: ~13/12 = [128.298, 138.573]

* 13- and 15-odd-limit diamond monotone and tradeoff: ~13/12 = [135.484, 136.364]

{{Optimal ET sequence|legend=1| 9, 35bd, 44, 53, 62, 115ef, 168eef }}

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* 11-odd-limit diamond monotone: ~7/6 = [270.968, 271.698] (7\31 to 12\53)

* 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]

* 11-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 271.698]

{{Optimal ET sequence|legend=1| 31, 84, 115, 376b, 491bd, 606bde }}

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== Sabric ==

The sabric temperament (53 &~~amp;~~ 190) tempers out the [[4375/4374|ragisma (4375/4374)]]. It is so named because it is closely related to the ''Sabra2 tuning'' (generator: 271.607278 cents).

The sabric temperament ({{nowrap| 53 & 190 }}) tempers out the [[4375/4374|ragisma (4375/4374)]]. It is so named because it is closely related to the ''Sabra2 tuning'' (generator: 271.607278 cents).

[[Subgroup]]: 2.3.5.7

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== Triwell ==

The triwell temperament (31 &~~amp;~~ 159) slices orwell major sixth ~128/75 into three generators, nine of which give the fifth harmonic.

The triwell temperament ({{nowrap| 31 & 159 }}) slices orwell major sixth ~128/75 into three generators, nine of which give the fifth harmonic.

[[Subgroup]]: 2.3.5.7

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== Quadrawell ==

The ''quadrawell'' temperament (31&~~amp;~~212) has an [[8/7]] generator of about 232 cents, twelve of which give the fifth harmonic.

The ''quadrawell'' temperament ({{nowrap| 31 & 212 }}) has an [[8/7]] generator of about 232 cents, twelve of which give the fifth harmonic.

[[Subgroup]]: 2.3.5.7

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== Rainwell ==

The ''rainwell'' temperament (31 &~~amp;~~ 265) tempers out the mirkwai comma, 16875/16807 and the [[rainy comma]], 2100875/2097152.

The ''rainwell'' temperament ({{nowrap| 31 & 265 }}) tempers out the mirkwai comma, 16875/16807 and the [[rainy comma]], 2100875/2097152.

[[Subgroup]]: 2.3.5.7

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== Quinwell ==

The quinwell temperament (22 &~~amp;~~ 243) slices orwell minor third into five generators and tempers out the wizma, 420175/419904.

The quinwell temperament ({{nowrap| 22 & 243 }}) slices orwell minor third into five generators and tempers out the wizma, 420175/419904.

[[Subgroup]]: 2.3.5.7

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-The 5-limit parent comma for the '''semicomma family''' is the [[semicomma]], 2109375/2097152 = {{monzo| -21 3 7 }}. This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor sixths.
+The [[5-limit]] parent [[comma]] for the '''semicomma family''' of [[regular temperament|temperaments]] is the [[semicomma]] ({{monzo|legend=1| -21 3 7 }}, [[ratio]]: 2109375/2097152). This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor sixths.
 == Orson ==
-'''Orson''', first discovered by [[Erv Wilson]], is the [[5-limit]] temperament tempering out the semicomma. It has a [[generator]] of [[75/64]], which is sharper than [[7/6]] by [[225/224]] when untempered, and less sharp than that in any good orson tempering, for example [[53edo]] or [[84edo]]. These give tunings to the generator which are sharp of 7/6 by less than five [[cent]]s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.
+'''Orson''', first discovered by [[Erv Wilson]]{{citation needed}}, is the [[5-limit]] temperament [[tempering out]] the semicomma. It has a [[generator]] of [[~]][[75/64]], seven of which give the [[3/1|perfect twelfth]]; its [[ploidacot]] is alpha-heptacot. The generator is sharper than [[7/6]] by [[225/224]] when untempered, and less sharp than that in any good orson tempering, for example [[53edo]] or [[84edo]]. These give tunings to the generator which are sharp of 7/6 by less than five [[cent]]s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.
 [[Subgroup]]: 2.3.5
@@ Line 17: / Line 17: @@
 * 5-odd-limit [[diamond monotone]]: ~75/64 = [257.143, 276.923] (3\14 to 3\13)
 * 5-odd-limit [[diamond tradeoff]]: ~75/64 = [271.229, 271.708] (1/3-comma to 2/7-comma)
-* 5-odd-limit diamond monotone and tradeoff: ~75/64 = [271.229, 271.708]
 {{Optimal ET sequence|legend=1| 22, 31, 53, 190, 243, 296, 645c }}
@@ Line 25: / Line 24: @@
 === Overview to extensions ===
 The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. Adding 65625/65536 (or 225/224) leads to orwell, but we could also add
-* 1029/1024, leading to the 31&amp;159 temperament (triwell) with wedgie {{multival| 21 -9 -7 -63 -70 9 }}, or
+* 1029/1024, leading to the {{nowrap| 31 & 159 }} temperament (triwell) with wedgie {{multival| 21 -9 -7 -63 -70 9 }}, or
-* 2401/2400, giving the 31&amp;243 temperament (quadrawell) with wedgie {{multival| 28 -12 1 -84 -77 36 }}, or
+* 2401/2400, giving the {{nowrap| 31 & 243 }} temperament (quadrawell) with wedgie {{multival| 28 -12 1 -84 -77 36 }}, or
-* 4375/4374, giving the 53&amp;243 temperament (sabric) with wedgie {{multival| 7 -3 61 -21 77 150 }}.
+* 4375/4374, giving the {{nowrap| 53 & 243 }} temperament (sabric) with wedgie {{multival| 7 -3 61 -21 77 150 }}.
 == Orwell ==
 {{Main| Orwell }}
-So called because 19\84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It is compatible with [[22edo|22]], [[31edo|31]], [[53edo|53]] and [[84edo|84]] equal, and may be described as the 22 &amp; 31 temperament. It is a good system in the [[7-limit]] and naturally extends into the [[11-limit]]. [[84edo]], with the 19\84 generator, provides a good tuning for the 5-, 7- and 11-limit, but it does use its second-closest approximation to 11. However, the 19\84 generator is remarkably close to the 11-limit [[POTE tuning]], as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. [[53edo]] might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of it slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out [[2430/2401]], the nuwell comma, [[1728/1715]], the orwellisma, [[225/224]], the septimal kleisma, and [[6144/6125]], the porwell comma.
+So called because 19\84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It is compatible with [[22edo|22]], [[31edo|31]], [[53edo|53]] and [[84edo|84]] equal, and may be described as the {{nowrap| 22 & 31 }} temperament. It is a good system in the [[7-limit]] and naturally extends into the [[11-limit]]. [[84edo]], with the 19\84 generator, provides a good tuning for the 5-, 7- and 11-limit, but it does use its second-closest approximation to 11. However, the 19\84 generator is remarkably close to the 11-limit [[POTE tuning]], as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. [[53edo]] might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of it slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out [[2430/2401]], the nuwell comma, [[1728/1715]], the orwellisma, [[225/224]], the septimal kleisma, and [[6144/6125]], the porwell comma.
-The 11-limit version of orwell tempers out [[99/98]], which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell.
+The 11-limit version of orwell tempers out [[99/98]], which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1–7/6–11/8–8/5 chord is natural to orwell.
 Orwell has [[mos scale]]s of size 9, 13, 22 and 31. The 9-note mos is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has [[Retuning 12edo to Orwell9|considerable harmonic resources]] despite its absence of 5-limit triads. The 13-note mos has those, and of course the 22- and 31-note mos are very well supplied with everything.
@@ Line 61: / Line 60: @@
 * 7-odd-limit [[diamond tradeoff]]: ~7/6 = [266.871, 271.708]
 * 9-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]
-* 7-odd-limit diamond monotone and tradeoff: ~7/6 = [266.871, 271.708]
-* 9-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 272.514]
 [[Algebraic generator]]: Sabra3, the real root of 12''x<sup>3</sup> - 7''x'' - 48.
@@ Line 89: / Line 86: @@
 * 11-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
 * 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]
-* 11-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 272.727]
 {{Optimal ET sequence|legend=1| 9, 22, 31, 53, 84e }}
@@ Line 107: / Line 103: @@
 * 13- and 15-odd-limit diamond monotone: ~7/6 = [270.968, 271.698] (7\31 to 12\53)
 * 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]
-* 13- and 15-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 271.698]
 {{Optimal ET sequence|legend=1| 22, 31, 53, 84e, 137e }}
@@ Line 138: / Line 133: @@
 * 13- and 15-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
 * 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 281.691]
-* 13- and 15-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 272.727]
 {{Optimal ET sequence|legend=1| 22f, 31 }}
@@ Line 158: / Line 152: @@
 * 13- and 15-odd-limit diamond monotone: ~13/12 = [135.484, 136.364] (7\62 to 5\44)
 * 13- and 15-odd-limit diamond tradeoff: ~13/12 = [128.298, 138.573]
-* 13- and 15-odd-limit diamond monotone and tradeoff: ~13/12 = [135.484, 136.364]
 {{Optimal ET sequence|legend=1| 9, 35bd, 44, 53, 62, 115ef, 168eef }}
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 * 11-odd-limit diamond monotone: ~7/6 = [270.968, 271.698] (7\31 to 12\53)
 * 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]
-* 11-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 271.698]
 {{Optimal ET sequence|legend=1| 31, 84, 115, 376b, 491bd, 606bde }}
@@ Line 196: / Line 188: @@
 == Sabric ==
-The sabric temperament (53 &amp; 190) tempers out the [[4375/4374|ragisma (4375/4374)]]. It is so named because it is closely related to the ''Sabra2 tuning'' (generator: 271.607278 cents).
+The sabric temperament ({{nowrap| 53 & 190 }}) tempers out the [[4375/4374|ragisma (4375/4374)]]. It is so named because it is closely related to the ''Sabra2 tuning'' (generator: 271.607278 cents).
 [[Subgroup]]: 2.3.5.7
@@ Line 213: / Line 205: @@
 == Triwell ==
-The triwell temperament (31 &amp; 159) slices orwell major sixth ~128/75 into three generators, nine of which give the fifth harmonic.
+The triwell temperament ({{nowrap| 31 & 159 }}) slices orwell major sixth ~128/75 into three generators, nine of which give the fifth harmonic.
 [[Subgroup]]: 2.3.5.7
@@ Line 243: / Line 235: @@
 == Quadrawell ==
-The ''quadrawell'' temperament (31&amp;212) has an [[8/7]] generator of about 232 cents, twelve of which give the fifth harmonic.
+The ''quadrawell'' temperament ({{nowrap| 31 & 212 }}) has an [[8/7]] generator of about 232 cents, twelve of which give the fifth harmonic.
 [[Subgroup]]: 2.3.5.7
@@ Line 273: / Line 265: @@
 == Rainwell ==
-The ''rainwell'' temperament (31 &amp; 265) tempers out the mirkwai comma, 16875/16807 and the [[rainy comma]], 2100875/2097152.
+The ''rainwell'' temperament ({{nowrap| 31 & 265 }}) tempers out the mirkwai comma, 16875/16807 and the [[rainy comma]], 2100875/2097152.
 [[Subgroup]]: 2.3.5.7
@@ Line 303: / Line 295: @@
 == Quinwell ==
-The quinwell temperament (22 &amp; 243) slices orwell minor third into five generators and tempers out the wizma, 420175/419904.
+The quinwell temperament ({{nowrap| 22 & 243 }}) slices orwell minor third into five generators and tempers out the wizma, 420175/419904.
 [[Subgroup]]: 2.3.5.7