Semicomma family: Difference between revisions

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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.

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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 {{nowrap| 31 & 159 }} temperament (triwell) ~~with wedgie {{multival| 21 -9 -7 -63 -70 9 }}~~, or

* 1029/1024, leading to the {{nowrap| 31 & 159 }} temperament (triwell), or

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

* 2401/2400, giving the {{nowrap| 31 & 243 }} temperament (quadrawell), or

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

* 4375/4374, giving the {{nowrap| 53 & 243 }} temperament (sabric).

== Orwell ==

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~~{{Multival|legend=1| 7 -3 8 -21 -7 27 }}~~

[[Optimal tuning]]s:

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* [[7-odd-limit]]: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}

: {{monzo list| 1 0 0 0 | 14/11 0 -7/11 7/11 | 27/11 0 3/11 -3/11 | 27/11 0 -8/11 8/11 }}

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.7/5

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5

* [[9-odd-limit]]: ~7/6 = {{monzo| 3/17 2/17 -1/17 }}

: {{monzo list| 1 0 0 0 | 21/17 14/17 -7/17 0 | 42/17 -6/17 3/17 0 | 41/17 16/17 -8/17 0 }}

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.9/5

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5

[[Tuning ranges]]:

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Mapping: {{mapping| 1 0 3 1 3 | 0 7 -3 8 2 }}

~~Wedgie: {{multival| 7 -3 8 2 -21 -7 -21 27 15 -22 }}~~

Optimal tunings:

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* 11-odd-limit: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 14/11 0 -7/11 7/11 0 }}, {{monzo| 27/11 0 3/11 -3/11 0 }}, {{monzo| 27/11 0 -8/11 8/11 0 }}, {{monzo| 37/11 0 -2/11 2/11 0 }}]

: ~~Eigenmonzo (unchanged~~-interval) basis: 2.7/5

: Unchanged-interval (eigenmonzo) basis: 2.7/5

Tuning ranges:

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~~{{Multival|legend=1| 7 -3 61 -21 77 150 }}~~

[[Optimal tuning]]s:

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: mapping generators: ~2, ~448/375

~~{{Multival|legend=1| 21 -9 -7 -63 -70 9 }}~~

[[Optimal tuning]]s:

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: mapping generators: ~2, ~8/7

~~{{Multival|legend=1| 28 -12 1 -84 -77 36 }}~~

[[Optimal tuning]]s:

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: mapping generators: ~2, ~2625/2048

~~{{Multival|legend=1| 35 -15 9 -105 -84 63 }}~~

[[Optimal tuning]]s:

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: mapping generators: ~2, ~405/392

~~{{Multival|legend=1| 35 -15 62 -105 0 186 }}~~

[[Optimal tuning]]s:

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[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Semicomma family| ]]

[[Category:Rank 2]]

[[Category:Orson]]

[[Category:Orwell]]

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+{{Technical data page}}
 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.
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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 {{nowrap| 31 & 159 }} temperament (triwell) with wedgie {{multival| 21 -9 -7 -63 -70 9 }}, or
+* 1029/1024, leading to the {{nowrap| 31 & 159 }} temperament (triwell), or
-* 2401/2400, giving the {{nowrap| 31 & 243 }} temperament (quadrawell) with wedgie {{multival| 28 -12 1 -84 -77 36 }}, or
+* 2401/2400, giving the {{nowrap| 31 & 243 }} temperament (quadrawell), or
-* 4375/4374, giving the {{nowrap| 53 & 243 }} temperament (sabric) with wedgie {{multival| 7 -3 61 -21 77 150 }}.
+* 4375/4374, giving the {{nowrap| 53 & 243 }} temperament (sabric).
 == Orwell ==
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 {{Mapping|legend=1| 1 0 3 1 | 0 7 -3 8 }}
-{{Multival|legend=1| 7 -3 8 -21 -7 27 }}
 [[Optimal tuning]]s:
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 * [[7-odd-limit]]: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}
 : {{monzo list| 1 0 0 0 | 14/11 0 -7/11 7/11 | 27/11 0 3/11 -3/11 | 27/11 0 -8/11 8/11 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
 * [[9-odd-limit]]: ~7/6 = {{monzo| 3/17 2/17 -1/17 }}
 : {{monzo list| 1 0 0 0 | 21/17 14/17 -7/17 0 | 42/17 -6/17 3/17 0 | 41/17 16/17 -8/17 0 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5
 [[Tuning ranges]]:
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 Mapping: {{mapping| 1 0 3 1 3 | 0 7 -3 8 2 }}
-Wedgie: {{multival| 7 -3 8 2 -21 -7 -21 27 15 -22 }}
 Optimal tunings:
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 * 11-odd-limit: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 14/11 0 -7/11 7/11 0 }}, {{monzo| 27/11 0 3/11 -3/11 0 }}, {{monzo| 27/11 0 -8/11 8/11 0 }}, {{monzo| 37/11 0 -2/11 2/11 0 }}]
-: Eigenmonzo (unchanged-interval) basis: 2.7/5
+: Unchanged-interval (eigenmonzo) basis: 2.7/5
 Tuning ranges:
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 {{Mapping|legend=1| 1 0 3 -11 | 0 7 -3 61 }}
-{{Multival|legend=1| 7 -3 61 -21 77 150 }}
 [[Optimal tuning]]s:
@@ Line 242: / Line 237: @@
 : mapping generators: ~2, ~448/375
-{{Multival|legend=1| 21 -9 -7 -63 -70 9 }}
 [[Optimal tuning]]s:
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 : mapping generators: ~2, ~8/7
-{{Multival|legend=1| 28 -12 1 -84 -77 36 }}
 [[Optimal tuning]]s:
@@ Line 318: / Line 309: @@
 : mapping generators: ~2, ~2625/2048
-{{Multival|legend=1| 35 -15 9 -105 -84 63 }}
 [[Optimal tuning]]s:
@@ Line 356: / Line 345: @@
 : mapping generators: ~2, ~405/392
-{{Multival|legend=1| 35 -15 62 -105 0 186 }}
 [[Optimal tuning]]s:
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 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Semicomma family| ]] <!-- main article -->
 [[Category:Rank 2]]
 [[Category:Orson]]
 [[Category:Orwell]]