Valinorsmic clan: Difference between revisions

Line 25:

=== Valimar ===

This extension to the no-threes 13-limit is motivated by extending the valinorsmic chain of 5/4s, where two form 11/7, so that four form 16/13. This logic is supported by rank-2s such as tridecimal didacus and the 13-limit [[magus]]/[[amigo]] extensions, as well as [[~~Orwell extensions|~~winston]] and [[~~Valentine extensions|~~valentino]]. This is also natural considering [[S-expression]]s, noting that 176/175 = S8/S10 is (nontrivially) equivalent to (~~S11*S12~~)/(~~S14*S15~~), and this extension tempers out ~~S11/S14 =~~ [[1573/1568]], and ~~S12/S15 =~~ [[3584/3575]]. It remains well-tuned in ~~ETs~~ with optimal valinorsmic 5/4s (such as [[37edo|37]], [[40edo|40]], and [[43edo|43]]).

This extension to the no-threes 13-limit is motivated by extending the valinorsmic chain of 5/4's, where two form 11/7, so that four form 16/13. This logic is supported by rank-2 temperaments such as tridecimal didacus and the 13-limit [[magus]]/[[amigo]] extensions, as well as [[winston]] and [[valentino]]. This is also natural considering [[S-expression]]s, noting that 176/175 (S8/S10) is nontrivially equivalent to (S11⋅S12)/(S14⋅S15), and this extension tempers out [[1573/1568]] (S11/S14), and [[3584/3575]] (S12/S15). It remains well-tuned in equal temperaments with optimal valinorsmic 5/4's (such as [[37edo|37]], [[40edo|40]], and [[43edo|43]]).

[[Subgroup]]: 2.5.7.11.13

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: error map: {{val| 0.000 +3.152 +2.691 -0.869 +1.607 }}

{{Optimal ET sequence|legend=1| 6, 9, 15f, 22f, 28, 31, 37, 231cd, 268cd }}

[[Badness]] (Sintel): 0.336

=== Valarin ===

This extension equates two, ideally slightly flattened, [[8/7]]s to [[13/10]], which is also natural given valinorsmic's tuning tendency towards sharpening [[7/4]]. This extension is supported by rank-2 temperaments including tridecimal didacus, [[~~Orwell~~ extensions|tridecimal orwell]], [[~~Valentine extensions|~~lupercalia]] (an extension of [[valentine]]), and [[llywelyn]].

This extension equates two, ideally slightly flattened, [[8/7]]'s to [[13/10]], which is also natural given valinorsmic's tuning tendency towards sharpening [[7/4]]. This extension is supported by rank-2 temperaments including tridecimal didacus, [[orwell extensions|tridecimal orwell]], [[lupercalia]] (an extension of [[valentine]]), and [[llywelyn]].

[[Subgroup]]: 2.5.7.11.13

Line 60:

: error map: {{val| 0.000 +2.875 +4.003 -0.110 +3.003 }}

{{Optimal ET sequence|legend=1| 6, 10e, 15, 16, 21, 31, 37, 206cdf, 243cdf, 280cdf, 317cddf, 354ccddff }}

[[Badness]] (Sintel): 0.227

@@ Line 25: / Line 25: @@
 === Valimar ===
-This extension to the no-threes 13-limit is motivated by extending the valinorsmic chain of 5/4s, where two form 11/7, so that four form 16/13. This logic is supported by rank-2s such as tridecimal didacus and the 13-limit [[magus]]/[[amigo]] extensions, as well as [[Orwell extensions|winston]] and [[Valentine extensions|valentino]]. This is also natural considering [[S-expression]]s, noting that 176/175 = S8/S10 is (nontrivially) equivalent to (S11*S12)/(S14*S15), and this extension tempers out S11/S14 = [[1573/1568]], and S12/S15 = [[3584/3575]]. It remains well-tuned in ETs with optimal valinorsmic 5/4s (such as [[37edo|37]], [[40edo|40]], and [[43edo|43]]).
+This extension to the no-threes 13-limit is motivated by extending the valinorsmic chain of 5/4's, where two form 11/7, so that four form 16/13. This logic is supported by rank-2 temperaments such as tridecimal didacus and the 13-limit [[magus]]/[[amigo]] extensions, as well as [[winston]] and [[valentino]]. This is also natural considering [[S-expression]]s, noting that 176/175 (S8/S10) is nontrivially equivalent to (S11⋅S12)/(S14⋅S15), and this extension tempers out [[1573/1568]] (S11/S14), and [[3584/3575]] (S12/S15). It remains well-tuned in equal temperaments with optimal valinorsmic 5/4's (such as [[37edo|37]], [[40edo|40]], and [[43edo|43]]).
 [[Subgroup]]: 2.5.7.11.13
@@ Line 40: / Line 40: @@
 : error map: {{val| 0.000 +3.152 +2.691 -0.869 +1.607 }}
-{{Optimal ET sequence|legend=1| TBD }}
+{{Optimal ET sequence|legend=1| 6, 9, 15f, 22f, 28, 31, 37, 231cd, 268cd }}
 [[Badness]] (Sintel): 0.336
 === Valarin ===
-This extension equates two, ideally slightly flattened, [[8/7]]s to [[13/10]], which is also natural given valinorsmic's tuning tendency towards sharpening [[7/4]]. This extension is supported by rank-2 temperaments including tridecimal didacus, [[Orwell extensions|tridecimal orwell]], [[Valentine extensions|lupercalia]] (an extension of [[valentine]]), and [[llywelyn]].
+This extension equates two, ideally slightly flattened, [[8/7]]'s to [[13/10]], which is also natural given valinorsmic's tuning tendency towards sharpening [[7/4]]. This extension is supported by rank-2 temperaments including tridecimal didacus, [[orwell extensions|tridecimal orwell]], [[lupercalia]] (an extension of [[valentine]]), and [[llywelyn]].
 [[Subgroup]]: 2.5.7.11.13
@@ Line 60: / Line 60: @@
 : error map: {{val| 0.000 +2.875 +4.003 -0.110 +3.003 }}
-{{Optimal ET sequence|legend=1| TBD }}
+{{Optimal ET sequence|legend=1| 6, 10e, 15, 16, 21, 31, 37, 206cdf, 243cdf, 280cdf, 317cddf, 354ccddff }}
 [[Badness]] (Sintel): 0.227