Dual-n: Difference between revisions

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A '''dual-n tuning''' is a tuning where two separate mappings are provided for a [[basis element]], turning the one basis element into two basis elements.

A '''dual-''n'' tuning''' is a tuning where two separate mappings are provided for a [[basis element]], turning the one basis element into two basis elements.

The most common type of dual-n tuning is [[dual-fifth tuning]], aka dual-3 tuning, where the subgroup 2.3 is replaced by 2.3+.3-

The most common type of dual-''n'' tuning is [[dual-fifth tuning]], aka dual-3 tuning, where the subgroup 2.3 is replaced by 2.3+.3−.

However it is also possible to do this for any other basis element. You could have a dual-2 tuning, dual-5 tuning, dual-7 tuning, etc.

One situation in which it might make sense to use a dual-n tuning is for use with an [[equal-step tuning]] ~~that’s good~~ on all primes except one~~, but you don’t want to leave~~ that prime ~~out~~. ~~You~~ could use a dual-n tuning where n is that prime in order to make the equal-step tuning workable.

== Use cases ==

One situation in which it might make sense to use a dual-''n'' tuning is for use with an [[equal-step tuning]] that has low [[relative error]] on all primes except one (call it ''n'').

But the tuning’s step size is small enough that one can't avoid hearing prime ''n'', so a no-''n''s tuning doesn’t make sense.

One could use a dual-''n'' tuning where ''n'' is that prime in order to make the equal-step tuning workable.

=== "Small enough" ===

How small is "small enough" is subjective and up to an individual composer's taste.

Some would say a step size of 30 cents is small enough, meaning that prime ''n'' must be within 15 cents.

Some would say a step size of 20 cents is small enough, meaning that prime ''n'' must be within 10 cents.

Some would say a step size of 10 cents is small enough, meaning that prime ''n'' must be within 5 cents.

Some would choose some other step size, while some would say the whole dual-''n'' concept isn’t for them and they would rather use no-''n'' no matter the step size.

It’s not a science and is up to a composer's discretion.

== Scales with dual-n interpretations ==

* Many large [[ed4]]s (''dual-2'')

* Many medium or large [[The Riemann zeta function and tuning#Local anti-zeta edos|local anti-zeta edos]]{{idiosyncratic}}

* Many of the [[sooty fox scale]]s{{idiosyncratic}}

~~{{stub}}~~

[[Category:Subgroup]]

@@ Line 1: / Line 1: @@
-A '''dual-n tuning''' is a tuning where two separate mappings are provided for a [[basis element]], turning the one basis element into two basis elements.
+{{DISPLAYTITLE:Dual-''n''}}
+A '''dual-''n'' tuning''' is a tuning where two separate mappings are provided for a [[basis element]], turning the one basis element into two basis elements.
-The most common type of dual-n tuning is [[dual-fifth tuning]], aka dual-3 tuning, where the subgroup 2.3 is replaced by 2.3+.3-
+The most common type of dual-''n'' tuning is [[dual-fifth tuning]], aka dual-3 tuning, where the subgroup 2.3 is replaced by 2.3+.3&minus;.
 However it is also possible to do this for any other basis element. You could have a dual-2 tuning, dual-5 tuning, dual-7 tuning, etc.
-One situation in which it might make sense to use a dual-n tuning is for use with an [[equal-step tuning]] that’s good on all primes except one, but you don’t want to leave that prime out. You could use a dual-n tuning where n is that prime in order to make the equal-step tuning workable.
+== Use cases ==
+One situation in which it might make sense to use a dual-''n'' tuning is for use with an [[equal-step tuning]] that has low [[relative error]] on all primes except one (call it ''n'').
+But the tuning’s step size is small enough that one can't avoid hearing prime ''n'', so a no-''n''s tuning doesn’t make sense.
+One could use a dual-''n'' tuning where ''n'' is that prime in order to make the equal-step tuning workable.
+=== "Small enough" ===
+How small is "small enough" is subjective and up to an individual composer's taste.
+Some would say a step size of 30 cents is small enough, meaning that prime ''n'' must be within 15 cents.
+Some would say a step size of 20 cents is small enough, meaning that prime ''n'' must be within 10 cents.
+Some would say a step size of 10 cents is small enough, meaning that prime ''n'' must be within 5 cents.
+Some would choose some other step size, while some would say the whole dual-''n'' concept isn’t for them and they would rather use no-''n'' no matter the step size.
+It’s not a science and is up to a composer's discretion.
+== Scales with dual-n interpretations ==
+* Many large [[ed4]]s (''dual-2'')
+* Many medium or large [[The Riemann zeta function and tuning#Local anti-zeta edos|local anti-zeta edos]]{{idiosyncratic}}
+* Many of the [[sooty fox scale]]s{{idiosyncratic}}
+{{todo|inline=1|complete list|add examples}}
-{{stub}}
 [[Category:Subgroup]]