Dual-fifth tuning: Difference between revisions

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A '''dual-fifth system''' is a(n octave-equivalent~~) tuning system~~ with two sizes of fifths, '''major fifth''' and '''minor fifth''' instead of a single perfect fifth, and accordingly two sizes fourths, '''major fourth''' and '''minor fourth''' instead of a single perfect fourth.

A '''dual-fifth tuning system''' is a [[tuning system]], often [[Octave equivalence|octave-equivalent]], with two sizes of fifths, '''major fifth''' and '''minor fifth''' instead of a single perfect fifth, and accordingly two sizes of fourths, '''major fourth''' and '''minor fourth''' instead of a single perfect fourth. The opposite of dual-fifth may be called ''plain-fifth''.

== Dual-fifth scales ==

[[Sixix]][7] can be regarded as the scale which is the most authentic representation of the "dual-fifth" phenomenon via its [[mode]]s, since it features both sharp and flat fifth on different modes, and the interval in this case occupies 5 staff positions. For example, in [[25edo]], sixix can take form of 4 3 4 3 4 3 4, where five staff positions occupy 18\25 (sharp fifth), but if the mode is 3 4 3 4 3 4 4, then five staff positions are equal to 17\25 (flat fifth). However, it should be noted that in better tunings of sixix, the flat fifth is Mavila-like in quality while the sharp fifth is a comparatively accurate [[superpyth]] diatonic fifth.

== Dual-fifth edos ==

[[~~18edo~~]] is ~~usually considered~~ the ~~quintessential~~ dual-fifth edo ~~by people who work in~~ dual-fifth ~~systems (which~~ are ~~admittedly few)~~, ~~which~~ has a 733.3¢ sharp fifth 31.4c sharp from pure [[3/2]], and a 666.7¢ flat fifth ~~almost equally off~~ as the ~~sharp one~~, 35.3 ~~cents~~ flat.

[[35edo]] is the [[equal temperament]] which can be said to most authentically represent the concept of "dual-fifth", since its fifths of 20\35 and 21\35 correspond to the bounds of the tuning range for the [[diatonic]] scale where the term ''fifth'' in the standard Western practice originates from. 35edo is the largest [[edo]] without a diatonic scale, and it is therefore the smallest whose sharp and flat fifth can be equally treated as being approximants of five staff positions of the diatonic scale.

Perhaps a more familiar dual-fifth system to many is [[18edo]]. It is the first system that has intervals that are close enough to 3/2 that they can be regarded as sharp and flat fifth, but also far enough to sound different. Its sharp fifth and flat fifth are almost equally off from just: it has a 733.3¢ sharp fifth 31.4¢ sharp from pure [[3/2]], and a 666.7¢ flat fifth is 35.3¢ flat.

Although edos like [[18edo]], [[23edo]] and [[25edo]] have been studied as dual-fifth, their corresponding dual-fifth intervals that are also often considered as [[2L 5s|antidiatonic]] generators or subminor sixths, and not every musical approach treats them as approximants of 3/2 or intervals playing the role of the fifth.

For a list of edos which could be considered dual-fifth, see:

* [[:Category:Dual-fifth temperaments]].

We may, heuristically, define dual-fifth edos as those whose [[relative error]] of the third harmonic is greater than 1/3. In that case 1/3 of all edos will be dual-fifth and the other 2/3 will be plain-fifth.

=== Stretched or compressed edos ===

[[Octave stretching]] or [[octave shrinking]] can be used to equally share the error between an edo's best and second-best fifths (i.e. bring them both to exactly 50% [[relative error]]).

Here is the amount of error the sharp and flat fifth will both exhibit in some possible dual-fifth edos after they have been stretched/compressed in this way:

{{Todo|correct maths|inline=1|text=This list gives half the unstretched step size for each edo. The correct values should be half of the step size after stretching, where the 9/4 reached by stacking both fifths is just.}}

'''>25{{c}} error'''

* [[16edo]]: 37.50{{c}}

* [[18edo]]: 33.33{{c}}

* [[20edo]]: 30.00{{c}}

* [[23edo]]: 26.09{{c}}

'''15-25{{c}} error'''

* [[25edo]]: 24.00{{c}}

* [[28edo]]: 21.43{{c}}

* [[30edo]]: 20.00{{c}}

* [[35edo]]: 17.14{{c}}

* [[37edo]]: 16.22{{c}}

* [[40edo]]: 15.00{{c}}

'''<15{{c}} error'''

* [[42edo]]: 14.29{{c}}

* [[47edo]]: 12.77{{c}}

* [[49edo]]: 12.25{{c}}

* [[52edo]]: 11.54{{c}}

* [[54edo]]: 11.11{{c}}

* [[57edo]]: 10.53{{c}}

* [[59edo]]: 10.17{{c}}

* [[64edo]]: 9.38{{c}}

* [[66edo]]: 9.09{{c}}

* [[69edo]]: 8.70{{c}}

* [[71edo]]: 8.45{{c}}

You can calculate this for any edo by subtracting the smaller fifth from the bigger fifth, then dividing the result by two (i.e., by dividing the size of one edo-step by two).

== Dual-fifth temperaments ==

Unlike conventional temperaments, "[[dual-fifth temperaments]]" do not attempt to optimize every interval to low-limit JI, but treat the "sharp 3" (3⁺) and the "flat 3" (3⁻) as distinct dimensions. The sharp 3 and the flat 3 are not meant to represent JI intervals by themselves, but satisfy 3⁺ × 3⁻ = 9 (representing [[9/1]] in JI); hence 2.3⁻.9 and 2.3⁻.3⁺ are the same subgroup.

Unlike conventional temperaments, "[[dual-fifth temperaments]]" do not attempt to optimize every [[interval]] to low-limit [[JI]], but treat the "sharp 3" (3⁺) and the "flat 3" (3⁻) as distinct dimensions. The sharp 3 and the flat 3 are not meant to represent JI intervals by themselves, but satisfy 3⁺ × 3⁻ = 9 (representing [[9/1]] in JI); hence 2.3⁻.9 and 2.3⁻.3⁺ are the same [[subgroup]].

For example, "dual-3 sixix" is a 2.3⁻.9.5 temperament with an optimal [[generator]] around 335.8¢ (optimizing only the 2.9.5 portion of the subgroup). Two generators up make the flat fifth, and five generators down make the flat fourth. Hence 3 generators down represent [[9/8]] and 6 generators down represent [[5/4]]. Hence dual-3 sixix tempers out [[81/80]] in the 2.9.5 subgroup, but only every third interval in the sixix generator chains represents a JI interval.

18edo is notable for supporting both dual-3 sixix and dual-3 [[A-Team]] with the 2.3⁻.3⁺.5 val {{val|18 28 29 42}}.

Alternatively, dual-fifth temperaments can be analyzed in a more conventional way as [[subgroup temperament]]s, where one of the fifths is mapped to [[3/2]] and the other is mapped to a nearby [[wolf interval|wolf fifth]] (such as [[64/43]], which is convenient since 2.3.43 is the same subgroup as 2.3.64/43).

For a list of dual-fifth temperaments and their properties, see:

* [[Dual-fifth temperaments]]

~~For example~~, ~~"dual-3 sixix"~~ is a ~~2.3⁻.9.5~~ temperament ~~with an optimal generator around 335.8¢~~ (~~optimizing only the 2.9.5 portion of the subgroup~~). ~~Two generators up make the flat fifth~~, ~~and five generators down make~~ the ~~flat fourth. Hence~~ 3 ~~generators down represent [[9/8]] and 6 generators down represent [[5/4]]. Hence dual-~~3 ~~sixix tempers out [[81~~/~~80]] in the 2.9.5 subgroup, but only every third interval in the sixix generator chains represents a JI~~ interval.

=== Multiple-fifth temperaments ===

By extension, it is also possible to consider a multiple fifth temperament where

:<math>\prod_{N=1}^{n} 3^{(N)} = 3^n</math>.

That is, all the different mappings of 3 align eventually at a 3''n'' interval.

~~18edo is notable for supporting both dual~~-3 ~~sixix~~ and ~~dual~~-3 ~~A-Team with the 2.3⁻.3⁺.5 val {{val|18 28 29 42}}~~.

For example, [[91edo]] has 3 usable fifths with their own functions - 52\91 (3-), 53\91 (3), and 54\91 (3+). Thus, if used this way they do not represent distinct dimensions, but rather correspond to 3 × 3- × 3+ = 27/1.

== Music ==

*[[File:Sixix Fugue.mp3|256x256px]] A fugue in [[18edo]] as a dual-fifth tuning (WIP)

[[Category:Dual-fifth| ]]

[[Category:Terms]]

~~[[Category:Dual-fifth|*]]~~

@@ Line 1: / Line 1: @@
-A '''dual-fifth system''' is a(n octave-equivalent) tuning system with two sizes of fifths, '''major fifth''' and '''minor fifth''' instead of a single perfect fifth, and accordingly two sizes fourths, '''major fourth''' and '''minor fourth''' instead of a single perfect fourth.
+A '''dual-fifth tuning system''' is a [[tuning system]], often [[Octave equivalence|octave-equivalent]], with two sizes of fifths, '''major fifth''' and '''minor fifth''' instead of a single perfect fifth, and accordingly two sizes of fourths, '''major fourth''' and '''minor fourth''' instead of a single perfect fourth. The opposite of dual-fifth may be called ''plain-fifth''.
+== Dual-fifth scales ==
+[[Sixix]][7] can be regarded as the scale which is the most authentic representation of the "dual-fifth" phenomenon via its [[mode]]s, since it features both sharp and flat fifth on different modes, and the interval in this case occupies 5 staff positions. For example, in [[25edo]], sixix can take form of 4 3 4 3 4 3 4, where five staff positions occupy 18\25 (sharp fifth), but if the mode is 3 4 3 4 3 4 4, then five staff positions are equal to 17\25 (flat fifth). However, it should be noted that in better tunings of sixix, the flat fifth is Mavila-like in quality while the sharp fifth is a comparatively accurate [[superpyth]] diatonic fifth.
 == Dual-fifth edos ==
-[[18edo]] is usually considered the quintessential dual-fifth edo by people who work in dual-fifth systems (which are admittedly few), which has a 733.3¢ sharp fifth 31.4c sharp from pure [[3/2]], and a 666.7¢ flat fifth almost equally off as the sharp one, 35.3 cents flat.
+[[35edo]] is the [[equal temperament]] which can be said to most authentically represent the concept of "dual-fifth", since its fifths of 20\35 and 21\35 correspond to the bounds of the tuning range for the [[diatonic]] scale where the term ''fifth'' in the standard Western practice originates from. 35edo is the largest [[edo]] without a diatonic scale, and it is therefore the smallest whose sharp and flat fifth can be equally treated as being approximants of five staff positions of the diatonic scale.
+Perhaps a more familiar dual-fifth system to many is [[18edo]]. It is the first system that has intervals that are close enough to 3/2 that they can be regarded as sharp and flat fifth, but also far enough to sound different. Its sharp fifth and flat fifth are almost equally off from just: it has a 733.3¢ sharp fifth 31.4¢ sharp from pure [[3/2]], and a 666.7¢ flat fifth is 35.3¢ flat.
+Although edos like [[18edo]], [[23edo]] and [[25edo]] have been studied as dual-fifth, their corresponding dual-fifth intervals that are also often considered as [[2L 5s|antidiatonic]] generators or subminor sixths, and not every musical approach treats them as approximants of 3/2 or intervals playing the role of the fifth.
+For a list of edos which could be considered dual-fifth, see:
+* [[:Category:Dual-fifth temperaments]].
+We may, heuristically, define dual-fifth edos as those whose [[relative error]] of the third harmonic is greater than 1/3. In that case 1/3 of all edos will be dual-fifth and the other 2/3 will be plain-fifth.
+=== Stretched or compressed edos ===
+[[Octave stretching]] or [[octave shrinking]] can be used to equally share the error between an edo's best and second-best fifths (i.e. bring them both to exactly 50% [[relative error]]).
+Here is the amount of error the sharp and flat fifth will both exhibit in some possible dual-fifth edos after they have been stretched/compressed in this way:
+{{Todo|correct maths|inline=1|text=This list gives half the unstretched step size for each edo. The correct values should be half of the step size after stretching, where the 9/4 reached by stacking both fifths is just.}}
+'''>25{{c}} error'''
+* [[16edo]]: 37.50{{c}}
+* [[18edo]]: 33.33{{c}}
+* [[20edo]]: 30.00{{c}}
+* [[23edo]]: 26.09{{c}}
+'''15-25{{c}} error'''
+* [[25edo]]: 24.00{{c}}
+* [[28edo]]: 21.43{{c}}
+* [[30edo]]: 20.00{{c}}
+* [[35edo]]: 17.14{{c}}
+* [[37edo]]: 16.22{{c}}
+* [[40edo]]: 15.00{{c}}
+'''<15{{c}} error'''
+* [[42edo]]: 14.29{{c}}
+* [[47edo]]: 12.77{{c}}
+* [[49edo]]: 12.25{{c}}
+* [[52edo]]: 11.54{{c}}
+* [[54edo]]: 11.11{{c}}
+* [[57edo]]: 10.53{{c}}
+* [[59edo]]: 10.17{{c}}
+* [[64edo]]: 9.38{{c}}
+* [[66edo]]: 9.09{{c}}
+* [[69edo]]: 8.70{{c}}
+* [[71edo]]: 8.45{{c}}
+You can calculate this for any edo by subtracting the smaller fifth from the bigger fifth, then dividing the result by two (i.e., by dividing the size of one edo-step by two).
 == Dual-fifth temperaments ==
-Unlike conventional temperaments, "[[dual-fifth temperaments]]" do not attempt to optimize every interval to low-limit JI, but treat the "sharp 3" (3⁺) and the "flat 3" (3⁻) as distinct dimensions. The sharp 3 and the flat 3 are not meant to represent JI intervals by themselves, but satisfy 3⁺ × 3⁻ = 9 (representing [[9/1]] in JI); hence 2.3⁻.9 and 2.3⁻.3⁺ are the same subgroup.
+Unlike conventional temperaments, "[[dual-fifth temperaments]]" do not attempt to optimize every [[interval]] to low-limit [[JI]], but treat the "sharp 3" (3⁺) and the "flat 3" (3⁻) as distinct dimensions. The sharp 3 and the flat 3 are not meant to represent JI intervals by themselves, but satisfy 3⁺ × 3⁻ = 9 (representing [[9/1]] in JI); hence 2.3⁻.9 and 2.3⁻.3⁺ are the same [[subgroup]].
+For example, "dual-3 sixix" is a 2.3⁻.9.5 temperament with an optimal [[generator]] around 335.8¢ (optimizing only the 2.9.5 portion of the subgroup). Two generators up make the flat fifth, and five generators down make the flat fourth. Hence 3 generators down represent [[9/8]] and 6 generators down represent [[5/4]]. Hence dual-3 sixix tempers out [[81/80]] in the 2.9.5 subgroup, but only every third interval in the sixix generator chains represents a JI interval.
+edo is notable for supporting both dual-3 sixix and dual-3 [[A-Team]] with the 2.3⁻.3⁺.5 val {{val|18 28 29 42}}.
+Alternatively, dual-fifth temperaments can be analyzed in a more conventional way as [[subgroup temperament]]s, where one of the fifths is mapped to [[3/2]] and the other is mapped to a nearby [[wolf interval|wolf fifth]] (such as [[64/43]], which is convenient since 2.3.43 is the same subgroup as 2.3.64/43).
+For a list of dual-fifth temperaments and their properties, see:
+* [[Dual-fifth temperaments]]
-For example, "dual-3 sixix" is a 2.3⁻.9.5 temperament with an optimal generator around 335.8¢ (optimizing only the 2.9.5 portion of the subgroup). Two generators up make the flat fifth, and five generators down make the flat fourth. Hence 3 generators down represent [[9/8]] and 6 generators down represent [[5/4]]. Hence dual-3 sixix tempers out [[81/80]] in the 2.9.5 subgroup, but only every third interval in the sixix generator chains represents a JI interval.
+=== Multiple-fifth temperaments ===
+By extension, it is also possible to consider a multiple fifth temperament where
+:<math>\prod_{N=1}^{n} 3^{(N)} = 3^n</math>.
+That is, all the different mappings of 3 align eventually at a 3<sup>''n''</sup> interval.
-edo is notable for supporting both dual-3 sixix and dual-3 A-Team with the 2.3⁻.3⁺.5 val {{val|18 28 29 42}}.
+For example, [[91edo]] has 3 usable fifths with their own functions - 52\91 (3<sup>-</sup>), 53\91 (3), and 54\91 (3<sup>+</sup>). Thus, if used this way they do not represent distinct dimensions, but rather correspond to 3 × 3<sup>-</sup> × 3<sup>+</sup> = 27/1.
+== Music ==
+*[[File:Sixix Fugue.mp3|256x256px]] A fugue in [[18edo]] as a dual-fifth tuning (WIP)
+[[Category:Dual-fifth| ]] <!-- main article -->
 [[Category:Terms]]
-[[Category:Dual-fifth|*]]