Ploidacot/Dicot: Difference between revisions

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{{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=0|Cots=2|Pergen=[P8, P5/2]|Forms=7, 10, 17|Title=Dicot|Wedgie=2}}

'''Dicot''' is a temperament archetype where the generator is a [[Neutral third (interval region)|neutral third]], two of which make a perfect fifth of [[3/2]], and the period is a [[2/1]] octave. Dicot temperaments usually generate the [[7L 3s]] MOS structure, fittingly named "dicoid", and one of its children [[10L 7s]] or [[7L 10s]]. Dicot temperaments ~~tend to~~ involve "neutral" intervals, ~~which~~ are ~~in-~~between ~~conventional diatonic intervals~~.

'''Dicot''' is a temperament archetype where the generator is a [[Neutral third (interval region)|neutral third]], two of which make a perfect fifth of [[3/2]], and the period is a [[2/1]] octave. Dicot temperaments usually generate the [[7L 3s]] MOS structure, fittingly named "dicoid", and one of its children [[10L 7s]] or [[7L 10s]]. Dicot temperaments involve "neutral" intervals - right in the middle of typical diatonic intervals. More precisely, they are ''interchromatic'', as the neutrals are found between [[Apotome|apotomes]] (chromatic semitones).

== Intervals and notation ==

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== Temperament interpretations ==

By definition, dicot temperaments ~~equate some interval~~ to ~~its fifth complement~~.

By definition, dicot temperaments map a pair of intervals to the neutral third, equating them.

=== ~~Dicot~~ ===

=== Low accuracy ===

The temperament named "[[dicot]]" is an exotemperament, equating the neutral third to 5/4. This means that 6/5 is the same interval, and the neutral sixth represents both 5/3 and 8/5. It is best tuned with either a sharpened generator of around 360{{c}} (optimizing for the tuning of 5/4) or a flattened generator of around 340{{c}} (optimizing for the tuning of 5/3).

=== ~~Neutral~~ ===

==== Dicot ====

[[~~Neutral (temperament)|Neutral~~]] is ~~the temperament equating [[11~~/~~9]]~~ with ~~[[27~~/~~22]]. This makes 11~~/~~9 the neutral third and [[11~~/~~8]]~~ the ~~semiaugmented fourth. [[Namo]] extends neutral so that [[16~~/~~13]] is found at~~ the ~~same neutral third. Namo is often used as an 11- and 13-limit extension~~ of ~~other temperaments~~.

The temperament named "[[dicot]]" is an exotemperament that equates 5/4 with 6/5, as well as 5/3 with 8/5 with It is well tuned either way with a sharpened generator of around 360{{c}} (optimizing for the tuning of 5/4) or a flattened generator of around 340{{c}} (optimizing for the tuning of 5/3).

~~=== Mohajira ===~~

Despite its noticeable innacuracy, it is the first "usable" dicot temperament, as the one before it would be the extreme temperament that "equates" 9/8 and 4/3.

~~When neutral~~ is ~~combined with [[meantone]] (which sets~~ the ~~major third equal to [[5/4]])~~, the ~~result is [[mohajira]], which tunes~~ the ~~generator of ~11~~/~~9 to about 348{{c}}~~ and ~~extends to the full 11-limit by setting 7~~/~~4 equal to the semidiminished seventh~~.

=== ~~Hemififths~~ ===

=== Medium accuracy ===

Fittingly to its name, [[hemififths]] divides the fifth evenly into two {{nowrap|[[49/40]]~[[60/49]]<nowiki/>s}}. 7/4 is the semiaugmented sixth, and consequently 5/4 is the sesqui-augmented (one-and-a-half augmented) second.

[[~~Category:Ploidacots~~|~~Dicot~~]]

==== Neutral ====

[[Neutral (temperament)|Neutral]] (AKA the 2.3.11 restriction of [[rastmic]]) is the temperament equating [[11/9]] with [[27/22]]. This maps 11/9 to the neutral third and [[11/8]] to the semiaugmented fourth. [[Namo]] extends neutral so that [[16/13]] is mapped at the same neutral third. Namo is often used as an 11- and 13-limit extension of other temperaments.

==== Mohajira ====

When neutral is combined with [[meantone]] (which maps the major third to [[5/4]]), the result is [[mohajira]], which tunes the generator of ~11/9 to about 348{{c}} and extends to the full 11-limit by setting 7/4 equal to the semidiminished seventh.

==== Hemififths ====

As its name implies, [[hemififths]] has hemififths as generators, equating [[49/40]] with [[60/49]]. 7/4 maps to a semiaugmented sixth, and consequently 5/4 maps to a sesqui-augmented (one-and-a-half augmented) second.

=== High accuracy ===

==== Newt ====

Much like hemififths, [[newt]] equates 49/40 with 60/49, but primes are found much deeper in the dicot chain, albeit its accuracy is astounding, solidifying it as a [[microtemperament]]. It maps, with incredible exactitude: 7/4 to a subminor seventh (doubly diminished octave), 11/8 to a hyperfourth (3x-augmented second), 5/4 at -28.5 hemififths, 13/8 at 40.5 fifths, 19/16 at -23.5 fifths.

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 {{Breadcrumb}}{{Infobox ploidacot|Ploids=1|Shears=0|Cots=2|Pergen=[P8, P5/2]|Forms=7, 10, 17|Title=Dicot|Wedgie=2}}
-'''Dicot''' is a temperament archetype where the generator is a [[Neutral third (interval region)|neutral third]], two of which make a perfect fifth of [[3/2]], and the period is a [[2/1]] octave. Dicot temperaments usually generate the [[7L&nbsp;3s]] MOS structure, fittingly named "dicoid", and one of its children [[10L&nbsp;7s]] or [[7L&nbsp;10s]]. Dicot temperaments tend to involve "neutral" intervals, which are in-between conventional diatonic intervals.
+'''Dicot''' is a temperament archetype where the generator is a [[Neutral third (interval region)|neutral third]], two of which make a perfect fifth of [[3/2]], and the period is a [[2/1]] octave. Dicot temperaments usually generate the [[7L&nbsp;3s]] MOS structure, fittingly named "dicoid", and one of its children [[10L&nbsp;7s]] or [[7L&nbsp;10s]]. Dicot temperaments involve "neutral" intervals - right in the middle of typical diatonic intervals. More precisely, they are ''interchromatic'', as the neutrals are found between [[Apotome|apotomes]] (chromatic semitones).
 == Intervals and notation ==
@@ Line 121: / Line 121: @@
 == Temperament interpretations ==
-By definition, dicot temperaments equate some interval to its fifth complement.
+By definition, dicot temperaments map a pair of intervals to the neutral third, equating them.
-=== Dicot ===
+=== Low accuracy ===
-The temperament named "[[dicot]]" is an exotemperament, equating the neutral third to 5/4. This means that 6/5 is the same interval, and the neutral sixth represents both 5/3 and 8/5. It is best tuned with either a sharpened generator of around 360{{c}} (optimizing for the tuning of 5/4) or a flattened generator of around 340{{c}} (optimizing for the tuning of 5/3).
-=== Neutral ===
+==== Dicot ====
-[[Neutral (temperament)|Neutral]] is the temperament equating [[11/9]] with [[27/22]]. This makes 11/9 the neutral third and [[11/8]] the semiaugmented fourth. [[Namo]] extends neutral so that [[16/13]] is found at the same neutral third. Namo is often used as an 11- and 13-limit extension of other temperaments.
+The temperament named "[[dicot]]" is an exotemperament that equates 5/4 with 6/5, as well as 5/3 with 8/5 with It is well tuned either way with a sharpened generator of around 360{{c}} (optimizing for the tuning of 5/4) or a flattened generator of around 340{{c}} (optimizing for the tuning of 5/3).
-=== Mohajira ===
+Despite its noticeable innacuracy, it is the first "usable" dicot temperament, as the one before it would be the extreme temperament that "equates" 9/8 and 4/3.
-When neutral is combined with [[meantone]] (which sets the major third equal to [[5/4]]), the result is [[mohajira]], which tunes the generator of ~11/9 to about 348{{c}} and extends to the full 11-limit by setting 7/4 equal to the semidiminished seventh.
-=== Hemififths ===
+=== Medium accuracy ===
-Fittingly to its name, [[hemififths]] divides the fifth evenly into two {{nowrap|[[49/40]]~[[60/49]]<nowiki/>s}}. 7/4 is the semiaugmented sixth, and consequently 5/4 is the sesqui-augmented (one-and-a-half augmented) second.
-[[Category:Ploidacots|Dicot]]
+==== Neutral ====
+[[Neutral (temperament)|Neutral]] (AKA the 2.3.11 restriction of [[rastmic]]) is the temperament equating [[11/9]] with [[27/22]]. This maps 11/9 to the neutral third and [[11/8]] to the semiaugmented fourth. [[Namo]] extends neutral so that [[16/13]] is mapped at the same neutral third. Namo is often used as an 11- and 13-limit extension of other temperaments.
+==== Mohajira ====
+When neutral is combined with [[meantone]] (which maps the major third to [[5/4]]), the result is [[mohajira]], which tunes the generator of ~11/9 to about 348{{c}} and extends to the full 11-limit by setting 7/4 equal to the semidiminished seventh.
+==== Hemififths ====
+As its name implies, [[hemififths]] has hemififths as generators, equating [[49/40]] with [[60/49]]. 7/4 maps to a semiaugmented sixth, and consequently 5/4 maps to a sesqui-augmented (one-and-a-half augmented) second.
+=== High accuracy ===
+==== Newt ====
+Much like hemififths, [[newt]] equates 49/40 with 60/49, but primes are found much deeper in the dicot chain, albeit its accuracy is astounding, solidifying it as a [[microtemperament]]. It maps, with incredible exactitude: 7/4 to a subminor seventh (doubly diminished octave), 11/8 to a hyperfourth (3x-augmented second), 5/4 at -28.5 hemififths, 13/8 at 40.5 fifths, 19/16 at -23.5 fifths.