Hemififths: Difference between revisions

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-The '''hemififths''' temperament tempers out the breedsma, [[2401/2400]], and the hemifamity comma, [[5120/5103]], and as the name suggests, uses a neutral-third generator. '''Hemif''' is the no-5 subgroup version of hemififths.
+'''Hemififths''' is the [[temperament]] [[tempering out]] the breedsma, [[2401/2400]], and the hemifamity comma, [[5120/5103]], and as the name suggests, uses a neutral-third generator. '''Hemif''' is the no-5 subgroup version of hemififths. It is supported by [[41edo|41-]], [[58edo|58-]], and [[99edo|99et]].
 See [[Breedsmic temperaments #Hemififths]] for more technical data.
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 |}
 <nowiki>*</nowiki> in 7-limit POTE tuning
+== Notation ==
+Hemififths can be notated in [[neutral circle-of-fifths notation]], in which case 5/4 is represented by a sesqui-augmented second (C-D#+), and 7/4 by a semi-augmented sixth (C-A+). In the 13-limit extension, 11/8 is represented by the semi-augmented fourth (C-F+), and 13/8 by the neutral sixth (C-Ad). This, of course, defies the tradition of tertian harmony. The just major triad on C is C-D#+-G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step. There are two solutions:
+# let an arrow represent a bend by the syntonic~septimal comma (17 gensteps, semidiminished second);
+# let an arrow represent a bend by the Pythagorean comma (24 gensteps, negative diminished second).
+Below is tabulated how to notate the prime harmonics with an arrow representing a syntonic~septimal comma.
+{| class="wikitable center-1 center-3"
+|+Hemififths nomenclature<br>for selected intervals
+! Ratio
+! Nominal
+! Example
+|-
+| 3/2
+| Perfect fifth
+| C-G
+|-
+| 5/4
+| Down major third
+| C-vE
+|-
+| 7/4
+| Down minor seventh
+| C-vBb
+|-
+| 11/8
+| Semi-augmented fourth
+| C-F+
+|-
+| 13/8
+| Neutral sixth
+| C-Ad
+|}
+Below is tabulated how to notate the prime harmonics with an arrow representing a Pythagorean comma.
+{| class="wikitable center-1 center-3"
+|+Hemififths nomenclature<br>for selected intervals
+! Ratio
+! Nominal
+! Example
+|-
+| 3/2
+| Perfect fifth
+| C-G
+|-
+| 5/4
+| Up neutral third
+| C-^Ed
+|-
+| 7/4
+| Up semidiminished seventh
+| C-^Bdb
+|-
+| 11/8
+| Semi-augmented fourth
+| C-F+
+|-
+| 13/8
+| Neutral sixth
+| C-Ad
+|}
 == Chords ==
@@ Line 157: / Line 218: @@
 Gencom: [2 11/9; 144/143 196/195 243/242 364/363]
-Gencom mapping: [{{val| 1 1 -5 -1 2 4 }}, {{val| 0 2 25 13 5 -1 }}]
+Gencom mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}
-{| class="wikitable center-all"
+{| class="wikitable center-1 center-2"
 |-
-! ET<br>generator
+! Edo<br>generator
-! [[eigenmonzo|eigenmonzo<br>(unchanged-interval]])
+! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
-! generator<br>(¢)
+! Generator<br>(¢)
-! comments
+! Comments
 |-
 |