Barium: Difference between revisions

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For technical data see: [[Fractional-octave temperaments#Barium]]

== Theory ==

An octave is equal to about <math>\frac{1}{\log{2}{\frac{81}{80}}} = 55.79763</math> syntonic commas, which when rounded to the closest integer yields 56. The associated comma in the 5-limit is {{monzo|-225 24 -56}}.

{{Infobox Interval|Monzo=-225 224 -56|Name=barium comma|Comma=yes}}

An octave is equal to about <math>\frac{1}{\log{2}{\frac{81}{80}}} = 55.79763</math> syntonic commas, which when rounded to the closest integer yields 56. The associated comma in the 5-limit is {{monzo|-225 224 -56}} (see infobox on the right). The comma is about 4 cents wide, but since each 81/80 is flattened by only about 0.07 cents as a consequence, barium is a very precise microtemperament.

Because the period is set to 81/80, interval stacking scheme works the same way as in [[meantone]], with the only difference being that the resulting intervals are represented in different 56ths of the octave. When the interval 3/2 is stacked 4 times, it also mirrors the pattern in every 1/56th of the octave, reaching [[5/4]] in 4 steps just as meantone would.

In the 7-limit, the reduced generator of barium is equal to the [[126/125]], a comma which together with the syntonic comma completes the basis for the [[septimal meantone]]. As such, barium can be interpreted this way as an "unfolding" of the septimal meantone into the fractional-octave temperament where one comma (81/80) is the period and the other (126/125) is the generator.

In the 7-limit, the reduced generator of barium is equal to the [[126/125]], a comma which together with the syntonic comma completes the basis for the [[septimal meantone]]. As such, barium can be interpreted this way as an "unfolding" of the septimal meantone into the fractional-octave temperament where one comma (81/80) is the period and the other (126/125) is the generator. Reading directly from the mapping, 7/4 is attained in 5 stacked intervals.

Barium in the 7-limit also tempers out the [[akjaysma]], meaning that 40 periods are set to [[105/64]].

@@ Line 3: / Line 3: @@
 For technical data see: [[Fractional-octave temperaments#Barium]]
 == Theory ==
-An octave is equal to about <math>\frac{1}{\log{2}{\frac{81}{80}}} = 55.79763</math> syntonic commas, which when rounded to the closest integer yields 56. The associated comma in the 5-limit is {{monzo|-225 24 -56}}.
+{{Infobox Interval|Monzo=-225 224 -56|Name=barium comma|Comma=yes}}
+An octave is equal to about <math>\frac{1}{\log{2}{\frac{81}{80}}} = 55.79763</math> syntonic commas, which when rounded to the closest integer yields 56. The associated comma in the 5-limit is {{monzo|-225 224 -56}} (see infobox on the right). The comma is about 4 cents wide, but since each 81/80 is flattened by only about 0.07 cents as a consequence, barium is a very precise microtemperament.
 Because the period is set to 81/80, interval stacking scheme works the same way as in [[meantone]], with the only difference being that the resulting intervals are represented in different 56ths of the octave. When the interval 3/2 is stacked 4 times, it also mirrors the pattern in every 1/56th of the octave, reaching [[5/4]] in 4 steps just as meantone would.
-In the 7-limit, the reduced generator of barium is equal to the [[126/125]], a comma which together with the syntonic comma completes the basis for the [[septimal meantone]]. As such, barium can be interpreted this way as an "unfolding" of the septimal meantone into the fractional-octave temperament where one comma (81/80) is the period and the other (126/125) is the generator.
+In the 7-limit, the reduced generator of barium is equal to the [[126/125]], a comma which together with the syntonic comma completes the basis for the [[septimal meantone]]. As such, barium can be interpreted this way as an "unfolding" of the septimal meantone into the fractional-octave temperament where one comma (81/80) is the period and the other (126/125) is the generator. Reading directly from the mapping, 7/4 is attained in 5 stacked intervals.
 Barium in the 7-limit also tempers out the [[akjaysma]], meaning that 40 periods are set to [[105/64]].