Barium: Difference between revisions

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For technical data see: [[56th-octave temperaments#Barium]]

== Theory ==

An octave is equal to <math>\frac{1}{\~~log~~{2}{\frac{81}{80}}} \approx 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}}, and therefore is tempered if and only if the EDO divides 56. 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.

An octave is equal to <math>\frac{1}{\log_{2}{\frac{81}{80}}} \approx 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}}, and therefore is tempered if and only if the EDO divides 56. 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.

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* [https://www.youtube.com/watch?v=Hmjx4wvLG7Q Uccellini - «Aria Sopra La Bergamasca» (1642), arranged for Organ, tuned into Adaptive Just Intonation] rendered by [[Claudi Meneghin]] (2024)

[[Category:~~Temperaments~~]]

[[Category:Barium| ]]

[[Category:Rank-2 temperaments]]

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 For technical data see: [[56th-octave temperaments#Barium]]
 == Theory ==
-An octave is equal to <math>\frac{1}{\log{2}{\frac{81}{80}}} \approx 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}}, and therefore is tempered if and only if the EDO divides 56. 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.
+An octave is equal to <math>\frac{1}{\log_{2}{\frac{81}{80}}} \approx 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}}, and therefore is tempered if and only if the EDO divides 56. 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.
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 * [https://www.youtube.com/watch?v=Hmjx4wvLG7Q Uccellini - «Aria Sopra La Bergamasca» (1642), arranged for Organ, tuned into Adaptive Just Intonation] rendered by [[Claudi Meneghin]] (2024)
-[[Category:Temperaments]]
+[[Category:Barium| ]] <!-- main article -->
 [[Category:Rank-2 temperaments]]