Barium: Difference between revisions

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'''Barium''' is a rank-2 temperament defined in the 5-limit by tempering out the comma which sets 56 syntonic commas equal to the octave. Extensions exist in the 7-limit and the 11-limit. It is named after the 56th chemical element.

'''Barium''' is a [[rank-2 temperament]] defined in the [[5-limit]] by [[tempering out]] the comma which sets 56 syntonic commas equal to the octave. Extensions exist in the 7-limit and the 11-limit. It is named after the 56th chemical element.

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

For technical data see: [[56th-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}}.

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.

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]].

[[Category:~~Temperaments~~]]

== Music ==

[[Category:Rank 2]]

Although full gamuts for barium start at 224 notes per octave, it is possible to use what is effectively a subset of barium temperament of much smaller gamut to produce the good major thirds of quarter-comma meantone while still getting good fifths, although any stacking of fifths will rapidly increase the size of gamut needed.

* [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:Barium| ]]

[[Category:Rank-2 temperaments]]

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-'''Barium''' is a rank-2 temperament defined in the 5-limit by tempering out the comma which sets 56 syntonic commas equal to the octave. Extensions exist in the 7-limit and the 11-limit. It is named after the 56th chemical element.
+'''Barium''' is a [[rank-2 temperament]] defined in the [[5-limit]] by [[tempering out]] the comma which sets 56 syntonic commas equal to the octave. Extensions exist in the 7-limit and the 11-limit. It is named after the 56th chemical element.
-For technical data see: [[Fractional-octave temperaments#Barium]]
+For technical data see: [[56th-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}}.
+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.
-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]].
-[[Category:Temperaments]]
+== Music ==
-[[Category:Rank 2]]
+Although full gamuts for barium start at 224 notes per octave, it is possible to use what is effectively a subset of barium temperament of much smaller gamut to produce the good major thirds of quarter-comma meantone while still getting good fifths, although any stacking of fifths will rapidly increase the size of gamut needed.
+* [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:Barium| ]] <!-- main article -->
+[[Category:Rank-2 temperaments]]