Barium - Revision history

Eliora: /* Theory */

2025-08-22T22:41:52Z

Theory

← Older revision		Revision as of 22:41, 22 August 2025
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	For technical data see: [[56th-octave temperaments#Barium]]		For technical data see: [[56th-octave temperaments#Barium]]
	== Theory ==		== 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.		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.

FloraC: - parent category

2025-04-28T13:47:38Z

- parent category

← Older revision		Revision as of 13:47, 28 April 2025
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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)		* [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]]		[[Category:Rank-2 temperaments]]

Lériendil at 22:59, 19 April 2025

2025-04-19T22:59:31Z

← Older revision		Revision as of 22:59, 19 April 2025
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	Barium in the 7-limit also tempers out the [[akjaysma]], meaning that 40 periods are set to [[105/64]].		Barium in the 7-limit also tempers out the [[akjaysma]], meaning that 40 periods are set to [[105/64]].

	~~[[Category:Temperaments]]~~
	~~[[Category:Rank 2]]~~

	== Music ==		== Music ==
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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)		* [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:Rank-2 temperaments]]

BudjarnLambeth: added links

2025-04-13T10:53:33Z

added links

← Older revision		Revision as of 10:53, 13 April 2025
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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: [[56th-octave temperaments#Barium]]		For technical data see: [[56th-octave temperaments#Barium]]

Lucius Chiaraviglio: Add section about and link to musical demonstration

2024-10-23T20:31:06Z

Add section about and link to musical demonstration

← Older revision		Revision as of 20:31, 23 October 2024
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	[[Category:Temperaments]]		[[Category:Temperaments]]
	[[Category:Rank 2]]		[[Category:Rank 2]]

			== Music ==

			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)

Eliora at 14:17, 12 July 2023

2023-07-12T14:17:55Z

← Older revision		Revision as of 14:17, 12 July 2023
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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 ==		== 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.

Eliora: /* Theory */

2023-01-03T16:47:51Z

Theory

← Older revision		Revision as of 16:47, 3 January 2023
Line 3:		Line 3:
	For technical data see: [[Fractional-octave temperaments#Barium]]		For technical data see: [[Fractional-octave temperaments#Barium]]
	== Theory ==		== 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\|-225 224 -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.		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.		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.

Eliora: Removing infobox interval because the template automatically adds categories.

2023-01-03T16:46:37Z

Removing infobox interval because the template automatically adds categories.

← Older revision		Revision as of 16:46, 3 January 2023
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	For technical data see: [[Fractional-octave temperaments#Barium]]		For technical data see: [[Fractional-octave temperaments#Barium]]
	== Theory ==		== Theory ==
	~~{{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.		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.

Eliora: /* Theory */ do we need a separate page for the 5-limit comma if it's in one-to-one correspondence with barium and it's not known for anything else besides that, as opposed to 81/80?

2022-12-25T14:49:00Z

Theory: do we need a separate page for the 5-limit comma if it's in one-to-one correspondence with barium and it's not known for anything else besides that, as opposed to 81/80?

← Older revision		Revision as of 14:49, 25 December 2022
Line 3:		Line 3:
	For technical data see: [[Fractional-octave temperaments#Barium]]		For technical data see: [[Fractional-octave temperaments#Barium]]
	== Theory ==		== 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.		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]].		Barium in the 7-limit also tempers out the [[akjaysma]], meaning that 40 periods are set to [[105/64]].

Eliora: Created page with "'''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 t..."

2022-12-25T14:34:16Z

Created page with "'''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 t..."

New page

'''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]]
== 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}}.

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.

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

[[Category:Temperaments]]
[[Category:Rank 2]]