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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== 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.

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 (and perhaps more importantly good minor thirds), 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~~)

=== Modern renderings ===

; {{W|Marco Uccellini}}

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

; {{W|Johann Pachelbel}}

* ''Canon in D, P 37'' (unknown date in range 1680 to 1706) — tuned into Adaptive Just Intonation and rendered by [[Claudi Meneghin]] (2026)

** [https://www.youtube.com/shorts/zh-obmZNSJM Arranged for organ]

** [https://www.youtube.com/shorts/EFaancQpYWY Arranged for 3 oboes and bassoon]

[[Category:Barium| ]]

[[Category:Rank-2 temperaments]]

@@ Line 3: / Line 3: @@
 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.
@@ Line 13: / Line 13: @@
 == 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.
+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 (and perhaps more importantly good minor thirds), 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)
+=== Modern renderings ===
+; {{W|Marco Uccellini}}
+* [https://www.youtube.com/watch?v=Hmjx4wvLG7Q ''«Aria Sopra La Bergamasca»''] (1642) — arranged for Organ and tuned into Adaptive Just Intonation, and rendered by [[Claudi Meneghin]] (2024)
+; {{W|Johann Pachelbel}}
+* ''Canon in D, P 37'' (unknown date in range 1680 to 1706) — tuned into Adaptive Just Intonation and rendered by [[Claudi Meneghin]] (2026)
+** [https://www.youtube.com/shorts/zh-obmZNSJM Arranged for organ]
+** [https://www.youtube.com/shorts/EFaancQpYWY Arranged for 3 oboes and bassoon]
 [[Category:Barium| ]] <!-- main article -->
 [[Category:Rank-2 temperaments]]