Mercurial comma: Difference between revisions

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'''557122275/556583944''', or the '''mercurial comma''', ~~has a value of~~ 1.673649 [[cent]]s~~, and is~~ the amount by which five justly tuned [[19/17]]s and two [[15/14]]s exceed an octave~~. As the ratio between these intervals is very close to phi~~, ~~this is a strong harmonic entropy minimum adjacent to the~~ [[~~golden meantone~~]] ~~sequence, particularly well suited to stringed instruments that are normally tuned with slight octave stretches due to the inharmonicity of their partials~~.

'''557122275/556583944''', or the '''mercurial comma''', is an unnoticeable comma measuring 1.673649 [[cent]]s. It represents the amount by which five justly tuned [[19/17]]s and two [[15/14]]s exceed an octave, thus being of importance in [[5L 2s]] diatonic scale.

== Etymology ==

It is named after the chemical element mercury due to its adjacency to gold, since mercury meantone is adjacent to golden meantone.

As the ratio between these intervals is very close to phi, a diatonic scale consisting of a 19/17 as the tone and 15/14 as the semitone represents a strong harmonic entropy minimum adjacent to the [[golden meantone]] sequence. This is particularly well suited to stringed instruments that are normally tuned with slight octave stretches due to the inharmonicity of their partials. Hence, the mercurial comma is named after the chemical element mercury due to its adjacency to gold, since mercury meantone is adjacent to golden meantone.

It is unrelated to the [[Mercury|80th-octave temperament]] by the same name.

It is worth noting that per principles of regular temperament theory, a temperament produced by the mercurial comma is a rank-7 19-limit temperament, which would naturally have the name mercurial. Alternately, there is a temperament produced in the 2.3.5.7.17.19 subgroup, which is a rank-5 temperament. This can be easily added to the commas of septimal meantone and [[256/255]] to produce a rank-2 temperament in the meantone family which supports the mercurial relation.

== Temperaments ==

Tempering this comma out in the full 19-limit produces the rank-19 '''mercurial''' temperament.

Subgroup:

Comma list: 557122275/556583944

[[Mapping]]: <br>

{| class="right-all"

|-

| [⟨ || 1 || 1 || 2 || 4 || 3 || -3 || 4 || 5 ||],

|-

| ⟨ || 0 || 1 || 0 || 1 || 0 || 0 || 0 || 0 ||],

|-

| ⟨ || 0 || 0 || 1 || 1 || 0 || 0 || 0 || 0 || ],

|-

| ⟨ || 0 || 0 || 0 || -5 || 0 || 0 || 0 || -2 || ],

|-

| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || 0 || 0 || ]],

|-

| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || 0 || 0 || ]],

|-

| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || 0 || ]],

|-

| ⟨ || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 1 || ]],

|}

: mapping generators: ~2, ~3, ~5, ~5415/4046, ~11, ~13, ~17

The interval 5415/4046 is within the range of meantone fourths. From this, 5415/4046 can be derived as the mercurial meantone fourth and respectively its inverse, 8092/5415, mercurial meantone fifth. The comma that separates each from the just versions is [[16245/16184]].

Despite sharing the name, mercurial temperament is unrelated to the [[Mercury|80th-octave temperament]] by the same name, though as a curiosity, the [[2960edo]], more specifically, the 2960dh val, is the unique tuning supporting them both.

[[Category:Meantone]]

[[Category:Mercurial]]

[[Category:Commas named after elements]]

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 }}
-'''557122275/556583944''', or the '''mercurial comma''', has a value of 1.673649 [[cent]]s, and is the amount by which five justly tuned [[19/17]]s and two [[15/14]]s exceed an octave. As the ratio between these intervals is very close to phi, this is a strong harmonic entropy minimum adjacent to the [[golden meantone]] sequence, particularly well suited to stringed instruments that are normally tuned with slight octave stretches due to the inharmonicity of their partials.
+'''557122275/556583944''', or the '''mercurial comma''', is an unnoticeable comma measuring 1.673649 [[cent]]s. It represents the amount by which five justly tuned [[19/17]]s and two [[15/14]]s exceed an octave, thus being of importance in [[5L 2s]] diatonic scale.
 == Etymology ==
-It is named after the chemical element mercury due to its adjacency to gold, since mercury meantone is adjacent to golden meantone.
+As the ratio between these intervals is very close to phi, a diatonic scale consisting of a 19/17 as the tone and 15/14 as the semitone represents a strong harmonic entropy minimum adjacent to the [[golden meantone]] sequence. This is particularly well suited to stringed instruments that are normally tuned with slight octave stretches due to the inharmonicity of their partials. Hence, the mercurial comma is named after the chemical element mercury due to its adjacency to gold, since mercury meantone is adjacent to golden meantone.
-It is unrelated to the [[Mercury|80th-octave temperament]] by the same name.
+It is worth noting that per principles of regular temperament theory, a temperament produced by the mercurial comma is a rank-7 19-limit temperament, which would naturally have the name mercurial. Alternately, there is a temperament produced in the 2.3.5.7.17.19 subgroup, which is a rank-5 temperament. This can be easily added to the commas of septimal meantone and [[256/255]] to produce a rank-2 temperament in the meantone family which supports the mercurial relation.
+== Temperaments ==
+Tempering this comma out in the full 19-limit produces the rank-19 '''mercurial''' temperament.
+Subgroup:
+Comma list: 557122275/556583944
+[[Mapping]]: <br>
+{| class="right-all"
+|-
+| [⟨ || 1 || 1 || 2 || 4 || 3 || -3 || 4 || 5 ||],
+|-
+| ⟨ || 0 || 1 || 0 || 1 || 0 || 0 || 0 || 0 ||],
+|-
+| ⟨ || 0 || 0 || 1 || 1 || 0 || 0 || 0 || 0 || ],
+|-
+| ⟨ || 0 || 0 || 0 || -5 || 0 || 0 || 0 || -2 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || 0 || 0 || ]],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || 0 || 0 || ]],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || 0 || ]],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 1 || ]],
+|}
+: mapping generators: ~2, ~3, ~5, ~5415/4046, ~11, ~13, ~17
+The interval 5415/4046 is within the range of meantone fourths. From this, 5415/4046 can be derived as the mercurial meantone fourth and respectively its inverse, 8092/5415, mercurial meantone fifth. The comma that separates each from the just versions is [[16245/16184]].
+Despite sharing the name, mercurial temperament is unrelated to the [[Mercury|80th-octave temperament]] by the same name, though as a curiosity, the [[2960edo]], more specifically, the 2960dh val, is the unique tuning supporting them both.
 [[Category:Meantone]]
 [[Category:Mercurial]]
+[[Category:Commas named after elements]]