Sensamagic dominant chord: Difference between revisions

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The '''sensamagic dominant chord''' (or '''Canovian chord''' since it was first explored by [[Flora Canou]]) refers to the tempered chord of ratios 1-[[9/7]]-[[~~140/81]], with steps 9~~/7~~-[[4/3~~]]-[[81/70]] ~~closing at~~ the ~~octave~~.

The '''sensamagic dominant chord''' (or '''Canovian chord''' since it was first explored by [[Flora Canou]]) refers to the sensamagic tempered chord of ratios 1-[[9/7]]-[[12/7]]. It features two instances of voice leading by [[28/27]] in the resolution to the tonic.

[[File:Canovian Chord 19-ET.png|thumb|right|Canovian chord notated in 19-ET]]

[[File:Canovian Chord FJS.png|thumb|right|Canovian chord notated in FJS]]

== ~~Components~~ ==

== Construction ==

The chord consists of a tempered 1-9/7-~~140~~/81, usually built on the fifth note of a diatonic scale – it is a dominant chord after all. ~~Hence, the~~ root is [[3/2]] above the tone to which it desires to resolve. The ~~third is a supermajor~~ 9/7~~, so it~~ is [[28/27]] below the tonic. ~~The seventh~~ is ~~an inframinor~~ 140/81, ~~so it~~ is 28/27 above ~~the mediant~~, the latter being [[5/4]] ~~above~~ the tonic. ~~Furthermore~~, ~~the third~~ and ~~the~~ seventh ~~form a just perfect fourth [[4~~/~~3]]~~, ~~which requires that~~ the ~~sensamagic comma [[245/243]] be tempered out~~.

The chord consists of a tempered 1-9/7-12/7, usually built on the fifth note of a diatonic scale – it is a dominant chord after all. The root is [[3/2]] above the tone to which it desires to resolve. The 9/7 is 28/27 below the tonic. By the tempering of sensamagic, the 12/7 is simultaneously 140/81, which is 28/27 above [[5/3]], the latter being [[5/4]] with respect to the tonic. For this reason the 12/7~140/81 is a supermajor sixth in terms of chord construction, and an inframinor seventh in terms of voice leading. The 140/81 spelling is arguably preferable for staff notation as it highlights the voice leading, shown on the right. The progression with respect to the tonic is

Not tempering out the comma ~~causes the interval between~~ the third ~~and the~~ seventh ~~to be a distinct~~ interval of [[980/729]] at 512 cents, called sensamagic fourth. It sounds only mildly wolf in JI, and [[12edo]] ears should be accustomed to a 14-cent-sharp interval anyway. But some tunings can make it much worse. For example, in [[31edo]], it is tuned to 542 cents – same as [[11/8]] – so the chord sounds highly dissonant and disturbing.

<math>\text {(Sensamagic) } 3/4–27/28–9/7 \rightarrow 1–5/4–3/2</math>

While the simplest ratios are presented here, it should be noted that the 9/7 is simultaneously 35/27, and the voice leading of 35/27 → 5/4 is characterized by 28/27, just as of 27/28 → 1.

Not tempering out the comma implies either giving up the voice leading by 28/27 or introducing wolf intervals. The lower voice leading calls for 9/7, a supermajor third. The upper voice leading calls for 140/81, an inframinor seventh. The interval between them is [[980/729]] at 512 cents, called the complex sensamagic fourth. It sounds only mildly wolf in JI, and [[12edo]] ears should be accustomed to a 14-cent-sharp interval anyway. But some tunings can make it much worse. For example, in [[31edo]], it is tuned to 542 cents – same as [[11/8]] – so the chord sounds highly dissonant and disturbing.

== Theory ==

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== Variations ==

=== Rotations ===

Like traditional chords, rotations can be used. Moving the third above the seventh produces a perfect fifth 3/2, or, if the comma is not tempered out, the sensamagic fifth [[729/490]] at 688 cents.

Like traditional chords, rotations can be used. Moving the third above the seventh produces a perfect fifth 3/2, or, if the comma is not tempered out, the complex sensamagic fifth [[729/490]] at 688 cents.

=== Negative harmony ===

The negative version of the chord consists of a tempered 1-7/6-[[243/140]], with ~~steps of~~ 7/6~~-729/490-~~[[~~280~~/~~243~~]] ~~closing at~~ the ~~octave~~.

The negative version of the chord consists of a tempered 1-[[7/6]]-[[7/4]] built on the subdominant, with 7/4 simultaneously acting as [[243/140]]. The 243/140 spelling is arguably preferable for staff notation for the same reason discussed above. The progression with respect to the tonic is

<math>\text {(Sensamagic) } 2/3–7/6–14/9 \rightarrow 1–6/5–3/2</math>

It should be noted that the 7/6 is simultaneously [[81/70]], and the voice leading of 81/70 → 6/5 is characterized by 28/27, just as of 14/9 → 3/2.

== External links ==

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-The '''sensamagic dominant chord''' (or '''Canovian chord''' since it was first explored by [[Flora Canou]]) refers to the tempered chord of ratios 1-[[9/7]]-[[140/81]], with steps 9/7-[[4/3]]-[[81/70]] closing at the octave.
+The '''sensamagic dominant chord''' (or '''Canovian chord''' since it was first explored by [[Flora Canou]]) refers to the sensamagic tempered chord of ratios 1-[[9/7]]-[[12/7]]. It features two instances of voice leading by [[28/27]] in the resolution to the tonic.
 [[File:Canovian Chord 19-ET.png|thumb|right|Canovian chord notated in 19-ET]]
 [[File:Canovian Chord FJS.png|thumb|right|Canovian chord notated in FJS]]
-== Components ==
+== Construction ==
-The chord consists of a tempered 1-9/7-140/81, usually built on the fifth note of a diatonic scale – it is a dominant chord after all. Hence, the root is [[3/2]] above the tone to which it desires to resolve. The third is a supermajor 9/7, so it is [[28/27]] below the tonic. The seventh is an inframinor 140/81, so it is 28/27 above the mediant, the latter being [[5/4]] above the tonic. Furthermore, the third and the seventh form a just perfect fourth [[4/3]], which requires that the sensamagic comma [[245/243]] be tempered out.
+The chord consists of a tempered 1-9/7-12/7, usually built on the fifth note of a diatonic scale – it is a dominant chord after all. The root is [[3/2]] above the tone to which it desires to resolve. The 9/7 is 28/27 below the tonic. By the tempering of sensamagic, the 12/7 is simultaneously 140/81, which is 28/27 above [[5/3]], the latter being [[5/4]] with respect to the tonic. For this reason the 12/7~140/81 is a supermajor sixth in terms of chord construction, and an inframinor seventh in terms of voice leading. The 140/81 spelling is arguably preferable for staff notation as it highlights the voice leading, shown on the right. The progression with respect to the tonic is
-Not tempering out the comma causes the interval between the third and the seventh to be a distinct interval of [[980/729]] at 512 cents, called sensamagic fourth. It sounds only mildly wolf in JI, and [[12edo]] ears should be accustomed to a 14-cent-sharp interval anyway. But some tunings can make it much worse. For example, in [[31edo]], it is tuned to 542 cents – same as [[11/8]] – so the chord sounds highly dissonant and disturbing.
+<math>\text {(Sensamagic) } 3/4–27/28–9/7 \rightarrow 1–5/4–3/2</math>
+While the simplest ratios are presented here, it should be noted that the 9/7 is simultaneously 35/27, and the voice leading of 35/27 → 5/4 is characterized by 28/27, just as of 27/28 → 1.
+Not tempering out the comma implies either giving up the voice leading by 28/27 or introducing wolf intervals. The lower voice leading calls for 9/7, a supermajor third. The upper voice leading calls for 140/81, an inframinor seventh. The interval between them is [[980/729]] at 512 cents, called the complex sensamagic fourth. It sounds only mildly wolf in JI, and [[12edo]] ears should be accustomed to a 14-cent-sharp interval anyway. But some tunings can make it much worse. For example, in [[31edo]], it is tuned to 542 cents – same as [[11/8]] – so the chord sounds highly dissonant and disturbing.
 == Theory ==
@@ Line 62: / Line 66: @@
 == Variations ==
 === Rotations ===
-Like traditional chords, rotations can be used. Moving the third above the seventh produces a perfect fifth 3/2, or, if the comma is not tempered out, the sensamagic fifth [[729/490]] at 688 cents.
+Like traditional chords, rotations can be used. Moving the third above the seventh produces a perfect fifth 3/2, or, if the comma is not tempered out, the complex sensamagic fifth [[729/490]] at 688 cents.
 === Negative harmony ===
-The negative version of the chord consists of a tempered 1-7/6-[[243/140]], with steps of 7/6-729/490-[[280/243]] closing at the octave.
+The negative version of the chord consists of a tempered 1-[[7/6]]-[[7/4]] built on the subdominant, with 7/4 simultaneously acting as [[243/140]]. The 243/140 spelling is arguably preferable for staff notation for the same reason discussed above. The progression with respect to the tonic is
+<math>\text {(Sensamagic) } 2/3–7/6–14/9 \rightarrow 1–6/5–3/2</math>
+It should be noted that the 7/6 is simultaneously [[81/70]], and the voice leading of 81/70 → 6/5 is characterized by 28/27, just as of 14/9 → 3/2.
 == External links ==