Sensamagic dominant chord: Difference between revisions

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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 a double subminor 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 fourth [[4/3]], which requires that the sensamagic comma [[245/243]] be tempered out.

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 septimal sesquidiminished fifth, or sensamagic fourth for short. It sounds only mildly wolf in JI, and [[12edo|12-ET]] ears should be accustomed to a 14-cent-sharp interval anyway. But some tunings can make it much worse. For example, in [[31edo|31-ET]], it is tuned to 542 cents – same as [[11/8]] – so the chord sounds highly dissonant and disturbing.

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 septimal sesquidiminished grave fifth, or sensamagic fourth for short. It sounds only mildly wolf in JI, and [[12edo|12-ET]] ears should be accustomed to a 14-cent-sharp interval anyway. But some tunings can make it much worse. For example, in [[31edo|31-ET]], it is tuned to 542 cents – same as [[11/8]] – so the chord sounds highly dissonant and disturbing.

== Theory ==

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=== Relationship to Essentially Tempered Dyadic Chords ===

The chord by itself is not a [[Dyadic chord|essentially tempered dyadic chord]] of sensamagic temperaments because 140/81 is an approximation of 12/7, and 1-9/7-12/7 is utonal. But this interval is invariable for voice leading, so 12/7 is not a legitimate substitute. These considerations imply a chord by putting the dominant and the tonic together, and the resultant essentially tempered dyadic chord is the 27-limit pentad with steps of 9/7-28/27-5/4-28/27-7/6.

The chord by itself is not a [[Dyadic chord|essentially tempered dyadic chord]] of sensamagic temperaments because 140/81 is an approximation of [[12/7]], and 1-9/7-12/7 is utonal. But this interval is invariable for voice leading, so 12/7 is not a legitimate substitute. These considerations imply a chord by putting the dominant and the tonic together, and the resultant essentially tempered dyadic chord is the 27-limit pentad with steps of 9/7-28/27-5/4-28/27-[[7/6]].

== Variations ==

=== Inversions ===

Like traditional chords, inversions can be used. Moving the third above the seventh produces a perfect fifth 3/2, or, if the comma is not tempered out, the septimal superaugmented acute fourth or 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 7/6-3/2-[[280/243]] closing at the octave.

== External Links ==

@@ Line 5: / Line 5: @@
 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 a double subminor 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 fourth [[4/3]], which requires that the sensamagic comma [[245/243]] be tempered out.
-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 septimal sesquidiminished fifth, or sensamagic fourth for short. It sounds only mildly wolf in JI, and [[12edo|12-ET]] ears should be accustomed to a 14-cent-sharp interval anyway. But some tunings can make it much worse. For example, in [[31edo|31-ET]], it is tuned to 542 cents – same as [[11/8]] – so the chord sounds highly dissonant and disturbing.
+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 septimal sesquidiminished grave fifth, or sensamagic fourth for short. It sounds only mildly wolf in JI, and [[12edo|12-ET]] ears should be accustomed to a 14-cent-sharp interval anyway. But some tunings can make it much worse. For example, in [[31edo|31-ET]], it is tuned to 542 cents – same as [[11/8]] – so the chord sounds highly dissonant and disturbing.
 == Theory ==
@@ Line 51: / Line 51: @@
 === Relationship to Essentially Tempered Dyadic Chords ===
-The chord by itself is not a [[Dyadic chord|essentially tempered dyadic chord]] of sensamagic temperaments because 140/81 is an approximation of 12/7, and 1-9/7-12/7 is utonal. But this interval is invariable for voice leading, so 12/7 is not a legitimate substitute. These considerations imply a chord by putting the dominant and the tonic together, and the resultant essentially tempered dyadic chord is the 27-limit pentad with steps of 9/7-28/27-5/4-28/27-7/6.
+The chord by itself is not a [[Dyadic chord|essentially tempered dyadic chord]] of sensamagic temperaments because 140/81 is an approximation of [[12/7]], and 1-9/7-12/7 is utonal. But this interval is invariable for voice leading, so 12/7 is not a legitimate substitute. These considerations imply a chord by putting the dominant and the tonic together, and the resultant essentially tempered dyadic chord is the 27-limit pentad with steps of 9/7-28/27-5/4-28/27-[[7/6]].
+== Variations ==
+=== Inversions ===
+Like traditional chords, inversions can be used. Moving the third above the seventh produces a perfect fifth 3/2, or, if the comma is not tempered out, the septimal superaugmented acute fourth or 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 7/6-3/2-[[280/243]] closing at the octave.
 == External Links ==