Sensamagic dominant chord

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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 built on the dominant. It features two instances of voice leading by 28/27 in the resolution to the tonic.

Canovian chord notated in 19-ET
Canovian chord notated in FJS

Construction

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 fifth of 3/2 can be added, omitted here for simplicity. The progression with respect to the tonic is

[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.

The negative harmony 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.

Theory

This section explains why the chord is what it is.

Original occurrence

The chord originally arose as 0-7-15 steps of 19et, used for the purpose of a stronger version of the traditional dominant chord.

Septimal voice leading

This section is transcluded from Flora's analysis on septimal voice leading

In Analysis on the 13-limit just intonation space, Flora Canou explained how 28/27 is suitable for the role of voice leading. To quickly show the background, we notice that just intonation can be viewed as an expansion of the Pythagorean tuning, where the interval classes are determined by pure fifths, and each has a number of varieties differing from each other by a formal comma. So the Pythagorean scale is thought of as the backbone, inflected by commas to add to its "colors". In 7-limit specifically, the formal commas are the syntonic comma, 81/80, and the septimal comma, 64/63.

81/80 translates a Pythagorean interval to a classical one. What is its septimal counterpart, which translates a Pythagorean interval to a septimal one? The answer is 64/63, the septimal comma.

Superpyth is the corresponding temperament of the septimal comma. It is the opposite of meantone in several ways. To send 81/80 to unison, meantone tunes the fifth flat. To send 64/63 to unison, superpyth tunes the fifth sharp. In septimal meantone, intervals of 5 are simpler than those of 7, whereas in septimal superpyth, intervals of 7 are simpler than those of 5, and their overall complexities are comparable. George Secor identified a few useful equal temperaments for meantone and superpyth. He noted 17, 22, and 27 to superpyth are what 12, 31, and 19 to meantone, respectively. I call those the six essential low-complexity equal temperaments.

The significance of the septimal comma is successfully recognized by notable notation systems including FJS, HEJI (Helmholtz–Ellis Just Intonation), and Sagittal. It corresponds to the following change of basis, in terms of generator steps.

[math] \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \rightarrow \begin{bmatrix} 1 & 1 & 0 & 4 \\ 0 & 1 & 4 & -2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} [/math]

Inflected by the commas introduced above, each interval class typically comes in three flavors: a Pythagorean one, a classical one, and a septimal one. The best example for this is the minor third, they are 32/27 (m3), 6/5 (m35), and 7/6 (m37).

Voice leading plays a significant role in traditional harmonies. It is customary to prefer the diatonic semitone to the chromatic semitone for this purpose. Consider 7-limit harmony, the class of diatonic semitones has three notable varieties. Besides 256/243 (m2), there are 16/15 (m25), sharp by 81/80, and 28/27 (m27), flat by 64/63. In 12et, the syntonic comma, the septimal comma and the Pythagorean comma are all tempered out, so all varieties of semitones are conflated as one, which is very adequate for voice leading. The classical diatonic semitone in just intonation, however, is larger. Consequently, the traditional dominant chord using this semitone would be very weak. The Pythagorean variant is not ideal either, since it lacks color and concordance. The septimal version is a much stronger choice.

A basic form of dominant–tonic progression is, therefore, a septimal major triad followed by a classical major triad:

[math]3/2–27/14–9/4 \rightarrow 1–5/4–3/2[/math]

where 27/14 resolves to 2/1.

21/20 (m275), the 5/7-kleismic diatonic semitone, is another possible candidate. Compound in color, however, it is not as easy to grasp as 28/27, nor is it as strong, since it is only flat of the Pythagorean version by 5120/5103, the 5/7-kleisma aka the hemififths–amity comma. In contrast, 28/27 creates more cathartic effects for voice leading.

Actually, septimal harmony entail different chord structures from classical ones, and 21/20 has a niche from this perspective. This will be discussed in Chapter VII.

Relation to essentially tempered chords

The chord by itself is utonal and not an essentially tempered chord of the sensamagic temperament, but the tempered essence is emergent if the chord is viewed relative to the tonic. The minimalist essence of this chord is the 27-odd-limit triad 1-28/27-9/7 with steps 28/27-5/4-14/9, and 1-28/27-27/14 with steps 28/27-15/8-28/27.

Variations

There is an undecimal variant, dubbed the semiporwellismic dominant chord, with the voice leading characterized by 33/32 instead. It works in undecimal sensamagic as well. It is built by 1-128/99-55/32. The progression with respect to the tonic is

[math]\text {(Semiporwellismic) } 3/4–32/33–128/99 \rightarrow 1–5/4–3/2[/math]

With the voice leading at 32/33 → 1 and 128/99 → 5/4 characterized by 33/32.

The negative harmony version is 1-64/55-96/55. The progression with respect to the tonic is

[math]\text {(Semiporwellismic) } 2/3–64/55–99/64 \rightarrow 1–6/5–3/2[/math]

With the same voice leading at 64/55 → 6/5 and 99/64 → 3/2.

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