Canovian chord

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The Canovian chord (or sensamagic dominant chord) 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 chord is named by Flora Canou.

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

Components

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

Theory

This section explains why the chord is the way it is.

Introduction

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

JI as an Extension of Pythagorean Tuning

As is explicitly stated in the Functional Just System, the entire just intonation can be viewed as an extension of the Pythagorean tuning, where the interval class are determined by pure fifths, and each has a number of varieties differing from each other by a formal comma. You can think of the Pythagorean scale as the backbone, and commas modifying it to add to its "colors". In 7-limit specifically, the formal commas are the syntonic comma 81/80 and septimal comma 64/63. For example, the major third is an interval class with the basic form M3, 81/64. Against this, there are two common varieties, M35, 5/4, lower by 81/80, and M37, 9/7, higher by 64/63.

In terms of tuning space, this is equivalent to changing the basis of 7-limit JI {2/1, 3/1, 5/1, 7/1}, which corresponds to a 4×4 identity matrix, to {2/1, 3/2, 80/81, 63/64}. The mapping matrix is shown below.

[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]

The tuning space remains the same because no comma is tempered out. Notice how the first two rows are the same as the dominant temperament. This basis is convenient in that most common intervals can be categorized into interval classes and accessed by a number of fifth shifts and one comma shift. Rarely is multiple shifts of a comma required.

7-limit Voice Leading

The voice leading plays a significant role in traditional harmonies. Consider 3-limit harmony, the diatonic semitone m2, 256/243, is usually preferred over the chromatic semitone A1, 2187/2048, for this purpose. In 7-limit harmony, the class of diatonic semitone has three common varieties. Besides m2, there are m25, 16/15, sharp by 81/80, and m27, 28/27, flat by 64/63. 12-ET has 81/80, 64/63 and the Pythagorean comma all tempered out, so the diatonic, the chromatic, and all varieties of them, are conflated. Its 100-cent interval is adequate for voice leading. In a meantone temperament like 19-ET, however, the Pythagorean~pental diatonic semitone 256/243~16/15 is large enough to sound too dull for voice leading, and consequently, the traditional dominant chord is also very weak. Some propose that the "chromatic semitone" should be used, but what is often meant by that is actually the septimal version 28/27.

In fact, the step size of 19-ET is 63.16 cents, so close to 62.96 cents of 28/27 that it is hard to interpret otherwise. It can be further constructed that 28/27 is meant for voice leading in any 7-limit intonations.

The Canovian chord is based on the theory that 28/27 is used for voice leading.

On 21/20

21/20 is another possible interpretation for voice leading. 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 a hemifamity, 5120/5103, the difference of 64/63 and 81/80. If interpreted this way, the case of 0-7-15 of 19-ET is 1-9/7-7/4 and requires 49/48 be tempered out – some do propose it in 19-ET, but that is another story.

Relationship to Essentially Tempered Dyadic Chords

The chord by itself is not a 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