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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 [[User:FloraC|Flora Canou]]. | | The '''sensamagic dominant chord''' (or '''Canovian chord''' since it was first explored by [[Flora Canou]]) is the [[sensamagic]] tempered chord of ratios 1–[[9/7]]–[[3/2]]–[[12/7]] built on the dominant. It features two instances of voice leading by [[28/27]] in the resolution to the tonic. |
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| == Components ==
| | [[File:Canovian Chord 19-ET.png|thumb|right|Sensamagic dominant chord notated in 19-ET]] |
| | [[File:Canovian Chord FJS.png|thumb|right|Sensamagic dominant chord notated in FJS]] |
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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. | | == Construction == |
| | The chord consists of a tempered [[14:18:21:24|1–9/7–3/2–12/7]], usually built on the fifth degree of a [[5L 2s|diatonic scale]]. 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 perfect fifth is omitted here for simplicity, so the progression with respect to the tonic is |
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| 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.
| | <math>\text {(Sensamagic) } 3/4–27/28–9/7 \rightarrow 1–5/4–3/2</math> |
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| == Theory ==
| | 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. |
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| | 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 retrosensamagic 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 wider. For example, in [[31edo]], it is tuned to 542 cents – same as [[11/8]] – so the chord sounds highly dissonant and disturbing. |
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| This section explains why the chord is the way it is.
| | The negative harmony version of the chord consists of a tempered 1–[[7/6]]–[[3/2]]–[[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 |
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| === Introduction ===
| | <math>\text {(Sensamagic) } 2/3–7/6–14/9 \rightarrow 1–6/5–3/2</math> |
| The chord arises as 0-7-15 steps of [[19edo|19-ET]], used for the purpose of a stronger version of the traditional dominant chord.
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| === JI as an Extension of Pythagorean Tuning ===
| | 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. |
| 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, M3<sup>5</sup>, 5/4, lower by 81/80, and M3<sub>7</sub>, 9/7, higher by 64/63.
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| 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.
| | == Theory == |
| | This section explains why the chord is what it is. |
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| <math>
| | === Original occurrence === |
| \begin{bmatrix}
| | The chord originally arose as 0–7–15 steps of [[19edo|19et]], used for the purpose of a stronger version of the traditional dominant chord. |
| 1 & 0 & 0 & 0\\
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| 0 & 1 & 0 & 0\\
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| 0 & 0 & 1 & 0\\
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| 0 & 0 & 0 & 1
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| \end{bmatrix}
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| \rightarrow
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| \begin{bmatrix}
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| 1 & 1 & 0 & 4\\
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| 0 & 1 & 4 & -2\\
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| 0 & 0 & 1 & 0\\
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| 0 & 0 & 0 & 1
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| \end{bmatrix}
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| </math>
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| The tuning space remains the same because no comma is tempered out. Notice how the first two rows are the same as the [[Meantone family #Dominant|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.
| | === Septimal voice leading === |
| | <small>This section is transcluded from [[Flora's analysis on septimal voice leading]]</small> |
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| === 7-limit Voice Leading ===
| | {{:Flora's analysis on septimal voice leading}} |
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| 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 m2<sub>5</sub>, [[16/15]], sharp by 81/80, and m2<sup>7</sup>, 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. | | === Relation to essentially tempered chords === |
| | The chord by itself is ambitonal and not an [[essentially tempered chord]] of the sensamagic temperament, but the tempered essence is emergent if the chord is viewed in the dominant–tonic progression. 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. |
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| 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.
| | == 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 |
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| The Canovian chord is based on the theory that 28/27 is used for voice leading.
| | <math>\text {(Semiporwellismic) } 3/4–32/33–128/99 \rightarrow 1–5/4–3/2</math> |
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| === On 21/20 ===
| | With the voice leading at 32/33 → 1 and 128/99 → 5/4 characterized by 33/32. |
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| [[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. | | The negative harmony version is 1–[[64/55]]–[[96/55]]. The progression with respect to the tonic is |
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| === Relationship to Essentially Tempered Dyadic Chords ===
| | <math>\text {(Semiporwellismic) } 2/3–64/55–99/64 \rightarrow 1–6/5–3/2</math> |
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| 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.
| | With the same voice leading at 64/55 → 6/5 and 99/64 → 3/2. |
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| == External Links == | | == External links == |
| * [https://www.reddit.com/r/microtonal/comments/h8wqhe/19et_theory_sensamagic_dominant_chord/ 19-ET theory: sensamagic dominant chord] - Reddit post | | * [https://www.reddit.com/r/microtonal/comments/h8wqhe/19et_theory_sensamagic_dominant_chord/ 19-ET theory: sensamagic dominant chord] – Reddit post |
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| [[Category:Chords]] | | [[Category:Dominant seventh chords]] |
| [[Category:Theory]]
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| [[Category:19edo]] | | [[Category:19edo]] |
| [[Category:Sensamagic]] | | [[Category:Sensamagic]] |