Sensamagic dominant chord: Difference between revisions
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This section explains why the chord is what it is. | This section explains why the chord is what it is. | ||
=== | === Original occurrence === | ||
The chord | 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. | ||
=== | === Septimal voice leading === | ||
In [[User:FloraC/Analysis on the 13-limit just intonation space|''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 class 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]]. | |||
<div style="font-style: italic; border: 1px solid silver; margin: 15px; padding: 15px;"> | |||
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> | <math> | ||
\begin{bmatrix} | \begin{bmatrix} | ||
1 & 0 & 0 & 0\\ | 1 & 0 & 0 & 0 \\ | ||
0 & 1 & 0 & 0\\ | 0 & 1 & 0 & 0 \\ | ||
0 & 0 & 1 & 0\\ | 0 & 0 & 1 & 0 \\ | ||
0 & 0 & 0 & 1 | 0 & 0 & 0 & 1 | ||
\end{bmatrix} | \end{bmatrix} | ||
\rightarrow | \rightarrow | ||
\begin{bmatrix} | \begin{bmatrix} | ||
1 & 1 & 0 & 4\\ | 1 & 1 & 0 & 4 \\ | ||
0 & 1 & 4 & -2\\ | 0 & 1 & 4 & -2 \\ | ||
0 & 0 & 1 & 0\\ | 0 & 0 & 1 & 0 \\ | ||
0 & 0 & 0 & 1 | 0 & 0 & 0 & 1 | ||
\end{bmatrix} | \end{bmatrix} | ||
</math> | </math> | ||
The | 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 (m3<sub>5</sub>), and 7/6 (m3<sup>7</sup>). | ||
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 (m2<sub>5</sub>), sharp by 81/80, and 28/27 (m2<sup>7</sup>), 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 (m2<sup>7</sup><sub>5</sub>), 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. | |||
</div> | |||
=== Relationship to essentially tempered chord === | === Relationship to essentially tempered chord === | ||
The chord by itself is not a [[Dyadic chord|essentially tempered chord]] of the sensamagic temperament because 140/81 is an approximation of [[12/7]], and 1-9/7-12/7 is utonal. 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. | The chord by itself is not a [[Dyadic chord|essentially tempered chord]] of the sensamagic temperament because 140/81 is an approximation of [[12/7]], and 1-9/7-12/7 is utonal. 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 == | == Variations == | ||
Revision as of 09:51, 8 February 2022
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.
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 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.
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.
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
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 class 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]\displaystyle{ \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]\displaystyle{ 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.
Relationship to essentially tempered chord
The chord by itself is not a essentially tempered chord of the sensamagic temperament because 140/81 is an approximation of 12/7, and 1-9/7-12/7 is utonal. 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
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.
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.
External links
- 19-ET theory: sensamagic dominant chord – Reddit post