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


=== Introduction ===
=== Original occurrence ===
The chord arises as 0-7-15 steps of [[19edo|19et]], used for the purpose of a stronger version of the traditional dominant 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.  


=== JI as an extension of Pythagorean tuning ===
=== Septimal voice leading ===
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.  
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]].  


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.  
<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 tuning space is isomorphic 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.
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>


=== 7-limit voice leading ===
where 27/14 resolves to 2/1.  
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. 12et 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 very adequate for voice leading. In a meantone temperament like 19et, however, the Pythagorean~classic 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 19et 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 sensamagic dominant chord is based on the theory that 28/27 is used for voice leading.  
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.  


=== On 21/20 ===
</div>
[[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 19et is 1-9/7-7/4 and requires [[49/48]] be tempered out – some do propose it in 19et, but that is another story.


=== 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 ==
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