Hemimage bleeding chord

A hemimage bleeding chord is any of the hemimage tempered chords built on the bleeding tones (see below). There are six chords in total. The standard form is of ratios 1–7/6–3/2–7/4 built on the upper bleeding tone. First explored by Flora Canou, it features up to four instances of voice leading by 28/27 in the resolution to the tonic.

Bleeding tone

In Mason Green's New Common Practice Notation, the bleeding tone is a subminor second or a supermajor seventh to the tonic. In this article, 28/27 will be dubbed the upper bleeding tone, and 27/14, the lower bleeding tone.

Construction

The standard form, the hemimage upper bleeding tetrad, consists of a tempered 1–7/6–3/2–7/4, usually built on the upper bleeding tone. The root is 28/27 above the tonic, and the fifth is 28/27 above the dominant. By the tempering of hemimage, the 7/6 is 98/81~135/112 with respect to the tonic, and 135/112 is 28/27 below 5/4. The 7/4 is 49/27~405/224 with respect to the tonic, and 405/224 is 28/27 below 15/8. For this reason the 7/6 works as a subminor third in terms of chord construction, and as an augmented second in terms of voice leading. The 7/4 works as a subminor seventh in terms of chord construction, and as a augmented sixth in terms of voice leading. The 135/112 and 405/224 spellings are arguably preferable for staff notation as they highlight the voice leading. The progression with respect to the tonic is

[math]\text {(Hemimage) } 28/27–98/81–14/9–49/27 \rightarrow 1–5/4–3/2–15/8[/math]

While the simplest ratios are presented here, it should be noted that the 98/81 is simultaneously 135/112, and that 49/27 is simultaneously 405/224. The voice leadings of 135/112 → 5/4 and 405/224 → 15/8 are characterized by 28/27, just as of 28/27 → 1 and of 14/9 → 3/2.

The negative harmony version, the hemimage lower bleeding tetrad, consists of a tempered 1–9/7–3/2–12/7, built on the lower bleeding tone, 27/28, with the 9/7 being 56/45~243/196, and 12/7 being 81/49~224/135 with respect to the tonic. The progression with respect to the tonic is

[math]\text {(Hemimage) } 27/28–56/45–81/56–81/49 \rightarrow 1–6/5–3/2–8/5[/math]

The 56/45 is simultaneously 243/196, and that 81/49 is simultaneously 224/135. The voice leadings of 243/196 → 6/5 and 224/135 → 8/5 are characterized by 28/27, just as of 27/28 → 1 and of 81/56 → 3/2.

As one might expect, the reverses work as well.

The hemimage reverse lower bleeding tetrad is 1–5/4–3/2–15/8 built on the lower bleeding tone.

[math]\text {(Hemimage) } 27/28–98/81–81/56–49/27 \rightarrow 1–7/6–3/2–7/4[/math]

with the same voice leading at 27/28 → 1, 98/81 → 7/6, 81/56 → 3/2, and 49/27 → 7/4.

The negative harmony version, the hemimage reverse upper bleeding tetrad, is 1–6/5–3/2–8/5 built on the upper bleeding tone.

[math]\text {(Hemimage) } 28/27–56/45–14/9–81/49 \rightarrow 1–9/7–3/2–12/7[/math]

with the same voice leading at 28/27 → 1, 243/196 → 9/7, 14/9 → 3/2, and 81/49 → 12/7.

There is a variant, which "bleeds" the tonic from above and the dominant from below. The negative harmony version does this also. Since that contrary motion is always present, the other two notes may be resolved either up or down.

The hemimage positively variant bleeding tetrad, is 1–7/6–7/5–7/4. This chord can be resolved to either 1–5/4–3/2–15/8 or 1–7/6–3/2–7/4.

[math] \begin {align} \text {(Hemimage) } 28/27–98/81–81/56–49/27 &\rightarrow 1–5/4–3/2–15/8 \\ &\rightarrow 1–7/6–3/2–7/4 \end {align} [/math]

In the first progression, voice leadings are present at 28/27 → 1, 135/112 → 5/4, 81/56 → 3/2, and 405/224 → 15/8. In the second progression, 28/27 → 1, 98/81 → 7/6, 81/56 → 3/2, and 49/27 → 7/4.

The hemimage negatively variant bleeding tetrad, is 1–6/5–7/5–8/5, again built on the upper bleeding tone. This chord can be resolved to either 1–6/5–3/2–8/5 or 1–9/7–3/2–12/7.

[math] \begin {align} \text {(Hemimage) } 28/27–56/45–81/56–81/49 &\rightarrow 1–6/5–3/2–8/5 \\ &\rightarrow 1–9/7–3/2–12/7 \end {align} [/math]

In the first progression, voice leadings are present at 28/27 → 1, 56/45 → 6/5, 81/56 → 3/2, and 224/135 → 8/5. In the second progression, 28/27 → 1, 243/196 → 9/7, 81/56 → 3/2, and 81/49 → 12/7.

Theory

This section explains why the chord is what it is.

Original occurrence

The chord originally arose as 1–4–11–15 steps of 19et, used for the purpose of a stronger version of the traditional leading 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: Episode I, 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 otonal and not an essentially tempered chord of the hemimage temperament, but the tempered essence is emergent if the chord is viewed in the bleed–tonic progression. The minimalist essence of this chord is the 27-odd-limit triad 1–28/27–27/14 with steps 28/27–28/15–28/27.