Hemimage bleeding chord: Difference between revisions

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The '''hemimage bleeding chord''' refers to ~~the~~ [[hemimage]] tempered ~~chord~~ of ratios 1-[[7/6]]-[[3/2]]-[[7/4]] built on the upper bleeding tone ~~(see below)~~. First explored by [[Flora Canou]], it features up to four instances of voice leading by [[28/27]] in the resolution to the tonic.

The '''hemimage bleeding chord''' refers to a collection of [[hemimage]] tempered chords built on the bleeding tones (see below). 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 ==

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== Construction ==

The ~~chord~~ 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

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>

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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 of the ~~chord~~ 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

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>

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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.

~~A possible variant is 1-7/6-[[7/5]]-7/4. The 7/5 goes up by 28/27 to~~ the ~~dominant~~. ~~The progression with respect to the tonic is~~

As one might expect, the reverse works as well.

~~<math>\text {(Hemimage) } 28~~/~~27–98~~/~~81–81/56–49/27 \rightarrow 1–5/4–3/2–15~~/8~~</math>~~

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

~~with the same voice leading at 28~~/~~27 → 1, 135~~/~~112 → 5~~/~~4, 81~~/~~56 → 3~~/~~2, and 405~~/~~224 → 15~~/8.

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

~~The negative harmony version is~~ 1-6/5-7/~~5-8~~/5, ~~again built on the ''upper'' bleeding tone~~.

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

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

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

with the same voice leading at 28/27 → 1, 56/45 → 6/5, 81/56 → 3/2, and 224/135 → 8/5.

<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 are four variants, all of 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 ==

@@ Line 1: / Line 1: @@
-The '''hemimage bleeding chord''' refers to the [[hemimage]] tempered chord of ratios 1-[[7/6]]-[[3/2]]-[[7/4]] built on the upper bleeding tone (see below). First explored by [[Flora Canou]], it features up to four instances of voice leading by [[28/27]] in the resolution to the tonic.
+The '''hemimage bleeding chord''' refers to a collection of [[hemimage]] tempered chords built on the bleeding tones (see below). 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 ==
@@ Line 5: / Line 5: @@
 == Construction ==
-The chord 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
+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>
@@ Line 11: / Line 11: @@
 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 of the chord 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
+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>
@@ Line 17: / Line 17: @@
 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.
-A possible variant is 1-7/6-[[7/5]]-7/4. The 7/5 goes up by 28/27 to the dominant. The progression with respect to the tonic is
+As one might expect, the reverse works as well.
-<math>\text {(Hemimage) } 28/27–98/81–81/56–49/27 \rightarrow 1–5/4–3/2–15/8</math>
+The '''hemimage reverse lower bleeding tetrad''' is 1-5/4-3/2-15/8 built on the lower bleeding tone.
-with the same voice leading at 28/27 → 1, 135/112 → 5/4, 81/56 → 3/2, and 405/224 → 15/8.
+<math>\text {(Hemimage) } 27/28–98/81–81/56–49/27 \rightarrow 1–7/6–3/2–7/4</math>
-The negative harmony version is 1-6/5-7/5-8/5, again built on the ''upper'' bleeding tone.
+with the same voice leading at 27/28 → 1, 98/81 → 7/6, 81/56 → 3/2, and 49/27 → 7/4.
-<math>\text {(Hemimage) } 28/27–56/45–81/56–81/49 \rightarrow 1–6/5–3/2–8/5</math>
+The negative harmony version, the '''hemimage reverse upper bleeding tetrad''', is 1-6/5-3/2-8/5 built on the upper bleeding tone.
-with the same voice leading at 28/27 → 1, 56/45 → 6/5, 81/56 → 3/2, and 224/135 → 8/5.
+<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 are four variants, all of 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 ==