Sensamagic dominant chord - Revision history

FloraC: Rework: add the fifth to the definition

2024-12-23T09:30:21Z

Rework: add the fifth to the definition

← Older revision		Revision as of 09:30, 23 December 2024
Line 1:		Line 1:
	The '''sensamagic dominant chord''' (or '''Canovian chord''' since it was first explored by [[Flora Canou]]) ~~refers to~~ the [[sensamagic]] tempered chord of ratios 1-[[9/7]]-[[12/7]] built on the dominant. It features two instances of voice leading by [[28/27]] in the resolution to the tonic.		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.

	[[File:Canovian Chord 19-ET.png\|thumb\|right\|~~Canovian~~ chord notated in 19-ET]]		[[File:Canovian Chord 19-ET.png\|thumb\|right\|Sensamagic dominant chord notated in 19-ET]]
	[[File:Canovian Chord FJS.png\|thumb\|right\|~~Canovian~~ chord notated in FJS]]		[[File:Canovian Chord FJS.png\|thumb\|right\|Sensamagic dominant chord notated in FJS]]

	== Construction ==		== Construction ==
	The chord consists of a tempered ~~1-9~~/~~7-12~~/7, usually built on the fifth ~~note~~ of a diatonic scale ~~– it is a dominant chord after all~~. 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 fifth ~~of 3/2 can be added,~~ omitted here for simplicity~~. The~~ progression with respect to the tonic is		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

	<math>\text {(Sensamagic) } 3/4–27/28–9/7 \rightarrow 1–5/4–3/2</math>		<math>\text {(Sensamagic) } 3/4–27/28–9/7 \rightarrow 1–5/4–3/2</math>
Line 11:		Line 11:
	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.		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.

	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 ~~complex 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.		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.

	The negative harmony version of the chord consists of a tempered 1-[[7/6]]-[[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		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

	<math>\text {(Sensamagic) } 2/3–7/6–14/9 \rightarrow 1–6/5–3/2</math>		<math>\text {(Sensamagic) } 2/3–7/6–14/9 \rightarrow 1–6/5–3/2</math>
Line 23:		Line 23:

	=== Original occurrence ===		=== Original occurrence ===
	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.		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 ===		=== Septimal voice leading ===
Line 31:		Line 31:

	=== Relation to essentially tempered chords ===		=== Relation to essentially tempered chords ===
	The chord by itself is ~~utonal~~ and not an [[essentially tempered chord]] of the sensamagic temperament, 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 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.

	== Variations ==		== 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		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

	<math>\text {(Semiporwellismic) } 3/4–32/33–128/99 \rightarrow 1–5/4–3/2</math>		<math>\text {(Semiporwellismic) } 3/4–32/33–128/99 \rightarrow 1–5/4–3/2</math>
Line 40:		Line 40:
	With the voice leading at 32/33 → 1 and 128/99 → 5/4 characterized by 33/32.		With the voice leading at 32/33 → 1 and 128/99 → 5/4 characterized by 33/32.

	The negative harmony version is 1-[[64/55]]-[[96/55]]. The progression with respect to the tonic is		The negative harmony version is 1–[[64/55]]–[[96/55]]. The progression with respect to the tonic is

	<math>\text {(Semiporwellismic) } 2/3–64/55–99/64 \rightarrow 1–6/5–3/2</math>		<math>\text {(Semiporwellismic) } 2/3–64/55–99/64 \rightarrow 1–6/5–3/2</math>
Line 49:		Line 49:
	* [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

	[[Category:~~Triads~~]]		[[Category:Dominant seventh chords]]
	[[Category:19edo]]		[[Category:19edo]]
	[[Category:Sensamagic]]		[[Category:Sensamagic]]

Fredg999: Categories

2024-12-22T22:39:04Z

Fredg999 category edits: Removing from Category:Theory using Cat-a-lot

2023-03-05T05:37:53Z

Removing from Category:Theory using Cat-a-lot

← Older revision		Revision as of 05:37, 5 March 2023
Line 50:		Line 50:

	[[Category:Chords]]		[[Category:Chords]]
	~~[[Category:Theory]]~~
	[[Category:19edo]]		[[Category:19edo]]
	[[Category:Sensamagic]]		[[Category:Sensamagic]]

FloraC: /* Theory */ update the essentially tempered chord section

2022-02-08T22:45:31Z

Theory: update the essentially tempered chord section

← Older revision		Revision as of 22:45, 8 February 2022
Line 30:		Line 30:
	{{:Flora's analysis on septimal voice leading}}		{{:Flora's analysis on septimal voice leading}}

	=== ~~Relationship~~ to essentially tempered ~~chord~~ ===		=== Relation to essentially tempered chords ===
	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 utonal and not an [[essentially tempered chord]] of the sensamagic temperament, 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 ==

FloraC: Adopt transclusion

2022-02-08T20:21:38Z

Adopt transclusion

@@ Line 26: / Line 26: @@
 === 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]].
+<small>This section is transcluded from [[Flora's analysis on septimal voice leading]]</small>
-<div style="font-style: italic; border: 1px solid silver; margin: 15px; padding: 15px;">
+{{:Flora's analysis on septimal voice leading}}
 === Relationship to essentially tempered chord ===

FloraC: Clarify

2022-02-08T17:53:16Z

Clarify

← Older revision		Revision as of 17:53, 8 February 2022
Line 1:		Line 1:
	The '''sensamagic dominant chord''' (or '''Canovian chord''' since it was first explored by [[Flora Canou]]) refers to the [[sensamagic]] tempered chord of ratios 1-[[9/7]]-[[12/7]]. It features two instances of voice leading by [[28/27]] in the resolution to the tonic.		The '''sensamagic dominant chord''' (or '''Canovian chord''' since it was first explored by [[Flora Canou]]) refers to the [[sensamagic]] tempered chord of ratios 1-[[9/7]]-[[12/7]] built on the dominant. It features two instances of voice leading by [[28/27]] in the resolution to the tonic.

	[[File:Canovian Chord 19-ET.png\|thumb\|right\|Canovian chord notated in 19-ET]]		[[File:Canovian Chord 19-ET.png\|thumb\|right\|Canovian chord notated in 19-ET]]
Line 5:		Line 5:

	== Construction ==		== Construction ==
	The chord consists of a tempered 1-9/7-12/7, usually built on the fifth note of a diatonic scale – it is a dominant chord after all. 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 progression with respect to the tonic is		The chord consists of a tempered 1-9/7-12/7, usually built on the fifth note of a diatonic scale – it is a dominant chord after all. 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 fifth of 3/2 can be added, omitted here for simplicity. The progression with respect to the tonic is

	<math>\text {(Sensamagic) } 3/4–27/28–9/7 \rightarrow 1–5/4–3/2</math>		<math>\text {(Sensamagic) } 3/4–27/28–9/7 \rightarrow 1–5/4–3/2</math>

FloraC: /* Variations */ +negative harmony version

2022-02-08T14:03:13Z

Variations: +negative harmony version

← Older revision		Revision as of 14:03, 8 February 2022
Line 74:		Line 74:
	<math>\text {(Semiporwellismic) } 3/4–32/33–128/99 \rightarrow 1–5/4–3/2</math>		<math>\text {(Semiporwellismic) } 3/4–32/33–128/99 \rightarrow 1–5/4–3/2</math>

	With the voice leading of 32/33 → 1 and 128/99 → 5/4 characterized by 33/32.		With the voice leading at 32/33 → 1 and 128/99 → 5/4 characterized by 33/32.

			The negative harmony version is 1-[[64/55]]-[[96/55]]. The progression with respect to the tonic is

			<math>\text {(Semiporwellismic) } 2/3–64/55–99/64 \rightarrow 1–6/5–3/2</math>

			With the same voice leading at 64/55 → 6/5 and 99/64 → 3/2.

	== External links ==		== External links ==

FloraC: I'd love to share an undecimal variant

2022-02-08T13:44:38Z

I'd love to share an undecimal variant

← Older revision		Revision as of 13:44, 8 February 2022
Line 1:		Line 1:
	The '''sensamagic dominant chord''' (or '''Canovian chord''' since it was first explored by [[Flora Canou]]) refers to the sensamagic tempered chord of ratios 1-[[9/7]]-[[12/7]]. It features two instances of voice leading by [[28/27]] in the resolution to the tonic.		The '''sensamagic dominant chord''' (or '''Canovian chord''' since it was first explored by [[Flora Canou]]) refers to the [[sensamagic]] tempered chord of ratios 1-[[9/7]]-[[12/7]]. It features two instances of voice leading by [[28/27]] in the resolution to the tonic.

	[[File:Canovian Chord 19-ET.png\|thumb\|right\|Canovian chord notated in 19-ET]]		[[File:Canovian Chord 19-ET.png\|thumb\|right\|Canovian chord notated in 19-ET]]
Line 5:		Line 5:

	== Construction ==		== Construction ==

	The chord consists of a tempered 1-9/7-12/7, usually built on the fifth note of a diatonic scale – it is a dominant chord after all. 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 progression with respect to the tonic is		The chord consists of a tempered 1-9/7-12/7, usually built on the fifth note of a diatonic scale – it is a dominant chord after all. 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 progression with respect to the tonic is

Line 13:		Line 12:

	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 complex 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.		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 complex 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.

			The negative harmony version of the chord consists of a tempered 1-[[7/6]]-[[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

			<math>\text {(Sensamagic) } 2/3–7/6–14/9 \rightarrow 1–6/5–3/2</math>

			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.

	== Theory ==		== Theory ==
Line 65:		Line 70:

	== Variations ==		== Variations ==
	~~=== Rotations ===~~		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
	~~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 complex sensamagic fifth~~ [[~~729~~/~~490~~]] ~~at 688 cents~~.

	~~=== Negative harmony ===~~
	~~The negative version of the chord consists of a tempered~~ 1-[[7/6]]-[[~~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

	<math>\text {(~~Sensamagic~~) } 2/~~3–7~~/~~6–14~~/9 \rightarrow ~~1–6~~/~~5–3~~/2</math>		<math>\text {(Semiporwellismic) } 3/4–32/33–128/99 \rightarrow 1–5/4–3/2</math>

	~~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.		With the voice leading of 32/33 → 1 and 128/99 → 5/4 characterized by 33/32.

	== External links ==		== External links ==

FloraC: Respell the 140/81 as 12/7 and elaborate on this equivalence

2022-02-08T13:22:22Z

Respell the 140/81 as 12/7 and elaborate on this equivalence

← Older revision		Revision as of 13:22, 8 February 2022
Line 1:		Line 1:
	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~~.		The '''sensamagic dominant chord''' (or '''Canovian chord''' since it was first explored by [[Flora Canou]]) refers to the sensamagic tempered chord of ratios 1-[[9/7]]-[[12/7]]. It features two instances of voice leading by [[28/27]] in the resolution to the tonic.

	[[File:Canovian Chord 19-ET.png\|thumb\|right\|Canovian chord notated in 19-ET]]		[[File:Canovian Chord 19-ET.png\|thumb\|right\|Canovian chord notated in 19-ET]]
	[[File:Canovian Chord FJS.png\|thumb\|right\|Canovian chord notated in FJS]]		[[File:Canovian Chord FJS.png\|thumb\|right\|Canovian chord notated in FJS]]

	== ~~Components~~ ==		== Construction ==

	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~~.		The chord consists of a tempered 1-9/7-12/7, usually built on the fifth note of a diatonic scale – it is a dominant chord after all. 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 progression with respect to the tonic is

	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.		<math>\text {(Sensamagic) } 3/4–27/28–9/7 \rightarrow 1–5/4–3/2</math>

			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.

			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 complex 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 ==		== Theory ==
Line 62:		Line 66:
	== Variations ==		== Variations ==
	=== Rotations ===		=== 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.		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 complex sensamagic fifth [[729/490]] at 688 cents.

	=== Negative harmony ===		=== 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~~.		The negative version of the chord consists of a tempered 1-[[7/6]]-[[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

			<math>\text {(Sensamagic) } 2/3–7/6–14/9 \rightarrow 1–6/5–3/2</math>

			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.

	== External links ==		== External links ==

FloraC: /* Theory */ I've decided to simply quote my essay

2022-02-08T09:51:30Z

Theory: I've decided to simply quote my essay

← Older revision		Revision as of 09:51, 8 February 2022
Line 13:		Line 13:
	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 ==

← Older revision		Revision as of 22:39, 22 December 2024
Line 49:		Line 49:
	* [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

	[[Category:~~Chords~~]]		[[Category:Triads]]
	[[Category:19edo]]		[[Category:19edo]]
	[[Category:Sensamagic]]		[[Category:Sensamagic]]