Marvel: Difference between revisions

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| Odd limit 2 = 11-limit 21 | Mistuning 2 = 3.86 | Complexity 2 = ?

}}

'''Marvel''' is a [[rank-3 ~~temperament|rank-3]] [[regular temperament|~~temperament]] with the same [[lattice]] structure as [[5-limit]] [[JI]], while identifying the [[7/4|harmonic seventh (7/4)]] as a stack of two [[15/8|classical major sevenths (15/8)]] [[octave reduction|octave-reduced]], [[tempering out]] [[225/224]]. It is the head of the [[marvel family]].

'''Marvel''' is a [[rank-3 temperament]] with the same [[lattice]] structure as [[5-limit]] [[JI]], while identifying the [[7/4|harmonic seventh (7/4)]] as a stack of two [[15/8|classical major sevenths (15/8)]] [[octave reduction|octave-reduced]], [[tempering out]] [[225/224]]. It is the head of the [[marvel family]].

Marvel tends to tune [[5/4]] flat, such that a stack of two such thirds give a sharp [[14/9]], and a ~~stack~~ of three ~~such~~ thirds ~~give~~ a ~~sub-octave of~~ [[35/18]] just short of the octave by a quartertone of [[36/35]]. The canonical [[11-limit]] [[extension]], sometimes aliased ''unimarv'', identifies the quartertone as [[33/32]], so that [[11/8]] is that plus a [[4/3|perfect fourth]]. This adds [[385/384]] and [[540/539]] to the comma list and makes it a member of both [[keenanismic temperaments|keenansimic]] and [[swetismic temperaments]].

Marvel tends to tune [[5/4]] flat, such that a stack of two such thirds give a sharp [[14/9]]. In other words, it has a 5/4, 5/4, 9/7 version of the [[augmented triad]]. Since this is a chord of [[meantone]] in wide use in Western {{w|common practice}} harmony long before [[12edo]] established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.

Another ~5/4 on top of ~14/9 gives a [[35/18]] sub-octave just short of the octave by a quartertone of [[36/35]]. The canonical [[11-limit]] [[extension]], sometimes aliased ''unimarv'', identifies the quartertone as [[33/32]], so that [[11/8]] is that plus a [[4/3|perfect fourth]]. This adds [[385/384]] and [[540/539]] to the comma list and makes it a member of both [[keenanismic temperaments|keenansimic]] and [[swetismic temperaments]].

The temperament was named by [[Gene Ward Smith]] in 2002–2003, when the 11-limit version was found first<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5145.html#5184 Yahoo! Tuning Group | ''Relative complexity and scale construction''] – first mention of ''marvel''.</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5687.html Yahoo! Tuning Group | ''Top 135 11-limit planar temperaments''] – establishment as an 11-limit temperament.</ref>. Gene carried it to the 7-limit restriction in 2004<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_50829.html Yahoo! Tuning Group | ''Marvel'']</ref>.

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== Notation ==

Marvel can be notated the same as [[5-limit]] [[just intonation]] since they share the same lattice structure. One way to do this is to take the conventional [[circle-of-fifths notation]] with an additional module of accidentals for the [[81/80]] comma. In this system, 5/4 is a major third, 7/4 an augmented sixth, and 11/8 a double diminished ~~5th~~.

Marvel can be notated the same as [[5-limit]] [[just intonation]] since they share the same lattice structure. One way to do this is to take the conventional [[circle-of-fifths notation]] with an additional module of accidentals for the [[81/80]] comma. In this system, 5/4 is a major third, 7/4 an augmented sixth, and 11/8 a double-diminished fifth.

{| class="wikitable center-1 center-3"

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| 3/2

| Perfect fifth

| ~~C-G~~

| C–G

|-

| 5/4

| Down major third

| ~~C-vE~~

| C–vE

|-

| 7/4

| Dudaugmented sixth

| ~~C-vvA#~~

| C–vvA♯

|-

| 11/8

| Trup double-diminished fifth

| C-^<sup>3</sup>~~Gbb~~

| C–^<sup>3</sup>G𝄫

|-

| 13/8

| Dup minor sixth

| C-^^Ab

| C–^^A♭

|-

| 19/16

| Minor third

| ~~C-Eb~~

| C–E♭

|}

Alternatively, it can be notated the same as full prime-limit just intonation, with a distinct accidental pair for each prime. That makes some intervals more intuitive, at the cost of hiding the structure of marvel tempering. For example, it is customary of the 5/4 to be a major third, and 7/4 to be a minor seventh. As a result, the fact that the 14/9 is a stack of two 5/4's is not revealed, and the related chords can be less convenient.

Alternatively, it can be notated the same as full prime-limit just intonation, with a distinct accidental pair for each prime. That makes some intervals more intuitive, at the cost of hiding the structure of marvel tempering. For example, it is customary of the 5/4 to be a major third and 7/4 to be a minor seventh. As a result, the fact that the 14/9 is a stack of two 5/4's is not revealed, and the related chords can be less convenient.

== Chords and harmony ==

Marvel enables [[essentially tempered chord]]s of [[marvel chords|marvel]], [[keenanismic chords|keenanismic]], [[swetismic chords|swetismic]], and [[undecimal marvel chords|undecimal marvel]].

Marvel enables [[essentially tempered chord]]s of [[marvel chords|marvel]], [[keenanismic chords|keenanismic]], [[swetismic chords|swetismic]], and [[undecimal marvel chords|undecimal marvel]]. It also allows for a tritone substitution whereby the [[dominant seventh chord]] formed by adding 16/9 above the root shares its tritone with the [[harmonic seventh chord]]. (The tritone of the dominant seventh is {{nowrap|(16/9)/(5/4) {{=}} 64/45}}. Setting this equal to 10/7 gives {{nowrap|(10/7)/(64/45) {{=}} 225/224}}.)

Extending the temperament to the 13-limit through 325/324, resulting in hecate, enables chords of [[marveltwin chords|marveltwin]] and [[squbemic chords|squbemic]]. [[Hecate hexad]] is a chord peculiar to this temperament. Alternatively, helios enables chords of [[ratwolfsmic chords|ratwolfsmic]].

@@ Line 19: / Line 19: @@
 | Odd limit 2 = 11-limit 21 | Mistuning 2 = 3.86 | Complexity 2 = ?
 }}
-'''Marvel''' is a [[rank-3 temperament|rank-3]] [[regular temperament|temperament]] with the same [[lattice]] structure as [[5-limit]] [[JI]], while identifying the [[7/4|harmonic seventh (7/4)]] as a stack of two [[15/8|classical major sevenths (15/8)]] [[octave reduction|octave-reduced]], [[tempering out]] [[225/224]]. It is the head of the [[marvel family]].
+'''Marvel''' is a [[rank-3 temperament]] with the same [[lattice]] structure as [[5-limit]] [[JI]], while identifying the [[7/4|harmonic seventh (7/4)]] as a stack of two [[15/8|classical major sevenths (15/8)]] [[octave reduction|octave-reduced]], [[tempering out]] [[225/224]]. It is the head of the [[marvel family]].
-Marvel tends to tune [[5/4]] flat, such that a stack of two such thirds give a sharp [[14/9]], and a stack of three such thirds give a sub-octave of [[35/18]] just short of the octave by a quartertone of [[36/35]]. The canonical [[11-limit]] [[extension]], sometimes aliased ''unimarv'', identifies the quartertone as [[33/32]], so that [[11/8]] is that plus a [[4/3|perfect fourth]]. This adds [[385/384]] and [[540/539]] to the comma list and makes it a member of both [[keenanismic temperaments|keenansimic]] and [[swetismic temperaments]].
+Marvel tends to tune [[5/4]] flat, such that a stack of two such thirds give a sharp [[14/9]]. In other words, it has a 5/4, 5/4, 9/7 version of the [[augmented triad]]. Since this is a chord of [[meantone]] in wide use in Western {{w|common practice}} harmony long before [[12edo]] established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
+Another ~5/4 on top of ~14/9 gives a [[35/18]] sub-octave just short of the octave by a quartertone of [[36/35]]. The canonical [[11-limit]] [[extension]], sometimes aliased ''unimarv'', identifies the quartertone as [[33/32]], so that [[11/8]] is that plus a [[4/3|perfect fourth]]. This adds [[385/384]] and [[540/539]] to the comma list and makes it a member of both [[keenanismic temperaments|keenansimic]] and [[swetismic temperaments]].
 The temperament was named by [[Gene Ward Smith]] in 2002–2003, when the 11-limit version was found first<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5145.html#5184 Yahoo! Tuning Group | ''Relative complexity and scale construction''] – first mention of ''marvel''.</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5687.html Yahoo! Tuning Group | ''Top 135 11-limit planar temperaments''] – establishment as an 11-limit temperament.</ref>. Gene carried it to the 7-limit restriction in 2004<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_50829.html Yahoo! Tuning Group | ''Marvel'']</ref>.
@@ Line 37: / Line 39: @@
 == Notation ==
-Marvel can be notated the same as [[5-limit]] [[just intonation]] since they share the same lattice structure. One way to do this is to take the conventional [[circle-of-fifths notation]] with an additional module of accidentals for the [[81/80]] comma. In this system, 5/4 is a major third, 7/4 an augmented sixth, and 11/8 a double diminished 5th.
+Marvel can be notated the same as [[5-limit]] [[just intonation]] since they share the same lattice structure. One way to do this is to take the conventional [[circle-of-fifths notation]] with an additional module of accidentals for the [[81/80]] comma. In this system, 5/4 is a major third, 7/4 an augmented sixth, and 11/8 a double-diminished fifth.
 {| class="wikitable center-1 center-3"
@@ Line 47: / Line 49: @@
 | 3/2
 | Perfect fifth
-| C-G
+| C–G
 |-
 | 5/4
 | Down major third
-| C-vE
+| C–vE
 |-
 | 7/4
 | Dudaugmented sixth
-| C-vvA#
+| C–vvA♯
 |-
 | 11/8
 | Trup double-diminished fifth
-| C-^<sup>3</sup>Gbb
+| C–^<sup>3</sup>G𝄫
 |-
 | 13/8
 | Dup minor sixth
-| C-^^Ab
+| C–^^A♭
 |-
 | 19/16
 | Minor third
-| C-Eb
+| C–E♭
 |}
-Alternatively, it can be notated the same as full prime-limit just intonation, with a distinct accidental pair for each prime. That makes some intervals more intuitive, at the cost of hiding the structure of marvel tempering. For example, it is customary of the 5/4 to be a major third, and 7/4 to be a minor seventh. As a result, the fact that the 14/9 is a stack of two 5/4's is not revealed, and the related chords can be less convenient.
+Alternatively, it can be notated the same as full prime-limit just intonation, with a distinct accidental pair for each prime. That makes some intervals more intuitive, at the cost of hiding the structure of marvel tempering. For example, it is customary of the 5/4 to be a major third and 7/4 to be a minor seventh. As a result, the fact that the 14/9 is a stack of two 5/4's is not revealed, and the related chords can be less convenient.
 == Chords and harmony ==
-Marvel enables [[essentially tempered chord]]s of [[marvel chords|marvel]], [[keenanismic chords|keenanismic]], [[swetismic chords|swetismic]], and [[undecimal marvel chords|undecimal marvel]].
+Marvel enables [[essentially tempered chord]]s of [[marvel chords|marvel]], [[keenanismic chords|keenanismic]], [[swetismic chords|swetismic]], and [[undecimal marvel chords|undecimal marvel]]. It also allows for a tritone substitution whereby the [[dominant seventh chord]] formed by adding 16/9 above the root shares its tritone with the [[harmonic seventh chord]]. (The tritone of the dominant seventh is {{nowrap|(16/9)/(5/4) {{=}} 64/45}}. Setting this equal to 10/7 gives {{nowrap|(10/7)/(64/45) {{=}} 225/224}}.)
 Extending the temperament to the 13-limit through 325/324, resulting in hecate, enables chords of [[marveltwin chords|marveltwin]] and [[squbemic chords|squbemic]]. [[Hecate hexad]] is a chord peculiar to this temperament. Alternatively, helios enables chords of [[ratwolfsmic chords|ratwolfsmic]].

Marvel: Difference between revisions

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