Marvel: Difference between revisions

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'''Marvel''' is ~~the~~ [[~~Rank~~-3 temperament|rank-3]] [[temperament]] [[tempering out]] [[225/224]], the ~~marvel comma. It has a~~ canonical 11-limit [[extension]] adding [[385/384]] and [[540/539]] to the comma list.

'''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]], and the canonical [[11-limit]] [[extension]] adding [[385/384]] and [[540/539]] to the comma list 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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* [[Pump17]]

* [[Pump18]]

* [[Diamond9plus-marvel]]

== Tunings ==

In the 7-limit, the optimal way such as that taken by [[TE]] and derivatives to close out the comma 225/224 is to tune primes 3 and 5 flat, and 2 and 7 sharp. If we tune the octave pure, the other inclinations remain~~, with prime 5 the most tempered in accordance with minimizing damage on 7-limit [[LCJI]] intervals~~. This indicates that the ~~supermajor second~~ [[~]][[8/7]] should be ~~flat~~ (towards [[~]][[~~256~~/~~225~~]]), the ~~subminor third~~ [[~]][[7/6]] be ~~sharp~~ (towards [[~]][[75/64]]), the ~~supermajor third~~ [[~]][[9/7]] be ~~flat~~ (towards [[~]][[32/25]]) and the tritone [[~]][[7/5]] be ~~sharp~~ (towards [[~]][[45/32]]), such that every [[7-limit]] [[9-odd-limit]] interval is tuned between itself and the [[5-limit]] interpretation it is separated from by [[225/224]]. If we take these as hard constraints, then [[53edo]] and [[84edo]] are the smallest edo tunings to ~~satify~~ them, but if overtempering is allowed, many smaller edos are possible, such as [[31edo|31-]] and [[41edo]]. Interestingly, [[72edo]] is overtempered for some of these constraints, whereas 53edo, though satisfying these constraints, tempers ~~in favor of tuning~~ the intervals closer to the more complex [[5-limit]] interpretations, though ~~because of their comparative simplicity (and thus lesser tuning fidelity),~~ the 7-limit concordances of the 9-odd-limit still clearly work ~~so that 53edo is [[consistent to distance]] 2 in the 9-odd-limit if we exclude 7/5 and 10/7 which are the most damaged~~. [[84edo]], another superset of 12edo, is an interesting edo to look at for its high performance in large odd-limits. Going up to larger edos, [[166edo|166-]], [[178edo|178-]], [[197edo|197-]], and [[240edo]] are all great choices with different intonational characteristics~~, though note that inconsistencies may often arise in the representation of the 7-limit lattice because [[225/224]] is itself larger than the size of the step of any of these edos~~.

In the 7-limit, the optimal way such as that taken by [[TE]] and derivatives to close out the comma 225/224 is to tune primes 3 and 5 flat, and 2 and 7 sharp. If we tune the octave pure, the other inclinations remain. This indicates that the diminished third [[~]][[256/225]] should be sharp (towards [[~]][[8/7]]), the augmented second [[~]][[75/64]] be flat (towards [[~]][[7/6]]), the diminished fourth [[~]][[32/25]] be sharp (towards [[~]][[9/7]]) and the tritone [[~]][[45/32]] be flat (towards [[~]][[7/5]]), such that every [[7-limit]] [[9-odd-limit]] interval is tuned between itself and the [[5-limit]] interpretation it is separated from by [[225/224]]. If we take these as hard constraints, then [[53edo]] and [[84edo]] are the smallest edo tunings to satisfy them, but if overtempering is allowed, many smaller edos are possible, such as [[31edo|31-]] and [[41edo]]. Interestingly, [[72edo]], though very performant as a 7- and 11-limit tuning, is overtempered for some of these constraints, whereas 53edo, though satisfying these constraints, tempers the intervals closer to the more complex [[5-limit]] interpretations, though the 7-limit concordances of the 9-odd-limit still clearly work. [[84edo]], another superset of 12edo, is an interesting edo to look at for its high performance in large odd-limits. Going up to larger edos, [[125edo|125-]], [[166edo|166-]], [[178edo|178-]], [[197edo|197-]], and [[240edo]] are all great choices with different intonational characteristics.

The marvel extension [[hecate]] has the no-17's [[19-limit]] as its subgroup, and tridecimal marvel, the extension chosen by [[Gene Ward Smith]], is in the 13-limit. They merge in the rank-2 temperament [[catakleismic]], which can be conceptualized as accepting both rank-3 marvel structures simultaneously. One such tuning is excellently given by [[125edo]]. If we are looking for a small edo tuning instead, 53edo and 72edo are also reasonable edo tunings for the full no-17's 19-limit catakleismic, though in 53edo the 11 and 19 are a little off and in 72edo the 13 and 19 are a little off instead; 72edo is positioned better as a full [[17-limit]] marvel system while 53edo is positioned better as a (potentially no-11's) [[13-limit]] marvel system. If we focus on the 11-limit of undecimal marvel (discarding the mapping of 13), 31edo and 41edo are the smallest to clearly succeed, though many accept 41edo's mapping of [[~]][[13/8]] to the neutral sixth and some accept that mapping for 31edo as contextually usable too.

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~~[[Category:Temperaments]]~~

[[Category:Marvel| ]]

[[Category:Rank-3 temperaments]]

[[Category:Marvel family]]

[[Category:Keenanismic temperaments]]

[[Category:Swetismic temperaments]]

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-'''Marvel''' is the [[Rank-3 temperament|rank-3]] [[temperament]] [[tempering out]] [[225/224]], the marvel comma. It has a canonical 11-limit [[extension]] adding [[385/384]] and [[540/539]] to the comma list.
+'''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]], and the canonical [[11-limit]] [[extension]] adding [[385/384]] and [[540/539]] to the comma list 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 78: / Line 78: @@
 * [[Pump17]]
 * [[Pump18]]
+* [[Diamond9plus-marvel]]
 == Tunings ==
-In the 7-limit, the optimal way such as that taken by [[TE]] and derivatives to close out the comma 225/224 is to tune primes 3 and 5 flat, and 2 and 7 sharp. If we tune the octave pure, the other inclinations remain, with prime 5 the most tempered in accordance with minimizing damage on 7-limit [[LCJI]] intervals. This indicates that the supermajor second [[~]][[8/7]] should be flat (towards [[~]][[256/225]]), the subminor third [[~]][[7/6]] be sharp (towards [[~]][[75/64]]), the supermajor third [[~]][[9/7]] be flat (towards [[~]][[32/25]]) and the tritone [[~]][[7/5]] be sharp (towards [[~]][[45/32]]), such that every [[7-limit]] [[9-odd-limit]] interval is tuned between itself and the [[5-limit]] interpretation it is separated from by [[225/224]]. If we take these as hard constraints, then [[53edo]] and [[84edo]] are the smallest edo tunings to satify them, but if overtempering is allowed, many smaller edos are possible, such as [[31edo|31-]] and [[41edo]]. Interestingly, [[72edo]] is overtempered for some of these constraints, whereas 53edo, though satisfying these constraints, tempers in favor of tuning the intervals closer to the more complex [[5-limit]] interpretations, though because of their comparative simplicity (and thus lesser tuning fidelity), the 7-limit concordances of the 9-odd-limit still clearly work so that 53edo is [[consistent to distance]] 2 in the 9-odd-limit if we exclude 7/5 and 10/7 which are the most damaged. [[84edo]], another superset of 12edo, is an interesting edo to look at for its high performance in large odd-limits. Going up to larger edos, [[166edo|166-]], [[178edo|178-]], [[197edo|197-]], and [[240edo]] are all great choices with different intonational characteristics, though note that inconsistencies may often arise in the representation of the 7-limit lattice because [[225/224]] is itself larger than the size of the step of any of these edos.
+In the 7-limit, the optimal way such as that taken by [[TE]] and derivatives to close out the comma 225/224 is to tune primes 3 and 5 flat, and 2 and 7 sharp. If we tune the octave pure, the other inclinations remain. This indicates that the diminished third [[~]][[256/225]] should be sharp (towards [[~]][[8/7]]), the augmented second [[~]][[75/64]] be flat (towards [[~]][[7/6]]), the diminished fourth [[~]][[32/25]] be sharp (towards [[~]][[9/7]]) and the tritone [[~]][[45/32]] be flat (towards [[~]][[7/5]]), such that every [[7-limit]] [[9-odd-limit]] interval is tuned between itself and the [[5-limit]] interpretation it is separated from by [[225/224]]. If we take these as hard constraints, then [[53edo]] and [[84edo]] are the smallest edo tunings to satisfy them, but if overtempering is allowed, many smaller edos are possible, such as [[31edo|31-]] and [[41edo]]. Interestingly, [[72edo]], though very performant as a 7- and 11-limit tuning, is overtempered for some of these constraints, whereas 53edo, though satisfying these constraints, tempers the intervals closer to the more complex [[5-limit]] interpretations, though the 7-limit concordances of the 9-odd-limit still clearly work. [[84edo]], another superset of 12edo, is an interesting edo to look at for its high performance in large odd-limits. Going up to larger edos, [[125edo|125-]], [[166edo|166-]], [[178edo|178-]], [[197edo|197-]], and [[240edo]] are all great choices with different intonational characteristics.
 The marvel extension [[hecate]] has the no-17's [[19-limit]] as its subgroup, and tridecimal marvel, the extension chosen by [[Gene Ward Smith]], is in the 13-limit. They merge in the rank-2 temperament [[catakleismic]], which can be conceptualized as accepting both rank-3 marvel structures simultaneously. One such tuning is excellently given by [[125edo]]. If we are looking for a small edo tuning instead, 53edo and 72edo are also reasonable edo tunings for the full no-17's 19-limit catakleismic, though in 53edo the 11 and 19 are a little off and in 72edo the 13 and 19 are a little off instead; 72edo is positioned better as a full [[17-limit]] marvel system while 53edo is positioned better as a (potentially no-11's) [[13-limit]] marvel system. If we focus on the 11-limit of undecimal marvel (discarding the mapping of 13), 31edo and 41edo are the smallest to clearly succeed, though many accept 41edo's mapping of [[~]][[13/8]] to the neutral sixth and some accept that mapping for 31edo as contextually usable too.
@@ Line 184: / Line 185: @@
 <references/>
-[[Category:Temperaments]]
 [[Category:Marvel| ]] <!-- main article -->
+[[Category:Rank-3 temperaments]]
 [[Category:Marvel family]]
 [[Category:Keenanismic temperaments]]
 [[Category:Swetismic temperaments]]