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}}
{{Infobox regtemp
| Title = Marvel
| Subgroups = 2.3.5.7, 2.3.5.7.11
| Comma basis = [[225/224]] (7-limit); <br>[[225/224]], [[385/384]] (11-limit)
| Edo join 1 = 19 | Edo join 2 = 22 | Edo join 3 = 31
| Mapping = 1; 1 0 2 -1; 0 1 2 -3
| Generators = 3/2, 5/4
| Generators tuning = 700.6, 383.5
| Optimization method = CWE
| Pergen = (P8, P5, ^1)
| Color name = Ruyoyoti
| Odd limit 1 = 9 | Mistuning 1 = 2.57 | Complexity 1 = ?
| 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 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]]. 

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>. 

Extending marvel to the 13-limit is not as obvious. Gene has chosen '''helios''', tempering out [[351/350]], as the canonical extension, but '''hecate''', tempering out [[325/324]] and [[729/728]], arguably makes more sense as it is closer in tuning<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101722.html Yahoo! Tuning Group | ''13-limit marvel'']</ref>. Hecate has a natural extension to the no-17 19-limit, by tempering out [[400/399]] and [[513/512]]. 

See [[Marvel family #Marvel]] for technical data.

== Interval lattice ==
<gallery>
File:Lattice Marvel.png|11-limit marvel
File:Lattice Marvel-hecate13.png|13-limit marvel/hecate
File:Lattice Marvel-hecate19.png|2.3.5.7.11.13.19 subgroup marvel/hecate
</gallery>

== 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. 

{| class="wikitable center-1 center-3"
|+Marvel/hecate nomenclature<br>for selected intervals
! Ratio
! Nominal
! Example
|-
| 3/2
| Perfect fifth
| C-G
|-
| 5/4
| Down major third
| C-vE
|-
| 7/4
| Dudaugmented sixth
| C-vvA#
|-
| 11/8
| Trup double-diminished fifth
| C-^<sup>3</sup>Gbb
|-
| 13/8
| Dup minor sixth
| C-^^Ab
|-
| 19/16
| Minor third
| C-Eb
|}

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]]. 

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]]. 

Alternative 11-limit extensions give different sets of chords. One notable example, tempering out [[441/440]], enables [[prodigy chords]].

== Scales ==
=== Marvel hobbit scales ===
* [[Marvel9]]
* [[Marvel10]]
* [[Marvel11]]
* [[Marvel12]]
* [[Marvel19]]
* [[Marvel22]]

=== Undecimal marvel hobbit scales ===
* [[Marvel22 11]]
* [[Unimarv19]]
* [[Unimarv22]]

=== Other marvel scales ===
* [[Pump12 1]]
* [[Pump12 2]]
* [[Pump13]]
* [[Pump14]]
* [[Pump15]]
* [[Pump16]]
* [[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. 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 helios 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.

=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" | 
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 700.9740{{c}}, ~5/4 = 384.2084{{c}}
| CWE: ~3/2 = 700.6222{{c}}, ~5/4 = 383.8540{{c}}
| POTE: ~3/2 = 700.4075{{c}}, ~5/4 = 383.6376{{c}}
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" | 
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 701.3729{{c}}, ~5/4 = 383.1461{{c}}
| CWE: ~3/2 = 700.6048{{c}}, ~5/4 = 383.4538{{c}}
| POTE: ~3/2 = 700.3887{{c}}, ~5/4 = 383.5403{{c}}
|}

=== Tuning spectrum ===
This spectrum assumes pure 2 and 7. 

{| class="wikitable center-all left-4"
|-
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! Perfect<br>fifth (¢)
! Classical<br>major third (¢)
! Comments
|-
| 5/4
| 698.099
| 386.314
| 
|-
| 6/5
| 700.027
| 384.386
| 7-odd-limit minimax
|-
| 15/11
| 700.351
| 384.062
| 
|-
| 10/9
| 700.670
| 383.743
| 9-odd-limit minimax
|-
| 11/10
| 700.885
| 383.528
| 
|-
| 15/13
| 700.916
| 383.497
| 15-odd-limit hecate minimax
|-
| 13/10
| 701.065
| 383.348
| 13-odd-limit hecate minimax
|-
| 13/11
| 701.199
| 383.214
| 
|-
| 18/13
| 701.361
| 383.052
| 
|-
| 13/12
| 701.480
| 382.933
| 
|-
| 16/13
| 701.559
| 382.854
| 
|-
| 4/3
| 701.955
| 382.458
| 
|-
| 14/11
| 702.278
| 382.135
| 
|-
| 11/8
| 702.278
| 382.135
| 
|-
| 12/11
| 702.602
| 381.811
| 
|}

== Music ==
; [[Billy Stiltner]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/marvel10_2.mp3 notJoelTaylorsTreeSpirit-Marvel10-2]{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/pump_tunings/pump1/daily20111029-a-pump1-woodwinds-no-verb.mp3 ''Pump1''] – in pump12 1, [[197edo]] tuning
* [http://chrisvaisvil.com/?p=314 ''Semimarvelous Blue Drawf''] (2010) – in Dwarf17marv, equal-beating tuning

== See also ==
* [[Marvel temperaments]], the collection of rank-2 temperaments that temper out the marvel comma

== References ==
<references/>

[[Category:Marvel| ]] <!-- main article -->
[[Category:Rank-3 temperaments]]
[[Category:Marvel family]]
[[Category:Keenanismic temperaments]]
[[Category:Swetismic temperaments]]