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'''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>. | |||
Extending marvel to the 13-limit is not as obvious. While Gene has chosen [[351/350]] as the canonical extension, '''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 == | |||
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. | |||
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 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. | |||
=== Tuning spectrum === | |||
This spectrum assumes pure 2 and 7. | |||
{| class="wikitable center-all" | |||
|- | |||
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]] | |||
! Fifth<br>(¢) | |||
! Major<br>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 | |||
== Notes == | |||
<references/> | |||
[[Category:Marvel| ]] <!-- main article --> | |||
[[Category:Rank-3 temperaments]] | |||
[[Category:Marvel family]] | |||
[[Category:Keenanismic temperaments]] | |||
[[Category:Swetismic temperaments]] | |||