Marvel: Difference between revisions
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'''Marvel''' is a [[rank-3 | {{Interwiki | ||
| en = Marvel | |||
| de = Marvel | |||
| es = | |||
| ja = | |||
}} | |||
{{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]] 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]]. | |||
The canonical [[11-limit]] [[extension]], sometimes aliased ''unimarv'', | 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>. | 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. | 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. | See [[Marvel family #Marvel]] for technical data. | ||
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== Notation == | == 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 | 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" | {| class="wikitable center-1 center-3" | ||
| Line 27: | Line 49: | ||
| 3/2 | | 3/2 | ||
| Perfect fifth | | Perfect fifth | ||
| | | C–G | ||
|- | |- | ||
| 5/4 | | 5/4 | ||
| Down major third | | Down major third | ||
| | | C–vE | ||
|- | |- | ||
| 7/4 | | 7/4 | ||
| Dudaugmented sixth | | Dudaugmented sixth | ||
| | | C–vvA♯ | ||
|- | |- | ||
| 11/8 | | 11/8 | ||
| Trup double-diminished fifth | | Trup double-diminished fifth | ||
| | | C–^<sup>3</sup>G𝄫 | ||
|- | |- | ||
| 13/8 | | 13/8 | ||
| Dup minor sixth | | Dup minor sixth | ||
| | | C–^^A♭ | ||
|- | |- | ||
| 19/16 | | 19/16 | ||
| Minor third | | Minor third | ||
| | | 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 | 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 == | == Chords and harmony == | ||
Marvel enables [[essentially tempered chord]]s of [[ | 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]]. | |||
Alternative 11-limit extensions give different sets of chords. One notable example, tempering out [[441/440]], enables [[prodigy chords]]. | Alternative 11-limit extensions give different sets of chords. One notable example, tempering out [[441/440]], enables [[prodigy chords]]. | ||
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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. | 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 | 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 === | === Norm-based tunings === | ||
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This spectrum assumes pure 2 and 7. | This spectrum assumes pure 2 and 7. | ||
{| class="wikitable center-all" | {| class="wikitable center-all left-4" | ||
|- | |- | ||
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]] | ! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]] | ||
! | ! Perfect<br>fifth (¢) | ||
! | ! Classical<br>major third (¢) | ||
! Comments | ! Comments | ||
|- | |- | ||