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.

{{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); [[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]].

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. ~~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]].

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.

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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 ==

== 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.

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

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

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 ==

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* [[Pump17]]

* [[Pump18]]

* [[Diamond9plus-marvel]]

== Tunings ==

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.

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}}

|}

~~[[53edo]] and [[84edo]] are the smallest [[edo]]s to tune the supermajor second [[~~~~~]][[8~~/~~7]] flat (towards [[~]][[256/225]])~~, ~~the subminor third [[~~~~~]][[7~~/~~6]] sharp (towards [[~~~~~]][[75~~/~~64]])~~, ~~the supermajor third [[~~~~~]][[9~~/~~7]] flat (towards [[~]][[32/25]]) and the tritone [[~]][[7/5]] 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]], though even if you allow overtempering, the only smaller edo to satisfy all of these constraints is [[12edo]], which is a trivial tuning of it (meaning it is very high-damage owing to conflating many intervals so that the lattice is oversimplified). ~~[[TE]]~~, ~~[[CTE]], [[CEE]] and [[CWE]] as well as the idea of tempering between pairs of~~ 5~~- and 7-limit intervals separated by 225~~/224 all implicate these tuning tendencies of these 7-limit [[LCJI]] intervals for optimized 7-limit marvel tunings. Interestingly, [[72edo]] fails some of these constraints, whereas 53edo though satisfying these constraints is obviously undertempered, tuning the intervals closer to the more complex [[5-limit]] interpretations despite that the very point of marvel is to use these intervals for 7-limit consonances. [[84edo]], another superset of [[12edo]], is an interesting edo to look at.

{| 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"

{| class="wikitable center-all left-4"

|-

! [[Eigenmonzo|Eigenmonzo (Unchanged-interval)]]

! ~~Fifth~~ (¢)

! Perfect fifth (¢)

! ~~Major~~ ~~Third~~ (¢)

! Classical major third (¢)

! Comments

|-

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

== ~~Notes~~ ==

== References ==

~~[[Category:Temperaments]]~~

[[Category:Marvel| ]]

[[Category:Rank-3 temperaments]]

[[Category:Marvel family]]

[[Category:Keenanismic temperaments]]

[[Category:Swetismic temperaments]]

@@ Line 1: / Line 1: @@
-'''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.
+{{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]].
+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. 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]].
+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.
@@ Line 50: / Line 72: @@
 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 ==
+== 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.
+Marvel enables [[essentially tempered chord]]s of [[marvel chords|marvel]], [[keenanismic chords|keenanismic]], [[swetismic chords|swetismic]], and [[undecimal marvel chords|undecimal marvel]].
-Alternative 11-limit extensions give different sets of chords. One notable example, tempering out [[441/440]], enables [[prodigy chords]].
+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 ==
@@ Line 78: / Line 102: @@
 * [[Pump17]]
 * [[Pump18]]
+* [[Diamond9plus-marvel]]
 == Tunings ==
-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.
+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}}
+|}
-[[53edo]] and [[84edo]] are the smallest [[edo]]s to tune the supermajor second [[~]][[8/7]] flat (towards [[~]][[256/225]]), the subminor third [[~]][[7/6]] sharp (towards [[~]][[75/64]]), the supermajor third [[~]][[9/7]] flat (towards [[~]][[32/25]]) and the tritone [[~]][[7/5]] 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]], though even if you allow overtempering, the only smaller edo to satisfy all of these constraints is [[12edo]], which is a trivial tuning of it (meaning it is very high-damage owing to conflating many intervals so that the lattice is oversimplified). [[TE]], [[CTE]], [[CEE]] and [[CWE]] as well as the idea of tempering between pairs of 5- and 7-limit intervals separated by 225/224 all implicate these tuning tendencies of these 7-limit [[LCJI]] intervals for optimized 7-limit marvel tunings. Interestingly, [[72edo]] fails some of these constraints, whereas 53edo though satisfying these constraints is obviously undertempered, tuning the intervals closer to the more complex [[5-limit]] interpretations despite that the very point of marvel is to use these intervals for 7-limit consonances. [[84edo]], another superset of [[12edo]], is an interesting edo to look at.
+{| 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"
+{| class="wikitable center-all left-4"
 |-
 ! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
-! Fifth<br>(¢)
+! Perfect<br>fifth (¢)
-! Major<br>Third (¢)
+! Classical<br>major third (¢)
 ! Comments
 |-
@@ Line 181: / Line 239: @@
 * [[Marvel temperaments]], the collection of rank-2 temperaments that temper out the marvel comma
-== Notes ==
+== References ==
 <references/>
-[[Category:Temperaments]]
 [[Category:Marvel| ]] <!-- main article -->
+[[Category:Rank-3 temperaments]]
 [[Category:Marvel family]]
 [[Category:Keenanismic temperaments]]
 [[Category:Swetismic temperaments]]