Marvel: Difference between revisions

(35 intermediate revisions by 8 users not shown)

Line 1:

See [[Marvel family #Marvel]] for ~~more information including~~ technical data and ~~links~~ to ~~scales~~.

'''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 ==

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 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-^3Gbb

|-

| 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 ==

~~See:~~

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 chords~~]]

* [[Undecimal marvel ~~chords~~]]

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

* [[~~Hecate hexad~~]]

* [[~~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.

== ~~Spectrum of marvel tunings by eigenmonzos~~ ==

=== Tuning spectrum ===

This spectrum assumes pure 2 and 7.

{| class="wikitable center-~~1 right-2 right-3~~"

{| class="wikitable center-all"

|-

! Eigenmonzo

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

! ~~3/2~~

! Fifth (¢)

! ~~5/4~~

! Major Third (¢)

! Comments

|-

Line 95:

Line 172:

== Music ==

* [http://~~chrisvaisvil~~.com/~~?p=314 Semimarvelous Blue Drawf~~] by [[Chris Vaisvil]]

; [[Billy Stiltner]]

* [http://micro.soonlabel.com/~~gene_ward_smith~~/~~Others~~/~~Stiltner~~/~~marvel10_2~~.~~mp3 notJoelTaylorsTreeSpirit~~-~~Marvel10-2] by~~ [[~~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 ==

[[Category:~~Temperament~~]]

[[Category:Marvel| ]]

[[Category:Marvel]]

[[Category:Rank-3 temperaments]]

[[Category:Marvel family]]

[[Category:Keenanismic temperaments]]

[[Category:Swetismic temperaments]]

@@ Line 1: / Line 1: @@
-See [[Marvel family #Marvel]] for more information including technical data and links to scales.
+'''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 ==
-See:
+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 chords]]
-* [[Undecimal marvel chords]]
+Alternative 11-limit extensions give different sets of chords. One notable example, tempering out [[441/440]], enables [[prodigy chords]].
-* [[Hecate hexad]]
-* [[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.
-== Spectrum of marvel tunings by eigenmonzos ==
+=== Tuning spectrum ===
 This spectrum assumes pure 2 and 7.
-{| class="wikitable center-1 right-2 right-3"
+{| class="wikitable center-all"
 |-
-! Eigenmonzo
+! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
-! 3/2
+! Fifth<br>(¢)
-! 5/4
+! Major<br>Third (¢)
 ! Comments
 |-
@@ Line 95: / Line 172: @@
 == Music ==
-* [http://chrisvaisvil.com/?p=314 Semimarvelous Blue Drawf] by [[Chris Vaisvil]]
+; [[Billy Stiltner]]
-* [http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/marvel10_2.mp3 notJoelTaylorsTreeSpirit-Marvel10-2] by [[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:Temperament]]
+[[Category:Marvel| ]] <!-- main article -->
-[[Category:Marvel]]
+[[Category:Rank-3 temperaments]]
+[[Category:Marvel family]]
+[[Category:Keenanismic temperaments]]
+[[Category:Swetismic temperaments]]