Marvel: Difference between revisions

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== 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, ~~it leaves~~ prime 7 ~~the only variable to be sharp~~. This indicates that the supermajor second [[~]][[8/7]] should be flat (towards [[~]][[256/225]]), the subminor third [[~]][[7/6]] be sharp (towards [[~]][[75/64]]), the supermajor third [[~]][[9/7]] be flat (towards [[~]][[32/25]]) and the tritone [[~]][[7/5]] be 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]]. If we take these as hard constraints, then [[53edo]] and [[84edo]] are the smallest edo tunings to satify them, but if overtempering is allowed, many smaller edos are possible, such as [[31edo|31-]] and [[41edo]]. Interestingly, [[72edo]] is ~~too~~ overtempered ~~to satisfy~~ some of these constraints, whereas 53edo though satisfying these constraints tempers in favor of tuning the intervals closer to the more complex [[5-limit]] interpretations, though because of their comparative simplicity (and thus lesser tuning fidelity), the 7-limit ~~consonances~~ of the 9-odd-limit still clearly work so that 53edo is [[consistent to distance]] 2 in the 9-odd-limit if we exclude 7/5 and 10/7 which are the most damaged. [[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, [[166edo|166-]], [[178edo|178-]], [[197edo|197-]], and [[240edo]] are all great choices with different intonational characteristics, though note that inconsistencies may often arise in the representation of the 7-limit lattice because [[225/224]] is itself larger than the size of the step of any of these edos.

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, with prime 5 the most tempered in accordance with minimizing damage on 7-limit [[LCJI]] intervals. This indicates that the supermajor second [[~]][[8/7]] should be flat (towards [[~]][[256/225]]), the subminor third [[~]][[7/6]] be sharp (towards [[~]][[75/64]]), the supermajor third [[~]][[9/7]] be flat (towards [[~]][[32/25]]) and the tritone [[~]][[7/5]] be 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]]. If we take these as hard constraints, then [[53edo]] and [[84edo]] are the smallest edo tunings to satify them, but if overtempering is allowed, many smaller edos are possible, such as [[31edo|31-]] and [[41edo]]. Interestingly, [[72edo]] is overtempered for some of these constraints, whereas 53edo, though satisfying these constraints, tempers in favor of tuning the intervals closer to the more complex [[5-limit]] interpretations, though because of their comparative simplicity (and thus lesser tuning fidelity), the 7-limit concordances of the 9-odd-limit still clearly work so that 53edo is [[consistent to distance]] 2 in the 9-odd-limit if we exclude 7/5 and 10/7 which are the most damaged. [[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, [[166edo|166-]], [[178edo|178-]], [[197edo|197-]], and [[240edo]] are all great choices with different intonational characteristics, though note that inconsistencies may often arise in the representation of the 7-limit lattice because [[225/224]] is itself larger than the size of the step of any of these edos.

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.

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 == 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, it leaves prime 7 the only variable to be sharp. This indicates that the supermajor second [[~]][[8/7]] should be flat (towards [[~]][[256/225]]), the subminor third [[~]][[7/6]] be sharp (towards [[~]][[75/64]]), the supermajor third [[~]][[9/7]] be flat (towards [[~]][[32/25]]) and the tritone [[~]][[7/5]] be 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]]. If we take these as hard constraints, then [[53edo]] and [[84edo]] are the smallest edo tunings to satify them, but if overtempering is allowed, many smaller edos are possible, such as [[31edo|31-]] and [[41edo]]. Interestingly, [[72edo]] is too overtempered to satisfy some of these constraints, whereas 53edo though satisfying these constraints tempers in favor of tuning the intervals closer to the more complex [[5-limit]] interpretations, though because of their comparative simplicity (and thus lesser tuning fidelity), the 7-limit consonances of the 9-odd-limit still clearly work so that 53edo is [[consistent to distance]] 2 in the 9-odd-limit if we exclude 7/5 and 10/7 which are the most damaged. [[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, [[166edo|166-]], [[178edo|178-]], [[197edo|197-]], and [[240edo]] are all great choices with different intonational characteristics, though note that inconsistencies may often arise in the representation of the 7-limit lattice because [[225/224]] is itself larger than the size of the step of any of these edos.
+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, with prime 5 the most tempered in accordance with minimizing damage on 7-limit [[LCJI]] intervals. This indicates that the supermajor second [[~]][[8/7]] should be flat (towards [[~]][[256/225]]), the subminor third [[~]][[7/6]] be sharp (towards [[~]][[75/64]]), the supermajor third [[~]][[9/7]] be flat (towards [[~]][[32/25]]) and the tritone [[~]][[7/5]] be 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]]. If we take these as hard constraints, then [[53edo]] and [[84edo]] are the smallest edo tunings to satify them, but if overtempering is allowed, many smaller edos are possible, such as [[31edo|31-]] and [[41edo]]. Interestingly, [[72edo]] is overtempered for some of these constraints, whereas 53edo, though satisfying these constraints, tempers in favor of tuning the intervals closer to the more complex [[5-limit]] interpretations, though because of their comparative simplicity (and thus lesser tuning fidelity), the 7-limit concordances of the 9-odd-limit still clearly work so that 53edo is [[consistent to distance]] 2 in the 9-odd-limit if we exclude 7/5 and 10/7 which are the most damaged. [[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, [[166edo|166-]], [[178edo|178-]], [[197edo|197-]], and [[240edo]] are all great choices with different intonational characteristics, though note that inconsistencies may often arise in the representation of the 7-limit lattice because [[225/224]] is itself larger than the size of the step of any of these edos.
 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.