225/224: Difference between revisions

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{{Infobox Interval

| Name = septimal kleisma~~, marvel comma~~

| Name = marvel comma, septimal kleisma

| Color name = ryy-2, ruyoyo negative 2nd,<br> Ruyoyo comma

| Comma = yes

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The interval of '''225/224''', the '''~~septimal kleisma~~''' or '''~~marvel comma~~''' is a [[7-limit]] [[superparticular]] ~~ratio~~. It pops up as the difference between ~~pairs of~~ 7-limit ~~ratios~~, ~~for example as (~~[[15/14]])/([[16/15]]) or ([[45/32]])/([[7/5]]).

The interval of '''225/224''', the '''marvel comma''', otherwise known as the '''septimal kleisma''', is a [[7-limit]] [[superparticular]] [[comma]]. It pops up as the difference between a 7-limit ratio and a 5-limit ratio. For example, it's the difference between [[16/15]] and [[15/14]], and between [[7/5]] and [[45/32]]. Moreover, it can be seen as the amount by which [[8/7]] exceeds a stack of two {{nowrap|[[16/15]]'s}}, or as the amount by which a stack of two {{nowrap|[[5/4]]'s}} exceeds [[14/9]]. It's also the difference between [[75/64]] and [[7/6]], and between [[25/24]], the classical chromatic semitone, and [[28/27]], the septimal third-tone.

~~Another~~ useful ~~relation is as the difference between~~ the [[~~25/24~~]]~~, the classical chromatic semitone, and~~ [[~~28/27~~]]~~, the septimal third-tone. Hence,~~ it ~~is also the difference between~~ [[32/~~25]] and [[9/7~~]], ~~and between~~ [[75/64]] and ~~[[7/6]]~~.

As a comma with a single power of 7 in it, it is tremendously useful in terms of bringing prime 7 into the framework of [[5-limit]] [[just intonation|JI]]; tempering it out maps [[7/4]] to the classic augmented sixth, [[225/128]] and enables all of the aforementioned equivalences.

In terms of commas, it is the difference between [[81/80]] and [[126/125]] and is tempered out alongside these two commas in [[septimal meantone]]. In the 11-limit it factors neatly into ([[385/384]])([[540/539]]), and in the 13-limit, ([[351/350]])([[625/624]]) or ([[325/324]])([[729/728]]).

== Temperaments ==

Tempering out this comma alone in the 7-limit leads to the [[marvel]] temperament, which enables [[marvel chords]]. See [[~~marvel~~ family]] for the family of rank-3 temperaments where it is tempered out. See [[~~marvel~~ temperaments]] for a collection of rank-2 temperaments where it is tempered out.

Tempering out this comma alone in the 7-limit leads to the [[marvel]] temperament, which enables [[marvel chords]]. See [[Marvel family]] for the family of rank-3 temperaments where it is tempered out. See [[Marvel temperaments]] for a collection of rank-2 temperaments where it is tempered out.

~~=== Canonical extensions of note ===~~

The marvel extension [[hecate]] has the no-17's [[19-limit]] as its subgroup, and [[undecimal marvel]] (aka unimarv), the extension chosen by [[Gene Ward Smith]], can be extended to the 13-limit. They merge in the rank 2 temperament [[catakleismic]] (which can be conceptualized as accepting both rank 3 marvel structures simultaneously), for which the smallest reasonable edo tuning for the full no-17's 19-limit is [[53edo]] followed by [[72edo]], 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.

~~=== [[53edo]] and [[84edo]] tunings of 7-limit marvel ===~~

[[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's 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 and is less optimized for others, in the sense that 53edo tunes closer to the more complex [[5-limit]] interpretations (which arguably need more tuning fidelity), which is something not taken into account by these tuning optimization schemes (so that they generally tune closer to [[LCJI]]). By contrast, [[84edo]], an overlooked superset of [[12edo]], has the benefit of being a high-limit performer in odd-limits 23 through 51 (inclusive). In fact, 53edo and 84edo are the '''only''' edos to satisfy all these constraints consistently when we include not overtempering to overshoot the 5-limit interval, and if we also require [[28/27]] to be sharp and [[25/24]] to be flat, 53edo is the only one, making it a uniquely optimized 7-limit marvel tuning; as far as the 9-odd-limit is concerned, the only intervals which are more than 25% off in 53edo are [[7/5]] and [[10/7]]], so that it is almost [[consistent to distance]] 2, and many more complex intervals of the 7-limit are [[consistent]] as well (barring the stacking of prime 7 more than once, so that 5 * 7 = 35 is fine but not 7 * 7 = 49, which causes inconsistencies in the 7-limited [[tonality diamond]]).

== Approximation ==

If we do not temper out this interval and instead repeatedly stack (and octave-reduce) it, we almost return to the starting point at the 311th step, meaning [[311edo]] is ~~practically~~ a [[consistent circle]] of 225/224's. Note that this is not true for 226/225 or 224/223, the adjacent superparticulars, as they accumulate too much error to close into a circle in 311edo.

If we do not temper out this interval and instead repeatedly stack (and octave-reduce) it, we almost return to the starting point at the 311th step, meaning [[311edo]] is a [[consistent circle]] of 225/224's. Note that this is not true for 226/225 or 224/223, the adjacent superparticulars, as they accumulate too much error to close into a circle in 311edo.

== Etymology ==

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== See also ==

* [[Marvel]]

* [[Small comma]]

* [[Gallery of just intervals]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Name = septimal kleisma, marvel comma
+| Name = marvel comma, septimal kleisma
 | Color name = ryy-2, ruyoyo negative 2nd,<br> Ruyoyo comma
 | Comma = yes
@@ Line 6: / Line 6: @@
 {{Wikipedia|Septimal kleisma}}
-The interval of '''225/224''', the '''septimal kleisma''' or '''marvel comma''' is a [[7-limit]] [[superparticular]] ratio. It pops up as the difference between pairs of 7-limit ratios, for example as ([[15/14]])/([[16/15]]) or ([[45/32]])/([[7/5]]).
+The interval of '''225/224''', the '''marvel comma''', otherwise known as the '''septimal kleisma''', is a [[7-limit]] [[superparticular]] [[comma]]. It pops up as the difference between a 7-limit ratio and a 5-limit ratio. For example, it's the difference between [[16/15]] and [[15/14]], and between [[7/5]] and [[45/32]]. Moreover, it can be seen as the amount by which [[8/7]] exceeds a stack of two {{nowrap|[[16/15]]'s}}, or as the amount by which a stack of two {{nowrap|[[5/4]]'s}} exceeds [[14/9]]. It's also the difference between [[75/64]] and [[7/6]], and between [[25/24]], the classical chromatic semitone, and [[28/27]], the septimal third-tone.
-Another useful relation is as the difference between the [[25/24]], the classical chromatic semitone, and [[28/27]], the septimal third-tone. Hence, it is also the difference between [[32/25]] and [[9/7]], and between [[75/64]] and [[7/6]].
+As a comma with a single power of 7 in it, it is tremendously useful in terms of bringing prime 7 into the framework of [[5-limit]] [[just intonation|JI]]; tempering it out maps [[7/4]] to the classic augmented sixth, [[225/128]] and enables all of the aforementioned equivalences.
 In terms of commas, it is the difference between [[81/80]] and [[126/125]] and is tempered out alongside these two commas in [[septimal meantone]]. In the 11-limit it factors neatly into ([[385/384]])([[540/539]]), and in the 13-limit, ([[351/350]])([[625/624]]) or ([[325/324]])([[729/728]]).
 == Temperaments ==
-Tempering out this comma alone in the 7-limit leads to the [[marvel]] temperament, which enables [[marvel chords]]. See [[marvel family]] for the family of rank-3 temperaments where it is tempered out. See [[marvel temperaments]] for a collection of rank-2 temperaments where it is tempered out.
+Tempering out this comma alone in the 7-limit leads to the [[marvel]] temperament, which enables [[marvel chords]]. See [[Marvel family]] for the family of rank-3 temperaments where it is tempered out. See [[Marvel temperaments]] for a collection of rank-2 temperaments where it is tempered out.
-=== Canonical extensions of note ===
-The marvel extension [[hecate]] has the no-17's [[19-limit]] as its subgroup, and [[undecimal marvel]] (aka unimarv), the extension chosen by [[Gene Ward Smith]], can be extended to the 13-limit. They merge in the rank 2 temperament [[catakleismic]] (which can be conceptualized as accepting both rank 3 marvel structures simultaneously), for which the smallest reasonable edo tuning for the full no-17's 19-limit is [[53edo]] followed by [[72edo]], 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.
-=== [[53edo]] and [[84edo]] tunings of 7-limit marvel ===
-[[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's 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 and is less optimized for others, in the sense that 53edo tunes closer to the more complex [[5-limit]] interpretations (which arguably need more tuning fidelity), which is something not taken into account by these tuning optimization schemes (so that they generally tune closer to [[LCJI]]). By contrast, [[84edo]], an overlooked superset of [[12edo]], has the benefit of being a high-limit performer in odd-limits 23 through 51 (inclusive). In fact, 53edo and 84edo are the '''only''' edos to satisfy all these constraints consistently when we include not overtempering to overshoot the 5-limit interval, and if we also require [[28/27]] to be sharp and [[25/24]] to be flat, 53edo is the only one, making it a uniquely optimized 7-limit marvel tuning; as far as the 9-odd-limit is concerned, the only intervals which are more than 25% off in 53edo are [[7/5]] and [[10/7]]], so that it is almost [[consistent to distance]] 2, and many more complex intervals of the 7-limit are [[consistent]] as well (barring the stacking of prime 7 more than once, so that 5 * 7 = 35 is fine but not 7 * 7 = 49, which causes inconsistencies in the 7-limited [[tonality diamond]]).
 == Approximation ==
-If we do not temper out this interval and instead repeatedly stack (and octave-reduce) it, we almost return to the starting point at the 311th step, meaning [[311edo]] is practically a [[consistent circle]] of 225/224's. Note that this is not true for 226/225 or 224/223, the adjacent superparticulars, as they accumulate too much error to close into a circle in 311edo.
+If we do not temper out this interval and instead repeatedly stack (and octave-reduce) it, we almost return to the starting point at the 311th step, meaning [[311edo]] is a [[consistent circle]] of 225/224's. Note that this is not true for 226/225 or 224/223, the adjacent superparticulars, as they accumulate too much error to close into a circle in 311edo.
 == Etymology ==
@@ Line 28: / Line 22: @@
 == See also ==
+* [[Marvel]]
 * [[Small comma]]
 * [[Gallery of just intervals]]