24576/24565: Difference between revisions

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'''24576/24565''', the '''mavka comma''' or '''archagallisma''', is an [[Unnoticeable comma|unnoticeable]] [[17-limit]] [[comma]] that represents the difference between ~~two adjacent~~ [[~~square superparticular~~]]~~s –~~ [[289/288]] ~~and~~ [[~~256/255~~]], making it an [[Square superparticular #Sk/S(k + 1) (ultraparticulars)|''ultraparticular'']], and identifies itself as the amount by which a stack of three [[17/16]]'s fall short of a [[6/5]] minor third. It is also the amount by which a stack of two [[128/85]]'s octave-reduced exceeds [[17/15]] and the amount by which a stack of three [[85/64]]'s octave-reduced falls short of [[75/64~~]]. What's more, it is the interval separating [[256/255]] and [[289/288~~]].

'''24576/24565''', the '''mavka comma''' or '''archagallisma''', is an [[Unnoticeable comma|unnoticeable]] [[17-limit]] [[comma]] that represents the difference between [[256/255]] and [[289/288]] – two adjacent [[square superparticular]]s, making it an [[Square superparticular #Sk/S(k + 1) (ultraparticulars)|''ultraparticular'']], and identifies itself as the amount by which a stack of three [[17/16]]'s fall short of a [[6/5]] minor third. It is also the amount by which a stack of two [[128/85]]'s octave-reduced exceeds [[17/15]] and the amount by which a stack of three [[85/64]]'s octave-reduced falls short of [[75/64]].

It can be factored into [[4096/4095]] × [[4914/4913]].

== Temperaments ==

Tempering out this comma in the full [[17-limit]] results in the rank-6 '''mavka ~~temperament~~''' a.k.a. '''archagallismic'''. Tempering it out in the 2.3.5.17 subgroup results in the rank-3 '''archagallic temperament'''. If we restrict it to the 2.75.85 subgroup, we get the rank-2 '''archagall temperament'''. You may find a list of good equal temperaments supporting them below. The rank-6 temperament can be thought of as being equivalent to the [[17-limit]] with the exception that 5/4 is reached by going down by 17/16 three times, starting at 3/2. In other words, [[5/4]] = ([[3/2]])/([[17/16]])<sup>3</sup>. Similarly, archagallic can be thought of as the 2.3.5.17 subgroup with that same equivalence (so that it is essentially being expressed through 2.3.17). Archagall has its own, more complex mapping of prime 5 at +13 archagall fourths ([[85/64]]'s) (octave-reduced; specifically: minus five octaves).

Tempering out this comma in the full [[17-limit]] results in the rank-6 '''mavka''' a.k.a. '''archagallismic temperament'''. Tempering it out in the 2.3.5.17 subgroup results in the rank-3 '''archagallic temperament'''. If we restrict it to the 2.75.85 subgroup, we get the rank-2 '''archagall temperament'''. You may find a list of good equal temperaments supporting them below. The rank-6 temperament can be thought of as being equivalent to the [[17-limit]] with the exception that 5/4 is reached by going down by 17/16 three times, starting at 3/2. In other words, [[5/4]] = ([[3/2]])/([[17/16]])<sup>3</sup>. Similarly, archagallic can be thought of as the 2.3.5.17 subgroup with that same equivalence (so that it is essentially being expressed through 2.3.17). Archagall has its own, more complex mapping of prime 5 at +13 archagall fourths ([[85/64]]'s) (octave-reduced; specifically: minus five octaves).

=== Mavka a.k.a. archagallismic ===

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-'''24576/24565''', the '''mavka comma''' or '''archagallisma''', is an [[Unnoticeable comma|unnoticeable]] [[17-limit]] [[comma]] that represents the difference between two adjacent [[square superparticular]]s – [[289/288]] and [[256/255]], making it an [[Square superparticular #Sk/S(k + 1) (ultraparticulars)|''ultraparticular'']], and identifies itself as the amount by which a stack of three [[17/16]]'s fall short of a [[6/5]] minor third. It is also the amount by which a stack of two [[128/85]]'s octave-reduced exceeds [[17/15]] and the amount by which a stack of three [[85/64]]'s octave-reduced falls short of [[75/64]]. What's more, it is the interval separating [[256/255]] and [[289/288]].
+'''24576/24565''', the '''mavka comma''' or '''archagallisma''', is an [[Unnoticeable comma|unnoticeable]] [[17-limit]] [[comma]] that represents the difference between [[256/255]] and [[289/288]] – two adjacent [[square superparticular]]s, making it an [[Square superparticular #Sk/S(k + 1) (ultraparticulars)|''ultraparticular'']], and identifies itself as the amount by which a stack of three [[17/16]]'s fall short of a [[6/5]] minor third. It is also the amount by which a stack of two [[128/85]]'s octave-reduced exceeds [[17/15]] and the amount by which a stack of three [[85/64]]'s octave-reduced falls short of [[75/64]].
 It can be factored into [[4096/4095]] × [[4914/4913]].
 == Temperaments ==
-Tempering out this comma in the full [[17-limit]] results in the rank-6 '''mavka temperament''' a.k.a. '''archagallismic'''. Tempering it out in the 2.3.5.17 subgroup results in the rank-3 '''archagallic temperament'''. If we restrict it to the 2.75.85 subgroup, we get the rank-2 '''archagall temperament'''. You may find a list of good equal temperaments supporting them below. The rank-6 temperament can be thought of as being equivalent to the [[17-limit]] with the exception that 5/4 is reached by going down by 17/16 three times, starting at 3/2. In other words, [[5/4]] = ([[3/2]])/([[17/16]])<sup>3</sup>. Similarly, archagallic can be thought of as the 2.3.5.17 subgroup with that same equivalence (so that it is essentially being expressed through 2.3.17). Archagall has its own, more complex mapping of prime 5 at +13 archagall fourths ([[85/64]]'s) (octave-reduced; specifically: minus five octaves).
+Tempering out this comma in the full [[17-limit]] results in the rank-6 '''mavka''' a.k.a. '''archagallismic temperament'''. Tempering it out in the 2.3.5.17 subgroup results in the rank-3 '''archagallic temperament'''. If we restrict it to the 2.75.85 subgroup, we get the rank-2 '''archagall temperament'''. You may find a list of good equal temperaments supporting them below. The rank-6 temperament can be thought of as being equivalent to the [[17-limit]] with the exception that 5/4 is reached by going down by 17/16 three times, starting at 3/2. In other words, [[5/4]] = ([[3/2]])/([[17/16]])<sup>3</sup>. Similarly, archagallic can be thought of as the 2.3.5.17 subgroup with that same equivalence (so that it is essentially being expressed through 2.3.17). Archagall has its own, more complex mapping of prime 5 at +13 archagall fourths ([[85/64]]'s) (octave-reduced; specifically: minus five octaves).
 === Mavka a.k.a. archagallismic ===