24576/24565: Difference between revisions

Line 1:

{{Infobox Interval

| Name = mavka comma

| Name = mavka comma, archagall comma

| Color name = trisu-agu negative 2nd

| Comma = yes

}}

'''24576/24565''', the '''mavka comma''', is an unnoticeable 17-limit comma that represents the difference between two adjacent [[square superparticular]]s – [[289/288]] and [[256/255]], [[Square superparticular#Sk/S(k + 1) (ultraparticulars)|therefore]], it is 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]].

'''24576/24565''', the '''mavka comma''' or '''archagallisma''', is an unnoticeable 17-limit comma that represents the difference between two adjacent [[square superparticular]]s – [[289/288]] and [[256/255]], [[Square superparticular#Sk/S(k + 1) (ultraparticulars)|therefore]], it is 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 ==

By tempering it out, the rank-6 '''mavka temperament''' is defined.

=== Archagall ===

By tempering it (S16/S17) out in the 2.75.85 subgroup, we have three [[85/64]]'s up and one octave down as a [[75/64]] and we have two [[128/85]]'s up and one octave down as a [[17/15]] whole tone. This is a highly accurate temperament that could be considered to be encoding the "high accuracy logic" of [[superpyth]] and which is inescapably related to the [[17L 5s]] scale form as it is the 17 & 22 temperament (or less accurately, the 5 & 17 temperament) in the following subgroup:

Subgroup: 2.75.85

Mapping: [ ⟨ 1 5 6 ] ⟨ 0 3 1 ] ⟩

Some good (relative to their size) EDOs supporting it: 5, 17, 22, 39, 61, 83

=== Srutal archagall ===

Named because this lower-accuracy temperament is also an extension of (the 5-limit) [[srutal]] temperament that adds prime 17 (and which thereby is able to express the harmonics 75 and 85 in their appropriate prime subgroup). It achieves this by equating [[85/64]] with [[4/3]] by tempering their difference of S16 = [[256/255]]. Therefore it also tempers S17 = [[289/288]] and thus equates [[17/15]] with [[9/8]] due to tempering S16*S17. It could be described as the 10 & 12 temperament (with strong emphasis on [[12edo]] being the better tuning hence the list of EDOs mentioned) on the following prior-discussed subgroup:

Subgroup: 2.3.5.17

Mapping: [ ⟨ 2 3 5 8 ] ⟨ 0 1 -2 1 ] ⟩

Some good (relative to their size) EDOs supporting it: 12, 22, 34, 46, 56, 58, 80

=== Rank 6 temperament ===

By tempering it out in the full [[17-limit]], the rank-6 '''mavka temperament''' is defined, to which extensions are also rank 6.

Subgroup: 2.3.5.7.11.13.17

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Name = mavka comma
+| Name = mavka comma, archagall comma
 | Color name = trisu-agu negative 2nd
 | Comma = yes
 }}
-'''24576/24565''', the '''mavka comma''', is an unnoticeable 17-limit comma that represents the difference between two adjacent [[square superparticular]]s – [[289/288]] and [[256/255]], [[Square superparticular#Sk/S(k + 1) (ultraparticulars)|therefore]], it is 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]].
+'''24576/24565''', the '''mavka comma''' or '''archagallisma''', is an unnoticeable 17-limit comma that represents the difference between two adjacent [[square superparticular]]s – [[289/288]] and [[256/255]], [[Square superparticular#Sk/S(k + 1) (ultraparticulars)|therefore]], it is 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 ==
-By tempering it out, the rank-6 '''mavka temperament''' is defined.
+=== Archagall ===
+By tempering it (S16/S17) out in the 2.75.85 subgroup, we have three [[85/64]]'s up and one octave down as a [[75/64]] and we have two [[128/85]]'s up and one octave down as a [[17/15]] whole tone. This is a highly accurate temperament that could be considered to be encoding the "high accuracy logic" of [[superpyth]] and which is inescapably related to the [[17L 5s]] scale form as it is the 17 & 22 temperament (or less accurately, the 5 & 17 temperament) in the following subgroup:
+Subgroup: 2.75.85
+Mapping:  [ ⟨	1	5	6	]      ⟨	0	3	1	] ⟩
+Some good (relative to their size) EDOs supporting it: 5, 17, 22, 39, 61, 83
+=== Srutal archagall ===
+Named because this lower-accuracy temperament is also an extension of (the 5-limit) [[srutal]] temperament that adds prime 17 (and which thereby is able to express the harmonics 75 and 85 in their appropriate prime subgroup). It achieves this by equating [[85/64]] with [[4/3]] by tempering their difference of S16 = [[256/255]]. Therefore it also tempers S17 = [[289/288]] and thus equates [[17/15]] with [[9/8]] due to tempering S16*S17. It could be described as the 10 & 12 temperament (with strong emphasis on [[12edo]] being the better tuning hence the list of EDOs mentioned) on the following prior-discussed subgroup:
+Subgroup: 2.3.5.17
+Mapping:  [ ⟨	2	3	5	8	]      ⟨	0	1	-2	1	] ⟩
+Some good (relative to their size) EDOs supporting it: 12, 22, 34, 46, 56, 58, 80
+=== Rank 6 temperament ===
+By tempering it out in the full [[17-limit]], the rank-6 '''mavka temperament''' is defined, to which extensions are also rank 6.
 Subgroup: 2.3.5.7.11.13.17