24576/24565: Difference between revisions

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

== Temperaments ==

= Temperaments =

=== Mavka ===

== Mavka ==

By tempering out this comma in the full [[17-limit]], the rank-6 '''mavka temperament''' is defined. You may find a list of good equal temperaments supporting it below.

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{{Val list|legend=1| 46, 58, 80, 103, 137, 149, 159, 171, 183, 217, 296, 320, 342f, 354, 400, 422, 525, 571, 581, 742, 764, 935, 1084, 1106, 1323, 1506, 3593g, 3947eg, 5053fgg, 6559defgg, 8065cdefggg, 10152cdeffgggg }}.

=== ~~Archagall~~ ===

== Archagall ==

By tempering it 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:

=== 2.75.85 (MVP archagall) ===

By tempering the comma S16/S17 = 24576/24565 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. (It is because of this combination of accuracy, efficiency and simplicity (mapping-wise) and its corresponding explanatory power in what this comma does that the comma has been named the "archagallisma".) The "MVP" stands for "Minimum Viable Product", as this is the core of what the archagall logic achieves, with further extensions adding to the subgroup while avoiding significantly impacting its accuracy. 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

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Optimal GPV sequence: {{Val list| 5, 17, 22, 61, 83 }}

=== 2.75.85.9/7 subgroup ===

A fairly natural way to extend archagall is by tempering S15/S17 which [[square superparticular|(because of how semiparticulars work)]] equates two [[17/15]]'s with [[9/7]] without much damage. As 9/7 was not previously in the subgroup, this does not decrease the rank of the temperament and qualifies a proper and natural extension. We can equally get the same temperament by tempering S15/S16 instead (equating three [[16/15]]'s with [[17/14]]), however it is unclear whether [[16/15]] can even be reached so it is preferred to think of it as adding S15/S17 = [[2025/2023]]. If you do want to reach [[16/15]] look to the next extension listed here that includes prime 5.

[[Mapping]]:<br>

[{{val| 1 2 5 6}}<br>

{{val| 0 -4 3 1 }}]

Commas: {{monzo| 13 1 -3 0 }} = 24576/24565, {{monzo 2 -2 0 1}} = 2025/2023

Some good (relative to their size) EDOs supporting it: 5, 12, 17, 22, 27, 39, 49, 61, 71, 83, 105, 127, 149, 159, 171

It should be noted that just because these are good for the generators given that does not mean that they are good for the broader 2.3.5.7.17 subgroup, so one may need to take supersets in that case, in which case again it is preferred to look at the next extension.

=== 2.3.5.7.17 subgroup (prime archagall) ===

We may observe that in a good tuning of archagall there is an accurate [[5/4]] at +13 fourths ([[85/64]]'s) minus five octaves ([[2/1]]'s). Because 75/25 = 3 and 85/5 = 17 this allows us to collapse it into its corresponding prime subgroup. This temperament is very closely related to [[171edo]] for which [[171edo]] is the tuning tempering {S49, S50, S18/S20} which is natural because this temperament tempers S49*S50 = S35 = [[1225/1224]] and (S18/S20)/S49 = [[5832/5831]] while not tempering any of {S49, S50, S18/S20} individually. Note that [[171edo]] is exceptionally efficient and accurate in the 2.3.5.7.17 subgroup, constituting a microtemperament for it.

Commas: 24576/24565 = S16/S17, 57375/57344 = S15/S16, 1225/1224 = S35

[[Mapping]]:<br>

[{{val| 1 11 -3 20 9 }}<br>

{{val| 0 -23 13 -42 -12 }}]

Some good (relative to their size) EDOs supporting it: 22, 149, 171, 193, 215, 320, 364

=== Srutal archagall ===

@@ Line 9: / Line 9: @@
 It can be factored into [[4096/4095]] × [[4914/4913]].
-== Temperaments ==
+= Temperaments =
-=== Mavka ===
+== Mavka ==
 By tempering out this comma in the full [[17-limit]], the rank-6 '''mavka temperament''' is defined. You may find a list of good equal temperaments supporting it below.
@@ Line 25: / Line 25: @@
 {{Val list|legend=1| 46, 58, 80, 103, 137, 149, 159, 171, 183, 217, 296, 320, 342f, 354, 400, 422, 525, 571, 581, 742, 764, 935, 1084, 1106, 1323, 1506, 3593g, 3947eg, 5053fgg, 6559defgg, 8065cdefggg, 10152cdeffgggg }}.
-=== Archagall ===
+== Archagall ==
-By tempering it 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:
+=== 2.75.85 (MVP archagall) ===
+By tempering the comma S16/S17 = 24576/24565 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. (It is because of this combination of accuracy, efficiency and simplicity (mapping-wise) and its corresponding explanatory power in what this comma does that the comma has been named the "archagallisma".) The "MVP" stands for "Minimum Viable Product", as this is the core of what the archagall logic achieves, with further extensions adding to the subgroup while avoiding significantly impacting its accuracy. 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
@@ Line 37: / Line 39: @@
 Optimal GPV sequence: {{Val list| 5, 17, 22, 61, 83 }}
+=== 2.75.85.9/7 subgroup ===
+A fairly natural way to extend archagall is by tempering S15/S17 which [[square superparticular|(because of how semiparticulars work)]] equates two [[17/15]]'s with [[9/7]] without much damage. As 9/7 was not previously in the subgroup, this does not decrease the rank of the temperament and qualifies a proper and natural extension. We can equally get the same temperament by tempering S15/S16 instead (equating three [[16/15]]'s with [[17/14]]), however it is unclear whether [[16/15]] can even be reached so it is preferred to think of it as adding S15/S17 = [[2025/2023]]. If you do want to reach [[16/15]] look to the next extension listed here that includes prime 5.
+[[Mapping]]:<br>
+[{{val| 1 2 5 6}}<br>
+{{val| 0 -4 3 1 }}]
+Commas: {{monzo| 13 1 -3 0 }} = 24576/24565, {{monzo 2 -2 0 1}} = 2025/2023
+Some good (relative to their size) EDOs supporting it: 5, 12, 17, 22, 27, 39, 49, 61, 71, 83, 105, 127, 149, 159, 171
+It should be noted that just because these are good for the generators given that does not mean that they are good for the broader 2.3.5.7.17 subgroup, so one may need to take supersets in that case, in which case again it is preferred to look at the next extension.
+=== 2.3.5.7.17 subgroup (prime archagall) ===
+We may observe that in a good tuning of archagall there is an accurate [[5/4]] at +13 fourths ([[85/64]]'s) minus five octaves ([[2/1]]'s). Because 75/25 = 3 and 85/5 = 17 this allows us to collapse it into its corresponding prime subgroup. This temperament is very closely related to [[171edo]] for which [[171edo]] is the tuning tempering {S49, S50, S18/S20} which is natural because this temperament tempers S49*S50 = S35 = [[1225/1224]] and (S18/S20)/S49 = [[5832/5831]] while not tempering any of {S49, S50, S18/S20} individually. Note that [[171edo]] is exceptionally efficient and accurate in the 2.3.5.7.17 subgroup, constituting a microtemperament for it.
+Commas: 24576/24565 = S16/S17, 57375/57344 = S15/S16, 1225/1224 = S35
+[[Mapping]]:<br>
+[{{val| 1 11 -3 20 9 }}<br>
+{{val| 0 -23 13 -42 -12 }}]
+Some good (relative to their size) EDOs supporting it: 22, 149, 171, 193, 215, 320, 364
 === Srutal archagall ===