24576/24565: Difference between revisions

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[[Tempering out]] this comma in the full [[17-limit]] results in the rank-6 '''mavka''' a.k.a. '''archagallismic''' temperament, or in the 2.3.5.17 subgroup, the rank-3 '''archagallic''' 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).

If we retract it to the 2.75.85 subgroup, we get the rank-2 ~~'''~~MVP archagall~~'''~~ temperament, which can be extended to ~~'''archagall''' in~~ the 2.75.9/7.85 subgroup. This has its own, more complex mapping of prime 5 at +13 charismic fourths ([[85/64]]'s) octave reduced, resulting in ~~'''~~prime archagall~~'''~~, an extension of [[fifthplus]].

If we retract it to the 2.75.85 subgroup, we get the rank-2 [[MVP archagall]] temperament, which can be extended to the 2.75.9/7.85 subgroup as [[archagall]]. This has its own, more complex mapping of prime 5 at +13 charismic fourths ([[85/64]]'s) octave reduced, resulting in [[prime archagall]], an extension of [[fifthplus]].

Finally, tempering out 256/255 and/or 289/288 results in a lower-accuracy temperament, [[srutal archagall]], of the [[diaschismic family]].

=== Archagallic ===

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[[Badness]] (Sintel): 15.8

~~=== MVP archagall (2.75.85) ===~~

By tempering out the comma 24576/24565 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 mapping-wise simplicity 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 2.75.85 subgroup.

~~[[Subgroup]]: 2.75.85~~

~~[[Comma list]]: 24576/24565 ({{monzo| 13 1 -3 }})~~

~~{{Mapping|legend=2| 1 2 5 | 0 3 1 }}~~

~~: mapping generators: ~2, ~85/32~~

~~[[Optimal tuning]]s:~~

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9692{{c}}, ~85/64 = 491.5853{{c}}

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~85/64 = 491.5794{{c}}

~~{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 61, 83, 310, 393, 476, 1345, 1821, 4118*, 5939*, 7760* }}~~

~~==== Archagall (2.75.9/7.85) ====~~

A fairly natural way to extend archagall is by tempering out [[2025/2023]] ([[S-expression|S15/S17]]), which equates a stack of 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 out S15/S16 instead (equating a stack of three [[16/15]]'s with [[17/14]]); however, [[16/15]] is not in the subgroup, so it is preferred to think of it as adding 2025/2023.

~~[[Subgroup]]: 2.75.9/7.85~~

~~[[Comma list]]: 2025/2023 ({{monzo| 2 -2 1 0 }}), 24576/24565 ({{monzo| 13 1 0 -3 }})~~

~~{{Mapping|legend=2| 1 2 6 5 | 0 3 -4 1 }}~~

~~: mapping generators: ~2, ~85/32~~

~~[[Optimal tuning]]s:~~

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0241{{c}}, ~85/64 = 491.3358{{c}}

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~85/64 = 491.3290{{c}}

~~{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 83, 105, 127, 403, 530, 657, 784, 1441* }}~~

~~=== 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.

~~Subgroup: 2.3.5.7.17~~

~~Comma list: 1225/1224, 24576/24565, 57375/57344~~

~~Subgroup-val mapping: {{mapping| 1 -12 10 -22 -3 | 0 23 -13 42 12 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.0516{{c}}, ~128/85 = 708.8057{{cent}}

* CWE: ~2 = 1200.0000{{c}}, ~128/85 = 708.7758{{cent}}

~~{{Optimal ET sequence|legend=0| 22, …, 149, 171, 1219, 1390 }}~~

~~Badness (Sintel): 0.421~~

~~==== Srutal archagall ====~~

This lower-accuracy temperament is an extension of [[srutal]] 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 on the 2.3.5.17 subgroup, implying ideal tunings of [[34edo]], [[46edo]] or [[80edo]].

~~See [[Diaschismic family #Srutal archagall]].~~

== Etymology ==

@@ Line 9: / Line 9: @@
 [[Tempering out]] this comma in the full [[17-limit]] results in the rank-6 '''mavka''' a.k.a. '''archagallismic''' temperament, or in the 2.3.5.17 subgroup, the rank-3 '''archagallic''' 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).
-If we retract it to the 2.75.85 subgroup, we get the rank-2 '''MVP archagall''' temperament, which can be extended to '''archagall''' in the 2.75.9/7.85 subgroup. This has its own, more complex mapping of prime 5 at +13 charismic fourths ([[85/64]]'s) octave reduced, resulting in '''prime archagall''', an extension of [[fifthplus]].
+If we retract it to the 2.75.85 subgroup, we get the rank-2 [[MVP archagall]] temperament, which can be extended to the 2.75.9/7.85 subgroup as [[archagall]]. This has its own, more complex mapping of prime 5 at +13 charismic fourths ([[85/64]]'s) octave reduced, resulting in [[prime archagall]], an extension of [[fifthplus]].
+Finally, tempering out 256/255 and/or 289/288 results in a lower-accuracy temperament, [[srutal archagall]], of the [[diaschismic family]].
 === Archagallic ===
@@ Line 53: / Line 55: @@
 [[Badness]] (Sintel): 15.8
-=== MVP archagall (2.75.85) ===
-By tempering out the comma 24576/24565 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 mapping-wise simplicity 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 2.75.85 subgroup.
-[[Subgroup]]: 2.75.85
-[[Comma list]]: 24576/24565 ({{monzo| 13 1 -3 }})
-{{Mapping|legend=2| 1 2 5 | 0 3 1 }}
-: mapping generators: ~2, ~85/32
-[[Optimal tuning]]s:
-* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9692{{c}}, ~85/64 = 491.5853{{c}}
-* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~85/64 = 491.5794{{c}}
-{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 61, 83, 310, 393, 476, 1345, 1821, 4118*, 5939*, 7760* }}
-==== Archagall (2.75.9/7.85) ====
-A fairly natural way to extend archagall is by tempering out [[2025/2023]] ([[S-expression|S15/S17]]), which equates a stack of 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 out S15/S16 instead (equating a stack of three [[16/15]]'s with [[17/14]]); however, [[16/15]] is not in the subgroup, so it is preferred to think of it as adding 2025/2023.
-[[Subgroup]]: 2.75.9/7.85
-[[Comma list]]: 2025/2023 ({{monzo| 2 -2 1 0 }}), 24576/24565 ({{monzo| 13 1 0 -3 }})
-{{Mapping|legend=2| 1 2 6 5 | 0 3 -4 1 }}
-: mapping generators: ~2, ~85/32
-[[Optimal tuning]]s:
-* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0241{{c}}, ~85/64 = 491.3358{{c}}
-* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~85/64 = 491.3290{{c}}
-{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 83, 105, 127, 403, 530, 657, 784, 1441* }}
-=== 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.
-Subgroup: 2.3.5.7.17
-Comma list: 1225/1224, 24576/24565, 57375/57344
-Subgroup-val mapping: {{mapping| 1 -12 10 -22 -3 | 0 23 -13 42 12 }}
-Optimal tunings:
-* WE: ~2 = 1200.0516{{c}}, ~128/85 = 708.8057{{cent}}
-* CWE: ~2 = 1200.0000{{c}}, ~128/85 = 708.7758{{cent}}
-{{Optimal ET sequence|legend=0| 22, …, 149, 171, 1219, 1390 }}
-Badness (Sintel): 0.421
-==== Srutal archagall ====
-This lower-accuracy temperament is an extension of [[srutal]] 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 on the 2.3.5.17 subgroup, implying ideal tunings of [[34edo]], [[46edo]] or [[80edo]].
-See [[Diaschismic family #Srutal archagall]].
 == Etymology ==