24576/24565: Difference between revisions

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== Temperaments ==

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]])3. 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, or in the 2.3.5.17 subgroup, 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]])3. 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 charismic fourths ([[85/64]]'s) octave reduced.

=== Archagallic ===

[[Subgroup]]: 2.3.5.17

[[Comma list]]: 24576/24565

: mapping generators: ~2, ~3, ~17/16

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9692{{c}}, ~3/2 = 701.9798{{c}}, ~17/16 = 105.1973{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9799{{c}}, ~17/16 = 105.2003{{c}}

{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 103, 115, 125, 137, 159, 171, 354, 376, 388, 559, 1882, 2441g, 3000g, 6559gg, 9559cggg }}

[[Badness]] (Sintel): 0.127

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

[[Subgroup]]: 2.3.5.7.11.13.17

[[~~Mapping~~]]:~~ ~~

[[Comma list]]: 24576/24565

~~[{{val| 1 0 1 0 0 0 4 }} ~~

~~{{val| 0 1 1 0 0 0 0 }} ~~

~~{{val| 0 0 -3 0 0 0 1 }} ~~

~~{{val| 0 0 0 1 0 0 0 }} ~~

~~{{val| 0 0 0 0 1 0 0 }} ~~

~~{{val| 0 0 0 0 0 1 0 }}]~~

~~Mapping~~ generators: ~2, ~3, ~17/16, ~7, ~11, ~13

{| class="right-all"

|-

| [⟨ || 1 || 0 || 1 || 0 || 0 || 0 || 4 || ],

|-

| ⟨ || 0 || 1 || 1 || 0 || 0 || 0 || 0 || ],

|-

| ⟨ || 0 || 0 || -3 || 0 || 0 || 0 || 1 || ],

|-

| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || 0 || ],

|-

| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]

|}

: mapping generators: ~2, ~3, ~17/16, ~7, ~11, ~13

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9692{{c}}, ~3/2 = 701.9798{{c}}, ~17/16 = 105.1973{{c}}, ~7/4 = 968.8875{{c}}, ~11/8 = 551.4103{{c}}, ~13/8 = 840.6200{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9799{{c}}, ~17/16 = 105.2003{{c}}, ~7/4 = 968.8842{{c}}, ~11/8 = 551.3897{{c}}, ~13/8 = 840.6045{{c}}

{{Optimal ET sequence|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 }}.

[[Badness]] (Sintel): 15.8

=== Archagall ===

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See [[Diaschismic family #Srutal archagall]].

~~=== Archagallic ===~~

~~Subgroup: 2.3.5.17~~

~~{{mapping|legend=1| 1 1 2 4 | 0 1 1 0 | 0 0 -3 1 }}~~

~~[[CTE]] generators: (2/1,) 3/2 = 701.943, 17/16 = 105.201~~

~~{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 103, 115, 125, 137, 159, 171, 354, 376, 388, 559, 1882, 2441g, 3000g, 6559gg, 9559cggg }}~~

~~Badness: 9.335 × 10-6~~

== Etymology ==

@@ Line 7: / Line 7: @@
 == Temperaments ==
-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).
+[[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. 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 charismic fourths ([[85/64]]'s) octave reduced.
+=== Archagallic ===
+[[Subgroup]]: 2.3.5.17
+[[Comma list]]: 24576/24565
+{{Mapping|legend=2| 1 1 2 4 | 0 1 1 0 | 0 0 -3 1 }}
+: mapping generators: ~2, ~3, ~17/16
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9692{{c}}, ~3/2 = 701.9798{{c}}, ~17/16 = 105.1973{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9799{{c}}, ~17/16 = 105.2003{{c}}
+{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 103, 115, 125, 137, 159, 171, 354, 376, 388, 559, 1882, 2441g, 3000g, 6559gg, 9559cggg }}
+[[Badness]] (Sintel): 0.127
 === Mavka a.k.a. archagallismic ===
 [[Subgroup]]: 2.3.5.7.11.13.17
-[[Mapping]]:<br>
+[[Comma list]]: 24576/24565
-[{{val| 1 0 1 0 0 0 4 }}<br>
-{{val| 0 1 1 0 0 0 0 }}<br>
-{{val| 0 0 -3 0 0 0 1 }}<br>
-{{val| 0 0 0 1 0 0 0 }}<br>
-{{val| 0 0 0 0 1 0 0 }}<br>
-{{val| 0 0 0 0 0 1 0 }}]
-Mapping generators: ~2, ~3, ~17/16, ~7, ~11, ~13
+{| class="right-all"
+|-
+| [⟨ || 1 || 0 || 1 || 0 || 0 || 0 || 4 || ],
+|-
+| ⟨ || 0 || 1 || 1 || 0 || 0 || 0 || 0 || ],
+|-
+| ⟨ || 0 || 0 || -3 || 0 || 0 || 0 || 1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || 0 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
+|}
+: mapping generators: ~2, ~3, ~17/16, ~7, ~11, ~13
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9692{{c}}, ~3/2 = 701.9798{{c}}, ~17/16 = 105.1973{{c}}, ~7/4 = 968.8875{{c}}, ~11/8 = 551.4103{{c}}, ~13/8 = 840.6200{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9799{{c}}, ~17/16 = 105.2003{{c}}, ~7/4 = 968.8842{{c}}, ~11/8 = 551.3897{{c}}, ~13/8 = 840.6045{{c}}
 {{Optimal ET sequence|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 }}.
+[[Badness]] (Sintel): 15.8
 === Archagall ===
@@ Line 70: / Line 98: @@
 See [[Diaschismic family #Srutal archagall]].
-=== Archagallic ===
-Subgroup: 2.3.5.17
-{{mapping|legend=1| 1 1 2 4 | 0 1 1 0 | 0 0 -3 1 }}
-[[CTE]] generators: (2/1,) 3/2 = 701.943, 17/16 = 105.201
-{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 103, 115, 125, 137, 159, 171, 354, 376, 388, 559, 1882, 2441g, 3000g, 6559gg, 9559cggg }}
-Badness: 9.335 × 10<sup>-6</sup>
 == Etymology ==