Horwell temperaments: Difference between revisions

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: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sesesix]].''

Fifthplus tempers out the [[wizma]] in addition to the horwell comma, and may be described as the {{nowrap| 22 & 171 }}. The name ''fifthplus'' means using a sharp fifth interval (such as a [[superpyth]] fifth) as a generator. ~~It is a restriction of [[24576/24565 #~~2.3.5.7.17 subgroup (prime archagall)~~|prime archagall~~]].

Fifthplus tempers out the [[wizma]] in addition to the horwell comma, and may be described as the {{nowrap| 22 & 171 }} temperament. The name ''fifthplus'' means using a sharp fifth interval (such as a [[superpyth]] fifth) as a generator.

Fifthplus may be extended to the 2.3.5.7.17 subgroup, called prime archagall, derived from the fact that it is an extension of [[archagall]] to a prime subgroup. In either the 7-limit or the add-17 subgroup, [[171edo]] is exceptionally efficient and accurate, but in the latter case 171edo is the tuning where [[2401/2400]] (S49), [[2500/2499]] (S50), and [[1701/1700]] (S18/S20) all vanish, which is natural because this temperament tempers out [[1225/1224]] (S35, S49⋅S50) and [[5832/5831]] ((S18/S20)/S49) while not tempering out any of above individually.

[[Subgroup]]: 2.3.5.7

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: error map: {{val| 0.000 -0.126 -0.391 -0.268 }}

{{Optimal ET sequence|legend=1| 22, 149, 171, 1903c, 2074c, …, 3613ccd }}

{{Optimal ET sequence|legend=1| 22, 105d, 127d, 149, 171, 1903c, 2074c, …, 3613ccd }}

[[Badness]] (Sintel): 0.654

=== Prime archagall ===

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, 105d, 127d, 149, 171, 1219, 1390 }}

Badness (Sintel): 0.421

== Mutt ==

@@ Line 25: / Line 25: @@
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sesesix]].''
-Fifthplus tempers out the [[wizma]] in addition to the horwell comma, and may be described as the {{nowrap| 22 & 171 }}. The name ''fifthplus'' means using a sharp fifth interval (such as a [[superpyth]] fifth) as a generator. It is a restriction of [[24576/24565 #2.3.5.7.17 subgroup (prime archagall)|prime archagall]].
+Fifthplus tempers out the [[wizma]] in addition to the horwell comma, and may be described as the {{nowrap| 22 & 171 }} temperament. The name ''fifthplus'' means using a sharp fifth interval (such as a [[superpyth]] fifth) as a generator.
+Fifthplus may be extended to the 2.3.5.7.17 subgroup, called prime archagall, derived from the fact that it is an extension of [[archagall]] to a prime subgroup. In either the 7-limit or the add-17 subgroup, [[171edo]] is exceptionally efficient and accurate, but in the latter case 171edo is the tuning where [[2401/2400]] (S49), [[2500/2499]] (S50), and [[1701/1700]] (S18/S20) all vanish, which is natural because this temperament tempers out [[1225/1224]] (S35, S49⋅S50) and [[5832/5831]] ((S18/S20)/S49) while not tempering out any of above individually.
 [[Subgroup]]: 2.3.5.7
@@ Line 40: / Line 42: @@
 : error map: {{val| 0.000 -0.126 -0.391 -0.268 }}
-{{Optimal ET sequence|legend=1| 22, 149, 171, 1903c, 2074c, …, 3613ccd }}
+{{Optimal ET sequence|legend=1| 22, 105d, 127d, 149, 171, 1903c, 2074c, …, 3613ccd }}
 [[Badness]] (Sintel): 0.654
+=== Prime archagall ===
+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, 105d, 127d, 149, 171, 1219, 1390 }}
+Badness (Sintel): 0.421
 == Mutt ==