Irrational-subgroup temperaments: Difference between revisions

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This page is a list of temperaments ~~which have an~~ irrational ~~subgroup~~ that cannot be expressed in surds.

This page is a list of [[regular temperament|temperaments]] of irrational subgroups that cannot be expressed in surds.

== Photosynthesis ==

Photosynthesis is a [[nonoctave]] temperament tempering out the comma 243/(5⋅ϕ8) in the ϕ.3.5 subgroup, equating five [[3/1]] with [[5/1]]. It has a period of [[acoustic phi]] and a generator of a major second just flat of 3/(ϕ2), the "phi-reduced 3rd harmonic". It possesses mos scales of the families 3L 1s<ϕ>, 4L 3s<ϕ>, 4L 7s<ϕ>, and 11L 4s<ϕ>.

[[Subgroup]]: ϕ.3.5

[[Comma list]]: 243/(5⋅ϕ8)

: mapping generators: ~ϕ, ~3

[[Optimal tuning]]s:

* [[CTE]]: ϕ = 833.0903{{c}}, ~3/ϕ = 1057.989{{c}}{{confirm}}

[[Edφ|Edϕ]]s: 11, 15, 7[+5], 19[-5], 26, 5[--5], 18[+5], 23[-3, -5], 34[-3, -5], 37, 41[-3], 27[-3, -5], 49[-3, -5], 31[-3, --5]

== Polyraider ==

The temperament below can also be written with [[34ed30]] nominals and cent-deviation (1\1171) accidentals. This temperament was found by [[User:2^67-1|2^67-1 (alias Cole Whitewood)]] in an attempt to figure out how to expand [[11edt]]'s approximations of this subgroup.

~~The temperament below can also be written with~~ [[~~34ed30~~]] ~~nominals and cent-deviation (1\1171) accidentals. This temperament was found by [[User~~:2~~^67-1|~~2~~^67-1 (alias Cole Whitewood)]] in an attempt to figure out how to expand [[11edt]]~~'~~s approximations of this subgroup.~~

[[Subgroup]]: 2.3.5.11.''f''2.''f''3 *

~~[[Subgroup]]~~: 2~~.3.5.11.f2.f3 where~~ ''~~f2, f3~~'' are the ~~second~~ and ~~third~~ modes of an ideal unsupported bar.

: <nowiki/>* ''f''2, ''f''3 are the 2nd and 3rd modes of an ideal unsupported bar.

[[Comma list]]: [[1771561/1771470]], 1310730070806386550964611/1310720000000000000000000, [-8 6 -3 2 -1 0>, [-11 10 -7 4 0 -1>

[[Comma list]]: [[1771561/1771470]], 1310730070806386550964611/1310720000000000000000000, {{monzo| -8 6 -3 2 -1 0 }}, {{monzo| -11 10 -7 4 0 -1 }}

{{Mapping|legend=1|1 65 172 148 162 27 | 0 −111 −297 −253 −281 −43}}

{{Mapping|legend=1| 1 65 172 148 162 27 | 0 −111 −297 −253 −281 −43 }}

: mapping generators: ~2, ~10460353203/7040000000

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.999{{c}}, ~10460353203/7040000000 = 685.567{{c}}

* [[WE]]: ~2 = 1199.999{{c}}, ~10460353203/7040000000 = 685.567{{c}}{{confirm}}

: [[error map]]: {{val| -0.001 -0.001 -0.002 -0.017 -0.024 0.122 }}

* [[CWE]]: ~2 = 1200.000{{c}}, ~10460353203/7040000000 = 685.568{{c}}

* [[CWE]]: ~2 = 1200.000{{c}}, ~10460353203/7040000000 = 685.568{{c}}{{confirm}}

: error map: {{val|0.000 0.000 -0.001 -0.014 -0.023 0.125 }}

@@ Line 1: / Line 1: @@
-This page is a list of temperaments which have an irrational subgroup that cannot be expressed in surds.
+This page is a list of [[regular temperament|temperaments]] of irrational subgroups that cannot be expressed in surds.
+== Photosynthesis ==
+Photosynthesis is a [[nonoctave]] temperament tempering out the comma 243/(5⋅ϕ<sup>8</sup>) in the ϕ.3.5 subgroup, equating five [[3/1]] with [[5/1]]. It has a period of [[acoustic phi]] and a generator of a major second just flat of 3/(ϕ<sup>2</sup>), the "phi-reduced 3rd harmonic". It possesses mos scales of the families 3L 1s<ϕ>, 4L 3s<ϕ>, 4L 7s<ϕ>, and 11L 4s<ϕ>.
+[[Subgroup]]: ϕ.3.5
+[[Comma list]]: 243/(5⋅ϕ<sup>8</sup>)
+{{Mapping|legend=3| 1 0 -8 | 0 1 5 }}
+: mapping generators: ~ϕ, ~3
+[[Optimal tuning]]s:
+* [[CTE]]: ϕ = 833.0903{{c}}, ~3/ϕ = 1057.989{{c}}{{confirm}}
+[[Edφ|Edϕ]]s: 11, 15, 7[+5], 19[-5], 26, 5[--5], 18[+5], 23[-3, -5], 34[-3, -5], 37, 41[-3], 27[-3, -5], 49[-3, -5], 31[-3, --5]
 == Polyraider ==
+The temperament below can also be written with [[34ed30]] nominals and cent-deviation (1\1171) accidentals. This temperament was found by [[User:2^67-1|2^67-1 (alias Cole Whitewood)]] in an attempt to figure out how to expand [[11edt]]'s approximations of this subgroup.
-The temperament below can also be written with [[34ed30]] nominals and cent-deviation (1\1171) accidentals. This temperament was found by [[User:2^67-1|2^67-1 (alias Cole Whitewood)]] in an attempt to figure out how to expand [[11edt]]'s approximations of this subgroup.
+[[Subgroup]]: 2.3.5.11.''f''<sub>2</sub>.''f''<sub>3</sub> *
-[[Subgroup]]: 2.3.5.11.f2.f3 where ''f2, f3'' are the second and third modes of an ideal unsupported bar.
+: <nowiki/>* ''f''<sub>2</sub>, ''f''<sub>3</sub> are the 2nd and 3rd modes of an ideal unsupported bar.
-[[Comma list]]: [[1771561/1771470]], 1310730070806386550964611/1310720000000000000000000, [-8 6 -3 2 -1 0>, [-11 10 -7 4 0 -1>
+[[Comma list]]: [[1771561/1771470]], 1310730070806386550964611/1310720000000000000000000, {{monzo| -8 6 -3 2 -1 0 }}, {{monzo| -11 10 -7 4 0 -1 }}
-{{Mapping|legend=1|1 65 172 148 162 27 | 0 −111 −297 −253 −281 −43}}
+{{Mapping|legend=1| 1 65 172 148 162 27 | 0 −111 −297 −253 −281 −43 }}
 : mapping generators: ~2, ~10460353203/7040000000
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.999{{c}}, ~10460353203/7040000000 = 685.567{{c}}
+* [[WE]]: ~2 = 1199.999{{c}}, ~10460353203/7040000000 = 685.567{{c}}{{confirm}}
-: [[error map]]: {{val| -0.001 -0.001 -0.002 -0.017 -0.024 0.122  }}
+: [[error map]]: {{val| -0.001 -0.001 -0.002 -0.017 -0.024 0.122 }}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~10460353203/7040000000 = 685.568{{c}}
+* [[CWE]]: ~2 = 1200.000{{c}}, ~10460353203/7040000000 = 685.568{{c}}{{confirm}}
 : error map: {{val|0.000 0.000 -0.001 -0.014 -0.023 0.125 }}