Irrational-subgroup temperaments: Difference between revisions
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This page is a list of temperaments | 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 == | == 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. | |||
[[Subgroup]]: 2.3.5.11.''f''<sub>2</sub>.''f''<sub>3</sub> * | |||
: <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, | [[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 | : mapping generators: ~2, ~10460353203/7040000000 | ||
[[Optimal tuning]]s: | [[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 }} | : error map: {{val|0.000 0.000 -0.001 -0.014 -0.023 0.125 }} | ||
Latest revision as of 09:38, 17 October 2026
This page is a list of 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)
Subgroup-val mapping: [⟨1 0 -8], ⟨0 1 5]]
- mapping generators: ~ϕ, ~3
- CTE: ϕ = 833.0903 ¢, ~3/ϕ = 1057.989 ¢[to be confirmed]
Edϕs: 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 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.f2.f3 *
- * f2, f3 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⟩
Mapping: [⟨1 65 172 148 162 27], ⟨0 −111 −297 −253 −281 −43]]
- mapping generators: ~2, ~10460353203/7040000000
- WE: ~2 = 1199.999 ¢, ~10460353203/7040000000 = 685.567 ¢[to be confirmed]
- error map: ⟨-0.001 -0.001 -0.002 -0.017 -0.024 0.122]
- CWE: ~2 = 1200.000 ¢, ~10460353203/7040000000 = 685.568 ¢[to be confirmed]
- error map: ⟨0.000 0.000 -0.001 -0.014 -0.023 0.125]
Optimal ET sequence: 7, 1164, 1171, 1178, 1185, 2335, 2349, 3506, 3520, 3527, 4691
Badness (Sintel): 0.655