Irrational-subgroup temperaments

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

Optimal tunings:

• 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

Optimal tunings:

• 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