(WIP, catalog of 3.5.7 subgroup rank two temperaments will eventually be redirected here)

This is a collection of subgroup temperaments which omit the prime harmonic of 2. Because of the absence of octaves, these are all nonoctave scales using a period of a tritave, or if harmonic 3 is also excluded, 5/1.

Overview by mapping of 5

Classified by focusing on the mapping of 5th harmonic, similar to Rank-2 temperaments by mapping of 3.

Arcturus, Aldebaran and Polaris are with 3/1 period and ~5/3 generator. There is one-to-one correspondence between 3.5 subgroup and mapped intervals.
BPS is with a ~9/7 generator, two of which give the ~5/3.
Sirius is with a ~25/21 generator, three of which give the ~5/3.
Deneb is with a ~11/9 generator, three of which give the ~9/5.
Canopus is with a ~7/5 generator, five of which give the ~27/5 (9/5 up a tritave).
Alnilam is with a ~81/55 generator, ten of which give the ~243/5 (9/5 up three tritaves).
Izar is with a ~16807/10125 generator, twelve of which give the ~2187/5 (9/5 up five tritaves).
Mintaka does not include the 5th harmonic, and has an ~11/7 generator, two of which give the ~27/11, and three of which give the ~27/7 (9/7 and a tritave).

3.5.7 subgroup temperaments

Arcturus

see Arcturus, Arcturus clan

Subgroup: 3.5.7

Comma list: 15625/15309

Sval mapping: [⟨1 0 -7], ⟨0 1 6]]

Sval mapping generators: ~3, ~5

POTE generator: ~5/3 = 878.042

Optimal ET sequence: b2, b11, b13

BPS

see Bohlen-Pierce-Stearns

Subgroup: 3.5.7

Comma list: 245/243

Sval mapping: [⟨1 1 2], ⟨0 -2 1]]

Sval mapping generators: ~3, ~9/7

Optimal tuning (POTE): ~3 = 1\1edt, ~9/7 = 440.4881

Optimal ET sequence: b4, b9, b13, b56, b69, b82, b95

Canopus

see Canopus

Subgroup: 3.5.7

Comma list: 16875/16807

Subgroup-val mapping: [⟨1 3 3], ⟨0 -5 -4]]

Sval mapping generators: ~3, ~7/5

Optimal tuning (CTE): ~3 = 1\1edt, ~7/5 = 584.017

Supporting ETs: b13, b10, b75, b62, b23, b49, b36, b16, b7cd, 29, b42c, b19cd, b33cd, b55c

Izar

Subgroup: 3.5.7

Comma list: 13841287201/13839609375

Subgroup-val mapping: [⟨1 7 5], ⟨0 -12 -7]]

Sval mapping generators: ~3, ~16807/10125

Optimal tuning (CTE): ~3 = 1\1edt, ~16807/10125 = 877.280

Supporting ETs: b13, b11cd, b193, b15cd, b180, b24c, b167, b37c, b154, 141, b50c, b28cd, b128, b63c

3.5.11 subgroup temperaments

Polaris

see Polaris

Polaris tempers out the comma 177147/171875, and thus equates 7 5/3's with 15/11, or equivalently 7 9/5's with 11/9.

Subgroup: 3.5.11

Comma list: 177147/171875

Gencom: [3/1 5/3; 177147/171875]

Mapping: [⟨1 2 1], ⟨0 1 -6]]

POTE generator: ~5/3 = 892.6

EDTs: 17, 15, 32, 49, 13[+11], 47, 19, 11[+11], 81, 66, 79[+11], 62[+11], 28[+11], 21[-11]

Deneb

see Deneb

Subgroup: 3.5.11

Comma list: 6655/6561

Gencom: [3/1 11/9; 6655/6561]

Mapping: [⟨1 2 2], ⟨0 -3 1]]

POTE generator: ~11/9 = 340.242

EDTs: 28, 11, 17, 6, 39, 5, 67, 45, 50, 16, 23, 73, 61, 62

Alnilam

Effectively a microtemperament, Alnilam takes a generator of an 81/55 flat fifth and equates 9 of them with 11/9. The name was given by CompactStar to continue with the theme of naming no-twos temperaments after proper star names, but also to indirectly reference mavila.

Subgroup: 3.5.11

Comma list: [0 -35 9 0 10⟩

Gencom: [3/1 81/55; [0 -35 9 0 10⟩]

Mapping: [[⟨1 5 -1], ⟨0 -10 9]]

CTE generator: ~81/55 = 672.410

EDTs: 99, 17, 82, 116, 181, 65, 14[-5], 280, 48, 215, 31, 133, 314, 263