Greenwoodmic temperaments

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of rank-2 greenwoodmic temperaments, which temper out 405/392, the greenwoodma.

Temperaments discussed elsewhere are

• Sidi (+25/24) → Dicot family
• August (+36/35) → Augmented family
• Superpelog (+49/48) → Semaphoresmic clan
• Injera (+50/49 or 81/80) → Meantone family
• Schism (+64/63) → Archytas clan
• Greenwood (+1323/1280) → Whitewood family

Considered below are secund and semishallowtone.

Secund

Secund tempers out the greendwoodma, the avicennma, and the orwellisma. It may be described as the 9 & 26 temperament, with a ploidacot signature of pentacot. It divides the perfect fifth into five ~16/15 generators, two for 7/6 and three for 9/7.

Subgroup: 2.3.5.7

Comma list: 405/392, 525/512

Mapping: [⟨1 1 3 2], ⟨0 5 -6 7]]

mapping generators: ~2, ~16/15

Optimal tunings:

• WE: ~2 = 1203.6786 ¢, ~16/15 = 138.3805 ¢

error map: ⟨+3.679 -6.374 -5.561 +7.195]

• CWE: ~2 = 1200.0000 ¢, ~16/15 = 138.0648 ¢

error map: ⟨0.000 -11.631 -14.702 -2.372]

Optimal ET sequence: 9, 17, 26, 61bc, 87bcc

Badness (Sintel): 2.27

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 99/98, 385/384

Mapping: [⟨1 1 3 2 3], ⟨0 5 -6 7 4]]

Optimal tunings:

• WE: ~2 = 1202.9415 ¢, ~12/11 = 138.2377 ¢
• CWE: ~2 = 1200.0000 ¢, ~12/11 = 137.9970 ¢

Optimal ET sequence: 9, 17, 26, 61bc

Badness (Sintel): 1.41

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 45/44, 99/98, 385/384

Mapping: [⟨1 1 3 2 3 2], ⟨0 5 -6 7 4 15]]

Optimal tunings:

• WE: ~2 = 1202.4465 ¢, ~12/11 = 137.2165 ¢
• CWE: ~2 = 1200.0000 ¢, ~12/11 = 137.0309 ¢

Optimal ET sequence: 9, 35

Badness (Sintel): 2.05

Secundly

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 65/64, 78/77, 99/98

Mapping: [⟨1 1 3 2 3 3], ⟨0 5 -6 7 4 6]]

Optimal tunings:

• WE: ~2 = 1203.0195 ¢, ~13/12 = 138.2642 ¢
• CWE: ~2 = 1200.0000 ¢, ~13/12 = 138.0317 ¢

Optimal ET sequence: 9, 17, 26, 61bcf

Badness (Sintel): 1.08

Semishallowtone

See also: Shallowtone

Subgroup: 2.3.5.7

Comma list: 405/392, 35721/32768

Mapping: [⟨2 0 36 15], ⟨0 1 -10 -3]]

mapping generators: ~189/128, ~3

Optimal tunings:

• WE: ~189/128 = 603.1653 ¢, ~3/2 = 686.6365 ¢ (~64/63 = 83.4712 ¢)

error map: ⟨+6.331 -8.988 -2.034 -0.248]

• CWE: ~189/128 = 600.0000 ¢, ~3/2 = 682.6498 ¢ (~64/63 = 82.6498 ¢)

error map: ⟨0.000 -19.305 -12.811 -16.775]

Optimal ET sequence: 14c, 44bd, 58bcd

Badness (Sintel): 8.42

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 99/98, 35721/32768

Mapping: [⟨2 0 36 15 32], ⟨0 1 -10 -3 -8]]

Optimal tunings:

• WE: ~189/128 = 603.0536 ¢, ~3/2 = 686.6175 ¢ (~64/63 = 83.5640 ¢)
• CWE: ~189/128 = 600.0000 ¢, ~3/2 = 682.7238 ¢ (~64/63 = 82.7238 ¢)

Optimal ET sequence: 14c, 44bd, 58bcde, 72bbccddee

Badness (Sintel): 4.15

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 45/44, 99/98, 35721/32768

Mapping: [⟨2 0 36 15 32 -2], ⟨0 1 -10 -3 -8 3]]

Optimal tunings:

• WE: ~91/64 = 601.8872 ¢, ~3/2 = 684.4774 ¢ (~64/63 = 82.5902 ¢)
• CWE: ~91/64 = 600.0000 ¢, ~3/2 = 682.1676 ¢ (~64/63 = 82.1676 ¢)

Optimal ET sequence: 14c, 30b

Badness (Sintel): 4.38