Semicomma family

The 5-limit parent comma for the semicomma family is the semicomma, 2109375/2097152 = [-21 3 7⟩. This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor sixths.

Orson

Orson, the 5-limit temperament tempering out the semicomma, has a generator of 75/64, which is sharper than 7/6 by 225/224 when untempered, and less sharp than that in any good orson tempering, for example 53edo or 84edo. These give tunings to the generator which are sharp of 7/6 by less than five cents, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.

Subgroup: 2.3.5

Comma list: 2109375/2097152

Mapping: [⟨1 0 3], ⟨0 7 -3]]

POTE generator: ~75/64 = 271.627

Tuning ranges:

• valid range: [257.143, 276.923] (3\14 to 3\13)
• nice range: [271.229, 271.708]
• strict range: [271.229, 271.708]

Badness: 0.0408

Seven limit children

The second comma of the normal comma list defines which 7-limit family member we are looking at. Adding 65536/65625 leads to orwell, but we could also add

• 1029/1024, leading to the 31&159 temperament with wedgie ⟨⟨ 21 -9 -7 -63 -70 9 ]], or
• 67528125/67108864, giving the 31&243 temperament with wedgie ⟨⟨ 28 -12 1 -84 -77 36 ]], or
• 4375/4374, giving the 53&243 temperament with wedgie ⟨⟨ 7 -3 61 -21 77 150 ]].

Orwell

Main article: Orwell

So called because 19\84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with 22, 31, 53 and 84 equal, and may be described as the 22&31 temperament, or ⟨⟨ 7 -3 8 -21 -7 27 ]]. It's a good system in the 7-limit and naturally extends into the 11-limit. 84edo, with the 19\84 generator, provides a good tuning for the 5, 7 and 11 limits, but it does use its second-best 11. However, the 19\84 generator is remarkably close to the 11-limit POTE tuning, as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. 53edo might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of it slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out 2430/2401, the nuwell comma, 1728/1715, the orwellisma, 225/224, the septimal kleisma, and 6144/6125, the porwell comma.

The 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell.

Orwell has MOS of size 9, 13, 22 and 31. The 9-note MOS is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has considerable harmonic resources despite its absence of 5-limit triads. The 13 note MOS has those, and of course the 22 and 31 note MOS are very well supplied with everything.

Subgroup: 2.3.5.7

Comma list: 225/224, 1728/1715

Mapping: [⟨1 0 3 1], ⟨0 7 -3 8]]

Wedgie: ⟨⟨ 7 -3 8 -21 -7 27 ]]

POTE generator: ~7/6 = 271.509

Minimax tuning:

• 7-odd-limit

[[1 0 0 0⟩, [14/11 0 -7/11 7/11⟩, [27/11 0 3/11 -3/11⟩, [27/11 0 -8/11 8/11⟩]

Eigenmonzos: 2, 7/5

• 9-odd-limit

[[1 0 0 0⟩, [21/17 14/17 -7/17 0⟩, [42/17 -6/17 3/17 0⟩, [41/17 16/17 -8/17 0⟩]

Eigenmonzos: 2, 10/9

Tuning ranges:

• valid range: [266.667, 272.727] (2\9 to 5\22)
• nice range: [266.871, 271.708]
• strict range: [266.871, 271.708]

Algebraic generator: Sabra3, the real root of 12x³ - 7x - 48.

Badness: 0.0207

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 121/120, 176/175

Mapping: [⟨1 0 3 1 3], ⟨0 7 -3 8 2]]

POTE generator: ~7/6 = 271.426

Minimax tuning:

• 11-odd-limit

[[1 0 0 0 0⟩, [14/11 0 -7/11 7/11 0⟩, [27/11 0 3/11 -3/11 0⟩, [27/11 0 -8/11 8/11 0⟩, [37/11 0 -2/11 2/11 0⟩]

Eigenmonzos: 2, 7/5

Tuning ranges:

• valid range: [270.968, 272.727] (7\31 to 5\22)
• nice range: [266.871, 275.659]
• strict range: [270.968, 272.727]

Badness: 0.0152

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 121/120, 176/175, 275/273

Mapping: [⟨1 0 3 1 3 8], ⟨0 7 -3 8 2 -19]]

POTE generator: ~7/6 = 271.546

Tuning ranges:

• valid range: [270.968, 271.698] (7\31 to 12\53)
• nice range: [266.871, 275.659]
• strict range: [270.968, 271.698]

Badness: 0.0197

Blair

Subgroup: 2.3.5.7.11.13

Comma list: 65/64, 78/77, 91/90, 99/98

Mapping: [⟨1 0 3 1 3 3], ⟨0 7 -3 8 2 3]]

POTE generator: ~7/6 = 271.301

Tuning ranges:

• valid range: []
• nice range: [265.357, 289.210]
• strict range: []

Badness: 0.0231

Winston

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 105/104, 121/120

Mapping: [⟨1 0 3 1 3 1], ⟨0 7 -3 8 2 12]]

POTE generator: ~7/6 = 271.088

Tuning ranges:

• valid range: [270.968, 272.727] (7\31 to 5\22)
• nice range: [266.871, 281.691]
• strict range: [270.968, 272.727]

Badness: 0.0199

Doublethink

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 121/120, 169/168, 176/175

Mapping: [⟨1 0 3 1 3 2], ⟨0 14 -6 16 4 15]]

POTE generator: ~13/12 = 135.723

Tuning ranges:

• valid range: [135.484, 136.364] (7\62 to 5\44)
• nice range: [128.298, 138.573]
• strict range: [135.484, 136.364]

Badness: 0.0271

Newspeak

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 1728/1715

Mapping: [⟨1 0 3 1 -4], ⟨0 7 -3 8 33]]

POTE generator: ~7/6 = 271.288

Tuning ranges:

• valid range: [270.968, 271.698] (7\31 to 12\53)
• nice range: [266.871, 272.514]
• strict range: [270.968, 271.698]

Badness: 0.0314

Borwell

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 1728/1715

Mapping: [⟨1 7 0 9 17], ⟨0 -14 6 -16 -35]]

POTE generator: ~55/36 = 735.752

Badness: 0.0384

Triwell

Subgroup: 2.3.5.7

Comma list: 1029/1024, 235298/234375

Mapping: [⟨1 7 0 1], ⟨0 -21 9 7]]

Wedgie: ⟨⟨ 21 -9 -7 -63 -70 9 ]]

POTE generator: ~448/375 = 309.472

Badness: 0.0806

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 456533/455625

Mapping: [⟨1 7 0 1 13], ⟨0 -21 9 7 -37]]

POTE generator: ~448/375 = 309.471

Badness: 0.0298