# Semicomma family

(Redirected from Winston)

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, first discovered by Erv Wilson, is the 5-limit temperament tempering out the semicomma. It 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]]

mapping generators: ~2, ~75/64

Optimal tuning (POTE): ~2 = 1\1, ~75/64 = 271.627

• 5-odd-limit diamond monotone: ~75/64 = [257.143, 276.923] (3\14 to 3\13)
• 5-odd-limit diamond tradeoff: ~75/64 = [271.229, 271.708] (1/3-comma to 2/7-comma)
• 5-odd-limit diamond monotone and tradeoff: ~75/64 = [271.229, 271.708]

### Overview to extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at. Adding 65625/65536 (or 225/224) leads to orwell, but we could also add

• 1029/1024, leading to the 31&159 temperament (triwell) with wedgie ⟨⟨21 -9 -7 -63 -70 9]], or
• 2401/2400, giving the 31&243 temperament (quadrawell) with wedgie ⟨⟨28 -12 1 -84 -77 36]], or
• 4375/4374, giving the 53&243 temperament (sabric) with wedgie ⟨⟨7 -3 61 -21 77 150]].

## 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 is compatible with 22, 31, 53 and 84 equal, and may be described as the 22 & 31 temperament. It is 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-limit, but it does use its second-closest approximation to 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 scales 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 271.509

[[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]
eigenmonzo (unchanged-interval) basis: 2.7/5
• 9-odd-limit: ~7/6 = [3/17 2/17 -1/17
[[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]
eigenmonzo (unchanged-interval) basis: 2.9/5
• 7-odd-limit diamond monotone: ~7/6 = [266.667, 272.727] (2\9 to 5\22)
• 9-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
• 7-odd-limit diamond tradeoff: ~7/6 = [266.871, 271.708]
• 9-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]
• 7-odd-limit diamond monotone and tradeoff: ~7/6 = [266.871, 271.708]
• 9-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 272.514]

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

### 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 271.426

Minimax tuning:

• 11-odd-limit: ~7/6 = [2/11 0 -1/11 1/11
[[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]
Eigenmonzo (unchanged-interval) basis: 2.7/5

Tuning ranges:

• 11-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
• 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]
• 11-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 272.727]

#### 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 271.546

Tuning ranges:

• 13- and 15-odd-limit diamond monotone: ~7/6 = [270.968, 271.698] (7\31 to 12\53)
• 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]
• 13- and 15-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 271.698]

#### 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 271.301

#### 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 271.088

Tuning ranges:

• 13- and 15-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
• 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 281.691]
• 13- and 15-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 272.727]

#### Doublethink

Doublethink is a weak extension of orwell to the 13-limit because it splits the generator of ~7/6 into two ~13/12~14/13's (by tempering their difference, 169/168).

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]]

Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 135.723

Tuning ranges:

• 13- and 15-odd-limit diamond monotone: ~13/12 = [135.484, 136.364] (7\62 to 5\44)
• 13- and 15-odd-limit diamond tradeoff: ~13/12 = [128.298, 138.573]
• 13- and 15-odd-limit diamond monotone and tradeoff: ~13/12 = [135.484, 136.364]

### 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]]

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 271.288

Tuning ranges:

• 11-odd-limit diamond monotone: ~7/6 = [270.968, 271.698] (7\31 to 12\53)
• 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]
• 11-odd-limit diamond monotone and tradeoff: ~7/6 = [270.968, 271.698]

### 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]]

Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.752

## Sabric

The sabric temperament (53 & 190) tempers out the ragisma (4375/4374). It is so named because it is closely related to the Sabra2 tuning (generator: 271.607278 cents).

Subgroup: 2.3.5.7

Comma list: 4375/4374, 2109375/2097152

Mapping[1 0 3 -11], 0 7 -3 61]]

Wedgie⟨⟨7 -3 61 -21 77 150]]

Optimal tuning (POTE): ~2 = 1\1, ~75/64 = 271.607

## Triwell

The triwell temperament (31 & 159) slices orwell major sixth ~128/75 into three generators, nine of which give the fifth harmonic.

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]]

Optimal tuning (POTE): ~2 = 1\1, ~448/375 = 309.472

### 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]]

Optimal tuning (POTE): ~2 = 1\1, ~448/375 = 309.471

The quadrawell temperament (31&212) has an 8/7 generator of about 232 cents, twelve of which give the fifth harmonic.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 2109375/2097152

Mapping[1 7 0 3], 0 -28 12 -1]]

Wedgie⟨⟨28 -12 1 -84 -77 36]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 232.094

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 14641/14580

Mapping: [1 7 0 3 11], 0 -28 12 -1 -39]]

Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 232.083

## Rainwell

The rainwell temperament (31 & 265) tempers out the mirkwai comma, 16875/16807 and the rainy comma, 2100875/2097152.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 2100875/2097152

Mapping[1 14 -3 6], 0 -35 15 -9]]

Wedgie⟨⟨35 -15 9 -105 -84 63]]

Optimal tuning (POTE): ~2 = 1\1, ~2625/2048 = 425.673

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 2100875/2097152

Mapping: [1 14 -3 6 29], 0 -35 15 -9 -72]]

Optimal tuning (POTE): ~2 = 1\1, ~2625/2048 = 425.679

## Quinwell

The quinwell temperament (22 & 243) slices orwell minor third into five generators and tempers out the wizma, 420175/419904.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 2109375/2097152

Mapping[1 0 3 0], 0 35 -15 62]]

Wedgie⟨⟨35 -15 62 -105 0 186]]

Optimal tuning (POTE): ~2 = 1\1, ~405/392 = 54.324

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4356, 2109375/2097152

Mapping: [1 0 3 0 5], 0 35 -15 62 -34]]

Optimal tuning (POTE): ~2 = 1\1, ~33/32 = 54.334

### Quinbetter

Subgroup: 2.3.5.7.11

Comma list: 385/384, 24057/24010, 43923/43750

Mapping: [1 0 3 0 4], 0 35 -15 62 -12]]

Optimal tuning (POTE): ~2 = 1\1, ~405/392 = 54.316