# Orson

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, the 5-limit temperament tempering it out, 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.

Comma: 2109375/2097152

valid range: [257.143, 276.923] (14b to 13)

nice range: [271.229, 271.708]

strict range: [271.229, 271.708]

POTE generator: ~75/64 = 271.627

Map: [<1 0 3|, <0 7 -3|]

EDOs: 22, 31, 53, 190, 243, 296, 645c

## Seven limit children

The second comma of the normal comma list defines which 7-limit family member we are looking at. Adding 65536/64625 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.

Commas: 225/224, 1728/1715

7-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-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

valid range: [266.667, 272.727] (9 to 22)

nice range: [266.871, 271.708]

strict range: [266.871, 271.708]

POTE generator: ~7/6 = 271.509

Algebraic generators: Sabra3, the real root of 12x^3-7x-48.

Map: [<1 0 3 1|, <0 7 -3 8|]

Wedgie: <<7 -3 8 -21 -7 27||

EDOs: 22, 31, 53, 84, 137, 221d, 358d

## 11-limit

Commas: 99/98, 121/120, 176/175

[|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

valid range: [270.968, 272.727] (31 to 22)

nice range: [266.871, 275.659]

strict range: [270.968, 272.727]

POTE generator: ~7/6 = 271.426

Map: [<1 0 3 1 3|, <0 7 -3 8 2|]

Edos: 22, 31, 53, 84e

## 13-limit

Commas: 99/98, 121/120, 176/175, 275/273

valid range: [270.968, 271.698] (31 to 53)

nice range: [266.871, 275.659]

strict range: [270.968, 271.698]

POTE generator: ~7/6 = 271.546

Map: [<1 0 3 1 3 8|, <0 7 -3 8 2 -19|]

EDOs: 22, 31, 53, 84e, 137e

## Blair

Commas: 65/64, 78/77, 91/90, 99/98

valid range: []

nice range: [265.357, 289.210]

strict range: []

POTE generator: ~7/6 = 271.301

Map: [<1 0 3 1 3 3|, <0 7 -3 8 2 3|]

EDOs: 9, 22, 31f

## Newspeak

Commas: 225/224, 441/440, 1728/1715

valid range: [270.968, 271.698] (31 to 53)

nice range: [266.871, 272.514]

strict range: [270.968, 271.698]

POTE tuning: ~7/6 = 271.288

Map: [<1 0 3 1 -4|, <0 7 -3 8 33|]

EDOs: 31, 84, 115, 376b, 491bd, 606bde

## Winston

Commas: 66/65, 99/98, 105/104, 121/120

valid range: [270.968, 272.727] (31 to 22f)

nice range: [266.871, 281.691]

strict range: [270.968, 272.727]

POTE generator: ~7/6 = 271.088

Map: [<1 0 3 1 3 1|, <0 7 -3 8 2 12|]

EDOs: 22f, 31

# Doublethink

Commas: 99/98, 121/120, 169/168, 176/175

valid range: [135.484, 136.364] (62 to 44)

nice range: [128.298, 138.573]

strict range: [135.484, 136.364]

POTE tuning: ~13/12 = 135.723

Map: [<1 0 3 1 3 2|, <0 14 -6 16 4 15|]

EDOs: 9, 35, 44, 53, 62, 115ef, 168ef

# Borwell

Commas: 225/224, 243/242, 1728/1715

POTE generator: ~55/36 = 735.752

Map: [<1 7 0 9 17|, <0 -14 6 -16 -35|]

EDOs: 31, 106, 137, 442bd

# Triwell

Commas: 1029/1024, 235298/234375

POTE generator: ~448/375 = 309.472

Map: [<1 7 0 1|, <0 -21 9 7]]

Wedgie: <<21 -9 -7 -63 -70 9||

EDOs: 31, 97, 128, 159, 190