Orwell

Orwell

Subgroups	2.3.5.7, 2.3.5.7.11
Reduced mapping	⟨1; 7 -3 8 2]
ET join	22 & 31
Generators (CWE)	~7/6 = 271.5¢
MOS scales	4L 1s, 4L 5s, 9L 4s, 9L 13s
Ploidacot	alpha-heptacot
Comma basis	225/224, 1728/1715 (7-limit); 99/98, 121/120, 176/175 (11-limit)
Pergen	(P8, cP5/7)
Minimax error	7-odd-limit: 4.27¢; 11-limit 21-odd-limit: 9.32¢
Target scale size	7-odd-limit: 13 notes; 11-limit 21-odd-limit: 22 notes

Orwell is a rank-2 temperament generated by a sharpened subminor third, representing 7/6. Three of these form 8/5, efficiently connecting to prime 5, and the result is then tuned such that it reaches 9/7 (an octave up) when stacked twice, so that seven generators in all form 3/1, a perfect twelfth. The equivalences that it makes are given by the commas 1728/1715 (the difference between 8/5 and (7/6)³), and 225/224 (the difference between 14/9 and (5/4)²). Orwell can also be interpreted in the 11-limit; in undecimal Orwell, two generators form an interval that simultaneously represents 15/11 and 11/8, while 9/7 is further equated to 14/11.

Melodically, Orwell's fundamental scale is a soft enneatonic, 4L 5s, and the notes of Orwell in further MOS scales (with 13, 22, ... notes) can be viewed as alterations from the basic 9-form by this scale's chroma, representing small intervals such as 36/35.

Overall, Orwell is quite efficient in covering the 7-limit as a whole in a way that does not closely adhere to diatonic structure, noting the substantial complexity of prime 3 in Orwell, compared to septal thirds and intervals of 5, 15, and 35. While undecimal Orwell's equivalences - represented by the commas 121/120 (the difference between 15/11 and 11/8), and 99/98 (the difference between 9/7 and 14/11) - are somewhat damaging to the structure of the 11-limit, they constitute useful simplifications, especially considering how readily available prime 11 is.

Orwell tends to select for primes 3 and 5 being tuned slightly flat, and 7 slightly sharp, with the minimax error of the 7-odd-limit being tunable to under 5¢. The most notable EDO tunings of Orwell include 22edo, 31edo, and 53edo, though it should also be mentioned that 84edo has a tuning generated by the interval 19\84 (from which the temperament derives its name). 40edo is another interesting tuning with a very flat fifth. All of these tune undecimal Orwell as well, though with a warted prime 11 in the case of 84edo.

Structural theory

General theory

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Notable features and related structures

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Guanyintet

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Interval chain

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Tunings and extensions

Tuning considerations

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Extensions

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Tuning spectrum

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