Orwell: Difference between revisions
Created page with "'''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)<sup>3</sup>), and 225/224 (the difference..." |
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{{Infobox regtemp | |||
| Title = Orwell | |||
| Subgroups = 2.3.5.7, 2.3.5.7.11 | |||
| Comma basis = [[225/224]], [[1728/1715]] (7-limit); <br> [[99/98]], [[121/120]], [[176/175]] (11-limit) | |||
| Edo join 1 = 22 | Edo join 2 = 31 | |||
| Mapping = 1; 7 -3 8 2 | |||
| Generators = 7/6 | Generators tuning = 271.5 | Optimization method = CWE | |||
| MOS scales = [[4L 1s]], [[4L 5s]], [[9L 4s]], [[9L 13s]] | |||
| Pergen = (P8, cP5/7) | |||
| Odd limit 1 = 7 | Mistuning 1 = 4.27 | Complexity 1 = 13 | |||
| Odd limit 2 = 11-limit 21 | Mistuning 2 = 9.32 | Complexity 2 = 22 | |||
}} | |||
'''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 [[comma]]s 1728/1715 (the difference between 8/5 and (7/6)<sup>3</sup>), and 225/224 (the difference between 14/9 and (5/4)<sup>2</sup>). 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]]. | '''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 [[comma]]s 1728/1715 (the difference between 8/5 and (7/6)<sup>3</sup>), and 225/224 (the difference between 14/9 and (5/4)<sup>2</sup>). 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]]. | ||
Revision as of 18:00, 1 March 2026
| Orwell |
99/98, 121/120, 176/175 (11-limit)
11-limit 21-odd-limit: 9.32¢
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.
