Orwell
| Orwell |
99/98, 121/120, 176/175 (11-limit)
11-limit 21-odd-limit: 9.32¢
11-limit 21-odd-limit: 22 notes
Orwell, [22 & 31], 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, or 12/11 and 11/10), 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
In the nine-note MOS scale of Orwell, the two primary step sizes are 35/32 and 16/15; these two steps are the main conjunctions from whence melodies can be derived in the tuning. Being generated by 7/6, Orwell temperament lends itself well to septimal triadic harmony based on 6:7:9 chords, as well as chthonic harmony based on 6:7:8 chords.
The augmented chthonic chord (a stack of two 7/6s, which in Orwell temperament make 11/8) built on the degree 4/3 above the tonic is one of the most useful dominant functions, as the 4/3 degree is shared between the two chords while the 11/6 (11/8 above 4/3) creates a satisfying motion to the tonic degree. If one desires more coherent voice leading, each chord may be extended to a four-note "cocytic" chord, so that the cocytic of the dominant chord aligns with the chthonic of the tonic chord and vice versa.
Notable features and related structures
If we take as Orwell's basic structure the equation of three sharpened 7/6s to 8/5 (tempering out 1728/1715 = S6/S7), we can also find that two 7/6s come close to 15/11 (tempering out 540/539 = S12/S14), and that two 5/4s form 14/11 (tempering out 176/175 = S8/S10). At five generators we have 12/11, and this implies the subgroup 2.5.7/3.11/3, without making the further tempering that places harmonics 3, 7, and 11 themselves on the generator chain.
This restricted version of Orwell is known as Guanyintet, and tunes well flatward from where Orwell is generally tuned. In particular, a generator of 270.13¢, close to 9\40 in 40edo, tunes both 15/14 and 12/11 nearly justly, where 15/14 comprises the small step and 12/11 the large step of the 9-note MOS.
(15/14)3 and (12/11)3 come very close to 16/13 and 13/10 respectively, and both equivalences can be made simultaneously, making Guanyintet definable as the 2.5.7/3.11/3.13 temperament that equates S11 = S12 = S14 = S15. Another prime that can be incorporated is 19, which can be found at 10 generators. The interval chain of this subgroup temperament is in #Guanyintet.
Interval chains
In the following table, odd harmonics and subharmonics 1–25 are labeled in bold. Cent values reflect pure-3/2 tuning.
| # | Cents | Approximate 7-limit ratios |
Approximate ratios of 11 |
|---|---|---|---|
| 0 | 0 | 1/1 | |
| 1 | 271.7 | 7/6 | |
| 2 | 543.4 | 48/35 | 11/8, 15/11 |
| 3 | 815.1 | 8/5 | 35/22 |
| 4 | 1086.8 | 15/8, 28/15 | 66/35 |
| 5 | 158.5 | 35/32, 49/45, 54/49 | 12/11, 11/10 |
| 6 | 430.2 | 9/7, 32/25 | 14/11 |
| 7 | 702.0 | 3/2 | 49/33 |
| 8 | 973.7 | 7/4 | 44/25 |
| 9 | 45.4 | 36/35, 49/48 | 33/32, 45/44, 56/55 |
| 10 | 317.1 | 6/5 | |
| 11 | 588.8 | 7/5, 45/32 | |
| 12 | 860.5 | 49/30 | 18/11, 33/20 |
| 13 | 1132.2 | 27/14, 48/25 | 21/11 |
Guanyintet
In this table, harmonics and subharmonics of odds and formal primes up to 25 are labeled in bold. Cent values reflect a tuning such that the 9-chroma is tuned to exactly 56/55.
| # | Cents | Approximate 2.5.7/3.11/3 ratios |
Approximate ratios of 13 and 19 |
|---|---|---|---|
| 0 | 0 | 1/1 | |
| 1 | 270.1 | 7/6 | 76/65 |
| 2 | 540.3 | 15/11, 48/35 | 26/19 |
| 3 | 810.4 | 8/5, 35/22 | |
| 4 | 1080.5 | 28/15 | |
| 5 | 150.7 | 12/11, 49/45 | |
| 6 | 420.8 | 14/11, 32/25 | |
| 7 | 690.9 | 49/33 | 95/64 |
| 8 | 961.1 | 96/55 | |
| 9 | 31.2 | 56/55 | 57/56, 65/64 |
| 10 | 301.3 | 19/16 | |
| 11 | 571.5 | 39/28 | |
| 12 | 841.6 | 13/8, 57/35 | |
| 13 | 1111.7 | 19/10 | |
| 14 | 181.9 | 39/35 | |
| 15 | 452.0 | 13/10, 57/44 | |
| 16 | 722.1 | 38/25 | |
| 17 | 992.2 | 39/22 | |
| 18 | 62.4 | 26/25, 57/55 |
Tunings and extensions
Tuning considerations
Extensions
Tuning spectrum
| View • Talk • EditRegular temperaments | |
|---|---|
| Rank-2 | |
| Acot | Blackwood (1/5-octave) • Whitewood (1/7-octave) • Compton (1/12-octave) |
| Monocot | Meantone • Schismic • Leapday • Archy |
| Complexity 2 | Diaschismic (diploid monocot) • Pajara (diploid monocot) • Injera (diploid monocot) • Rastmatic (dicot) • Mohajira (dicot) • Intertridecimal (dicot) • Interseptimal (alpha-dicot) |
| Complexity 3 | Augmented (triploid) • Misty (triploid) • Slendric (tricot) • Porcupine (omega-tricot) |
| Complexity 4 | Diminished (tetraploid) • Tetracot (tetracot) • Buzzard (alpha-tetracot) • Squares (beta-tetracot) • Negri (omega-tetracot) |
| Complexity 5-6 | Magic (alpha-pentacot) • Amity (gamma-pentacot) • Kleismic (alpha-hexacot) • Miracle (hexacot) |
| Higher complexity | Orwell (alpha-heptacot) • Sensi (beta-heptacot) • Octacot (octacot) • Wurschmidt (beta-octacot) • Valentine (enneacot) • Ammonite (epsilon-enneacot) • Myna (beta-decacot) • Ennealimmal (enneaploid dicot) |
| Straddle-3 | A-Team (alter-tricot) • Machine (alter-monocot) |
| No-3 | Trismegistus (alpha-triseph) • Orgone (trimech) • Didacus (diseph) |
| No-octaves | Sensamagic (monogem) |
| Exotemperament | Dicot • Mavila • Father |
| Higher-rank | |
| Rank-3 | Hemifamity • Marvel • Parapyth |
