Orwell
Orwell – so named because 19 steps of 84edo, i.e. 19\84, is a possible generator – is an excellent 7-limit temperament and an amazing 11-limit temperament because of the simplicity of harmonic 11.
In orwell, the just perfect twelfth (3/1) is divided into 7 equal steps. One of these steps represents 7/6; three represent 8/5. Alternately, the 5th harmonic (5/1) divided into 3 equal steps also makes a good orwell generator, being ~12/7.
In the 11-limit, two generators are equated to 11/8 (meaning 99/98 is tempered out). This means that three stacked generators makes the orwell tetrad 1–7/6–11/8–8/5, a chord in which every interval is a (tempered) 11-odd-limit consonance. Other such chords in orwell are the keenanismic chords and the swetismic chords.
Compatible equal temperaments include 22edo, 31edo, 53edo, and 84edo. Orwell is in better tune in lower limits than higher ones; the optimal patent val is 296edo in the 5-limit, 137edo in the 7-limit, and 53edo in the 11-limit. It tempers out the semicomma in the 5-limit, and so belongs to the semicomma family. In the 7-limit it tempers out 225/224, 1728/1715, 2430/2401 and 6144/6125, and in the 11-limit, 99/98, 121/120, 176/175, 385/384 and 540/539. By adding 275/273 to the list of commas it can be extended to the 13-limit as tridecimal orwell, and by adding instead 66/65, winston temperament.
See Semicomma family #Orwell for technical details.
Interval chain
Odd harmonics 1–21 and their inverses are in bold.
# | Cents* | Approximate ratios | |||
---|---|---|---|---|---|
11-limit | 13-limit extensions | ||||
Orwell mapping | Winston mapping | Blair mapping | |||
0 | 0.00 | 1/1 | |||
1 | 271.46 | 7/6 | 13/11, 15/13 | ||
2 | 542.91 | 11/8, 15/11 | 18/13 | 35/26, 39/28 | |
3 | 814.37 | 8/5 | 21/13, 52/33 | 13/8 | |
4 | 1085.82 | 15/8, 28/15 | 13/7 | 24/13 | |
5 | 157.28 | 12/11, 11/10, 35/32 | 13/12 | 14/13 | |
6 | 428.73 | 14/11, 9/7, 32/25 | 13/10, 33/26 | ||
7 | 700.19 | 3/2 | 52/35 | ||
8 | 971.64 | 7/4 | 26/15 | ||
9 | 43.10 | 49/48, 36/35, 33/32 | 40/39 | 27/26 | 26/25 |
10 | 314.55 | 6/5 | 13/11 | 39/32 | |
11 | 586.01 | 7/5 | 39/28 | 18/13 | |
12 | 857.46 | 18/11 | 64/39 | 13/8 | 21/13 |
13 | 1128.92 | 21/11, 27/14, 48/25 | 25/13 | 39/20 | |
14 | 200.37 | 9/8, 28/25 | |||
15 | 471.83 | 21/16 | 13/10 | ||
16 | 743.28 | 49/32, 54/35 | 20/13 | ||
17 | 1014.74 | 9/5 | |||
18 | 86.19 | 21/20 | 26/25 | 27/26 | |
19 | 357.65 | 27/22, 49/40 | 16/13 | 39/32 | |
20 | 629.10 | 36/25 | 56/39 | ||
21 | 900.56 | 27/16, 42/25 | 22/13 | ||
22 | 1172.01 | 63/32 | 39/20 |
* in 11-limit CWE tuning
Chords and harmony
List of chords
Functional harmony
The generator, ~7/6, is a septimal interval, so the most basic chords could be 1–7/6–3/2 (0–1–7) and 1–9/7–3/2 (0–6–7), or 1–7/4–3 (0–8–7) and 1–12/7–3 (0–(-1)–7), or tetrads such as 1–7/6–3/2–7/4 (0–1–7–8); this is similar, but also in clear contrast to the 1–5/4–3/2 (0–4–1) and 1–6/5–3/2 (0–(−3)–1) of meantone. Two approaches to functional harmony thus arise.
First, we suggest using the septimal chords above as the basis of harmony, but swapping the roles of 3 and 7 according to their temperamental complexities (number of generator steps). Thus a "dominant" chord is either 7/6 or 12/7 over tonic; a "subdominant" chord is either 7/6 or 12/7 under tonic. This leads to an approach closely adherent to mos scales. The 9-tone mos contains a tonic and a "dominant" triad. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions, triads and tetrads alike.
Second, we suggest using the same chords as the basis of harmony, and keeping the role of the chain of fifths as the spine on which the functions are defined. This means dominant is still 3/2 over tonic, for example. A consequence is we must step out of the logic of mos scales, as they are often too restrictive without the many fifths to stack. This is essentially working in JI, but using the commas tempered out in some way to lock into the identity of the temperament.
Scales
MOS scales
- 9-tone scales (sLsLsLsLs, proper)
- Orwell9 – 84edo tuning
- Orwell9-12 – 7-limit POTE tuning, mapped to 12-tones
in POTE tuning
in 22edo
in 53edo
Small ("minor") interval | 114.29 | 228.59 | 385.72 | 500.02 | 657.15 | 771.44 | 928.57 | 1042.87 |
JI intervals represented | 15/14~16/15 | 8/7 | 5/4 | 4/3 | 16/11 | 14/9~11/7 | 12/7 | 11/6 |
Large ("major") interval | 157.13 | 271.43 | 428.56 | 542.85 | 699.98 | 814.28 | 971.41 | 1085.71 |
JI intervals represented | 12/11~11/10 | 7/6 | 14/11~9/7 | 11/8 | 3/2 | 8/5 | 7/4 | 15/8 |
- 13-tone scales (LsLLsLLLsLLsL, improper)
- Orwell13 – 84edo tuning
- Orwellwoo13 – [6 5/2] eigenmonzo (unchanged-interval) tuning
Small ("minor") interval | 42.84 | 157.13 | 271.43 | 314.26 | 428.56 | 542.85 | 585.69 | 699.98 | 814.28 | 857 | 971.41 | 1085.71 |
JI intervals represented | 12/11~11/10 | 7/6 | 6/5 | 14/11~9/7 | 11/8 | 7/5 | 3/2 | 8/5 | 18/11 | 7/4 | 15/8 | |
Large ("major") interval | 114.29 | 228.59 | 342.88 | 385.72 | 500.02 | 614.31 | 657.15 | 771.44 | 885.74 | 928.57 | 1042.87 | 1157.16 |
JI intervals represented | 15/14~16/15 | 8/7 | 11/9 | 5/4 | 4/3 | 10/7 | 16/11 | 14/9~11/7 | 5/3 | 12/7 | 11/6 |
- 22-tone scales
- Orwell22
- Orwellwoo22 – [6 5/2] eigenmonzo (unchanged-interval) tuning
Transversal scales
Others
- Orwell-graham – 13-tone modmos in 53edo tuning
- Orwell13-modmos-containing-minerva12 – 13-tone modmos in POTE tuning
- Minerva12-orwell-tempered – minerva[12] tempered to orwell
Tunings
Weight-skew\order | Euclidean |
---|---|
Tenney | CTE: ~7/6 = 271.5130¢ |
Weil | CWE: ~7/6 = 271.5097¢ |
Equilateral | CEE: ~7/6 = 271.3553¢ |
Skewed-equilateral | CSEE: ~7/6 = 271.3339¢ |
Benedetti/Wilson | CBE: ~7/6 = 271.5725¢ |
Skewed-Benedetti/Wilson | CSBE: ~7/6 = 271.5741¢ |
Weight-skew\order | Euclidean |
---|---|
Tenney | CTE: ~7/6 = 271.5597¢ |
Weil | CWE: ~7/6 = 271.4552¢ |
Equilateral | CEE: ~7/6 = 271.4920¢ |
Skewed-equilateral | CSEE: ~7/6 = 271.3038¢ |
Benedetti/Wilson | CBE: ~7/6 = 271.5915¢ |
Skewed-Benedetti/Wilson | CSBE: ~7/6 = 271.5302¢ |
Tuning spectrum
Edo generator |
Eigenmonzo (unchanged-interval)* |
Generator (¢) | Comments |
---|---|---|---|
2\9 | 266.667 | Lower bound of 7-odd-limit diamond monotone | |
7/6 | 266.871 | ||
15/11 | 268.475 | ||
11/7 | 269.585 | ||
11/6 | 270.127 | ||
15/14 | 270.139 | ||
49/48 | 270.633 | ||
21/11 | 270.728 | ||
7\31 | 270.968 | Lower bound of 9- and 11-odd-limit diamond monotone | |
11/9 | 271.049 | ||
7/4 | 271.103 | ||
7/5 | 271.137 | 7- and 11-odd-limit minimax | |
5/4 | 271.229 | ||
21/20 | 271.359 | ||
21/16 | 271.385 | ||
19\84 | 271.429 | 84e val | |
25/24 | 271.487 | ||
64/63 | 271.488 | ||
5/3 | 271.564 | 5-odd-limit minimax | |
9/5 | 271.623 | 9-odd-limit minimax | |
81/80 | 271.661 | ||
12\53 | 271.698 | ||
3/2 | 271.708 | ||
17\75 | 272.000 | ||
15/8 | 272.067 | ||
36/35 | 272.086 | ||
9/7 | 272.514 | ||
5\22 | 272.727 | Upper bound of 7-, 9- and 11-odd-limit diamond monotone | |
11/10 | 273.001 | ||
11/8 | 275.659 |
* Besides the octave
Non-octave settings
Watcher
By switching the roles of the period and generator, we end up with a nonoctave temperament that is to orwell what angel and devadoot are to meantone and magic, respectively. There is an interesting mos with 7 notes per period; if this is derived as a subset of 84edt (which has 12 notes per period, and is almost identical to 53edo), the resulting mos has the same structure as the 12edo diatonic scale, only compressed so that the period is ~272 cents rather than an octave. Thus, a piano keyboard for this mos would look exactly the same as a typical keyboard, only what looks like an octave would not be one anymore. This temperament could be called watcher, a reference to a class of angels whose very name carries Orwellian connotations. The 12-integer-limit otonality (1::12) and utonality (1/(1::12)) both have complexity 4. If we consider these to be the fundamental consonances, then using the 7-note-per-period mos, there are exactly 3 of each type per period, which again is analogous to the diatonic scale. While angel and devadoot do not perform well past the 10-integer-limit, watcher handles the 12-integer-limit with ease. Straight-fretted watcher guitars could be built as long as the strings were all tuned to period-equivalent notes.
Rank-3 temperaments
Following is a list of rank-3, or planar temperaments that are supported by orwell temperament.
Rank-3 temperament | Among others, rank-3 temperament is also supported by… | ||||
---|---|---|---|---|---|
7-limit | 11-limit Extension |
9tet | 22tet | 31tet | 53tet |
Marvel | Negri, septimin, august, amavil, enneaportent |
Magic, pajara, wizard, porky | Meantone, miracle, tritonic, slender, würschmidt |
Garibaldi, catakleismic | |
Marvel | Negri, septimin, enneaportent | Magic, pajarous, wizard | Meanpop, miracle, tritoni, slender | Garibaldi, catakleismic | |
Minerva | Negric, august, amavil | Telepathy, pajara | Meantone, revelation, würschmidt | Cataclysmic | |
Artemis* | Wilsec | Divination, hemipaj, porky | Migration, oracle, tritonic | ||
Hewuermity | Triforce, armodue, twothirdtonic |
Porcupine, astrology, shrutar, hendecatonic, septisuperfourth |
Hemiwürschmidt, valentine, mohajira, grendel |
Amity, hemischis, hemikleismic | |
Zeus | Triforce, armodue, twothirdtonic |
Porcupine, astrology, shrutar, hendecatonic |
Hemiwur, valentine, mohajira | Hitchcock, hemikleismic | |
Jupiter | Septisuperfourth | Hemiwürschmidt, grendel | Amity, hemischis | ||
Orwellismic | Beep, secund, infraorwell, niner |
Superpyth, doublewide, echidna |
Myna, mothra, sentinel, semisept |
Quartonic, buzzard | |
Orwellian | Pentoid, secund | Suprapyth, doublewide | Myno, mothra, sentinel | ||
Guanyin | Infraorwell, niner | Superpyth, fleetwood, echidna | Myna, mosura, semisept | Quartonic, buzzard | |
Nuwell | Progression, superpelog | Quasisuper, hedgehog | Squares, nusecond | Tricot, hamity | |
Big brother | Progression, superpelog | Quasisupra, hedgehog | Squares, nusecond | Tricot, hamity | |
Horwell | Bisupermajor, escaped, fifthplus |
Hemithirds, worschmidt, tertiaseptal |
Countercata, pontiac | ||
Zelda | Bisupermajor, sensa | Hemithirds, worschmidt, tertia | Countercata |
* weak extension (one or more generators from the parent temperament are split)
Music
- Mountain Villiage (2013) – play | SoundCloud – Orwell[9]
- Earwig (2012) – play – in 31edo tuning
- Elf Dine on Ho Ho (2012) – play – in 53edo tuning
- Spun (2012) – play – Orwell[13]
- one drop of rain
- i've come with a bucket of roses
- my own house
- Schizo Blue – play | SoundCloud ^{[dead link]}
- Sejaliscos (2013) – play | SoundCloud – Orwell[9] in 22edo tuning
- Trio in Orwell – details | play
- Swing in Orwell-9
Keyboards
To play interactive versions of these keyboards, check out Vito Sicurella's plugin, which works with REAPER: