Orwell

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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, 8/5 is divided into three equal steps, each of which represent 7/6 (alternatively, the 5th harmonic (5/1) is divided into 3 equal steps that represent ~12/7), so that 1728/1715 = S6/S7 is tempered out. After two 8/5s (six generators), 9/7 is found by tempering out the marvel comma, 225/224, and thus the just perfect twelfth (3/1) is divided into 7 equal steps.

In the 11-limit, two generators are equated to 15/11 and 11/8 (meaning 99/98 and 121/120 are 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 undecimal orwell are the keenanismic chords and the swetismic chords. (A far more complicated mapping of 11 at 33 generators, tempering out 441/440 instead, is also possible and is known as newspeak temperament.)

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.

See Semicomma family #Orwell for technical details. See Orwell extensions for details about 13-limit extensions.

Theory

Interval chain

Odd harmonics 1–21 and their inverses are in bold.

#	Cents*	Approximate ratios
0	0.00	1/1
1	271.46	7/6
2	542.91	11/8, 15/11
3	814.37	8/5
4	1085.82	15/8, 28/15
5	157.28	11/10, 12/11, 35/32
6	428.73	9/7, 14/11, 32/25
7	700.19	3/2
8	971.64	7/4
9	43.10	33/32, 36/35, 49/48
10	314.55	6/5
11	586.01	7/5
12	857.46	18/11
13	1128.92	21/11, 27/14, 48/25
14	200.37	9/8, 28/25
15	471.83	21/16
16	743.28	49/32, 54/35
17	1014.74	9/5
18	86.19	21/20
19	357.65	27/22, 49/40
20	629.10	36/25
21	900.56	27/16, 42/25
22	1172.01	63/32

* In 11-limit CWE tuning, octave reduced

Chords and harmony

Main article: Chords of orwell

See also: Functional harmony in rank-2 temperaments

The fundamental otonal consonance of orwell, voiced in a roughly tertian manner, is 4:5:6:7:9:11. In terms of generator steps this is 0–(−3)–7–8–14–2, only available in a 22-tone mos. However, some subsets of this chord are way simpler, such as 8:11:12:14, which is 1–11/8–3/2–7/4 (0–2–7–8).

The generator, ~7/6, is a septimal interval, so chords could instead be built around it as 1–7/6–3/2 (0–1–7), 1–7/4–3 (0–8–7), or tetrads such as 1–7/6–3/2–7/4 (0–1–7–8).

To 1–7/6–3/2–7/4 we may add 11/8, or to 1–11/8–3/2–7/4 we may add 7/6, to form an essentially tempered pentad, 1–7/6–11/8–3/2–7/4 (0–1–2–7–8). Its inverse is 1–12/11–9/7–3/2–12/7 (0–5–6–7–(−1)), which can serve as a minor counterpart. 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) chords of meantone. Two approaches to functional harmony thus arise.

First, we can treat 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 to pentads alike.

Second, we can treat 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

Main article: Orwell scales

Mos scales

Orwell5

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] unchanged-interval (eigenmonzo) 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] unchanged-interval (eigenmonzo) 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

7-limit norm-based tunings
	Euclidean
	Constrained	Constrained & skewed
Equilateral	CEE: ~7/6 = 271.3553 ¢	CSEE: ~7/6 = 271.3339 ¢
Tenney	CTE: ~7/6 = 271.5130 ¢	CWE: ~7/6 = 271.5097 ¢
Benedetti, Wilson	CBE: ~7/6 = 271.5725 ¢	CSBE: ~7/6 = 271.5741 ¢

11-limit norm-based tunings
	Euclidean
	Constrained	Constrained & skewed
Equilateral	CEE: ~7/6 = 271.4920 ¢	CSEE: ~7/6 = 271.3038 ¢
Tenney	CTE: ~7/6 = 271.5597 ¢	CWE: ~7/6 = 271.4552 ¢
Benedetti, Wilson	CBE: ~7/6 = 271.5915 ¢	CSBE: ~7/6 = 271.5302 ¢

DR and equal-beating tunings
Optimized chord	Generator value	Polynomial	Further notes
3:4:5 (+1 +1)	~7/6 = 272.890 ¢	f¹⁰ − 8f³ + 8 = 0	1–3–5 equal-beating tuning
4:5:6 (+1 +1)	~7/6 = 271.508 ¢	f¹⁰ + 2f³ - 8 = 0	1–3–5 equal-beating tuning

Tuning spectrum

Edo generator	Unchanged interval (eigenmonzo)*	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