Commas and Generators
In terms of octave and generator, Orwell is defined by a generator which is a somewhat sharp (four or five cents worth) subminor third. While of some interest as a five-limit temperament, with a comma of 2109375/2097152, it really comes into its own as a seven-limit temperament, where it joins that important class of temperaments (including Meantone, Magic, and Kleismic) whose generators (in this case the approximate 7/6) are consonances of the system. As a seven-limit temperament, it is defined by the commas <1728/1715, 225/224>. The 1728/1715 comma is of particular significance for Orwell, since it tells us that three Orwell generators are an approximate 8/5, and so Orwell is closely allied to the planar temperament this comma defines. It also does well as an eleven-limit temperament, where in its best incarnation it is defined by the commas <99/98,121/120,176/175>. Here 99/98 is of particular significance, telling us that two generators give us our approximate 11/8.
Melodic Properties of Orwell
Orwell has a nine-note MOS defined by a chain of eight generators. It may be compared to the diatonic scale of Meantone in the sense that it strikes a happy medium between the blandness of the ten-note Miracle MOS and the unevenness of the ten-note Magic MOS, and its good melodic properties are one of the best features of Orwell.If we call the large step of this MOS L, and the small step s, then the MOS has step sizes LsLsLsLss, where s is a flat secor (in the sense that it serves as both a 16/15 and a 15/14) and the L, in the eleven-limit version, can be regarded as an 11/10; in any case Ls gives us an Orwell generator of about 7/6. The seven and nine limit harmonic resources of this scale may be considerably improved by permuting its steps; of particular interest here are the variant scales LssLLLsss, LssLsssLsL and LsLssLssL.
Mapping to Primes
Here's the matrix defining the mapping to primes for Orwell:
(1 0)
(0 7)
(3 -3)
(1 8)
(3 2)
To get a complete 7- or 11-limit chord, you need a string of at least eleven generators.
Equal Temperaments covering Orwell
Orwell in its five- and seven-limit versions is done very well by the generator 19/84 (and hence the name.) For the eleven-limit, 12/53 is preferable. The sequence of generators
7/31 < 19/84 < 12/53 < 17/75 < 5/22
shows the range of Orwell equal temperament generators worth considering.
An Example for Orwell
For an audible example of Orwell, you may listen to the Trio for Clarinet, English Horn and Banjo
Except for a brief excursion into chromaticism, it is in the nine-note Orwell MOS, in the 12/53 version.