In part two we will build a tenuous bridge between the composer John Cage and writer George Orwell. I already started with the banner picture of Cage performing on a TV show called Good Morning Mr. Orwell, from 1984. Orwell, of course is most famous for his dystopian book 1984 and the previous article introduced Cage’s late system of composing with microtones derived from adding six new notes between normal semitones, resulting in 84 equally-spaced tones per octave instead of the usual 12.
I doubt there is an International Standardization Organization definition of the semitone but we shall define it as a musical interval with the frequency ratio equal to the 12th root of the 2nd harmonic, and we shall define 1 cent (¢) as 1/100th of this semitone, giving 1200 cents per 2nd harmonic or octave, and 100 cents per semitone.
If we divide 1200¢ by 84 steps we get about 14.3¢ per step, which is quite small, close to the ‘just noticeable difference’ where it’s hard to distinguish one tone from the next. The frequency ratio (which we don’t need now but should know how to calculate) is the 84th root of 2, or 2 to the power of 1/84 ≈ 1.00828589.
Bear with me because things are about to get more interesting. With this tiny musical interval of 14.3¢ we could build a long, fine scale and reach the octave after 84 steps. Or we could skip around in some manner and make do with fewer but larger steps using multiples of 14.3 cents. We shall do this with single leaps of nineteen of these tiny intervals and see & hear what kinds of scales this leads to.
The motif today is George Orwell so, after the title of his famous book, we’ll take the interval of 19\84ths of an octave and, to make this concept easier to follow, I’ll compare the process side-by-side with our familiar, normal scales. We’re going to cook up two flavours each of pentatonic, diatonic and chromatic scales!
1a) We can build our regular scale (the chromatic scale) with a string of twelve semitones, or we can leap around by some other interval, either by 7 semitones (the ‘perfect fifth’) or 5 semitones (the ‘perfect fourth’—they are inversions of each other). Let’s start on the note middle-C and go up two perfect fifths: G, and D. Next, start on C and go down two fifths: F, and Bb. Finally let’s transpose the notes by octaves so that they are all assembled within one octave, from C to C’. We now have a pentatonic scale: C, D, F, G, Bb, and C’—five notes (not including the repeated C shown with an apostrophe) and, in this case, arranged symmetrically. (I’m using ‘Bb’ to represent ‘B-flat’.)
We could have made a scale with more or fewer notes but this one is interesting because there are only two types of intervals between the notes: either a distance of 2 semitones (major second) or 3 semitones (minor third or augmented second). Since the scale is also symmetrical the theorist Erv Wilson would call this a Moment of Symmetry (MOS). The actual presented scale, from its starting note to ending note, doesn’t need to be symmetrical, just so long as some particular rotation of the scale is.
1b) Let’s add two more notes by going up and down an extra perfect fifth. From C this gives G, D and A; and F, Bb and Eb. Re-ordering and assembling into a compact scale gives C, D, Eb, F, G, A, Bb, C’. This collection of seven notes is not an official diatonic scale but considered one of the diatonic modes, the dorian mode. It is essentially the B-flat major scale but starting on the second degree rather than the first. I did this on purpose to preserve a symmetrical form. You will notice that there are, again, only two types of intervals between notes, this time smaller: the whole tone and the semitone.
1c) Now we shall complete the process by starting on C and going up & down five or six perfect fifths. From C again: (going up) G, D, A, E, B, and maybe F#; (going down) F, Bb, Eb, Ab, Db, and maybe Gb. The notes F# and Gb sound the same (in equal-tempered tuning) so it won’t matter which we pick. The final scale is the chromatic scale of 12 notes: C, Db, D, Eb, E, F, F#/Gb, G, Ab, A, Bb, B, and C’.
The circle is complete; there are no more notes that we could add. (Although in the just intonation system the circle would be incomplete… but that’s another story.) Just as F# and Gb sound the same so do many other notes, e.g. Db can be enharmonically paired with C#. Have a listen to these familiar scales before we move on to the Orwell system to see & hear how it compares.
2a) Instead of leaping around by 7\12ths of an octave we shall leap around by the honorific 19\84ths of an octave. We shall use the backslash to distinguish fractions used in equal-temperament, from forward-slashes used with frequency ratios, such as 3/2 or 7/6. This shall become clearer later. Whereas above we built a scale from a chain of perfect fifths, this time we’ll forge a chain of septimal minor thirds—the just interval most strongly identified with 19\84ths of octave.
Going up from C, the first two leaps bring you to D# −1/7th-tone, then F +3/14ths-tone. Going down from C two such leaps will take you to A +1/7th-tone and G −3/7ths-tone. This is a very different pentatonic scale from earlier, but the method is identical. As before, there are two sizes of intervals between notes: instead of 200 and 300 cents, these are 271.4 and 114.3 cents (the smaller of them falling centre, between F and G).
2b) Continuing on, we need to add more notes than before to arrive at a Moment of Symmetry, 9 notes intead of 7. Four leaps of our septimal minor thirds upward and four downward yield a scale with the neat pattern of alternating small (’s’) and large (‘L’) in-between intervals: sLsLsLsLs. (You may substitute ’s’ with ‘k’ or ‘p’, and ‘L’ with ‘G’ if you prefer to think of them in German or French.) Their sizes are 114.3¢ and 157.1¢ respectively, or 8x and 11x the smallest unit of 14.29¢ which, remember, came from dividing the octave of 1200¢ by 84 steps.
It’s important to understand that we’re not leaping around by pure septimal minor thirds; we’re using something very slightly sharper. If we had used pure, or ‘just’ septimal minor thirds (ratio 7/6 ≈ 266.87¢) then our efforts would stop here with a 9-note equidistant scale, or virtually so because the scale would be closed. Instead we are using a sharp version of the third, ratio 2^(19/84) ≈ 271.43¢, which has here yielded a 9-note un-equidistant scale.
And you’re calling this a diatonic scale?!
Well, it’s similar in the sense that it is somewhat between a quasi-pentatonic scale and, soon to be revealed, a quasi-chromatic scale. But there is no dubious historical association with the modes of ancient Greece, nor any connection to European ecclesiastical modes. The Xenharmonic Alliance of music/math theorists and enthusiasts call this an albitonic scale from the Latin word for ‘white’, referring to the white keys on a piano keyboard. Gene Ward Smith has much to say regarding Orwell theory, so please seek his articles to delve deeper, e.g. here: http://xenharmonic.wikispaces.com/Orwell
2c) Finally we shall make a comparison to the regular chromatic scale of 12 steps by extending our Orwellian scale to 13 notes. Six leaps up and six down reveal the scale pattern LsLLsLLLsLLsL, where ‘L’ is 8x the smallest unit and ’s’ is 3x (114.3¢ and 42.9¢). Alternatively, one could forego the use of ‘L’ and ‘s’ and write 8388388838838, although the relative sizes and similar shapes to indicate them are not as easily discernible. As before, this is a symmetrical arrangement of a scale with only two sizes of intervals, hence it is a MOS. The distribution of this 13-note scale, however, is not as evenly distributed as the 9-note scale arrived at previously.
The normal 12-note chromatic scale is perfectly equidistant and is a closed system, but this last scale that we generated is not the end of the story. One could continue and catalogue further and finer moments of symmetry (22, for example, which could be comparable to the conventional quarter-tone scale of 24 notes).
There are only a handful of short pieces written in these scales. If you are inspired to compose your own please share, we would love to hear it. Some music to whet your appetite can be found here.
[Caution: the following information may cause temporary discomfort behind the eyes.]
In closing we consider the convention of using back- and forward-slashes to differentiate between two kinds of ratios, e.g. 7\12 vs. 3/2 which, in a tuning shorthand, actually approximate each other; likewise with 19\84 and 7/6. How so?
Assuming that the number 2 is implicit as a base, we use the nomenclature ‘7\12’ as an exponential shorthand to stand in for ‘two to the power of seven-over-twelve’, i.e. 2^(7/12), which approximately equals 1.4983 and is a frequency ratio. This ratio is supposed to be, in fact, a neat 700¢ wide and called the equal-tempered perfect fifth. The just intonation perfect fifth is simpler: 3/2 = 1.5 ≈ 701.955¢. You would hear these intervals if you could generate tones at 100 and 150 Hz (just intonation), or 100 and 149.83 Hz (equal-tempered). The tiny discrepancy between 150 and 149.83 is negligible at only 2¢.
(By the way, a handy formula for converting to cents is 1200 * log(p/q) / log(2). Where p/q is a frequency ratio, e.g. 3/2. You may substitute ‘log’ with the symbol for natural logarithm ‘ln’. Try it in the Google search bar or, if you have a Mac, as a Spotlight search term.)
In today’s Orwellian exercise we used the interval 19\84 as a short-hand for 2^(19/84), which I said is close to the septimal minor third of frequency ratio 7/6. The latter is also a just intonation interval, e.g. the difference between 600 and 700 Hz. Furthermore, 7÷6 ≈ 1.1667, and 2^(19/84) ≈ 1.1697. They are very close, only about 4½ cents apart.
But what about trying other intervals, like 18\84?
Sure, why not? Anything is possible or at least worth investigating. I can tell you now that 18\84 would lead to an equal-tempered 14-tone scale—notoriously dissonant—if you add a remaining note. Infinite possibilities… As a friend and composer once said, “So many microtonal scales. So little time.” Now go, and create some music!