84ed2 (2): George Orwell and Microtonality

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!

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84ed2 (1): John Cage and Microtonality

Cage’s final period was notable, unsurprisingly, in at least two or three respects: he devised a system of time bracket notation with varying degrees of rigidity and looseness, he entitled his works with a dry, basic numbering system, and three of his last works were microtonal.

His time bracket notation concerns rhythm in a loose manner. Instead of using standard notation with quarter-notes, half-notes etc., and instead of using graphic notation as he had for much of his output, where 1 horizontal inch might represent a time value of, say, 1 second, he devised a compact form whereby bars begin and end at the performer’s discretion within values given by the composer; some freedom is allotted to the performer.

For example, Section A–Part 1 from Four, for string quartet, begins anytime in the first twenty or so seconds: the first bar contains two whole notes (A and B, above middle-C) but no conventional indication of meter or tempo. Instead there is a beginning bracket, 0’00″–0’22.5″, and an ending bracket, 0’15″–0’37.5″. The player must begin and end within these ranges (and the player must not end before he has begun). In this work each of four parts and each of three sections has identical brackets but different material within.

The third bar, or system, has no range but, instead, specific starting and ending times: 1’07.5″ and 1’22.5″. Cage differentiates these ‘fixed’ brackets from the other ‘flexible’ brackets. There are many articles that explain this with more examples, or that offer other details (such as his compositional periods with and without computer software as an aid—see here), and there’s even an article out there showing geometrical properties of this treatment of time.

His titles became austere in that he wrote many works with simple titles such as One7, the ‘7’ being a superscript [not possible to display with internet/HTML]. Rather than “1 to the power of 7” he means a seventh piece for one performer. We shall look at an aspect in common within the series Two4, Two5, Two6, and Ten, that they are not only some of his last but, more relevantly, his microtonal pieces.

I’m mainly going to point to others’ writings but before I do so I want to clarify that Cage wrote microtonal music with very small intervals. I perused some CD booklets at the local library and most simply described this music as “microtonal” and one unhelpfully said that he wrote with “quarter-tones.” Often people use ‘quarter-tone’ as a blanket term to mean ‘anything but normal tones.’ I also looked online, of course, and saw that the John Cage Trust describes the microtonal writing as having ‘six notes, degrees or steps between each half step.’ Yes but… potentially misleading and I would like to clarify.

It would be better to say that Cage divided each semitone into 7 steps, meaning there are twice as many per whole tone, 14 steps. You may call it 1/14th-tone music (but not 1/4-tone).

What I find remarkable is that the other—more normal?—choices for dividing a semitone so finely are somewhat more established, as much as microtonal praxis can be. 1/16th-tone music, i.e. dividing each semitone into 8 tiny steps, was—believe it or not—being done to pianos a century ago.

The Mexican composer and inventor (and Nobel prize nominee) Julián Carillo patented pianos able to play in various tunings, the 1/16th-tone piano being one example that was manufactured. Instead of 88 keys it requires 97, and that’s just to play one octave! (I tried one belonging to Bruce Mather for a few moments at the conservatory in Montreal. You play a chord… and a single, warbling tone is the result.) Sixteenth-tone music was commissioned in the roaring ‘20s and played by Stokowski and the Philadelphia Symphony Orchestra. It could have become a new standard of tuning but then musical developments evolved in other directions and such microtonal music faded out of the picture.

The other choice, 6 tiny steps per semitone was also not chosen by Cage. This scale is very popular (as much as microtonal music can be). Georg Friedrich Haas writes in this scale and they teach this scale in Boston in sight-singing classes! There are also at least two good notation systems for representing it. Twelfth-tones, better known as the 72ed2 scale.

But Cage marches to the beat of a different drummer. Below is text from a festival programme in Frankfurt twenty years ago, which I enjoyed translating (as an exercise for my German language acquisition—please indulge me):


“In July 1991 John Cage composed three works for the japanese shō instrument, a mouth organ using reed pipes: One9 for shō, alone; Two3 for shō and 1 percussionist, who plays five water-filled mussels [shells]; and—a commission from the McKim Fund in the Library of Congress—Two4 for shō and violin, a piece in which a piano can partake instead of the shō. The musical outcome’s material, that Cage employs, is entirely different for both instruments: The shō has 17 pipes and, from them, up to 6 can be blown simultaneously. The composer uses here all possibilities from single tones up to hexachords, and builds rather vast phrases in which the tones either detach from one another through breath-marks or, alternatively, are linked through legato arcs. By way of contrast the violin plays a maximum of 3 tone pitches per system but, instead, with constantly varying timbre.

“Remaining is the violin part—like all the voices besides piano and percussion in Ten (October 1991)—written in a fourteenth-tone system:


“From the notation’s kind and manner one can distinctly discern how immediate are the sonorous characteristics of this work by Cage, coming at once from the possibilities of the refined notation system: The decision for 14 intervals per whole tone is not owing to melodious considerations or general, theoretical deliberations, but because Cage uses upward- or downward-pointing arrows as notation symbols: This allows one to apply, without optically taxing the player, three different positions (high, middle, low) before a conventional note; also two notes come without arrows, those boundaries marked by semitones; so 7 intervals per semitone, 14 intervals per whole tone, and 84 intervals per octave emerge. The dynamic of the half-hour work stays in the range of pp.

Martin Erdmann

[transl. Todd Harrop]

(Rausch, Ulrike and Martin Erdmann. “4. Number Pieces II”, pp. 198–200. Schädler, Stefan and Walter Zimmerman, editors. John Cage: Anarchic Harmony. Frankfurt Feste ’92/Alte Oper Frankfurt. Mainz: Schott, 1992.)

Cage continued writing microtonal writing in Two5, Two6 and Ten with the same scale, 84 tones per octave, hence the title of this article, “84ed2”, meaning 84 divisions of the 2nd harmonic (which is the octave, a.k.a. 84edo). So, Part 1 has introduced the full, ‘chromatic’ set of pitches and one composer’s employment of them. (See this page for more information on the 84-tone scale and YouTube links to performances of Cage’s works.) The next article will compare this microtonal scale with the standard 12-note scale by constructing pentatonic- and diatonic-like subsets inspired by the writer George Orwell.

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Fish Out of Water

__A friend of mine (let’s call him Jimmy) wrote a play which is in rehearsal now. He’s an actor and I was reminded of when we met, when we were both in a play nearly ten years ago. This story concerns music-theatre, which is going to be a focus of mine for a while, and one example of the audition process and of crossing disciplines.
__Jimmy had auditioned for R. Murray Schafer, the composer, for a music-theatre show, probably one of his large-scale outdoor promenade operas. When he finished his monologue, some text from straight-up dramatic theatre, the panel asked him to make some weird sounds and to use his body and voice in a grotesque, expressionistic manner.
__Jimmy dug deeply into his soul and his psyche, to root up those experiences that haunt him when he’s not consciously keeping his demons locked in the cellar of his mind. He uttered forth some expectorant, toxic, gurgling, half-strangled scream that ripped through his shuddering body and spewed into the theatre, leaving him panting and nearly sweating with the effort.
__“That’s fine,” said someone, unseen. “But could you please do that again, only this time louder and longer, and maybe go from some low notes to high, then back again?” Continue reading

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Arduino variable types

Happy New Year, Bonne année, and Frohes neues Jahr!

2014 will see me fooling around with an Arduino microcontroller. I made a rudimentary ‘light organ’ using photosensors and Max/MSP. You play music by covering the sensors with your fingers, and allowing various amounts of light to shine between them. Darkness for low notes, brightness for high notes.

First problem: the Arduino code could be more optimized by using the right type of variable to hold the sensor values. The code that I downloaded uses ‘floats’ for each sensor but I think that that is overkill. Float is the nickname for floating-point number, i.e., a number with a decimal point that can move to the left or right to show more or fewer digits. They are complicated types of numbers in computing and, hence take up more memory and power. On the Arduino, like old-school 8-bit computers of yore, memory and power are precious!

The result was that my organ could only respond to movements of my fingers three times per second. Much too sluggish for a tocatta & fugue. There could be other problems dragging down the code execution too, so if I find them I’ll post ‘em here.

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Spectropol Records

Many thanks to Bruce Hamilton for including one of my works on this album! I am happy to be heard amongst my colleagues.

Possible Worlds Volume Two continues Spectropol’s showcase of contemporary microtonal/xenharmonic music and sound art from around the world.

“There are many approaches to pitch use outside non-12-tone equal temperament on display here, conveyed through a wide variety of musical styles. Artists in this collection, which include well-established experts in microtonal practice, explore just intonation, the harmonic series, free & mixed tunings, extended playing techniques, invented instruments, and an emphasis on various equal divisions of the octave.”

About the scale used in Fifteen Short Pieces:
Banff scale 15 short pieces
There are different ways to describe this. It is a 15-note, symmetrical subset of 30ed2 (30 equal divisions of the octave), or a 15-note, symmetrical subset of a 1/5th-tone scale. The pattern of small and large scale steps is ssLss ssLss ssLss, where ‘s’ is 1 fifth-tone and ‘L’ is 6 fifth-tones. I arrived at this by taking the augmented triad of D-F#-A# and transposing copies of it by 40 cents and 80 cents higher and lower.

The scale approximates exotic, 13-limit just intonation rather well. The cleanest group of intervals, within 5 cents’ accuracy, includes 13/8, 15/7, 13/9, 5/3 major sixth (and 6/5 minor third), 9/7 and 11/10. Although there is a minor third there is no major third, nor perfect fifth, so one cannot write a triad unless it’s a diminished one.

The next group of intervals deviates from just from between 7 and 9 cents—still an excellent approximation! Among the more exotic intervals there are also a few from the same harmonic series: 7/4, 11/8, and 15/8. (13/8 was in the first group and the most accurately represented of all.)

What I’ve learned from this experiment in 30 EDO is to not be pulled in by the gravity of more obvious choices, such as the venerable and distinguished 31 EDO scale. With some imagination and an open ear one can make do with unusual harmonic materials, and express something seldom ever heard.

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Bayreuth Composers’ Commune

I travelled down to Bavaria last August to participate in the inaugural Composers’ Commune, a new initiative of the venerable Festival Junger Künstler Bayreuth. The host organisation has always had a political mandate to bring artists from the ‘east’ and the ‘west’ together to make music. There were ensembles from different corners of the world and I was encouraged to attend many of their concerts in some very sublime venues. I especially liked a show of blistering drums, movement and electronics by Joss Turnbull, percussionist, and Kaveh Ghaemi, a contemporary dancer from Iran. Originally, I think, east & west meant post-war Germany; today, it means the Middle East and Western culture. My colleagues were from Serbia, England, Palestine and Syria. Helmut Erdmann had brought us together and we introduced ourselves by playing a sample of our music, live or recorded. Continue reading

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For the spring 2013 semester I spent many hours piloting an EMS Synthi AKS analog synthesizer. It’s a solid machine, still made in Cornwall, England, invented in the ’70s, and was a robust instrument for electronic music pedagogy. It was also used in music by Brian Eno, Jean-Michel Jarre and Pink Floyd, as well as for the TV series Doctor Who. My sessions on this vintage instrument took me to the European Live Electronic Centre in the charming village of Lüneburg.

The machine comes in a handy suitcase and can be set up within minutes. There are dials on the face, a pin matrix instead of a patch bay (i.e., using pins instead of wires, as in the American synthesizers by Moog and Buchla). Inside the suitcase lid is a keyboard that responds to the fingers’ natural capacitance.

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