The Ross (née Heisenberg) Uncertainty Principle, and Other Musical Dichotomies

Most readers of this blog are, I am sure, well acquainted with the Heisenberg uncertainty principle, the gist of which is that there are certain pairs of physical properties in quantum mechanics (e.g., position and momentum) where the more precisely one is measured, the less precisely the other can be measured, represented as follows:

As every musician knows, the uncertainty principle can be applied to Fourier transforms of complex waveforms:

Now, you may be saying, "Not so fast, cowboy!  I am not at all acquainted with Heisenberg's uncertainty principle (nor, I suspect, are most most readers of this blog!), and all of these mathematical formulae are making my head spin!  Besides, what does any of this have to do with musical composition?"

I will respond by admitting that  (i) I am not actually a cowboy, and (ii) my opening statements are mischievous attempts to be provocative (or vice versa; I'm not really sure).  I realize that Heisenberg and complex mathematical equations are not common areas of study for most musicians, but, whether you understand them or not, those equations certainly catch the eye, do they not? ;)  In any event, please do not worry; there will only be one further mathematical equation in this blog, but I will explain it with disturbing clarity, or concise obfuscation, depending on my mood at the time.  Or not…

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One of the most important dichotomies to be found in most classical music is certainty versus uncertainty; today's blog is about the value of uncertainty, in a very general sense, within musical compositions.

Here are some examples of how this can work:

Certainty	Uncertainty
Theme; a recognizable melodic idea.	Development; use of familiar motives in unfamiliar contexts; transformation of motives in order to create new material.  Some aspects of the material may be recognizable, but the listener may be unsure as to where it is going. Transition; the beginning of a transition often sounds like a continuation or repetition of previous thematic material, but it soon becomes apparent that something different is going on, as  modulation takes place, and the material is taken in a different direction, creating uncertainty.
Key/Modality/Pitch Center; a section is in a particular key, or modality, or, if non-tonal, it may be centered on a particular pitch class.	What key are we in?  Development sections, transitions and retransitions, cadenzas, and even some coda sections (notably Beethoven's) all move between key areas, creating harmonic instability.  Even tonicizations within more stable key areas can create some harmonic uncertainty.
Form; I recognize this form! I therefore have a pretty good sense of what is likely to happen next. If the form includes a recapitulation (and most do), then I have a very good sense of what to expect for the last section of the piece.	Form? Um... What's going on here? I don't recognize the form.  Or, I thought I recognized the form, but the composer has thrown in unexpected elements (such as a coda that is longer than the development, or an unusually long transition, or a cadenza thrown into a piano sonata (as in Mozart's K.333, III), or a new theme in the development section).  Is it sonata form, or rondo, or sonata rondo, or sui generis, etc.?

My musical "uncertainty principle" is this: It is at least as important to have sections that give rise to a sense of uncertainty in a composition as it is to have sections of certainty. 


Fortunately, this can be represented by the following equation, which makes composing extraordinarily easy (if you're a physicist); x = quantity of uncertainty (measured in photon energy), p = mass of uncertainty (e.g., any Mass movement, such as kyrie, gloria, credo, etc.), and the h-bar is, of course, Planck's constant (I'm guessing this is a reference to Planck's faithful canine companion, Helmut):

Why?  The excitement of a roller-coaster ride — not an emotional one, an actual one! — is probably related to both the ascents and descents; going up the big hill that tends to be right at the start creates a sense of Heisenbergian uncertainty (some might call it "dread"), as you wonder what lies in store for you once you reach the top (Is this thing safe?  Why did I think this would be fun? Do I have a legal will?), and going down creates a sense of certainty (I am going to die! I know it for sure! Whee!), mixed with uncertainty (how much longer? Why is it so dark in here? Will I toss my cookies?).  People who love these rides, I would guess, love both the uncertainty and the certainty of the experience, but especially the former.  At least I do...

But a musical composition isn't a roller-coaster ride, is it?

Well, perhaps not, but I was making a rather loose analogy.  A musical composition can be compared to a journey, and, if this analogy makes sense to you, then it is a wonderful example of the old saying that what matters most in life is the journey, not the destination. How much fun would life be if you knew exactly what was going to happen at every stage?  How enjoyable would a musical composition be if you knew beforehand exactly what would happen at every stage?


→ Uncertainty; don't leave home without it!

What about compositions that are memorized?  They are enjoyable, even when I know exactly what will happen next!  Good point!  But I think what may be taking place here is that even when you know exactly what notes are going to be played before they are actually played, I am not sure that you know exactly what your emotional response to those notes will be, so here again, I suspect that part of the attraction to the composition may be based on uncertainty.  This too is analogous to a roller-coaster ride; even if you've been on it numerous times, you might respond slightly differently to it each ride.

I encourage you to look for opportunities within your compositions to try this idea out.

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This topic is an offshoot of the predictability/unpredictability dichotomy that I have mentioned in class and written about in past blogs. Predictability within a musical composition, like routines in life, can be comforting and reassuring at times, but too much can quickly bore the listener; a great composition seems to have a perfect balance of the two.

Below are links to two blog entries relating to this one, FYI:

• Two musical dichotomies: Familiar vs. Unfamiliar, and Expected vs. Unexpected
• More musical dichotomies

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Questions:

1. Besides those already discussed, to what other musical parameters can this certainty/uncertainty dichotomy be applied?

2. What are some of the ways in which it can be applied to the composition on which you are currently working?

3. Is this a useful way to think about music?

4. What are some other dichotomies to be found in music?

5. Can the certainty/uncertainty dichotomy be applied to other genres of music, such as popular, jazz, folk, or world?

Fun With Scales and Modes

If you have been following my blog entries on tonality (Why Atonal?, Atonality — Even the Word Sounds Unpleasant!, Atonality = Noise?), you will know that I encourage (i.e., require) student composers to explore harmonic/melodic systems that move beyond conventional tonality, where tonality is defined as " the system of musical organization of the common practice period, and of Western-influenced popular music throughout much of the world today" (from Wikipedia). One way to do this would be to experiment with any of the thousands of scales and modes that either already exist or that you can create yourself, hence the title of today's entry.

→ Here is an "octatonic" scale, also known as a "diminished" scale because a diminished seventh chord is formed from every second note. The intervalic pattern consists of alternating half- and whole-steps (or vice-versa):

Here is a short waltz based on this octatonic scale (click on it to enlarge):

^{Octatonic Scale Waltz:}

→ Here is a Hirajoshi scale:

Here is the first phrase of the waltz, this time based on the Hirajoshi scale:

^{Hirajoshi Scale Waltz:}

→ Here is one form of a blues scale:

… and here is the first phrase of the waltz based on that blues scale:

^{Blues Scale Waltz:}

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Discussion:

• We tend to limit ourselves to the use major and minor scales if composing tonal music, but there are thousands of other scale possibilities that have unique nuances and harmonic implications. If you have fun (i.e., experiment) with even a few of them, you may discover that every different scale gives your compositions a slightly (or even radically) different feel.
• Of these, there are a number of commonly-used alternatives to major and minor scales, such as anhemitonic (which just means "no semitones") pentatonic scales (5-note scales whose pattern can be found by playing only the black notes on a piano, any of which can be the tonic), the blues scale (there are different permutations, but all are derived from the form of the black-note pentatonic scale beginning on Eb, or La-Do-Re-Me-So-La), the Hirajoshi scale (another pentatonic scale, from Japan, but unlike the previous penatonic scale this one has two semitones (which means it is hemitonic): La-Ti-Do-Mi-Fa-La), or the octatonic scale (used in some Russian folk melodies and by some Russian composers such as Stravinsky and Scriabin, as well as by Bartok, and also used in jazz).
• You can make up your own scales and modes; Messiaen created scales with repeated patterns that he called "Modes of Limited Transposition," such as:
  
  Tone-Semitone-Semitone-Tone-Semitone-Semitone-Tone-Semitone-Semitone (which he called his third mode):
  
  
  or Semitone-Semitone-Minor Third-Semitone-Semitone-Semitone-Minor Third-Semitone (which he called his fourth mode):
  
• Messiaen's Modes of Limited Transposition are all based on repeating patterns within equal subdivisions of the octave, but in making up your own modes or scales, you do not need to be limited in this way. You could, for example, create a scale with a repeating pattern that spans a major sixth. After four such pattern repetitions, you would have spanned three octaves and the overall, three-octave, pattern would then repeat. But the pitch patterns in each of the three octaves would be slightly different. Such as this, for example:

• The following scale is a mirror around the pitch F#, but you could also create a scale with few or no pattern repetitions in it:

• Another approach, suggested by my friend and former colleague Dr. Scott Godin, is to construct a few (2-3) atonal chords that you wish to use as the basis for a composition, then construct a scale containing all or most of those notes. You can then use that scale to create additional harmonies if you wish.
• Once you choose or create some scales with which you want to work (play), you could make charts of the triads and "seventh" chords formed on each scale degree. However, you need not build these chords in the same way as is done with major and minor scales (in thirds); you could form chords based on unusual patterns, such as chord I comprising the 1^st, 2^nd, 4^th, and 5^th scale degrees; chord II comprising the 2^nd, 3^rd, 5^th, and 6^th scale degrees, etc. Remember that there are no rules here, so you can form chords in whatever way you like.
• All of these approaches create harmonic and melodic sound worlds that are distinct from major/minor tonality, but which can yield some fresh and attractive results.
• Consider this: "Tonality" refers to far more than mere scales and the chords formed from those scales; it mostly refers to the relationships between the notes and chords in the scales. When using alternative scales and modes, you may notice some relationships between notes and chords that seem "natural" to you, and you are of course free to use them, but bear in mind that often what seems "natural" in these scales and modes are the aspects that are most closely tied to tonality, such as dominant-tonic relationships or leading tones. It can be fruitful to explore note and chord relationships that are not similar to the more familiar aspects of tonality.
• Incidentally, the objective, when using something like a blues or Hirajoshi scale, is not necessarily to create blues music or Japanese music (although it obviously can be if you wish); it is to write compositions that may sound to the listener as though they are related the kinds of music from which the scales originated, but with your own unique spin on them. For example, my blues-scale waltz fragment above does not sound particularly bluesy, because the F#-F-F#-G in the 3^rd bar is not characteristic of blues music. More bluesy in that bar would have been F-Eb-F-G, or even Gb-F-Eb-G, because the F#/Gb in that scale is usually treated as an inflection of F or G.
• And finally, don't forget that the concepts of "non-harmonic tones" and "modulation" can be borrowed from tonality and applied to any music you compose using these alternative scales.
• Have fun!