Post-Tonal Harmony Ideas (1)

1. Post-Tonal Harmony

"Post-Tonal" Harmony refers to harmonic practices not based on tonality that emerged since the end of the nineteenth century.

Basically, this includes any variety of atonality, such as free (i.e., non-serial) atonality and serialism, but, at least in my definition, it could also include music based on Messiaen's modes of limited transposition (or other constructed modes), quartal and quintal harmony, bitonality (provided it does not sound like tonality with chord extensions), and even the use of chords borrowed from tonality, but not used in a tonal context. A longer, but by no means comprehensive, list can be found in an earlier blog post I wrote (A Sampling of Post-Tonal Techniques and Ideas for Composition), and there are many on-line sites with information on this topic.

2. Atonal Harmony

"Atonality" which can be thought of as a sub-genre of post-tonality, tends to be defined more narrowly. Here is the opening paragraph of Wikipedia's article on Atonality:

Atonality in its broadest sense is music that lacks a tonal center, or key. Atonality, in this sense, usually describes compositions written from about 1908 to the present day where a hierarchy of pitches focusing on a single, central tone is not used, and the notes of the chromatic scale function independently of one another (Kennedy 1994). More narrowly, the term atonality describes music that does not conform to the system of tonal hierarchies that characterized classical European music between the seventeenth and nineteenth centuries (Lansky, Perle, and Headlam 2001). "The repertory of atonal music is characterized by the occurrence of pitches in novel combinations, as well as by the occurrence of familiar pitch combinations in unfamiliar environments" (Forte 1977).

To be clear, my definition of post-tonality is considerably less restrictive than the opening sentences of the Wikipedia article above, which appear to preclude the possibility of pitch-centres in atonal music.

3. Pitch Centricity

The idea of "pitch centricity" – music that is based in some way on a pitch centre – is inherent to tonal and modal music, but many (click this link, and/or Google the term) argue that it is also relatively common in atonal/post-tonal music.

I agree with this, which is why I often encourage students to write some variety of post-tonal music with pitch centres, and to move between different pitch centres within a composition, borrowing from tonality the concept of departure from, and return to, a "home" pitch centre, using various "modulations" along the way. A fellow composer and long-time friend of mine, Omar Daniel (who teaches at Western University), once told me something along the lines of, "one of the biggest problems I see in student compositions is an unwillingness to modulate," by which he meant change pitch centre, not change key. I think.

4. Can Post-Tonal Music use Triadic structures from Tonality?

Quick answer: Yes, it is fine to use harmonies borrowed from tonality (e.g., major, minor, diminished, dominant sevenths, etc.) in post-tonal music, as long as they are removed from their hierarchical/functional context within tonality. Indeed, that is the main topic of today's blog, and if you want to skip ahead for examples of how this can be done, scroll down to #8 below.

If part of our definition of post-tonal harmony is "harmonic practices not based on tonality," it would be useful to understand what we mean by tonality.

5. Tonal Harmony

Tonality refers to a systematic approach to musical composition using major and minor scales, based on:

1. Hierarchical chord-progression practices involving chord functions (e.g. pre-dominant to dominant to tonic class; );
2. Relationships between notes, such as contextual attractions or tendencies (e.g., leading-tone resolution in dominant harmony (^7-^8));
3. Resolutions of perceived instabilities (e.g., chord 7ths, suspensions, and other non-chord tones).

6. Common Chord Progressions Found in Tonal Music; A Chord-Flow Chart

"Hierarchical chord-progression practices" in tonality refers most generally to the chords that establish a key, namely dominant – tonic harmony, and predominant – dominant – tonic harmony. This is the basis of the following chord-flow chart, as found in Tonal Harmony, by Stefan Kostka and Dorothy Payne (McGraw-Hill):

By the way, this is a diatonic version of the chart for major keys, but it is virtually identical in minor keys. Chromatic variants of the above chords usually function as their diatonic versions, so bIII functions as iii, bVII functions as vii°, bVI functions as vi, etc. Also, there are exceptions to this chart found in the music of many composers of tonal music; the chart is a pedagogical tool, meant to represent the chord-flow options that are usually found in tonal music.

7. Does this mean chord progressions that do not follow the above chart are post-tonal?

Not necessarily; V - IV - I is a relatively common pop and blues chord progression that is clearly tonal, and yet V to IV is not available in the chart, and there are other exceptions as well (another common one is bVII - IV - I).

The following progression, in which every chord after the third does not follow the above chart, is clearly tonal. It consists of a descending C-major scale with a first-inversion triad on every note. This is an example of "parallel-sixth chords," wherein passing sonorities are not considered to be functional; the underlying functional harmony would be I6 - V6 - I6:

8. Finally! Some Post-Tonal Options: Combining and Recontextualizing Chords to produce Post-Tonal Sonorities; You won't BELIEVE #3!

As stated previously, my definition of post-tonality is fairly open; harmonic practices that came after tonality and are not tonal can be considered to be post-tonal. This would include post-tonal music that combines triads (or seventh chords, or ninths, etc.) found in tonality in such as way as to produce sonorities that are clearly not tonal and are not used within a tonal context.

Here are some examples; play the audio file below each example to hear what they sound like:

1. This is based on the combined C major and F# major chords (i.e., two major chords whose roots are a tritone apart) found in Stravinsky's Petrushka (1911); I added a three-note figure with two additional pitches (Ab and D) at the end, because I like the sound:

2. This begins with a C7 chord, upon which four additional pitches based on a B° triad are added:

3. The next example starts with a D7 chord that becomes a D9 on the third beat of the bar; a G#m chord with a major seventh and major ninth is superimposed:

• Examples 2 and 3 above, which began with dominant seventh chords, could be used in a tonal context, if the dominants resolved to their expected tonics within tonal music. #2 could therefore resolve to an F chord, and #3 could resolve to a G chord. Try this yourself, if you can access a keyboard, to hear what this would sound like.

• Therefore, in order for the above examples to truly be post-tonal, they should not progress to any chords that could be interpreted as constituting a progression of functional harmony.   

 4. The next example uses quartal harmony, but, instead of stacking a series of perfect fourths on top of one another, which creates a pleasant-but-static quality, I stacked two perfect fourths, then went down by whole tone and stacked two more perfect fourths on that note, then repeated it a third time, finishing with three stacked fourths instead of two. The result is very different than just stacking fourths on top of one another until you run out of notes:

Try superimposing different chord combinations, notating any you like and/or find to be of potential use in your compositions. Feel free to borrow any of the examples above as well. You don't need to limit yourself to chords, either; you can start with a chord and then add to it different notes or scalar passages that happen to sound good, and help recontextualize the chord so that it no longer sounds like a traditional tonal sonority.

9. But Wait! There's More! Tonal Chords Progressing in a Non-Tonal Way

Another way to present tonal chords in a post-tonal context is to create progressions that consistently and deliberately do not follow the above chord-flow chart, and do so in a way that prevents any suggestion of a clear tonic chord and functional harmony. If you try this, you may find that it is a surprisingly difficult task to create a chord progression that doesn't sound "wrong" to your ears.

This may be due to the strongly tonal association each individual chord has, since each individual chord in such a progression is typically major or minor; when recognizable chord-types do not "behave" (i.e., progress) as we expect them to, it can be disconcerting. In the section 8 examples above, where different chords were superimposed, the resulting vertical structures were not traditional tonal chords, and thus created fewer expectations that they "ought" to progress in a tonal way.

– – – – –

Giant Steps is a John Coltrane jazz composition so seminal that its chord progression is known as the "Coltrane changes;" it is required learning for any gigging jazz musician. Although it is tonal, it uses some unexpected chord changes: BMaj7 to D7, GMaj7 to B♭, and E♭Maj7 to F#7; these are somewhat unusual progressions in tonal music, although they are common enough that there is a name for them: Each chord pair forms a chromatic-mediant relationship. Not only that, and this is probably what makes it sound so unusual, but the first chord of each of the chromatic-mediant pairs also forms a chromatic-mediant relationship with the first chord of the next pair, and the same is true for the second chord of each pair as well.

It also uses some very common progressions, notably, several V7-I tonicizations. However, each tonicized chord (G, E♭, and B) is a major third from the next one, which means that together, they outline an augmented triad; this is highly-unusual! It is usually played very quickly, which helps make the augmented triad of tonicized roots even more evident:

Again, Giant Steps is tonal, but you can explore the possibility of using tonal triadic structures (i.e., major, minor, diminished, etc. chords, possibly with chord extensions like 7ths, 9ths, etc.) in a post-tonal context by writing chord progressions that do not follow our chord-flow chart above, taking particular care to avoid any hint of ii - V - I progressions, which are used to establish keys in tonal music. As mentioned above, You may find this a challenging task, but if you do come up with any you'd be willing to share, please do so in the comments section!

Here's one attempt; some of it uses double-chromatic-mediant relationships (two triads a third apart with no notes in common), some uses chromatic-mediant relationships (two triads a third apart with one note in common), and there are some non-tertian root movements as well. To my ears, it succeeds in avoiding being tonal (at least in any obvious way), but does it succeed as a musically-useful chord progression?

10. And That's Not All!

While this last approach above can produce useful results, I find that a much more satisfying and rewarding approach is to write a progression of non-tonal harmonies, each of which would be the result of sitting at a piano and just trying different harmonic sonorities until you find one you like or consider to be useful, and repeating this until you have perhaps 12-16 chords. If you'd like learn more about this approach, it is described in greater detail in this blog post: Project 1: Writing an Atonal Theme and Variations. In my experience as a teacher, it has produced some of the best work I have heard from early-stage composition students.

One of the keys to growth and improvement as a composer is to be willing to try new things; I encourage you to experiment with these approaches and many others.

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!

Atonal — Even the word sounds unpleasant!

I find above all that the expression, 'atonal music,' is most unfortunate--it is on a par with calling flying 'the art of not falling,' or swimming 'the art of not drowning.' Arnold Schoenberg, Style and Idea, p.210

Do you have an unpleasant connotation with the word, "atonal?"

If you do, you're not alone; many people who have some understanding of what "tonality" means don't seem to feel very warm and fuzzy about the concept of atonality.

But, strictly speaking, all that is meant by the word is that the music in question is not based on tonality. It doesn't really tell us anything about what the music is based on.

A quick primer on tonality, from Wikipedia:

Tonality is a system of music in which specific hierarchical pitch relationships are based on a key "center" or tonic. The term tonalité originated with Alexandre-Étienne Choron (1810) and was borrowed by François-Joseph Fétis in 1840 (Reti, 1958; Simms 1975, 119; Judd, 1998; Dahlhaus 1990). Although Fétis used it as a general term for a system of musical organization and spoke of "types de tonalités" rather than a single system, today the term is most often used to refer to Major-Minor tonality (also called diatonic tonality, common practice tonality, or functional tonality), the system of musical organization of the common practice period, and of Western-influenced popular music throughout much of the world today. [Emphasis mine.]

The "common practice period" is generally understood to refer to the baroque, classical, and romantic periods of European music history, roughly spanning 1600 to 1900. So, using this definition, "atonal" could be applied to medieval plainchant, renaissance masses, most Debussy piano preludes, etc., or it could be referring to a work that involves hitting the keys of a piano with an aluminum (so as not to mar the surface of the bat) baseball bat with reckless abandon. It could be referring to "Le Marteau Sans Maitre" by Pierre Boulez, or to "L'histoire du Soldat" by Igor Stravinsky, or to minimalist works by Steve Reich, Philip Glass, and John Adams, or even parts of "Toontown Follies" by yours truly.

It could refer to music that is as deeply moving and beautiful as any music ever composed, or it could be applied to very harsh, disturbing music.

"Atonal" doesn't necessarily equate with "highly dissonant" any more than "tonal" automatically means "consonant." Dissonance and consonance are essential aspects of tonality, but they are essential aspects of much atonal music as well.

Explore the various sonorities that can be created by the scales and modes that you created for our first project of the semester. There is no ban on the use of major or minor triads; I am hoping that your scales will lead you to discover other sonorities that you like and feel can be used in your compositions, and if some of the sonorities happen to be major or minor, so be it! No problem!

But just try to use them in ways that go beyond their use in the context of tonality.