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Pedagogy of Harmony

The title of this post comes from a phrase coined by Stephen Downes in a Mastodon conversation where he said:

“This and the related discussion led me to think of a ‘pedagogy of harmony’ as my own perspective (as opposed to pedagogy of small, say, or pedagogy of slow – buy also, on reflection, as opposed to Friere’s pedagogy of the oppressed (and later pedagogy of hope’)).

What is a ‘pedagogy of harmony’? I’m not exactly sure, but it combines a feeling of well-being and comfort and inclusion.” (source link to the full thread here)

A day or two later, I received an email from Matthias Melcher suggesting perhaps the concept could perhaps be explained by melodic dissonances and maybe with an audible demonstration. This unexpected email sparked a firework of ideas in my mind. I’ll do my best to put a few of them in an understandable order here.

My frame has to do with painting different images of harmony, how we can practically understand it, and what it has to do with people and pedagogy/learning (that ‘p’ word is a good one, but laden with baggage). Humour me with explaining and dancing around a topic that is as big as history itself, well nearly.

Within music there are sound waves, and just like ripples in the pond, sometimes notes interact and literally (physically) interact. With any one pitch, there is an associated sine wave, and when notes interact, the waves aggregate. We can see this in some very fun explorations that are very based in science. The fun stuff has to do with making patterns in the waves’ interactions visible. This first one is just pure fun, and lets us see the physicality of sound waves.

This can be made more apparent when looking at Cymatics

The visual helps as a starter, especially for those who are not coming from a musical background. Each of the notes making a different pattern in these examples is part of the harmonic series, which is something that happens in nature.

In the middle of the 1800s Helmholtz wrote an excellent book On the Sensations of Tone in which he devised instruments to illustrate the principles of the interactions of notes and he analysed these in a great deal of scholarly depth. On p.14 he explains about the relationship of how notes when they occur simultaneously, as intervals:

“Long before anything was known of pitch numbers, or the means of counting them, Pythagoras discovered that if a string be divided into two parts by a bridge, in such a way as to give two consonant musical tones when struck, the length of these parts must be in the ratio of these whole numbers. If the bridge is so placed that 2/3 of the string lie to the right, and 1/3 on the left. so that the two lengths are in the ratio of 2:1, they produce the interval of the Octave, the greater length giving the deeper tone.”

In practical terms he is talking about would be heard, say, as two notes of C, one higher and one lower. The sine waves will directly fit into one another. Helmholtz goes on to explain:

“If the bridge is so placed that 3/5 of the string lie on the right and 2/5 on the left, the ratio of the two lengths is 3:2, and the interval is a Fifth.

These measurements had been executed with great precision by the Greek musicians, and had given rise to a system of tones, contrived by considerable art.” p.14

I cannot expect to explain everything about intervals, but allow for the idea that commonly harmony happens when there is interaction of more than one sound or pitch. Within the scale there are 8 notes: think of the Do a Deer song: do, re, me, fa, sol, la ti, do. In the simplest form, these can be imagined as the white notes on the piano. The important thing here is that there are some black notes between some of them, but not all of them. The result is not a symmetrical division, but it does allow for the creation of various intervalic ratios. For a video version of some of the scientific theory, see this 9 min vid by B. Crowell. (the video is longer, but the rest is references! It is ALL properly referenced, which is FANTASTIC.)

Making intervals:

We can begin to really understand the differences between intervals when we physically experience the intervals as sensation. Just like the geometric patterns, the most pure intervals create a certain feel that is consonance. It is smooth like glass. Try it starting with matching a note. Here are a few examples. In each there are short bits of playing for each interval. Here’s how they work:

  • I play the first note which is our reference. It is the tonic or key note. You join in singing that same note. I pause but you continue with the note.
  • while I glide to the second note. (You are still singing the first note) This has the effect that you can ‘feel’ when it settles on being in tune. It also means that you only need be able to match the first note and do not have to be musically trained at all to ‘get it’.
  • For people who would like a challenge, I then go back to playing the first note again and this time you can sing the second note (the note I previously moved to) to create the interval.

Here are examples for each of the intervals of the major scale, starting with an Octave:

Perfect 5th

Perfect 4th

Major 3rd

Major 6th

Major 7th

Major 2nd

With these intervals, there are gradations of dissonance, the unison and octave being the most pure, followed by the 4th and 5th (the reverse of one another, but historically very distinct and the 5th was definitely more pure), next are the 3rd and 6th which produce a velvety sort of dissonance, and then the 7th and 2nd which are distinctly crunchy- perhaps like grinding eggshells, and they are definitely the most challenging to do in terms of ‘holding your note’ and not correcting it and drifting onto the dissonant note.

It is an exercise you can also do with a friend, where you both sing. I like to sing and play the cello, and I can feel the interactions of the vibrations between me and the cello. If you like intervals, you can test yourself on this website, and there are further fun things via the home tab on that page.

Harmony is not exclusively built on the pure or consonant intervals, but allows for all sorts, with notes that may seem outside the dictated specification of ‘harmony’. When Metthias wrote to me, he asked what about ‘melodic dissonances’ like changing note, suspension, anticipation 
and passing tone (in German: Wechselnote, Vorhalt, Antizipation, and  Durchgang)

Here’s an explanation of the underlying concept of these ‘non-harmonic’ tones:

And in contrast, here is a piece without these colourful non-chord notes: Fugue in A Major by Shostakovich

It is interesting that both are possible. It is possible to have a completely harmonious piece, but- but, it has to be masterfully crafted to be interesting and beautiful. It would be akin to writing a story without the use of adjectives. It is possible, but the more advanced suggestion would have to be woven in such a way… For example, I strongly suspect that if one were to analyse the chord progressions in the Fugue above, there would be suggestions in the harmonic reduction of harmonic melodic motion (moving chord-by-chord) that implied and even explicitly used non-chord tones.

Pedagogy? (or is this all music theory?)

In teaching, in learning, in human interaction there are times when people gel, and somehow are easy to fit with. There are other times when aspects of the people, subject matter, juxtaposition of ideas is slightly more dissonant. In a pedagogy of harmony perhaps there can  be a place for those that instantly seem to fit and for notes that are out of place, and also a place for exploration. It makes me think of using musical forms that have some parameters defined and then the added variables are what take learning outside the pages of any textbook. Musically, examples could include Sonata Form, which is like an architectural framework. You can think of it like a house. We know all houses have certain things, like a kitchen and bathroom, and the walls are fixed, but the decor is up to the habitants (or composers). Then there are other examples like the Blues where the harmonic progression, the series of chords, is the defined factor and there is a stylistic expectation, but the instruments, the melodies, the content that is poured into and onto those chords is completely up for grabs.

Within learning, there are some underlying principles, just like the way the sine wave of the top note in an octave fits neatly into the wave of the lower note with the interval, or like the fixed parameters of various musical forms. Ideally there is engagement from the learners, at least two to make an octave, and more to make a chorus or symphony. Sometimes there is a conductor, and other times there may be antiphonal musicians or even whole ensembles that are dotted about the place and interact in a loosely or tightly organised fashion.

One of the important things with harmony is the perception of the listener. The way I hear music is different to how my great-great ancestors did, not because my ears have mutated, but because of the context in which we each live(d). I am used to dissonances as commonplace and 200 years ago there would not have been the same dissonances present – think of how we hear traffic noise and how differently someone pre-automobile would react to the sound of a city-jam. People had not conceived of using musical notes in the ways that are possible now. It is a bit like moving from realism to impressionism in art; they have a different palate of expression.

The relationships of the notes, the ratios and intervals found within the natural harmonic series have not changed over the years, but the capabilities of reproducing the notes on manmade instruments has, and the accuracy and discernment available to us through the various instruments has improved. (For example, before the addition of valves, trumpets could not reliably play all the chromatic notes in the scale, but modern instruments can do this with ease.)

What has also changed is our tolerance for adding new ideas to the conception of harmony. It does not have to be always only a triad, but can be something that includes every note – if the perspective and purpose is understood. Intentionality plays a part. and sometimes we need to allow for the strangeness of various combinations, that on paper may not look so categorically pleasing, with 7ths and 9ths (which are really 2nds that are extended by an octave) to allow for an overall picture that is harmonious. In this last example you can hear the delicious notes outside the simple triadic harmonies, played both in the piano, and in the voice- with the leaning, decorative, and expressive lines.


Stephen said:

I  think that learning is the process of adjusting our expectations to align with experience so we are not taken by surprise and thrown off balance by what comes next.

A pedagogy of harmony would involve bringing the learner into this picture, such that they understand why they experience disruption and disharmony in their lives, and how learning helps them to resolve that through predictions and actions.

When we interact, we are disturbances on the surface of one another’s waters, whether geometric or otherwise intersectional, and the way we make waves and receive the waves has to do with our allowances for a harmonious existence. The notes in the example above would be completely out of place and abrasive in a Mozart Symphony, but there is a mutual intention and support in the way they are constructed and delivered. What it does mean, this pedagogy of harmony, is that you cannot exist within it and be unaffected by the others in it. It is not a system of autocracy. It requires more than an acknowledgement that the teacher is (or was) a learner, it requires acceptance at a fundamental level that we are of the same stuff, and the impact of one can be as great as the impact of another.

That turns the tables of teaching.

When Stephen first posted his comment about ‘pedagogy of harmony’ it made me think of the patterns of sound, and how there is often more than we expect. I shared this video about the extra notes that we hear when two musical tones interact – The extra note, known as a Tartini Tone, isn’t physically there. I described it saying, ‘their disruptions weave into and impact our teaching – and sometimes something unexpected happens’.

My favourite Tartini tone is the sub-octave when playing a 10th on the cello. If you play the open C string (lowest string) and the E a tenth above it at the same time, a note an octave lower than the C string sounds, and it is lush. It is like a lovely resonant organ glowing from nowhere.

It is important for me that the pedagogy of harmony is not something bound by rules of counterpoint, or some other form of composition. As with anything physical there are parameters we cannot change and those bind us in some ways, but as for the possibilities – to say we know what they are is a limit I am not willing to impress. New things, new integrations, new sounds and sensations: new learning.

Featured image CC BY-SA by Andrew Gustar

3 thoughts on “Pedagogy of Harmony”

  1. adding links to the other posts/comments that have been shared as a response, ongoing discussion (but somehow pingbacks have not tracked here…)
    Matthias Melcher’s post 19 Dec, 2017
    Stephen Downes post in OLDaily: Expectations make the difference
    Jenny Mackness’ thoughts in her post Hope and harmony as pedagogies for 2018:

    and Stephen’s reply on G+:

    and Kevin’s compilation post:

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