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In my previous post on the so called “overtone scale,” I questioned whether any scale defined in the context of 12-tone equal temperament can be said to mimic the harmonic series.  I provided an audio example that shows how accurately tuned harmonic partials seem to fuse into a single tone, whereas when those partials are altered to match equal temperament, the composite sound is rough and unstable.  My point was that the special perceptual properties of the harmonic series depend on accurate intonation, an appropriate pattern of amplitude decay, and appropriate registration; we should be cautious about assuming that any scale constructed in equal temperament will somehow inherit the special properties of the harmonic series by virtue of an incomplete resemblance to it.  My aim in the post was not to call into question the musical worth of any particular scale (and certainly not to address George Russell’s Lydian Chromatic Concept as a whole), but only to point out that the Lydian Dominant scale shouldn’t be called an “overtone scale” when played on the piano or any tempered instrument: that name is misleading.

In this post I’d like to share another simple audio example that might help readers form their own judgments on the matter.  If the scale 1, 2, 3, #4, 5, 6, b7 really does invoke the harmonic series in our minds, even when we hear it on a tempered instrument, how would that scale sound if its pitches were brought into exact alignment with the harmonic series?  Would that tuning bring us closer to the essence of the scale, or would it conflict with what we want to hear?

In the first audio example below, you will hear the Lydian Dominant scale rendered on an equal-tempered organ.  The ascending and descending scale is followed by a short tune that I wrote in the scale.  In the second audio example, you’ll hear the Lydian Dominant scale tuned so its pitches match the harmonic series, followed again by the example tune.  The just intonation ratios used in the second clip are: 1/1, 9/8, 5/4, 11/8, 3/2, 13/8, 7/4.  You will hear that the second, third, and fifth degrees of scale sound similar to what you hear in equal-temperament, while the flat seven, the sharp four, and the natural six are quite different.  (The flat seven is tuned to the seventh partial, the sharp four is tuned to the eleventh partial, and the natural six is tuned to the thirteenth partial.)  Which tuning do you prefer?

 

Clip 1 — Lydian Dominant Scale in Equal Temperament:

 

Clip 2 — Lydian Dominant Scale in Harmonic Series Tuning:

 

In this clip I sing a Hindustani-style vocal alap in Rag Marwa, accompanying myself on tanpura.  Marwa is a difficult raga to sing as phrases often avoid sa (the home note) and seem to establish dha or re as an alternate tonic.  When the true sa is finally heard, the experience is more of a surprise than a comforting return home.  In the recording here I’m attempting to sing with the specific intonation I learned from my teacher.  The tanpura sometimes takes me an hour to really get in tune, as does the voice, and the raga should be sung at dusk, so it’s taken me many evenings of practice to get a usable recording. This one is far from perfect but it felt good to make and I hope you enjoy listening.

At my voice class yesterday I sang Whither Must I Wander? by R. Vaughan Williams for my teacher and some other students.  This was my first time running through it with the wonderful collaborative pianist J. Marzan and I learned some new things about the piece from our interaction.  I took a recording on my phone and thought I’d share the clip:

Home no more home to me, whither must I wander?
Hunger my driver, I go where I must.
Cold blows the winter wind over hill and heather:
Thick drives the rain and my roof is in the dust.
Loved of wise men was the shade of my roof-tree,
The true word of welcome was spoken in the door -
Dear days of old with the faces in the firelight,
Kind folks of old, you come again no more.

Home was home then, my dear, full of kindly faces,
Home was home then, my dear, happy for the child.
Fire and the windows bright glittered on the moorland;
Song, tuneful song, built a palace in the wild.
Now when day dawns on the brow of the moorland,
Lone stands the house, and the chimney-stone is cold.
Lone let it stand, now the friends are all departed,
The kind hearts, the true hearts, that loved the place of old.

Spring shall come, come again, calling up the moorfowl,
Spring shall bring the sun and rain, bring the bees and flowers;
Red shall the heather bloom over hill and valley,
Soft flow the stream through the even-flowing hours.
Fair the day shine as it shone on my childhood -
Fair shine the day on the house with open door;
Birds come and cry there and twitter in the chimney -
But I go for ever and come again no more.

R. L. Stevenson

Time and time again I find that technical challenges in music amount to the same basic problem: how to achieve independence between coupled parameters so that you can control one parameter without interference from the other.  For example, one “parameter” might be pitch, and another might be volume.  The question is: can you sing higher without getting louder?  Of course, there may be times when you want to do a crescendo on an ascending run, but what if a diminuendo on that rising line would sound better – can you sing high and soft at the same time?  The challenge is to break the coupling or correlation between higher pitch and higher volume so that you can explore different relationships between those attributes, all in search of the best expressive choice.

Such musical challenges remind me of the more general issue of prejudice and how to overcome it.  Without delving too deep into the psychology of prejudice, I think it’s fair to say that prejudice often involves a belief and a behavior that are mutually reinforcing.   For example, let’s say you’re a generally shy person and you think parties are annoying.  You have a prejudicial belief about parties (that they’re annoying because they’re always filled with loud people who gossip about trivialities) which supports a prejudicial behavior (you avoid parties and try to leave early whenever you’re forced to go to a party) which reinforces the belief (you never have the opportunity to enjoy a party so it’s easy to continue thinking they’re all bad).  In many cases the belief has some merit (many parties are annoying for the very reasons you’ve identified) and the behavior has some merit too (avoiding parties shields you from discomfort).  However, the belief is incomplete (some parties are actually filled with people you’d enjoy meeting) and the behavior has significant disadvantages (avoiding parties means you miss all the good ones).

An example of prejudice in music might be that you think high notes are hard to sing.  You believe they’re hard because you’ve often experienced tension and vocal cracking when you reach into your upper register.  And so you behave anxiously when it comes time to hit a high note, either shying away from it and under-supporting the note or else applying too much breath pressure and punching the note – in either case, the unpleasant outcome reinforces your belief that high notes are difficult.  Indeed there’s some merit to the belief (high notes do require a more coordinated technique than those in the comfortable middle of your range) and there’s some merit in the behavior too (you’re right to avoid a note that you really can’t sing, and when you do sing a high note you’re right to support it with strong breath).  But the belief is incomplete (high notes can in fact be easy when executed with a relaxed but precise technique) and the behavior is limiting (avoiding high notes prevents you from learning to sing them, whereas exaggerating them prevents you from singing them beautifully).  Your challenge as a musician is to break your prejudice about high notes – a system of mutually reinforcing beliefs and behaviors – that you may have developed through years of experience in handling this parameter in a certain way.

In some musical scenarios, the prejudice that impedes you may be of more psychological kind, where beliefs and assumptions play a strong role, as in the example we’ve just seen of a singer who’s anxious about high notes.  In other cases, the prejudice is less rooted in higher-order psychology and more in physicality – in your tendencies and habits regarding how you use body.  For example, in playing any instrument that requires two hands (like piano or guitar) you need to learn to control your hands separately –  to make each hand perform a complex motion that is coordinated with the other hand’s motion but still completely different from it.  One hand should not prejudice the other.   But even if there are no beliefs or fears in the way — even if you don’t hold the preconception that your left and right hands should always act similarly — you might still discover that you have a physical inclination to move your hands in dependent way, so that when one is playing loud or exerting a lot of pressure, the other hand tends to follow suit.  “Habit” or “predisposition” might be a better description of what you face here than “prejudice,” but the challenge to overcome is similar.  Can you make one hand play loud and the other play soft, and then have them switch dynamics?  Can you make one hand play a triplet rhythm while the other plays in duples?

If you want a systematic way to get good at playing a piece, try listing the most significant parameters in its performance, and then come up with small exercises or experiments in controlling those parameters independently.  Ask which of your own biases make the piece difficult to perform — which biases inhibit independence?  Consider whether those biases are rooted mainly in belief, or mainly in your physicality, or perhaps in both – and find ways to disrupt them.  Break unnecessary couplings!

Can you play more smoothly without getting slower?

Can you sing more softly and tenderly without getting breathy, or louder without getting shrill?

Can you keep a strict rhythm in one hand while playing a loose rhythm in the other?

Can you play guitar louder with the right hand without exerting more pressure on the fretboard with the left hand?

Can you sing passionately without gesturing with your hands?

Can you sing higher without looking upward?

Can you play one note very loud without also playing the next note loud?

Can you switch from playing one guitar string to another without a change in tone?

Can you compose two melodic lines that sound good when played simultaneously even as they maintain distinct personalities?

Please share your own examples.

This post is about the idea of using silent or imagined vocalizations as an aid in performing instrumental music – it’s about the idea of “singing in one’s mind” as one plays.

In vocal music, or even in instrumental transcriptions of vocal music, the performer can rely on both the meaning and the structure of the text as basis for interpreting the notes themselves.  But when there are no words at play – when the written music is a step removed from the voice and spoken language – the instrumentalist has more decisions to make, and perhaps a greater challenge in making the music “sing.”

There are some common ways a performer might rely on inner vocalization as an aid to interpretation, even when playing instrumental music with no associated text.  When working on the rhythmic structure of a piece, a musician might imagine rhythmic syllables like “1 e & a 2 e & a 3 e & a 4 e & a” or the “ta ke di mi” syllables used in Indian classical music.  When working on the pitch content of a piece, a musician might imagine solfège syllables like “do re mi fa so la ti do” or “sa re ga ma pa da ni sa.”  When focusing on the dramatic aspects of a piece, a musician might imagine that there is some story or plot behind it, although the “text” of that story might not be matched to the music note-for-note.  And when performing a piece on an instrument, a musician might imagine herself singing at the same time, possibly using nonce or scat syllables (“da ba da ba dee ba”) or a simple hum (“mm mmm mmm”).

What’s less common – in fact I haven’t found any explicit discussions of this approach – is to take a piece of instrumental music and actually create very specific imaginary lyrics for the music so that each note is matched to a specific word.  It’s this idea of singing a specific text in one’s mind that I’d like to focus on here: not just using scat syllables, and not just maintaining a general sense of some story that’s transpiring as the piece goes on, but actually setting the music to words in one’s mind and singing those words internally as one plays.

There are two ways to do this.  First, one can treat it as a serious literary exercise, where an effort is made to find lyrics that fit the mood and style of the music – lyrics that would sound good if the performer really did sing them aloud.  This is a fascinating and challenging project to undertake, and it deserves to be discussed separately – it leads into all the complexities and possibilities of lyric writing as an art form.

Here I’d like to focus on a second approach which is much simpler and easier to experiment with, but still valuable.  The idea is to pick a phrase which one treats as a “mantra” that gets repeated throughout the duration of the piece.  For example, if you’re playing or improvising a blues piece, your mantra could be the phrase “I’ve got the blues.”  If you’re going to play a line with a simple rhythmic structure – “da da da da” – you would say “I’ve got the blues” in your mind as you play the four notes of the line, with each note being matched to one word of the mantra.  If you want to play a line that’s twice the length – “da da da da da da da da” – you could simply say “I’ve got the blues” twice.  But what if you want to play a line with a different rhythmic structure, like “da dada da da”?  Here, you can vary the mantra using any of the linguistic operations that we typically use to embellish sentences.  For example, you could say “I’ve really got the blues” with the word “really” matched to the “dada” rhythm.  What if you want to play a line with five beats like “da da da da da”?   You could say “I’ve got the blue blues,” or “I’ve got the blues bad.”  The important points are that 1) you’re always repeating the mantra in your mind and matching its words to the notes you’re playing, and 2) you’re embellishing the mantra-sentence as necessary to accommodate rhythmic variations and complexities in the lines that you’re playing.

Since you’re not actually singing the mantra aloud, you can be quite free with how you embellish it and you don’t need stick to things that are tasteful or pleasant.  In performing a longer musical phrase you might come up with a mantra embellishment like “I really really really really really really really-really got the blue blues bad, I do.”  You’re the only one who’s going to hear it.

It’s nice if there’s some connection between the mantra and the music you’re playing (i.e. “I’ve got the blues” is a good mantra for actually performing the blues) but it’s not entirely necessary.  You could use the “I’ve got the blues” mantra while performing Bach, for example, or you could change the words to “I like your smile” or “I want some cheese” and many of the effects of using the mantra would remain the same.

So what are those effects?  Why bother using a mantra if it’s going to lead to repetitive and convoluted variations like “I really really really really really really really-really got the blue blues bad, I do”?  Wouldn’t one be better off just humming silently or using scat syllables like “Ba daba daba daba daba daba daba dabadaba ba da boo bop boo bop”?

In my own experiments I’ve found there’s something very powerful about mentally pronouncing an actual sentence as one plays, as opposed to simply imagining scat syllables that are not constrained by any kind of linguistic grammar.  No matter how ugly the mantra becomes as a sentence through expansion and embellishment, it’s still a sentence, while a sequence of scat syllables is not.  For me the value of using a mantra is that it connects the part of my mind that “knows” how to form and interpret sentences with the part of my mind that knows how to play my instrument and manipulate pitches.  Having an actual sentence in mind as I play heightens my sense of music as a kind of speech, even if I’m not focusing on the meaning of the sentence but only exploiting its grammatical structure, the fact that it has a beginning, middle, and end, that it has clauses, that it has nouns, verbs, modifiers, and so on.

These are some specific advantages I’ve found to using a mantra:

  • *When I’m playing a pre-composed piece, the mantra forces me to pay attention and prevents me from “zoning out” even if I’ve rehearsed the piece hundreds of times, because I need to constantly take care to align the mantra text with the notes I’m playing.  I tend not to write out the mantra text beforehand.  I keep the text-to-music association as something that I create on the fly.  That text-to-music association (the way I embellish the mantra to make it fit the music) comes out a bit differently each time, keeping the performance fresh.
  • When I’m improvising, the mantra sometimes gives me rhythmic ideas, in that I find myself playing the rhythm created by a certain linguistic embellishment of the mantra-sentence.
  • When I’m improvising, the mantra encourages me to build phrases with discernible arcs, and to take “breaths” between phrases, in the same way the mantra-sentence has its own “shape” and I would take breaths between sentences if actually speaking.  The mantra reminds me that phrases need to end, like sentences do.
  • The mantra gives me ideas ideas about articulation, in that I might imagine saying a word from the mantra in a particular way and then apply that same gesture to the note I’m playing.  For example if I imagine pronouncing the word “blues” with a sigh, I might then look for a way to create a sighing effect in the note or phrase that I’m playing at the same time.
  • When I’m improvising, the mantra creates a kind of linkage between successive phrases: I will imagine myself pronouncing the same mantra-sentence in different ways, and the musical phrases that I’m playing then come out sounding like contrasting variations on a theme.
  • If I’m using a mantra that connects with the music on a semantic level, the mantra helps me stay focused on the feeling I’m trying to express.

Of course, there’s no reason why one should need to stick with the same mantra for an entire piece; you might experiment with a different mantra for each section, and you can experiment by different mantras in the same section and seeing how the choice of mantra influences your interpretation.

This is material that I’ve been dabbling with for a couple of years, and I’m writing this post partially as a note-to-self to remind me to keep exploring it, and as an invitation to others to try it out and share their experiences with it.  I hope it gives you some new ideas to work with; if you already practice something like this, please tell me about it!

This is my contribution to the First Climate Message Video Festival, an online event organized by vocalist, composer, and teacher Warren Senders.  The idea is to record yourself making a little music, then stop and say a short message urging the listener to learn about climate change, and then continue with the music.  If you’re involved in music you can participate too.  See the Facebook Event for the video festival and the project website.

The Tonnetz is a graphical depiction of chord relationships that was developed by 19th-century music theorists building on work by the mathematician Leonhard Euler. If you pick any major or minor triad, you can use the Tonnetz to quickly find all of the triads that share one or two of its tones. If you’re building a chord progression, you can use the Tonnetz to find extended sequences of chords that are connected by common tones.

When I first came across the Tonnetz I found it amazingly elegant but also a bit mysterious: “Yeah, it works,” I thought, “But how on earth did anyone come up with this?” In this post I’d like to share a step-by-step process for building a Tonnetz. This is not necessarily the fastest way to construct the diagram, and I’m neither claiming that this approach is original nor that it matches the process that was used historically. This is just a way that made sense to me in my own study and I hope it might help the reader gain insight into how the Tonnetz works.

Step 1.  We’ll start by drawing a simple grid:

tonnetz1Step 2.  Next we’ll turn our grid into a representation of musical space by placing a note at each grid intersection point.  If we were building a so-called just-intonation lattice, we would represent notes by their precise frequency ratios, but here we are going to use scale degrees from an equal-tempered octave: 1, ♭2, 2, ♭3, 3, 4, ♯4, 5, ♭6, 6, ♭7, 7.  We’ll lay the notes out so that stepping right in the grid corresponds to ascending by a perfect fifth and stepping up corresponds to ascending by a major third.  So, if we put 1 at an arbitrary intersection, we must put 5 to its right, and 3 above it.  Now let’s look at 5 — what should its neighbors be?  Following the rules, we must put 2 to its right and 7 above it.  Continuing like this we fill up the whole grid.  Each horizontal line (reading from left to right) is a sequence of ascending fifths and each vertical line (reading from bottom to top) is a sequence of ascending major thirds:

tonnetz2

Step 3. Next we’ll add diagonal lines to our grid.  These lines represent the minor third / major sixth relationship.  Starting at 1, for example, and taking a diagonal hop down and to the right, we end up at ♭3, a minor third higher in musical space.  Going in the other direction graphically (up and to the left) we land at 6, a minor third lower or a major sixth higher in musical space:

tonnetz3

Step 4.  Notice that each of the triangles formed by adding the diagonals is in fact a triad in the musical sense. Take a look at 1, for example: above it we have 3 and to its right we have 5.  All three points are now connected by an edge — together, these edges form a right triangle that we can take to represent the major triad rooted at 1. We can also find an upside-down right triangle in the points 1, ♭3, 5, representing the minor triad rooted at 1.  Let’s label all the chord-triangles in the diagram using Roman numeral notation: the 1, 3, 5 triangle gets the label I and the 1, ♭3, 5 triangle gets the label i:

tonnetz4

Step 5. Notice that if two triangles share an edge, the corresponding chords have two tones in common.  If two triangles share a single point (i.e. they meet at a single intersection in the grid), the corresponding chords have one tone in common.  In preparation to make these relationships more explicit, we’ll remove the black grid lines, leaving only the chord-triangles in place:

tonnetz5

Step 6.  Finally we’ll draw new lines connecting all of the chords that have two common tones.  This will create a background pattern of rectangular tiles.  We can see that any six triangles that share a rectangle represent six chords that share a common tone — the tone written in the center of the rectangle.  The diagram can be prettified by stretching the rectangular tiles into hexagonal ones, but we’ll leave that for next time.

tonnetz6

Here’s a bigger version:

tonnetz7

PS. The diagrams here have had limited proof-reading so far — if you catch a typo, please let me know.

I like musical exercises that are extremely simple but unexpectedly challenging.  One such exercise in the domain of rhythm is to set your metronome really slow — 10 or 20 beats per minute — and try to tap along.  We often associate difficulty with high speeds but high speeds present more of a motor challenge than a perceptual one.  In the case of very low speeds it’s difficult to tap accurately because it’s difficult to feel the beat as such — the very slow metronome clicks may seem like disconnected events instead of being part of a continuous rhythmic thread that we can follow.  (Well, that’s my own subjective account of the difficulty, but it turns out that tapping has been the focus of extensive research in psychology — see for example a paper by B. Repp titled Sensorimotor Synchronization: A review of the tapping literature.)

When you try to tap at 10 beats per minute you might find yourself playing a game of chicken with the metronome.  You don’t want to tap too early, so you wait, and wait some more, and then you suddenly hear the metronome click and you rush to tap.  If your reflexes are fast enough you can make it seem like you’re tapping in sync with the metronome but really what you’re doing is tapping in response to it, and each tap is ever so slightly late.  One way to break out of this pattern is with plain mental fortitude: try to make a clear decision to tap and then execute it completely without second guessing, but if you’re way too late and you hear the metronome click before you’ve even decided to tap, just skip that beat, don’t rush to tap in response.  Another way to break out of the pattern is to change the exercise so you aim to tap slightly before each beat instead of right on the beat.  This way you can’t use the sound of the metronome as a cue.  An advantage of practicing this way is that you’ll notice the brief duration between your tap and the following metronome click, and you can take this as feedback.  Try to make those durations as consistent as possible so that you’re always anticipating the beat by the same amount each time — easier said than done.

Of course you can greatly improve your accuracy by subdividing the beat in your mind, and actually counting to yourself “One and two and three and four…” as you might do in standard musical practice.  But that’s not the point of this exercise — the point is to see how close you can get to perceiving the slow metronome clicks as a fundamental beat without relying on a faster beat that you’re tracking internally.  So give it a shot without counting.

In experimenting with this ultra-slow rhythmic practice I’ve noticed that it can make an interesting sort of meditation.  Tapping is useful from a meditative standpoint since it requires attention, but it doesn’t really require thought — it forces you to stay present and not zone out.  For a meditative exercise that combines pitch and rhythm try this: turn on a drone (electric tanpura, etc.) and keep singing its pitch on a steady, continuous “aaah” vowel; meanwhile turn on a metronome at 10 BPS and tap to it as you sing.

If you’re like me you’ll probably want to get the tapping right and you’ll experience a little bit of frustration every time you tap too early or too late.  Can you isolate that frustration, notice any ways that it might manifest physically, and learn to dissolve it, even as you keep making mistakes?  And when you get it right can you move on without becoming distracted?

There’s an idea floating around that the Lydian Dominant scale (1, 2, 3, #4, 5, 6, b7), defined in the context of twelve-tone equal temperament, is somehow special because it emulates a harmonic series. Sometimes the Lydian Dominant scale is called the “acoustic scale” or the “overtone scale.”  Those names strike me as misleading. The idea is that if we look at the harmonic series of C2, for example, we find something roughly similar to these notes from the piano: C2, C3, G3, C4, E4, G4, Bb4, C5, D5, E5, F#5, G5, A5. Arranging the distinct notes from that sequence into an octave starting at C gives us C, D, E, F#, G, A, Bb. This scale might have some wonderful properties, but can those properties really be explained by the scale’s resemblance to the harmonic series?

In fact it’s a stretch to say this scale is “based on the harmonic series” at all because the tuning discrepancies are so vast. The members of the harmonic series do not match the piano’s C, D, E, F#, G, A, Bb — you can’t play any portion of the harmonic series accurately on a conventionally tuned piano. The piano’s G and D are reasonably close to the third and ninth partials of C; E is far off from the fifth partial; Bb and F# are still further off from the seventh and eleventh partials; and A is even further off from the thirteenth partial (which is actually closer to Ab).

Alright then, even if we acknowledge the discrepancy between a true harmonic series and the notes we can access on the piano, might there still be something useful or interesting about approximating the harmonic series through our 1, 2, 3, #4, 5, 6, b7 scale — might the ear tolerate the discrepancies and still hear some special cohesion in those notes on the piano because of the way they mimic, if not exactly match the harmonic series?

One way to get at this question is to flip it around. We know that the ear does perceive a special cohesion in a set of simultaneous pitches arranged in an exact harmonic series with decaying amplitudes — we generally perceive this phenomenon as a single note. So what would happen if we were to take the harmonic series of C and adjust its pitches so that they aligned with the piano’s C, D, E, F#, G, A, Bb — would these “tempered harmonics” still seem to fuse into a single note?

Here is a sound composed of the first eleven partials of C2 (65.4 hz), where the amplitudes decay as 1/n where n is the number of the partial:

And here is a sound built from the equal-tempered pitches of the Lydian Dominant scale, arranged as an approximate harmonic series — C2, C3, G3, C4, E4, G4, Bb4, C5, D5, E5, F#5 — with the same decay pattern as before.  In other words, this how our C2 sounds if all its partials are tuned to be playable on a piano:

If your perceptions are at all like mine, you’ll hear the first example as a clearly defined steady pitch. The second example is less steady (there’s some wobbling or beating) and if you listen closely you can begin picking out individual components of the sound which don’t quite fuse into a seamless whole — the texture is “messy” and/or “chaotic.”

What can we conclude from this? I think it’s safe to say that the special perceptual properties of the harmonic series start breaking down when the harmonics are mistuned. If the mistuning is carried so far as to bring the harmonics in line with pitches of twelve-tone equal temperament, the difference in sound is quite drastic: the components cease to fuse.  There may be great creative value in taking inspiration from the harmonic series and trying to build structures in twelve-tone equal temperament that mimic it, but if the result sounds good, we shouldn’t be too quick to assume the goodness comes directly from some resemblance to the harmonic series; any such resemblance is limited by a very significant difference in tuning.

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