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:


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:


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:


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:


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.


Here’s a bigger version:


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” and 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. Now this scale might have some wonderful properties, but do any of these properties specifically derive from a connection 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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This diagram shows the fretboard layout for a 6-string guitar in All Major Thirds tuning, assuming the open strings are tuned to E, G#, C, E, G#, C as recommended by Ralph Patt.  I made the diagram because I’m beginning to learn this nonstandard tuning and I wanted a study aid that emphasized the amazing regularity of the system.

Notice that because the three open bass strings are tuned the same as the three open treble strings (modulo an octave), the entire pattern of notes among the bass strings is repeated among the trebles — the left and right halves of the diagram are identical.

Each of the four colors used in the diagram indicates one of four possible augmented triads (modulo inversion and enharmonic respelling).  Notice, for example, that two copies the F Augmented triad (F A C♯) occur along the first fret and are shown with a light green background.  At the fifth fret the same set of notes occurs in a different inversion — now the notes are ordered A C♯ F; again they are shown with a light green background.   Finally, at the ninth fret, the notes occur in the order C♯ F A.

The “fret dots” on the left are positioned according to most common inlay pattern for standard tuning.  For simplicity, notes are always spelled using sharps instead of flats, though of course all the notes in the diagram could be written in multiple ways.

Major 3rds Tuning For Guitar: Fretboard LayoutAn intriguing property of this layout is that any block of notes spanning three strings and four frets can be considered as a “tile” that repeats across the entire fretboard, in a way where the tiles don’t overlap and also don’t leave any gaps.  In the image directly below, I started by drawing a box around the notes across the bass strings at frets zero through three; next, I placed boxes around all other instances of that same pattern.  (The notes with a gray background aren’t actually on the fretboard, of course — I included them to make the pattern clear.)


This next diagram is similar to the previous one, except I outlined a different block of notes:


Other regular tunings like All Fourths also give rise to tiling patterns like the ones above, but in the case of All Fourths, non-overlapping tiles won’t form nice, simple rectangles, and it’s not possible to “fit” as many complete tiles on the fretboard.  Here’s one way of tiling a fretboard tuned to E, A, D, G, C, F:


A recent voice lesson gave me the chance to reflect on my path in music and to notice how a major stumbling block of the past now seems easier to manage.

My teacher asked me to stop thinking about how I was singing and instead “let it come naturally.” We were working on Gute Nacht, the opening piece from Schubert’s Winterreise cycle.

In the many years that I listened to Winterreise as a teenager and then a twenty-something (collecting well over a dozen recordings along the way), I believed I would never be able to sing. I had played guitar since age 15, but my great secret wish was to make music with my own voice. Fear kept me from pursuing it: I thought I’d never be able to hold a steady pitch or make a pleasing tone.  But two years ago I finally signed up for voice lessons. These days it’s startling to realize that I’m singing the music I had once admired from afar, and it’s gratifying to see how a circumstance of self-doubt has turned into an opportunity to improve through practice.

My teacher read me a quote about running: according to research in sports psychology, the best runners don’t think at all while they’re running. They operate on auto-pilot.

Perhaps because my mind is so often abuzz, I sometimes receive this advice from teachers: “Don’t think so hard.” It’s good advice, but one faces a conundrum in applying it. Sure, the best athletes and musicians don’t need to think because they’ve practiced so long that good technique is now automatic, but what do you do before you reach that point? Sometimes you may have progressed further than you realize – all that’s needed to “cash in” on your practice is to step back, relinquish control, and let the good habits you’ve built now work for themselves. But there are other times when you try to “let go” and find that old habits come rushing back: without the oversight of your conscious mind, you regress. You can still benefit from taking a calmer approach, with less mental chatter, but you’re not quite ready for auto-pilot.

Of course, you don’t know what’s going to happen until you try. So, I took my teacher’s suggestion: I resolved to stop thinking about my singing and just enjoy the music. I let myself gesture freely and gave up my concerns about intonation and projection, jaw position and diction – all the things I had been studying in class. I was skeptical at first, but something magical happened within a few bars: I felt I had become a character in the play I had been watching all those years. Now I was that hapless wanderer, shivering as he departs his maiden’s house in that bleak snowy night. The music seemed to pour forth from me, and the dynamics fell into place: softer here, more forceful there – there was no need to consciously “interpret” the piece now that I was experiencing its drama first hand. The German text had become my own.

Fremd bin ich eingezogen,
Fremd zieh' ich wieder aus...

(A stranger I came,
and a stranger I depart...)

When the piece came to an end I felt I had said what I needed to say, nothing less, nothing more. The performance had been a transcendent moment for me – the reason I wanted to study music in the first place – and a terrifying moment too, as I had entered the psyche of Schubert and Müller’s frightful character.

There was silence as I came out of the “scene” and finally looked at my teacher. I was still swept up in the storm and cold of the piece and not quite ready to speak. She had been accompanying me on piano, and I thought she too might need a moment to rest after such intense music-making.

“That was nice…” she said. “It was nice… but… it might be time for you to take this piece to the next level… to make some more sensitive dramatic choices… to really start conveying the text. And also… I don’t want to nitpick, because the German was great overall, but there were a few places where the consonants got lost.”

My teacher is wonderfully encouraging, and she’s praised my Schubert before, so coming from her, this lukewarm response amounted to something like a C+.

The best way to vent my inner turmoil in that moment would have been to sing more Winterreise – but no, I thought, apparently my soulless Winterreise doesn’t convey any emotion so there would be no point in doing that!

While I experienced great emotional contrasts (from tenderness to rage) in the performance and thought I was communicating them, what actually came across to my teacher on this particular run-through was a narrow dramatic range, not enough variety between sections. Also, she thought my physicality could be more relaxed (less shifting back and forth and conducting with my hands) and I should take calmer breaths earlier, rather than gasping right before phrases began. And there were places where I could have rolled my German r’s with more vigor.

My teacher’s technical comments did not surprise me – these were precisely the things I had chosen not to worry about during my uninhibited performance, but I expected I’d still have to go back and work on them. The more confusing thing was that all the inner passion which I assumed must necessarily manifest in my singing just hadn’t come through. This was one of the most fervid moments I remembered having in the voice studio, and I felt I had taken a real risk in laying myself bare like that. For someone who’s usually reserved, these times of exposure don’t come often. How could my experience of performing the piece be so at odds with what my teacher perceived?

At one point in my life, a disconnect like this would have been more than confusing, it would have been crippling, sending me into a spiral of questioning and doubt. As an audience member – on the one hand – I’ve always reveled in the mystery of artistic expression. While we can analyze a performance and talk about its features, there’s no way to systematically predict what will move a listener, or how communication between artist and audience will unfold – and that keeps the game interesting. But when it comes to my own performances (whether in singing, playing guitar, speaking, or any other medium) I’ve always wanted there to be a clear causal relation between inner experience and external response. I want to know that what the audience hears will be somehow connected to what I feel, or at least that when I have a great inner moment, something rare and transcendent, when I think I’m at my very best, it won’t all turn out to be a fantasy! When that hope has failed, I’ve often become obsessed with trying to understand why. Where did the communication go wrong? Was I deluded, or was the listener in the wrong, or was something strange happening in the air between us?

In this particular context I began to wonder whether my teacher and I read the Schubert score differently — perhaps our interpretations were simply irreconcilable? Or maybe she was concentrating mainly on technical points as she listened? Or could it just have been that I was “off” without knowing it? But then how could it have felt so right? Endless questions sprung up, but I was able to walk away from them before too long, and that’s a choice that would have been difficult for me to make earlier in my life. It’s frustrating when there’s a disconnect and then… no, you don’t go and brood over it for hours… you go on and sing the next piece.

The way I look at things now is like this: as you perform, you might be moved by the music you’re making, or you might be unmoved, as if you’re executing the mechanics without true participation; likewise, the audience might be moved, or they might be unmoved. Of course, this is an extreme simplification of what’s possible. The important thing to realize is that all combinations of inner experience and external response can happen: you might be moved and the audience might be moved too – that’s great. Or you might be unmoved and the audience might be unmoved as well. That’s unfortunate, but at least it makes some kind of “sense.” In both cases, you and the audience appear to be in sync. But there are two other scenarios that make less intuitive sense and yet they happen all the time: you might be moved but the audience is unmoved, they just don’t “get it.” And on the other hand, you might be unmoved but the audience turns out to be deeply moved by what you’re doing – somehow! There’s really no way to know for certain, or to fully control, which combination will arise — the best you can do is influence it by practicing and trying your best every time.

And what do you know? In this particular class, we talked about a few other things I could work on, I thanked my teacher for the comments, and then went on to sing a couple of other pieces in a different vein, including Cole Porter’s So In Love and Donaudy’s O Del Mio Amato Ben, both of which she thought were spot on.

If you’ve followed my guitar posts here, you’ll know that I like the All Fourths tuning (E A D G C F) because it imposes a regularity on the fretboard that allows a player to shift chord and scale patterns across strings without fingering adjustments.  It’s also a comfortable tuning to explore if you’re familiar with standard tuning, since only the highest two strings are changed: you can reuse any chord or scale pattern that you’ve learned on the lower four strings, which are already tuned in fourths.

I had been cautiously avoiding other nonstandard tunings since switching to All Fourths around a year ago — I didn’t want to spread myself too thin.  But a blog visitor recently asked why I hadn’t considered All Major Thirds tuning here, and I couldn’t resist the invitation to experiment.  Now, after exploring M3 for a week, I’d like to share some initial observations.  (Alexandre, thanks again for the question that prompted this!)

One way to implement M3 is to keep the guitar’s lowest string at E and tune major thirds above that, giving E, G#, C, E, G#, C.  This is the 6-string setup recommended by Ralph Patt, who is considered to be the originator of M3 tuning.  Notice that while P4 tuning expands the guitar’s range by a semitone, M3 narrows it by a major third (the highest string drops from E down to C), which is why some M3 players prefer a 7-string guitar.  And while P4 only requires two strings to be retuned, M3 requires five retunings, which makes it a very different beast from standard tuning.

One of the first observations people make about M3 tuning is that you can play an entire 12-note chromatic scale in a span of four frets, with the same finger always playing the same fret, without shifting hand position.  Since any octave-repeating scale is a subset of the chromatic scale, this means you can play any scale you want without a position shift or stretch.  (If you’ve grown up adjusting to the shifts and stretches of standard tuning, it’s worth taking a moment to consider how remarkable this is.)

A nuance I haven’t seen emphasized elsewhere is that all this holds true regardless of which finger you use for the root note.  You can start the scale with your index finger or your pinky, and in each case there’s no need to move your hand or stretch beyond four frets.  The diagram below shows four different fingerings of the chromatic scale, corresponding to the four fingers you could use to play the root.  The numbers indicate which finger is used to play the notes at the corresponding fret — in the first example we play the root with the index finger (1), in the second example we play it with the middle finger (2), and so on.

M3ChromaticFingeringsJust as the chromatic scale can be played with any starting finger, without a position shift, so too can any scale be played with any starting finger, without a position shift.  What’s more, the four single-position fingerings of any given scale bear noticeable similarities to each other, as they are composed of the same building blocks — it’s easy to learn all four fingering patterns at the same time!  In the remainder of this post I’ll elaborate on this point with the major scale as an example.

In working out 7-note scales in M3 tuning I’ve found it useful to think of these scales as stacked tetrachords.  For our purposes a tetrachord is any sequence of four notes spanning a fourth (alternatively you could think of a tetrachord as a sequence of three intervals that add up to a fourth); here we’ll only look at tetrachords that span a perfect fourth but in a followup post we’ll consider tetrachords that span an augmented fourth.  The major scale is nicely regular in that it can be seen as two stacked copies of the same whole-whole-semi tetrachord pattern.  Starting at the root, say C, and traversing a whole tone, another whole tone, and finally a semitone gives us C, D, E, F.   Starting at the fifth, G, and applying the whole-whole-semi pattern again gives us G, A, B, C.  Put them together and you have the entire major scale: C, D, E, F, G, A, B, C.  The diagram below shows how the whole-whole-semi tetrachord pattern is fingered in M3 tuning, with all possible starting fingers.  Notice that the first two examples have the same shape though they employ the fingers differently.

M3WholeWholeSemiFingeringsNow we’re ready to finger the major scale itself, not just in one way but in four ways that are interrelated.  Since the major scale consists of two copies of the whole-whole-semi tetrachord, each single-position fingering of the major scale can be understood as a pairing of two of the whole-whole-semi fingerings we saw above.  Let’s say we want to play the major scale starting with our index finger on the root.  First we’d play the whole-whole-semi tetrachord pattern starting from the index finger (I’ve colored this pattern dark blue in the diagrams).  Notice how the pattern ends with the second finger playing the fourth degree of the scale.  Next we need to skip a whole step up to the fifth degree of the scale, which falls under the pinky.  Keeping the hand position fixed and applying the whole-whole-semi fingering starting from the pinky (I’ve colored this patten cyan) completes the scale.  What if we wanted to start playing the major scale with the pinky instead of the index finger on the root?  The reasoning is similar: first play the whole-whole-semi tetrachord pattern starting from the pinky; then, since the pattern ends on the first finger, skip a whole step up to the third finger and play the tetrachord patten that starts from the third finger (green). The diagram below shows all four fingerings of the major scale as combinations of the whole-whole-semi tetrachord fingerings from the previous diagram:

M3MajorFingeringsOf course it’s possible to conceive of scale fingerings in terms of tetrachords in other tuning systems, but M3 is the only system I know where it works so well — where you can mix and match tetrachord fingerings as we’ve seen to build scales that stay entirely within four frets.  In a followup post we’ll take a look at other tetrachord patterns (like semi-whole-whole, whole-semi-whole, etc.) and how they can be used in M3 to finger the natural minor scale, the melodic minor scale, and pretty much any scale you could imagine.

In my previous post on W. A. Mathieu’s idea of “virtual return,” we looked at a chord progression that drifts flat by a syntonic comma when rendered in just intonation. Here we’ll look at a shorter progression that drifts flat by an even greater amount.

Our present example is a sequence of major triads with their roots ascending in major thirds, with common tones sustained. Depending on whether we think of this sequence as a chromatic “expansion” of C major — starting and ending on the same chord — or as a progression from C to somewhere else in harmonic space, we could notate it as C – E – Ab – C or as C – E – G# – B#:

C-E-Ab-C C-E-Gsharp-Bsharp

In equal temperament with no intonational liberties, C and B# are played at the same pitch, and it’s only the musical context that determines whether the listener hears the progression as a “return” or a “departure.”  However, if we tune the passage in just intonation, so as to achieve pure major thirds and perfect fifths in each triad, with no pitch adjustment across ties, then all the pitches in the final chord will turn out flat of their counterparts in the opening chord by almost half a semitone — the diesis — and no matter whether the listener perceives the discrepancy, the sequence is technically not a return.

The diagram below shows how the progression is tuned so as to keep each triad just: we start at the bottom of the “ladder” and climb upwards.  Our opening C-E-G, at the bottom, is tuned at 1/1, 5/4, and 3/2.  The 5/4 E is sustained between measures 1 and 2, and if this E is to be used as the root of a just major triad in measure 2, we need to tune G# at 5/4×5/4=25/16 and B at 5/4×3/2=15/8 — a step upwards in the ladder.  G# is then sustained between measures 2 and 3, and hence it dictates the tuning of B# and D# in measure 3.  Finally, the sustained B# dictates the tuning of D## and F## in measure 4, the top of the ladder, where the pitches are flat of their opening counterparts by roughly 41 cents.


The following sound clips offer a few ways to examine what’s happening. There’s one clip using just intonation as described, and another using equal temperament.  In each clip, you will first hear the four bar progression played once. After a pause, you’ll then hear eight bars where the progression is stated twice, back to back, without any modifications: listen for the contrast (or lack thereof) between the end of the first statement and the beginning of the second statement. After another pause, you will hear a comparison passage that plays the closing chord, the opening chord, the closing chord, and then both at once. The entire set of examples is then repeated an octave lower, so you can see whether the range affects how you hear it.

Diesis Progression — Just Intonation:

Diesis Progression — Equal Temperament:


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