Learning difficult music

May 11

Equal Temperament

Equal temperament or ET is the current tuning framework for most Western music. It is a kind of acoustical compromise, compared with the pure mathematical relationships of just intonation (JI). No intervals are ‘true’ in the system, but the equality of half-steps allows for free modulation to any key, ensuring that each would be as viable as any other. Pitches which in JI would be derived from the lowest primes are generally the best approximated pitches in ET: perfect fifths (3/2 in JI) are 2 cents low in ET, major thirds (5/4) are 14c+, and minor (7/4) sevenths are 31c+. In JI, all pitches are related intervalically to a fundamental (1/1); in ET, pitches are derived as equal logarithmic subdivisions of an interval, most usually the octave. (An interesting exception in this case is the Bohlen-Pierce system, which divides a perfect 12th into 13 equal steps) Thus, instead of the simple ratios involved in JI, the size of each ET halfstep is derived from the twelveth root of 2. Any ET division of the octave can be reached this way. For example, a 24 note scale’s smallest interval can be derived from the twenty-fourth root of 2.

The ET system used in most Western music is 12 note ET, also called 12ET. However, since the early twentieth century (and with a few notable exceptions, hundreds of years before), composers have worked in other equal subidvisions of the octave. 24ET introduces the quarter tone, 36 the sixth tone, 48 the eighth tone, and 72 the twelfth tone. These are the most common divisions, though there are many musics, composers and cultures who use different divisions (Klaus Huber, for example, in his later works, uses 18ET, creating an equal tempered scale of third tones). 19ET has been used as a better compromise for true JI intervals in tonal music than 12ET, differentiating between sharps and flats as differently tuned. 31ET is a system which was approximated by instrument makers and theorists in Italy in the 16th century via a kind of mean-tone temperament. It allows for diatonic (white note) chromatic (accidentals both sharp and flat) and enharmonic (double sharps and flats) genera, extending the range of possible harmonies greatly. Ekmeles will be experimenting in 31ET tuning in the performance of Gesualdo madrigals this Fall, as historical records indicate that Scipone Stella, a composer in Gesualdo’s court, built replicas of Vicentino’s 31ET keyboard instruments.

Non-exhaustive list of composers using ET microtones

Charles Ives (24ET), Alois Hába (24ET, 36ET, 72ET) Julián Carillo (18ET, 24ET, 30ET, 36ET, 42ET, 48ET, 54ET, 60ET, 66ET, 12ET, 78ET, 84ET, 90ET, 96ET [if you don’t know him, you should really check him out!]). James Dillon, Brian Ferneyhough, Liza Lim, and many other second modern or complexist composers make liberal use of ET microtones.

Learning ET microtones

Without the aid of rote learning, ET microtones can be exceptionally difficult to find. Acoustically, further divisions of 12ET rarely become more consonant, with the exception of 11th partial relationships which lie only a few cents away from a quarter tone. I reccommend the use of computer models, and have made use of several. I have occasionally used simple software synths for learning quarter tones. I reprogrammed a fine-tuning knob built into the synth to instead move only in gradations of 50c, and altered the pitches by hand on the fly. This is useful for melodic work, but makes harmonic hearing of quarter tones impossible. OpenMusic is an IRCAM-developed program made for computer assisted composition. A companion program, microplayer, can handle up to 72ET playback in multiple channels. To hear the score of an ET microtonal piece, I can’t just sit down and play it at the piano, so I enter it into OpenMusic, and can hear a completely accurate version of it, harmonically and melodically. When you have a limited amount of time to rehearse with an ensemble for a difficult piece, practicing with a computer model can allow you to devote that rehearsal time to music making, and not to panicking over whether you’re singing the right notes.

Apr 11

Just Intonation

Just Intonation is the tuning of pitches related by whole number ratios. The following will serve as a brief overview and introduction to the system’s theory and practice. The harmonic series is a good place to start when discussing Just Intonation (henceforth JI).

A harmonic series on low C

Partial numbers inside the staff, frequencies below

From a fundamental frequency (here the low C at ~65.4 Hz), the harmonic series ascends in multiples. If we refer to the partials of the tone, rather than the overtones, we can more easily see the math behind the present frequencies. Numbering the fundamental frequency as the 1st partial, the 2nd multiplies the frequency by 2, the 3rd partial by 3, ad infinitum. What does all this have to do with JI? JI deals in tuning intervals by simple whole number frequency ratios, and since we have demonstrated that the number of a partial is a multiplier of the fundamental frequency, we can use the harmonic series to find the intervals of these ratios. 2/1 is an octave, 3/2 the perfect fifth, 4/3 the perfect 4th, 5/4 the major third, 6/5 the minor third. What is significant about these intervals is that they deviate from the tempered intervals one finds on the modern piano. The simpler ratios sound beatless, and ‘pure’. If you are a choral singer or brass player, you probably are already used to finding this beatless sound by tuning wider fifths and lower thirds and sevenths in chords.

JI systems are sometimes referred to as “x-limit” systems, where x is some prime number. For example, a 5-limit system includes no prime numbers higher than 5 in any ratios, allowing for pure major thirds, but not true septimal or 7-limit consonances. 3-limit tuning is often referred to as Pythagorean tuning, and is composed entirely of just fifths. 5-limit tuning can approximate the major-minor system of Western music very well.

A few notable composers using JI

Americans: Ben Johnston, Harry Partch, LaMonte Young, James Tenney. Europeans: Gérard Grisey, Georg Friedrich Haas, György Ligeti. This is of course, no exhaustive list of composers, but a guide for the novice to what might be more familiar music, and an easier entry into the system. For a further discussion of the motivation to use JI, see Colin Holter’s fantastic paper “The Spiritual Construction of Tuning in American Experimental Music” at the Search Journal.

Notation of JI

The notation of JI varies depending on the composer and the circumstances. Ligeti often notated JI intervals simply by the fundamental on which the horn was to play (as in his last work, Hamburg Concerto). James Tenney sometimes notated JI intervals by writing cents deviation from ET. Perhaps the most exhaustive and most widely embraced system yet devised is Ben Johnston’s. Beginning from the assumption that the C Major scale is to be built from interlocking C F and G major triads, all tuned 6:5:4, Johnson introduces novel accidentals to shift these notes to different relationships. For example, in his C major scale, the supertonic D is ~4 cents higher than ET, while the A is ~16 cents low. If we are to use a properly tuned triad based on D, this requires an accidental to raise the A approximately 21 cents, Johnston’s +. Alternately, the D could be lowered, using -. The interval expressed by + or – is called the syntonic comma (81:80), and is the difference between a Pythagorean (or 3-limit) major third (81:64) and a 5:4 (5-limit) major third. Johnston’s system of accidentals continues similarly, with each successive accidental expressing a kind of fundamental shift related to a higher prime ratio.

Learning JI

Just Intonation tunings appear in traditional drone-based musics, like North-Indian Classical music, and can be easily practiced over a drone. If you play any string instrument, you can accompany yourself with a drone for practice, use a recording of a tambura, or even an electronic tone (preferably one rich in harmonics). Singing a just interval correctly feels ‘anchored’ in the fundamental tone and its harmonic spectrum. Aside from going by feel, if you are a string player of any kind you can also use the open harmonics of a string to learn simple just intervals. Being a computer enthusiast, I prefer to practice my microtones with the aid of software. Rote learning is the basis of the oral traditions that function within a JI framework, and is an indispensable tool. I use a program called Scordatura for the playback of JI microtones, and it is extremely flexible. Using CSE, a companion program to Scordatura, I can designate tunings wholly via ratio from any given fundamental at any tuning. It has made the task as simple as entering ratios in scalar order, and assigning them to keys on my midi keyboard. From this point on, I work by transcribing the pitch notation of the score to the re-mapped microtones of the midi keyboard (one octave of pitch extended over more than 4 octaves of keyboard (!) in my most recent JI undertaking, Johnston’s arrangement of Partch’s Barstow). I have a similar setup prepared for learning the tuning of Randy Gibson‘s upcoming work for ekmeles.

Of course, this is but a simplified overview – there are manifold internet resources for learning more about JI theory and practice. I’m happy to address questions or requests for elaboration on any of these points (and to accept corrections from those more deeply involved in JI than I). But if you’re interested, I would recommend as a first step getting your hands on a CD – or better yet, attending a concert – featuring JI music and hearing the difference a few cents here or there can make! Ben Johnston’s fourth String Quartet is a melodic and beautifully lucid introduction to JI.

Mar 11


Minimalist composer Randy Gibson, curator of the Avant Music Festival, is currently working on a piece for Ekmeles. I’m just starting to explore the sound-world he’s developed, and wanted to share some of the unique aspects of the tuning systems.

Randy is working with what he calls “pillars” of just intonation intervals to build scalar tunings. Just intonation, for the uninitiated, is a tuning system which involves small whole number ratios. These were originally developed by the ancients, and the ratios corresponded to string lengths of simple monochord instruments. Pythagoras is credited with the discovery of these ratio tunings, the simplest of which are derived from the lowest primes. Taking 1 (or 1/1) as the tuning frequency, multiplying by 2 will yield the octave, 3 (expressed as 3/2 to put it within the range of the octave) the pure fifth, 5 (again, 5/4 to drop it into the proper octave) will yield the just major third. You can also think of these intervals as being derived from the overtone series; the 2nd partial is the octave, the third partial the fifth, the fifth partial the major third etc. These perfect intervals were, for various reasons, gradually left behind for tempered tuning systems, which eventually led to the 12-note equal temperament we find on modern keyboard instruments.

The tuning of the piece for Ekmeles focuses on the prime number 7, a ratio favored by Randy’s teacher and mentor, La Monte Young. Most of Young’s seminal work The Well Tuned Piano is in septimal tunings, and the use of 7 is prominent in his work generally. Most simply, the seventh scale degree in this upcoming work will be tuned as 7/4. This is the natural, lowered minor seventh we encounter in natural brass instruments. The majority of the rest of the scale is constructed by building a “pillar” of 7/4 ratios on top of this first 7/4 ratio, yielding, in descending scalar order, 49/32, 343/256, and 2401/2048. Note the powers of seven as numerators (the denominators again function to move these ratios down within the octave range). Much like the simpler ratios noted above, the numerator of the properly reduced fraction also represents the partial to which the note corresponds. This pillar also means that the interval between 7/4 and 2/1, the septimal second, is also repeated between 49/32 and 7/4, 343/256 and 49/32, and 2401/2048 and 343/256. Randy fills out the scale by including several lower primes, 9/8 (a true major second above the tonic) and 3/2 (the perfect fifth, which lies between the 2401/2048 and 343/256 in the scale).

Intervals constructed from lower primes are easier to hear, but Randy’s scale is actually very singable! He’s provided the group with a sine wave drone, which includes most of the pitches from the scale in various registers, so much like Indian classical music, properly tuning these notes is a matter of resonating with the drone. The experience of singing just-tuned music is a physical one, in a way that is difficult to describe. Each interval takes on a unique character borne of its ratio, and the way the waves interfere. The piece is in its earliest stages now, and I look forward to spending the time internalizing this unique tuning system.

Feb 11

Complexity and failure

“How joyous the notion that, try as we may, we cannot do other than fail and fail absolutely and that the task will remain always before us, like a meaning for our lives.” – Donald Barthelme ‘Nothing: A Preliminary Account’

The physical manifestation of the fear of failure is surely an element in the performative practice of complex music. Concentration on many levels of notation flying by at breakneck speed almost guarantees a slipup here and there. Is learning and performing a complex score really then a Sisyphean task? I would like to argue that complex scores require a redefinition of, or at least a reconsideration of the idea of failure in performance. A certain kind of failure is a necessary and aesthetically important part of all performance. The details of just how we fail and why are worth examining.

Firstly I would propose that the moral character of the performer is engaged by complex scores. Acknowledging the inevitability of failure does not absolve the performer from the responsibility to realize the score as faithfully as possible, using all the tools at his disposal. Likewise, if the composer intended for his work to be realized graphically or approached from a quasi-improvised angle, one would hope he would be clever enough to use notation which suggests as much, rather than the extremely detailed instructions we find in many complex scores.

The information density of complex scores represents in some ways not an evolution and continuation of notational practice, but a break which implies important performative consequences. The performer is transformed from an imperfect conduit for the composer’s ideal vision, into an integral and indispensible contributor to the work.

I recently read an article in Perspectives of New Music entitled “Re-Complexifying the Function(s) of Notation in the Music of Brian Ferneyhough and the ‘New Complexity,’” written by Stuart Paul Duncan. My writing here is in many ways a response to that article. It got me thinking about what the article refers to as the “High-Modernist” model of musical performance, and the break from that tradition that many complex scores represent. Some people take the progressive-historicist view that music notation, beginning at first as mere mnemonics, moving to a two line, four line, five line staff, specifying instrumentation, dynamics etc., moves along a straight line towards its goal, which is the ultimate specification of all parametric information. Certainly scores have become more and more prescriptive as time has gone on, but, as Duncan argues, not always for the same reasons. The “High-Modernist” model of notation and performance might be defined as a direct and absolutely prescriptive communication from the composer to the performer. The score is the work in an ideal form, towards which each performer strives; what is desired above all is accuracy and fidelity. With the increasing density of information communicated by complex scores, it’s easy to see how this kind of accuracy becomes more and more difficult, or even beyond the capacity of the performer, if not the instrument itself.

What to make then, of this extreme density of information? After all, if the composer is after such a complicated web of musical ideas, he could just as easily program it into a computer, rather than spend the time communicating it to us fallible human performers. Paradoxically, the completely prescriptive nature of complex scores involves the performers in a deeper way than a more traditional “High-Modernist” score might. In addition to the hours and hours of engagement with the score required by complexities of notation, rhythm, pitch, or other parametric information, which deepen the performer’s relationship to the work, the conflict between the various physical performative demands requires that the performer function as a kind of filter. If something really is “impossible” as written, whether because of context, or simply the performer’s human limitations, in some cases the performer simply must make a choice of what to do, and what not to do. Conflicting and competing physiological demands in the score can also create unstable and unpredictable sounds. Is this a kind of fakery that proves the illegitimacy of complex notation? I think not. Any human performance contains some artifact of that performer’s humanity; complex music highlights these artifacts, and elevates them, elevating thereby the performance to stand with the score as more of an equal. The late great Milton Babbitt wrote that his scores are indeed intended to be precisely realized and perceived by some ideal performer and ideal listener. In terms of information density, his scores are simpler than say, Ferneyhough’s, but certainly not simple. Does this mean that any imperfect performance (read: at a high enough resolution, absolutely every performance) of his music is a failure? Anyone who has had the joy to hear great interpreters perform his music knows that it most certainly is not. Surely, something about the involvement of actual human performers is necessary for the success of the musical enterprise, despite the fact that an objectively perfect performance which reflects in every way the notation is, depending on your view, either exceedingly rare, or totally impossible. A fascination with this failure and its subtleties is at the core of what we find appealing in any musical performance; the notational practice of complex scores only brings our attention to that failure’s inevitability.

I’ll leave the last word here to Cage: “Composing’s one thing, performing’s another, listening’s a third.”

Jan 11

Tuplets and polyrhythms

A section of John Cage's "Living Room Music" which features quintuplets

How do I do this accurately at a slow tempo?

5 for 4 over a full bar of 4/4 at quarter=60? How do you do anything but fake that? All it takes is a little math! Slow tuplets can be a real killer, and if you try to perform them like you would an eighth note triplet (probably by feel), you’ll be all over the place. The least common multiple is your friend! In the case of our example (from John Cage’s Living Room Music, if you’re playing along at home), that would be 5×4 = 20. This number is the number of even pulses the bar is divided into that can accomodate both the quintuplet and the normal quarter note pulses. We could also have come to this from another angle: if the rules and standards of notation are preserved (tuplets over 3 beats need to have dots. Very few composers follow this standard; Carter does), tuplet notes will subdivide just like normal ones. This means a quintuplet quarter note split into 4 will yield 4 quintuplet sixteenth notes, just like its non-tupleted cousin will split into 4 normal sixteenths. Each normal quarter note in the bar could be divided into 5 quintuplet sixteenths. Either way we get there, 5×4 or 4×5 results in 20 quintuplet sixteenths, four of which add up to a quintuplet quarter. Knowing this, we can re-notate the rhythm as follows.

The quintuplets renotated

First the pulse, subdivided by accents, then written with ties

Now instead of a mysterious 5 floating somewhere in the bar we have attacks in relationship to the quarter note pulse, easily realized with a facility in subdivision. The purely mechanical accuracy of this method is, in many cases, only a first step; ideally, a tuplet like this should be realized without the syncopation accents implied by the re-notation. When you know exactly where each note lands in relation to the regular pulse of the piece, you can perform the tuplet smoothly, knowing proper points of reference throughout.

Any interaction of pulses can be rationalized in this manner – a more complicated example yielding a similar method: 5 for 3 over 3/32 in Aaron Cassidy’s I, purples, spat blood, laugh of beautiful lips. We can use our knowledge of the standards of tuplet notation to find how the pulses interact. First, we can imagine each iteration of the the 5 over 6 to have dots, which would make the first note of the quintuplet 9 pulses long, and the second note 6 pulses long. This 15 part division of the bar is in fact the LCM, allowing us to easily place the 32nd note every 5 pulses (15/3) and the 32nd note of the quintuplet on the 10th iteration of this pulse, leaving us with attacks on iterations 1, 6, 10 and 11 of a 3/32 measure hypothetically subdivided by 15 128th note quintuplets!

The same method can be used in the abstract to learn a polyrhythm. What does 5 over 3 in general sound like? Find the LCM (15), write out the digits 0 – 14 on a piece of paper, and circle every multiple of 3 and put a square around every multiple of 5. Following the numbers as a regular pulse, tap both hands at 0, the left at circled numbers and the right at numbers with a square. Voila, 5 over 3, mathematically done! Practice feeling each number as the ‘main pulse’, and each as the ‘cross rhythm’.

Two ways of feeling a 5:3 polyrhythm

Two ways of feeling a 5:3 polyrhythm

Generally, feeling the higher prime as a pulse will be easier, as it will feel subdivided by a lower prime. Take the above example: the 5/4 measure requires accurate placement of only triplets, while the 3/4 measure would require quintuplet level accuracy. An ability to shift between the two feels can help with accurate performances of polyrhythms and tuplets.

Oct 10

Learning “I, purples, spat blood, laugh of beautiful lips”

Aaron Cassidy’s I, purples, spat blood, laugh of beautiful lips (henceforth I, purples) intrigued me from the second I saw the notation. As a man who sings contemporary music, my repertoire is limited compared to the high soprano’s. An especial gap in repertoire was in the category of complexist scores – excepting Ferneyhough’s Shadowtime, which is a less-than-practical undertaking to put together. (A lovely exception is Liza Lim’s Chang-O, which was a delight to give the US Premiere of.) So I was very pleased to see in Cassidy’s piece a new kind of notation for the voice, as part of a coherent and new musical idea. It would be a few months before I would really get working on the score – nothing like a deadline to make you dig in! As the most complex score I had yet worked with, I, purples demanded a kind of rigorous procedural approach in learning. There was to be no hacking through the score at the piano – for one, there aren’t any notes on the page!

Through years of working on difficult scores I’ve tried many different approaches, but one aspect of my method is always consistent: decoupling aspects of the score for the purposes of learning. This idea is especially useful when the work itself, like many pieces by Ferneyhough and composers of the “Second Modernity,” decouples aspects of traditional performance techniques. Below, I will outline the general sketch of the procedure that I have found most useful, followed by the specific work I did in learning I, purples.

  1. Deal with the text. If it’s in a foreign language, translate it and write out the IPA, if you need to. Be able to fluently speak it.
  2. Speak the text in the notated rhythms.
  3. Speak the text with notated rhythm and dynamics.
  4. Speak the text with notated rhythm, dynamics, and articulations.
  5. Learn the notes out of time
  6. Put it all together

At each step, it is often useful to prioritize the new information. For example, step 3 could also include practicing just the text and dynamics, adding in the rhythm later; step 4 could involve speaking the text in rhythm with articulations, ignoring the dynamics, etc. What follows is a more specific outline of my method for learning I, purples.

  1. Text – translation was not an issue since the piece contains all the translations I need. The cross-cut texts made it difficult to see what word (and even language) a phoneme belonged to, so I wrote the IPA into the score. I then practiced speaking/intoning the text in a rough semblance of the the notated proportions.
  2. Rhythm – I looked for polyrhythms that were solvable by a least common multiple method (something I’ll write about in an upcoming post on rhythm), and learned the general feel of those. (The first bar, for example, is a 5:3 polyrhythm.) This way I could practice pure ratio relationships, and know where things fall at a reasonable speed before they become too fast to be comprehensible. I then programmed the whole thing into Sibelius to learn by rote, section by section, beginning as I am wont to do with the end, assuming the beginning gets extra attention all the time anyways. Starting at less than half tempo, I gradually worked my way up to 75% of tempo in each section.
  3. Dynamics – I then removed the difficulty of the rhythmic aspect of the work to focus on the dynamic contour paired with the text and a rough approximation of the rhythmic structure. Knowing what syllables are louder, and the shape of each word or phonetic cluster dynamically helped to define the gesture. Later in my work on dynamics, I continued to practice with the rhythm track from the computer, gradually moving from slower tempos upward.
  4. Articulations – I had been gradually adding in the contour of the gesture-defined notes, and now added those to an otherwise steadily intoned pitch. I added accents, staccatti, mordents, breathy and scratchy voice et al. Always, as before starting slow and speeding up gradually, occasionally stopping to take the gesture out of time to define its shape and characteristics more specifically, still working with the computer rhythm track.
  5. Notes (In this case, the patch) – The notes for I, purples come via an earbud and a Max/MSP patch which generates a randomly changing glissando. The first time I practiced with the patch I thought I was going to have to cancel the performance, or at least fake it. By the end of the day I was much more confident. It’s an odd skill of listening, but I found it helpful to start off define the ranges of the voice rather small, and practice at a low tempo, and expand from there. A general lesson emerges at this point: simplify and reduce the principle demanded of you as far as you can without corrupting the gestalt, then move step by step towards the goal.
  6. Back to basics – for a few checkups, I went back to my computer track of the rhythm and just spoke along to ensure I hadn’t slipped too far away from accuracy in my internalizing of the gesture.

Add to the density of the page the fact that you have to be reacting live to a different pitch glissando each time, and you have to be very close to memorized. I will experiment in the future with mnemonic page markings such as color-coded dynamics, but usually find that adding more information to a page like this would be distracting. On the other hand, a simple device like a colored highlight could engender an automatic reaction during performance, ensuring recognition of a certain parameter that might otherwise go unperformed.