Intrepid soprano Christie Finn has written a lovely blog post about her relationship to absolute pitch and the tuning fork, specifically in our work on the upcoming Quando Stanno Morendo. I can verify both the chopstick difficulties and the tuning fork bruise.
Learning difficult music
16
Apr 12
Nono in extremis
Preparing our upcoming performance of Nono’s “Quando stanno morendo” has been, as the piece is, a study in extremes. The work’s glacial tempi (down to quarter = 30) destroy the listener’s (and often the performer’s) sense of meter and time. If that weren’t enough, these tempos are coupled with a system of numbered fermatas wherein each digit indicates the duration of the note in quarter notes. This means even a sixteenth note quintuplet at quarter = 40, with a ’5′ fermata, would last more than 7 seconds. This dilation of time has the paradoxical effect of heightening the intensity of its passing. Placing a note on the third triplet eighth of a beat in quarter = 120 is a simple matter of feel. Placing the same note at quarter = 40 puts the whole process under a magnifying glass. Any discrepancy between performers is thrown into relief.
The singers dealing with these broad tempos are also grappling with extremely wide ranging parts, marked pppp and quieter. Aside from these technical considerations, there is a deep political and emotional background to the work, which was written during the onset of martial law in Poland. These extreme emotions of both the context in which the piece was created, and the texts it sets are balanced by a dynamic restraint. With the difficult registers and dynamics bordering on silence, Nono very literally evokes the stifled voices of political dissidents.
This is a piece that on one hand appears simple, often consisting a single line shared between voices, or chords moving in rhythmic unison. These voices however, may be projected through a 10 speaker array, rotating around the audience both clockwise every 10 seconds, and counterclockwise every 7 seconds with 2 seconds of delay. The cellist plays only open strings in one large section of the work. These however, are open strings on one of three cellos string with 4 identical strings and tuned to 4 near-unison tones, played with two bows simultaneously. These amazing juxtapositions of simplicity and complexity, of technical virtuosity and emotional directness, are what allowed Nono to go to such extremes and still create a balanced and coherent masterpiece.
28
Nov 11
Tuning – Vertical vs. Horizontal
I was talking with Sasha Zamler-Carhart, director of Ascoli Ensemble, about the ways that our ensembles approach tuning. Ekmeles’s approach to tuning Gesualdo in 31-note equal temperament is a mostly harmonically focused – 31ET being a keyboard temperament -and is about aiming for pure verticalities. Sasha’s group, specializing in Medieval music, and often reading from original manuscript parts, approaches tuning in an entirely linear sense. Of course they make sure they begin lines and cadence together, but reading from parts and singing in Pythagorean tuning – which is beautifully melodic, but only harmonically satisfying for major seconds fourths and fifths – has led them to consider tuning in this way.
Our most recent project, the premiere of Randy Gibson‘s “Circular Trance“, was an almost totally vertical experience. Scored for an array of sine wave drones in addition to the seven singers required, the piece’s complex just intonation tuning system requires us to constantly listen vertically, and to subsume our voices into the tuning of the drone. Perhaps the most linearly conceived work that I’ve ever performed is the first movement of Johannes Schöllhorn‘s “Madrigali a Dio”. The pitches for the singers are graphically specified on a 3 line staff representing the full compass of the voice, so that the pitches are only determined relatively within each voice, and are free to interact with the other voices at any interval, tempered or otherwise.
Despite these extreme examples I think I do my best work tuning diagonally, imagining both the melodic contour of my own individual part, and the way it will interact with the other parts as they go along. In reality, tuning with an ensemble of voices is a constant game of listening and subtle adjustments. Rules and approaches to tuning are a jumping-off point and a reference; but in practice, the voice is both a producer of an infinite continuum of pitch, and fallibly organic.
10
Oct 11
The new continuo?
Ekmeles is currently preparing a performance of several of Gesualdo’s madrigals, applying a tuning that is a combination of historical fact and conjecture – Vicentino’s 31-note division of the octave. There is a surfeit of forgotten theories of the tuning of musical instruments and performances, including many that were likely never used in performance. Nicola Vicentino (1511-1575) went a step further than many theorists, actually building and designing instruments capable of producing the scalar divisions he proposed mathematically. He devised the archiorgano, and the archicembalo, respectively an organ and a harpsichord capable of playing 31 (roughly) equal divisions of the octave, allowing free modulation through the keys in a mean-tone tuning, and application of the ancient Greek enharmonic genus. Scipione Stella, a composer at Gesualdo’s court, made a copy of the archicembalo – thus our historical conjecture.
Vicentino himself was a madrigalist, though it is recorded that his enharmonic vocal works were never performed without the harmonic support of a player at the archicembalo. This idea of needing continuo in the context of difficult intonation reminded me of the place singers of contemporary music often find ourselves – ears attached to computer synthesized tracks of our pitches. As readers of the blog will know, I am an advocate for making use of all technological tools possible in the course of learning difficult music. What I am interested in exploring is performing with these computer crutches. Of course, in some cases (like Martin Iddon‘s commission for Ekmeles, Hamadryads, or Aaron Cassidy‘s I, purples, spat blood, laugh of beautiful lips) the in-ear pitch component of the piece is a considered and integral part of the piece.
But what about when the composer hasn’t asked that a pitch track be used, and precise intonation is just too difficult, whether because of short rehearsal time, vocal considerations, or extremely small divisions of the octave? Is performing with a pitch track in our ears just cheating or is it the new continuo? Is the vitality and authenticity of a performance threatened by adherence to a mechanical version of the work which, by literally blocking the ears, supersedes the natural interaction of the performers? Thanks are due to a 16th-century Italian composer for raising these very modern questions – but more importantly, what do you think?
31
May 11
Irrational meters
I was going to write a blog post about what are called “irrational meters” (time signatures with denominators that are not powers of 2) – then I re-read a fantastic post by Helen Bledsoe, flutist for MusikFabrik (among others), and realized I should just link to her! She very lucidly explains the mathematical workings of Ferneyhough-style rhythm, complete with “irrational meters”, which really pose very little additional challenge, if you know how to interpret them.
The most basic point to remember about these meters is that the denominator, just like in more familiar time signatures, indicates the number of notes it will take to fill a whole note. 4/8 indicates 4 of something it takes 8 of to fill a whole note (namely eighth notes). A denominator of 5 would indicate that quarter note quintuplets are the basic unit of the bar, and the numerator, as in familiar time signatures, indicates the number of units in the bar. Thus, 3/5 would be a bar of 3 quarter note quintuplets! Of course, you can, instead, treat these changes of denominator as metric modulations and tempo changes – but I’ll leave some of the specifics to Ms. Bledsoe’s lucid explanations!
Head on over to her blog, Flutin’ High, for the full post!
23
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 twelveth 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 13ET, 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.
11
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).
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.
21
Mar 11
Apparitions
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.
14
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.”
10
Jan 11
Tuplets and polyrhythms
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.
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
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.