Mar 14


Ekmeles performs Karlheinz Stockhausen’s immortal masterpiece for vocal ensemble, preceded by a 6:45PM talk by director Jeffrey Gavett and Stockhausen scholar Paul V. Miller.

  • Karlheinz Stockhausen – Stimmung (1968)

Personnel for concert

This performance is made possible in part with public funds from the Manhattan Community Arts Fund, supported by the New York City Department of Cultural Affairs in partnership with the City Council and administered by Lower Manhattan Cultural Council.

Stimmung acknowledgment

Nov 13

JI Composers: Taylor Brook

This is the second in a series of conversations with composers who work in just intonation, and other microtonal systems. Questions from Ekmeles are bolded, the composer’s responses follow. This is a special edition, since we’re premiering Brook’s Motorman Sextet this Friday, November 15th, at the DiMenna Center.

Composer Taylor Brook

Composer Taylor Brook

Why not 12-note equal temperament?

I do sometimes use 12tet, however I consider 12tet as one possibility among many. Perhaps a better question for most composers would be: why use any temperament at all? I look to just intonation as a default tuning system because it is not a temperament, but the most direct way to think about intervals. Indeed, all temperaments are in relation to just intonation, a deformation of JI that serves an end that may have to do with the music or the construction of the instrument. With this in mind, if I choose to use a temperament then it should be a meaningful part of the composition. For example, in my new work for Ekmeles, Motorman Sextet, I employed pythagorean tuning in quotations of an organ prelude by Buxtehude in order to better invoke how the tuning of a Buxtehude piece would be realized by voices as well as create a division between the quotations of Buxtehude and my music.

Taylor Brook - Motorman Sextet score excerpt

Taylor Brook – Motorman Sextet score excerpt

Why the systems and pitches you use?

I tend to approach each new piece I write from as fundamental a place as possible. Most of my pieces begin with an idea that has nothing to do with pitch, and so pitch will then be controlled to serve some larger goal. Not until I have a clear idea of what I want to do with a piece, will I begin to think about how I will use pitch. This means that many of my pieces use very different systems or sometimes no system at all for pitch. With this said, I always think about pitch from the basis of just intonation. It’s important to remember that just intonation isn’t a system, just a way to measure the acoustic consonance of an interval, and so one must carve out a system from it as many composers do with 12tet.

What was your first encounter with microtones?

The first piece I heard with microtones was James Tenney’s Bridge and Flocking for two pianos tuned a quarter-tone apart. I was in high-school at the time and I was good friends with Tenney’s daughter and so was curious about his music and bought a CD of his works. I must say that I didn’t really understand or enjoy his music until years later and this encounter was not so meaningful to me other than bringing forward the idea that such a thing would be possible.

What piece of microtonal music that you didn’t write is most important to you?

I’m not sure about it being the most important, but the most meaningful microtonal piece for me is LaMonte Young’s The Well-Tuned Piano. During my undergraduate studies I spent a lot of time in the electronic music studio with another composer named Jacob Sudol. Jacob was doing his masters and gave me a lot of great recordings. One time he played The Well-Tuned Piano for my on the super-hi-fi speakers at the studio and it had a very strong effect on me. This piece is very direct and allows for enough time to deeply feel all of the pitch relationships as they are presented in different combinations and textures. It somehow invents and completes a genre of music all its own. Many composers have attempted to write pieces like this, but in my opinion, The Well-Tuned Piano is one of these truly unique works.

Nov 13

JI Composers: Christopher Trapani

This is the first in a series of conversations with composers who work in just intonation, and other microtonal systems. Questions from Ekmeles are bolded, the composer’s responses follow.

Why not 12-note equal temperament?

It comes down to this question, one I ask myself often: What exactly enchants me as a listener in the music I love? More often than not, it’s small expressive details that capture my attention, the fleeting surface gestures or stylistic inflections that make a performance distinctive. Replicating that kind of expressivity requires a full palette of pitch material; my works often run the gamut, mixing highly-detailed microtonality with tempered passages. Microtones are a vital component of almost every musical tradition: jazz, blues, country, electronic music, many folk styles, early music — almost everywhere except in the mainstream art music of the last few centuries.

Speaking more broadly, I’m captivated by the idea of consonance, and devoted to pushing the boundaries of what can be considered to “sound good.” Just intonation sparks my imagination because what appears to be a complex network of pitch relationships can be boiled down to multiples of whole numbers, simplest ratios that require extreme precision in tuning. I’m very attracted to this idea that simplicity and complexity can be a matter of perspective.

Why the systems and pitches you use?

My approach is simple: the system has to fit the project. I’m very concerned with making my music practical form a performance standpoint, so that the microtones can be reliably performed — whatever that might mean in a given context. I’ve often used the trick of retuning winds, plucked strings (guitar, mandolin, autoharp), or bowed strings (playing only harmonics and open strings) down by a quarter-tone, so that a player can use “normal” fingerings but still play reliable microtones.

I’ve also written for instruments which are specifically designed to produce microtones, in which case the system is more or less decided for me. I’ve worked extensively with the qanûn for instance, a middle-eastern zither equipped with small levers under the string that can produce microtones by changing the string length. I’ve used both the Syrian version (in Üsküdar and Widening Circles) which gives tempered quarter-tones, as well as a just intonation qanûn designed by Julien Weiss, for whom I wrote a solo part tailored to his particular tuning system in Cognitive Consonance. Or another example: the Fokker Organ, a MIDI-controlled microtonal organ in the Amsterdam Muziekgebouw, for which I also composed a short piece.

Another approach I’ve used is to write strictly tempered music for tempered instruments complemented by electronically created microtonal sounds, aiming for a fusion of live sound and synthesis or retuned samples that sounds like a single microtonal instrument. And there’s also my hexaphonic electric guitar, whose strings can be electronically retuned by a Max/MSP patch. I find that working with electronics offers the broadest range of possibility and precision, and a lot of my most fantastical pitch-related ideas are best realized in that medium.

A qanûn, showing the levers used for tuning

A qanûn, showing the levers used for tuning

What was your first encounter with microtones?

Wow, it’s coming up on a decade now… Shortly after I first moved to Paris in 2003, I caught a complete performance of Gérard Grisey’s Les Espaces Acoustiques. I’d heard Partiels (the third piece in the series) about a year earlier, and (weird to think of this in retrospect) only the theatrics at the end made an impression. But being enveloped by those lush harmonies in the concert hall was life-changing. Microtones were the key, it dawned on me around then, and I started devouring all the French microtonal music I could find, starting with Murail, Hurel, Leroux… For my first piece with microtones — a quartet for four clarinets written in 2004 — I decided I’d consciously train myself to hear microtonal intervals, and started constructing chords from slices of the harmonic series. But even in this first piece, I wasn’t doing just intonation drones, but working towards a richer polyphony, focusing on the voice-leading interactions between multiple microtonal lines.

My second microtonal revelation came when I discovered Turkish music. Often you hear that microtonal music has to be slow to be effective, that it takes the ear a while to attune itself to “unfamiliar” intervals. But in Ottoman classical music I discovered an entire tradition of fast and yet extremely precise microtones. The Turks rely on a combination of specially designed instruments and an internalized tradition that divides the octave (approximately but not quite exactly or consistently) into 53 parts. The idea that a microtonal theoretical framework could exist in tandem with a practice that tolerations deviations (which is in fact the way most supposedly tempered music is performed!) was illuminating for me and a source of inspiration for several pieces to follow.

Excerpt from Trapani's Cognitive Consonance

Excerpt from Trapani’s Cognitive Consonance

What piece of microtonal music that you didn’t write is most important to you?

I’ll go with either Harry Partch’s And on the Seventh Day the Petals Fell in Petaluma or Blind Willie Johnson’s version of “Dark was the Night, Cold was the Ground”

Aug 12

Just Intonation in Renaissance Music

Just a little mid-Summer check-in here! We’ll be back on a more regular schedule as we come into our season in the Fall, which kicks off mid-October.

I’ve just been totally engrossed by Ross Duffin’s fantastic article about the theory and practice of Just intonation in Renaissance music. While Ekmeles has performed music explicitly written for and formulated to work in Just intonation, I’ve always been fascinated with quantifying the more intuitive method of tuning employed by choirs and vocal ensembles performing older music.

Before we get into the nuts and bolts, a few terminology refreshers: The octave is divided into 1200 cents, making each half-step in 12 note equal temperament 100 cents. Just intonation is more easily expressed in terms of frequency ratios, which will appear as a ratio between 1:1 and 2:1, expressing an interval smaller than an octave.

Anyone who’s performed in an ensemble knows the sound of a just perfect fifth, and its solid, grounded feeling due to the elimination of beating caused by overtone dissonances. It can be expressed as the ratio 3:2, or 702 cents, putting it 2 cents, or 2 100ths of a halfstep wider than the fifth on the piano. Just thirds, the major’s ratio 5:4, the minor’s 6:5, are slightly less intuitive to modern ears, as they are 386 cents and 316 cents respectively, deviating by 14 and 16 cents from the piano’s tempered thirds.  However, performers of Renaissance music are familiar with the sound of these mellower thirds. Why this discussion about applying Just intonation? Why not just sing these pure fifths and thirds and be done with it? The short answer: polyphony.

The long answer is dealt with beautifully in Duffin’s essay. Tuning purely homorhythmic vertical chords one after another is a simple matter, relatively speaking. When a composer writes notes that hold while other notes move, the moving notes have to tune to the held note (assuming the held note isn’t sliding around). This sometimes means pure fifths will have to be tuned downward from a held note that was tuned as, say, a pure major third. Now a third tuned above that lower note will be doubly low, and you can imagine how things go from there! Duffin deals with Renaissance theoretical puzzles in which the chains of intervals tend to push the pitch in one direction or another, and devises wonderfully musical solutions to the practical problems of keeping in tune.

As a practical example, Lassus’s Ave Regina coelorum is exceptional. If all the intervals were to be tuned justly, the pitch would droop by somewhat less than a quarter tone (43 cents) over the first 5 measures! This example is included in the text of Duffin’s essay, along with several possible tunings.

Below you can hear a synthesized version of this excerpt in a pure (and totally impractical, given the aforementioned drift) Just intonation tuning, followed by a different kind of mechanical solution, performed using an H-Pi Instruments keyboard. Incidentally, I use their exceptional Scordatura software for realizing microtonal scores. Below that video is a human performance of this beautiful piece which doesn’t drop a quarter tone – you can really hear them fighting that downward tendency though, when your ears are attuned to it!

Jun 11

The mathematics of tuning

Since we’re about to embark on a season of non-standard tunings, ranging from columns of septimal JI intervals (thanks to Mr. Randy Gibson) to 31-note ET (a speculative historical journey with everyone’s favorite musical murderer, Don Carlo Gesualdo da Venosa), I thought it might be interesting to have a quick review of the mathematics of tuning.

We’ve recently discussed JI and ET tunings, and they differ in mathematical foundations. Absurdly briefly put: JI is based on ratios; ET is based on logarithms.

I highly recommend Kyle Gann’s Just Intonation Explained for a basic background on ratio tunings, and to be able to hear exactly what all those numbers mean.

And though I hesitate to link to it for obvious reasons, Wikipedia actually has a very clear writeup on the math of equal temperament! I’ve linked you past the rambling and somewhat questionable narrative section straight to the goods, starting with the Chinese discovery of the logarithmic solution to equal temperament.

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.