Thanks to Ken Ueno, you can hear a beautiful combination of a few of my favorite things! (So they’re not Tuvan. But it’s still pretty fun!)
One of my absolute favorite new music blogs is Tim Rutherford-Johnson’s “The Rambler”. Truly intelligent writing about contemporary music and the community surrounding it. The comments on his posts are a real goldmine, as his readership includes many eminent composers and performers; Ian Pace and Liza Lim both made recent appearances in discussions of complex music and the publishing business respectively.
Of especial interest to Americans – now that the service is available to us – might be his Spotify playlist of contemporary music, which are a great way of discovering new things!
I’ve just stumbled across the Vimeo page of Daryl Buckley, guitarist and artistic director of ELISION Ensemble, and among the many treasures to be found there, recommend this recent Liza Lim piece. I had the pleasure of singing the US Premiere of her piece Chang-O and love the timbral detail in her music.
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.
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.
If you’re not knowing him, I’m here to let you know. Georges Aperghis has developed one of the most distinct styles of vocal writing of any composer, and is rather prolific to boot! Check out his website and blog, do some listening at UbuWeb to his famous Recitations and see below for several YouTube clips of his Machinations.
Despite the fact that we work mostly with the voice in its classically trained sense, I still have a deep appreciation and love for many non-classical voices. If you don’t know the following, get to know them.
Alfred G. Karnes, preacher and gospel singer;
Mike Patton, erstwhile frontman of Mr. Bungle, Faith No More, collaborator with Zorn;
Tom Waits, singer/songwriter with the most expressive chronic laryngitis you’ll ever hear;
Demetrio Stratos, Greek prog-rocker, researcher, and experimentalist;
and Roy Hart, South African actor who was the star student of avant-garde teacher Alfred Wolfsohn and for whom Eight Songs for a Mad King was written.
I’d like to direct you to listen to some more music by Martin Iddon, the composer of an incredible piece we commissioned earlier this year, hamadryads. Below is an excerpt of his string quartet Mohl ip, as performed by the Kairos Quartett. You can find the score to both the string quartet and the ekmeles commission at this site.
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).
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.
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.
Here’s a lovely recording of some early music for voices and electronics (not Early Music) called Extended Voices by a motley assortment of 60s luminaries, all conducted by Alvin Lucier. Also included are some magnificent non-electronic works by Feldman for chorus that could stand to see the light of day more often! I might also point you to the links on the left of the page to access the rest of the fabulous resource that is UbuWeb.