Floating in Free Pitch Space

Microtonal theorist Timothy Johnson, of whose theoretical skills and even more his work ethic I stand in awe, has sent me the MIDI file he made of the first 30 measures of the final movement of Ben Johnston’s Seventh String Quartet, of which I wrote in my last post. At 2:41, this represents about a sixth of the third movement, which must total 16 minutes. I can’t listen to it enough: exotic consonances floating in a totally free, gridless pitch space. This is truly the music of the distant future. He made the file with piano sounds, since MIDI string sounds are vulgarly inadequate, so you’ll have to imagine this played by a string quartet. I wish I thought I would live long enough to write music like this, but I’m too pragmatic, not visionary enough. The score is published by Smith Publications, if you’re interested in studying it yourself.



  1. Herb Levy says

    I’m not getting anything at the URL for the midi-file. Is that a problem at my end of things or yours?
    KG replies: It works for me. Anyone else have troubles?

  2. Ernest Ambrus says

    Thank you for posting that file. I’ve never heard anything like it.
    I didn’t have trouble with it; I right clicked, and chose ‘save link as’ right to my desktop.

  3. Juhani Nuorvala says

    (The midi file works here)
    Hi Kyle,
    this is so fascinating! Thank you for the reports from the colloqium, as well as for the score and audio samples.
    I read the first 1,5 bars of the 7th Quartet, and – ever so slowly – played them on my Tonal Plexus keyboard (with 205 tones/keys per octave): the first viola note in the second bar landed two keys above the first C!
    I confess I have a hard time of accepting utonal chords (the inverted harmonic series) as consonant, with the exception of the minor triad, of course. In Johnston’s tonal practice, the consonance of such chords is often assumed.
    You write that in the second bar the violist is supposed to find the 11th subharmonic below G7b-. But in addition to the G7b-, a bunch of other notes are sounding: the 1st, 5th, 3th and 9th subharmonic of G7b- in close position. Now, I don’t hear much difference in the consonance, or in-tune-ness, of the chord whether the viola note is tuned to the 11the subharmonic or not – this is acoustically such a complex chord. I also think it’s extremely demanding to find the D7bv- by ear when all those other notes are sounding.
    I suppose one should have the sound of the subharmonic series in one’s ear.
    It would be very interesting to hear about your experiences in learning to hear utonal chords. Has the way you hear them or think about them changed over the years?
    KG replies: Hi Juhani! I feel exactly the same way you do about the subharmonic series, I never use it. It just sounds too thick to me, and doesn’t balance the harmonic series well. And yet Ben uses utonalities to good advantage, and so did Partch, and I don’t understand why they sound so sour when I try it. I thought maybe it was because I work with synthesized sounds, but McLaren told me that was bullshit, that other people do it with synthesizers and it’s fine. I just can’t make my ear accept it. But I do grow and change, and maybe I’ll get the hang of it someday.

  4. mclaren says

    In all fairness, I don’t think I really said it was bullshit — I think what I mentioned about the subharmonic series is that it can’t possibly be true that the subharmonic series sounds more acoustically rough or less musically consonant than the harmonic series. And the reason is that every single vertical complex (let’s not call ’em chords — as Erv Wilson remarks, we don’t want to limit ourselves that way) in the harmonic series can be constructed from members of the subharmonic series, and every single vertical complex in the subharmonic series can be constructed from members of the harmonic series.
    The just minor triad, for instance, which is often described as 1/4:1/5:1/6 (that is, members 4 and 5 and 6 of the subharmonic series) is perfectly represented by the identical pitches 10:12:15 in the harmonic series. Likewise, the major just triad 4:5:6 is perfectly represented by pitches which sound the identical frequencies in the subharmonic series, but are gapped farther apart and found in a different place in the subharmonic series. In fact, the major just triad 4:5:6 is represented by the subharmonic series members 1/10:1/12:/1/15. To see that this is true, note that 10*12*15 = 1800, and 1800/10 = 180, 1800/12 = 150, and 1800/15 = 120. Now observe that 180/120 = 3:2, while 150/120 = 5:4, et voila! There’s your 4:5:6 just major harmonic triad, which can equally well be viewed as the 1/10:1/12:1/15 subharmonic triad. Yes, Virginia, the 4:5:6 is a utonality, just as the 1/4:1/5:1/6 is an otonality. It’s both a floor wax and a dessert topping! Which explains why Partch’s terminology of “otonality” and “utonlity” deters musical understanding and contradicts the observed audible reality. In actual musical fact, any otonality is also a utonality and vice versa. It is simply a matter of representation — a map, if you will. And, as Alfred Korzybski constantly reminded us, the map is not the territory.
    I think all I mentioned to you is that my musical experience tends to bear out these basic mathematical facts out. YMMV.
    In any case it’s simply a basic mathematical reality (which various math folks have proved) that any set of melodic pitches or any vertical complex you can find in the subharmonic series has its exact equivalent in the harmonic series. The only difference is that if the subharmonic series members are close together in the subharmonic representation they will always be gapped in the harmonic series representation. (And vice versa.)
    Likewise, any set of melodic pitches or any vertical complex you can find in the harmonic series has its exact equivalent, with precisely the same frequencies, in the subharmonic series — provided you adjust what Fokker called the “guide tone,” that is, the fundamental of the subharmonic series, to the proper pitch. Just as you can transpose any set of harmonic series JI pitches by moving the fundamental 1/1 up or down, you can just as easily transpose any set of subharmonic series JI pitches by moving the guide tone /1:/1 up or down. So any harmonic series vertical complex or melody can be exactly and precisely duplicated with no difference whatsoever merely by moving the guide tone up or down, typically up.
    The upshot of all this? Since every pitch or vertical complex you can sound in the harmonic series can just as easily be represented by a member or members of the subharmonic series, it’s absolutely impossible that the subharmonic series somehow sounds “worse” or or “more sour” or “muddier” or “more out of tune” or “more dissonant” than the harmonic series. The 4:5:6 just major triad is also the subharmonic triad 1/10:1/12:1/15– how can it be true that subharmonic vertical complexes are “murky” or “sour” if the 4:5:6 triad can be represented equally well as a subharmonic triad? Does the 4:5:6 triad sound “murky”? Clearly not. Yet it is indisputably a subharmonic triad (as well as a harmonic triad, of course).
    This tells us that the subharmonic series merely behaves slightly differently than the more familiar harmonic series. The subharmonic series exhibits increasingly large gaps as you move farther along the series, as opposed to the increasingly smaller gaps as you move farther along the harmonic series. As Ivor Darreg put it, the subharmonic series and the harmonic series are like card games, where you have deuces wild in one game but jacks are wild in the other game. Exact same cards, just different rules.
    Bill Wesley, who has done more work than almost anyone in exploring subharmonic series both in pitch and rhythmically, puts it this way: the overall “sound” or “mood” or “sonic fingerprint” of the harmonic series is a plain bland whitebread John Philip Sousa kind of sound. Whereas the overall “sound” or “mood” or “sonic fingerpint” of the subharmonic series is a jazzy kind of kick-ass rock ‘n roll type of sound. The subharmonic series, melodically, sounds funky and sultry and downtown; the harmonic series, melodically, sounds square and prim and uptown. The reason for this difference in overall “sound” or “mood” of the two series, however, is entirely due to the arrangement with which they present their pitches to you. Any pitch (or chord) available in the subharmonic series is available in the harmonic series, and vice versa…just in a different place.
    So the real issue is merely knowing where to look in the subharmonic series to find the pitches or chords you want. All our instincts from dealing with the harmonic series must be inverted: it requires lateral thinking to deal with the subharmonic series. It’s very much akin to that famous psychology experiment (first conducted by George Stratton in the 1890s!) in which people got fitted with glasses that used a reflecting mirror to reverse the view of the world upside down. At first, experimental subjects stumbled and fumbled, unable to do the simplest things, like open a door, or tie their shoes. But after a short time, their brains rewired themselves and adjust to the reversed view of the world and soon the subjects had no problem doing everything they had done before, even riding a bicycle or writing on a blackboard. Then, when the reversing glasses came off, the subjects once again found themselves thrown into chaos until their brains adjusted back to the customary right-side-up view of the world.
    Dealing with the subharmonic series throws you for a loop musically, in the same way as those reversing glasses do visually. Psychologists call it “perceptual adaptation” in vision. It’s only temporary if you keep at it because your brain soon rewires, and you get the feel of it and pretty soon it’s as easy to make music with all subharmonic pitches as it is to make music with harmonic series pitches.
    The issue of harmonic series rhythms as opposed to subharmonic series rhythms, however, presents much more profound differences than subharmonic as opposed to harmonic pitches. Here, you not only change the rules, you also change the cards, to use Ivor’s metaphor. Subharmonic polyrhythms really do differ from harmonic polyrhythms. Once again Bill Wesley has worked at the forefront of subharmonic polyrhythms, and in combining subharmonic with harmonic polyrhythms — a field he calls “microtemporality.” (My own compositions also explored subharmonic polyrhythms extensively as well as combining subharmonic with harmonic polyrhythms, though my work duplicated Bill’s without either of us knowing it for quite a while. In my case, the reason for using subharmonic polyrhythms is that when you use a MIDI sequencer it’s actually much easier to get subharmonic polyrhythms than harmonic polyrhythms, because when you copy a track and multiply all its time values by some percentage, like 120% or 140% or 175%, you’re creating a subharmonic polyrhythm. 120% is 4 in the time of 5, 140% is 5 in the time of 7, 175% is 4 in the time of 7, and so on. It’s much harder to multiply the length of all notes in a MIDI track by, say, 112.5%, which is the harmonic series polythythm 9 in the time of 8, because you have to multiply the length of all notes by 125% and then reduce their length by 50%, and as you go up to higher harmonic series rhythms, these sequencer math percentage gyrations grow increasingly elaborate and difficult, whereas getting subharmonic polyrhythms is trivially easy with a MIDI sequencer.)
    As Bill points out in relating subharmonic and harmonic polyrhythms to subharmonic and harmonic pitches: “I suspect subharmonic relationships FOCUS RHYTHMIC ENERGY in the subaudio or periodic range, but in the audio or frequency range they SCATTER CHORDAL ENERGY. I also suspect that harmonic relationships SCATTER RHYTHMIC ENERGY in the subaudio range, but in the audio range they FOCUS CHORDAL ENERGY. The scattering effect is a bit ‘against the grain’ and therefore it tends to be more thrilling. The focusing effect is a bit more ‘comfortable’ and therefore it tends to be a bit more soothing
    The use of harmonics in rhythms has a certain kind of heightened drama that equates to the use of suharmonics in chords.”
    (Bill Wesley, personal correspondence.)
    Or, to put it another way, the overall “sound” or “mood” of harmonic series polyrhythms is jittery and hyper, and cranks you up and makes you want to move around and shake your booty. The overall “sound” or “mood” of subharmonic series polyrhythms, by contrast, is relentless and implacable and gives the overall feeling of ever-increasing inevitability and solidity as you add more and ever-slower subharmonic polyrhythms. Harmonic series polyrhythms would be great for a Rio de Janeiro carnival, while subharmonic polyrhythms would be ideal for Gandhi’s salt march.
    As you add more and more increasingly fast harmonic series polyrhythms, say, if you start with something simple like 4 against 5 against 6 and then add 7 in the time of 4 and 11 in the time of 4 and 13 in the time of 4, it gets harder and harder to determine where the downbeat falls. But as you add more and more increasingly slow subharmonic series polyrhthms, say, starting with something elementary like 3 in the time of 4 and add 3 in the time of 5 against 4/4, and you then add 3 in the time of 7 and 3 in the time of 11 and 3 in the time of 13 all against 4/4, the downbeat gets progressively reinforced and a slow relentless “sound” develops that practically forces you to play (or dance) in lockstep.
    The combination of subharmonic polyrhythms with harmonic polyrhythms, and the use of subharmonic-series-ordered accents and their combination with harmonic-series-ordered accents, remain subjects too large to discuss here, inasmuch as accents can be considered “negative rhythms” and thus polyaccents have entirely different musical properties from polyrhythms.
    Bill Wesley remains the real mastermind of that stuff. By comparison with him, I come off like a mentally retarded ignoramus when discussing these topics, which is probably why my email to you got misinterpreted. Unfortunately, Bill is too busy building amazing xenharmonic instruments and creating astounding polyrhythmic and polytemporal and polyaccent music to go online and talk about all this, so it’s left to slow-witted dolts like me to pick up, and make obtusely inane comments about, a few of the less interesting sonic seashells on the shore of the vast ocean of music where people like Bill go voyaging.
    It also goes without saying that Bill and I qualify as ants by comparison with a musical giant like Ben Johnston, of whose work I have little to say because I’m not remotely smart enough or knowledgeable enough.
    KG replies: Well, I dunno, still sounds like you mean it’s bullshit to me. The inversion of the so called “dominant” seventh 4:5:6:7 is 60:70:84:105, which is *way* up there in the harmonic series. I’ll take your word for it, but I still need to learn to hear it. Like it takes me some time to figure out that YMMV = Your Mileage May Vary, and time = time in any case.

  5. Bob Gilmore says

    There’s a tremendous piece by Marc Sabat, Three Chorales for Harry Partch, for violin and viola, which uses a single Utonality on A. It’s a great way to really hear a Utonal pitch world in the context of an actual piece. It’s hopefully coming out on Mode Records sometime but you can also check it out on Marc’s page on his Plainsound website.

  6. Juhani Nuorvala says

    OK, let’s just say then that the vertical complex 45:55:99:165:495 on the downbeat of the second bar sounds jazzy kind of kick-ass rock ‘n roll compared to those in the first bar.
    WIll look into the Sabat piece for utonal ear-training.